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Normal functions of product varieties Lewis, James Dominic 1981

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NORMAL FUNCTIONS OF PRODUCT VARIETIES by James Dominic Lewis .Sc., The University of B r i t i s h Columbia, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Faculty of Graduate Studies (Department of Mathematics) We accept t h i s t hesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1980 © James Dominic Lewis, 1980 In presenting th i s thes is in pa r t i a l fu l f i lment of the r e q u i r e m e n t s f o r an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree t h a t the L ibrary shal l make it f ree ly ava i lab le for r e f e r e n c e and study . I fur ther agree that permission for extensive copying o f t h i s t h e s i s for scho la r l y purposes may be granted by the Head o f my Department or by his representat ives . It is understood that c o p y i n g o r p u b l i c a t i o n o f th is thes is for f inanc ia l gain sha l l not be allowed without my writ ten permission. Department of Mathematics The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date I W e ^ W 1^ I 7 S O Leaf ( i i ) does not e x i s t . ( i i i ) Supervisor: Dr. L.G. Roberts Abstract The work of t h i s thesis i s to motivate the following: Statement: The Hodge conjecture holds f o r products of v a r i e t i e s Z = XxC where (i) X i s smooth, pr o j e c t i v e of dimension 2m-l, ( i i ) C i s a smooth curve. diagram: r-1 The b a s i c s e t t i n g of t h i s thesis i s depicted by the following — 1, k"x(u) = Z° c y Z P -E = U -><P where (i) k (t) = Z = X xC, {x } a fc t€P Lefschetz p e n c i l of hyperplane sections of X ( i i ) £ i s the singular set of k, i . e . , k = k i s smooth and proper. Corresponding to t h i s diagram are the extended Hodge bundle U H (Z . C) with integrable connection V , and the family of t t ? 1 t intermediate Jacobians. U JCZ ) with corresponding normal functions t ^ P 1 v : P""" *• U J(Z ). Now V induces an operator (also denoted by V) t € p a on the normal functions, and those normal functions v s a t i s f y i n g the d i f f e r e n t i a l equation Vv = 0 are labeled h o r i z o n t a l , which includes those normal functions a r i s i n g from the p r i m i t i v e algebraic cocycles (iv) i n H (Z). Now the known generalization of Lefschetz's techniques state that every p r i m i t i v e i n t e g r a l class of type Cm,m) i n H (.Z) comes from a h o r i z o n t a l normal function i n some natural way, so that what's needed to prove the above statement i s some way of converting a normal function to an algebraic cocycle. We motivate t h i s statement by proving some r e s u l t s about the group of normal functions, i n p a r t i c u l a r our main r e s u l t : Theorem: The group of normal functions are h o r i z o n t a l . To prove t h i s theorem, we e x h i b i t Vv as a global section of some holomorphic vector bundle over , and then show that there are no non-zero global sections of t h i s vector bundle. The main idea i s to compare the quasi-canonical extensions of c e r t a i n holomorphic vector bundles with integrable connection with those extensions a r i s i n g from algebra (hypercohomology), by c a l c u l a t i n g c e r t a i n periods of growth. Once t h i s comparison i s made precise, we apply a vanishing theorem statement about the global sections of the algebraic extensions to our geometric extensions, thus concluding the proof of the theorem. :(.v). Table •' of •' Contents Abstract ( i i i ) Acknowledgement (vi) Introduction 1 Chapter 0. The Preliminaries 6 Chapter 1. The Hodge Structure on E^' 2™ ^ (k) 2 0 Chapter 2. The Intermediate Jacobian J(Z^) 29 Chapter 3. A Vanishing Theorem and i t s C o r o l l a r i e s 42 Chapter 4. A Summary on the Normal Functions 66 Bibliography 92 Appendix 95 (vi) Acknowledgement I am indebted to my supervisor, Dr. L. G. Roberts, f o r the endless hours of h i s encouraging support and the necessary guidance i n making t h i s t h e s i s p o s s i b l e . I would also very much l i k e to thank Dr. J . B. C a r r e l l f o r his encouragement and h e l p f u l suggestions i n the course of preparing my t h e s i s . Special thanks goes to Dr. P. J . Kiernan for the encouragement he provided during the various stages of my work. I would also l i k e to thank the Natural Sciences and Engineering Research Council, and the University of B r i t i s h Columbia f o r t h e i r f i n a n c i a l support during the course of my studies. F i n a l l y , I wish to thank my wife, Jane, f o r her support i n providing the desired environment to enable me to complete my t h e s i s . 1 I n t r o d u c t i o n The modern t e c h n i q u e s used i n the work o f the Hodge c o n j e c t u r e stems from the i d e a s i n t r o d u c e d by P o i n c a r e - and L e f s c h e t z , where Poincare' o r i g i n a l l y i n t r o d u c e d the n o t i o n of a normal f u n c t i o n , and L e f s c h e t z s u b s e q u e n t l y used i t l a t e r t o s o l v e t h e Hodge c o n j e c t u r e f o r s u r f a c e s . P o i n c a r e ' s m o t i v a t i o n f o r u s i n g normal f u n c t i o n s came from h i s study o f c u r v e s on a complex, a l g e b r a i c s u r f a c e S, where he a s s o c i a t e d a l g e b r a i c f a m i l i e s o f c u r v e s w i t h normal f u n c t i o n s and hence was a b l e t o determine t h e d i m e n s i o n o f th e "now c a l l e d " P i c a r d v a r i e t y , P i c (S) o f S, by means o f some l i n e a r d a t a , namely dim P i c ( S ) = d i m ^ H 1 ( S ,6 g ) . We remark t h a t t h i s r e s u l t i s not e n t i r e l y v a l i d i n c h a r a c t e r i s t i c p>0. In h i g h e r d i m e n s i o n s , the Hodge c o n j e c t u r e i s o n l y known f o r 5 v e r y s p e c i a l c a s e s , such as f o r h y p e r s u r f a c e s Z i n P o f degree <_ 5 ( [2] and [30]) , and Fermat v a r i e t i e s o f c e r t a i n dimensions and degrees ( [24]) , where t h e knowledge o f t h e geometry o f such Z i s r e q u i r e d i n p r o v i n g t h e c o n j e c t u r e . The s t a t e o f a r t i n a t t e m p t i n g t o prove the Hodge c o n j e c t u r e f o r a g i v e n Z o f d i m e n s i o n 2m i s t o use a g e n e r a l i z e d form o f L e f s c h e t z ' s t e c h n i q u e s , and t o do t h i s one f i r s t a s s o c i a t e s t o e v e r y p r i m i t i v e , i n t e g r a l cohomology c l a s s Y e P r i m m , m ( Z ,7/J a complex, a n a l y t i c o b j e c t , namely a normal f u n c t i o n v , and s e c o n d l y f i n d some way o f t r a n s f o r m i n g v i n t o an a l g e b r a i c c y c l e . As mentioned above, when m = 1 t h i s c o n j e c t u r e u n f o l d s , and i s due to the .Jaeobi inversion theorem f or curves (cf (0-29)). In higher dimensions there i s no longer necessarily such a strong r e l a t i o n s h i p between a v a r i e t y and i t s Hodge structure as i n the case of a curve, and consequently we are l e f t with the need to study the normal functions i n more d e t a i l , i n p a r t i c u l a r i n terms of t h e i r i n f i n i t e s i m a l properties (cf [10]). The work of t h i s t h e s i s i s to extend some of the known generalizations of Lefschetz's techniques (such as i n [28]) to the case of the product v a r i e t i e s Z = XxC, where X i s a smooth, pr o j e c t i v e v a r i e t y of dimension 2m-l, and C i s a smooth curve. We focus on a p a r t i c u l a r f i b e r i n g of Z, namely a p e n c i l of d i v i s o r s of the form {Zt= x t x C ^ t g p l ' where ' f x t ^ e p l l s a Lefschetz p e n c i l of hyperplane sections of X. Assoc--iated to t h i s p e n c i l are the extended 1 Hodge bundle . u H 2 m 1(Z t,C) t e p 1 with extended connection V, the Hodge f i l t r a t i o n subbundles u F pH 2 i n 1 ( Z 4 t e p 1 •p * 2 in—1 ,C) (p^O) , t h e i r duals u F >' H- (Z ,C) , .-and-the.corresponding, family. tep 1'' ': -fc • ' " '-m * 2m—1' of intermediate Jacobians u J ( Z , ) , where g e n e r i c a l l y J(Z ) = F ' H ( t e P 1 t Z t,C) / H 2 r n - 1(Z t,7£) . Now the connection V does not preserve the Hodge f i l t r a t i o n above, but rather s a t i s f i e s a so c a l l e d i n f i n i t e s i m a l period r e l a t i o n . As a consequence of t h i s r e l a t i o n and the known e x p l i c i t d e s c r i p t i o n of t h i s connection,V induces an operator ( s t i l l denoted byV ) on the holo-morphic cross sections (normal functions) V: P 1 > u ^ J ( Z t ) . In chapter teP "'"Extended r e f e r r i n g to a quasi-compactification over-the singular f i b r e s . 3 4, we prove that the Abel-Jacobi mapping (defined i n chapter 0) i s meromorphic, and as a consequence i t can be shown that a p r i m i t i v e , algebraic cocycle Y e Prim™' m (Z ,7Z) induces a normal function v : p^ " u J(Z ). Now i t i s a standard f a c t that such v s a t i s f y the d i f f e r e n t i a l t e p 1 equation Vv = o (the h o r i z o n t a l i t y condition), and we prove i n t h i s t h e s i s that a l l our normal functions s a t i s f y t h i s h o r i z o n t a l i t y condition, thus resembling the s i t u a t i o n of a Lefschetz p e n c i l of hyperplane sections of Z ( [28]) . The plan of t h i s t hesis i s roughly as follows: In chapter 1 we check that the important part of the i n t e g r a l cohomology P r i m m , m ( Z ,TCi l i e s i n the image of the cohomology classes of the normal functions. In chapter 2, we ri g o r o u s l y define the generalized intermediate Jacobians J(Z f c) and i d e n t i f y a subgroup l y i n g i n the image of the Abel-Jacobi morphism. For a c e r t a i n r e s t r i c t e d c l a s s of Z, t h i s subgroup i s a l l of J ( z t ) so that using the techniques i n chapter 4, we are able to mimic a modern form of Lefschetz's proof to obtain the Hodge conjecture f o r such Z. Although the knowledge of the conjecture f o r such Z i s not s i g n i f i c a n t l y newer than what's already known, the known techniques for proving the conjecture f o r these Z requires the understanding of the geometry of X, whereas i n t h i s t hesis the emphasis i s on the geometry of the hyperplane sections X of X. In chapter 3, we deduce our h o r i z o n t a l i t y statement on the normal functions from a vanishing theorem r e s u l t , thus providing a step i n the study of the normal functions i n terms of t h e i r i n f i n i t e s i m a l 4 properties. In addition to t h i s , we a r r i v e at an e x p l i c i t d e s c r i p t i o n of the kernel of the cohomology cla s s homomorphism 6 : normal functions * cohomology classes (defined i n chapter 2). The ideas on proving the h o r i z o n t a l i t y statement f o r the normal functions comes from a close study of [28,§4], where the d i f f e r e n c e s i n some proofs are accounted for by the difference i n l o c a l monodromy between the case ^ z ^ . ^ t e p l a n c^ that of a Lefschetz p e n c i l ( [28]). We also u t i l i z e the knowledge of the i r r e d u c i b i l i t y of the monodromy group action on the r a t i o n a l , vanishing cohomology associated to the {x.}. _n , and e x p l o i t t h i s f a c t to i t s f u l l e s t to obtain the t t^P-1-desired h o r i z o n t a l i t y r e s u l t . F i n a l l y , i n chapter 4 we summarize our r e s u l t s on the normal functions, and make some general remarks i n t h i s d i r e c t i o n . A summary of the l o g i c a l sequence of steps i n t h i s thesis i s given i n the chart below. 5 E x p l i c i t l y d e s c r i b e the E l , 2 m 1 t e r m ^ W h i c h c o n t a i n s t h e cohomology c l a s s e s o f t h e normal f u n c t i o n s . C o n s t r u c t t h e g e n e r a l i z e d i n t e r m e d i a t e J a c o b i a n s J ( Z f c ) and compute a subgroup J ( Z ) i n J ( Z f c ) which l i e s i n t h e image o f t h e A b e l - J a c o b i morphism. (i ) E x p l i c i t l y d e s c r i b e t h e k e r n e l o f S : normal f x n s — — > cohomology c l a s s e s i n E ^ ' 2 m - 1 ( k . ) . ( i i ) C o r o l l a r i e s . Deduce t h a t a l l t h e normal f u n c t i o n s a r e h o r i z o n t a l from a v a n i s h i n g theorem r e s u l t . E v e r y normal f u n c -t i o n w i t h v a l u e s i n the J A ( z t ) g i v e s r i s e t o an a l g e b r a i c c o c y c l e . T h i s c o r r e s p o n d s t o t h e Hodge c o n j e c t u r e f o r case 1 i n c h . 4. Prove t h e meromor-p h i c i t y o f t h e A b e l -J a c o b i mapping. (i ) D i s c u s s t h o s e normal f u n c t i o n s which do not t a k e t h e i r v a l u e s i n t h e J A ( z t ) • T n e d i f f i c u l t y i n p r o v i n g t h e Hodge c o n j e c t u r e f o r Z l i e s h e r e . The t e c h n i q u e s i n t h i s t h e s i s a r e i n s u f f i c i e n t t o p r o v e t h e c o n j e c t u r e at. t h i s s t a g e . ( i i ) Some c o n j e c t u r a l statements made. 6 Chapter 0. The Pr e l i m i n a r i e s . The subject of t h i s thesis i s centered around an a n a l y t i c r e s u l t about a s p e c i a l class of manifolds, namely the Hodge conjecture for products of v a r i e t i e s based on c e r t a i n inductive assumptions. The general statement of the conjecture i s for a l l p r o j e c t i v e , algebraic manifolds. The s t r i k i n g phenomena about such manifolds Z i s the re l a t i o n s h i p between t h e i r a n a l y t i c and top o l o g i c a l properties, formally expressed i n the following way: (0.1) H m(Z,C) = © H P , q(Z,C) m= 0,1,2,... p+q=m where Hm(Z,(E) , HP,q;(Z,<C) are the deRham, respectively Dolbeault cohomology with complex c o e f f i c i e n t s . This r e l a t i o n s h i p holds more generally f o r compact, complex manifolds Z which are known to be Kahler ( [11]) ... In doing analysis on such manifolds, one i s frequently confronted with both the to p o l o g i c a l and a n a l y t i c aspects of the subject, as r e f l e c t e d by the above r e l a t i o n s h i p (.0.1) . In t h i s t h e s i s , the t o p o l o g i c a l aspects w i l l be the weak and strong Lefschetz theorems, the p r i m i t i v e decomposition theorem, Lefschetz p e n c i l s , the Leray s p e c t r a l sequence, whereas for the a n a l y t i c aspect we have the Hodge f i l t r a t i o n s subbundles, the Gauss-Manin connection and the r e g u l a r i t y theorem, and hypercohomology. Both these aspects play a role i n reducing the problem of the Hodge conjecture to something more manageable as w i l l be discussed below. 7 The statement of the Hodge conjecture f or p r o j e c t i v e , algebraic manifolds Z i s the following ( l e t (g = r a t i o n a l numbers) : (0.2) Hodge P , P(Z,g): The fundamental class homomorphism, $ O C^Z) — • H p' p(Z,Q ) i s an epimorphism, where C P(Z) i s the group of codimension p algebraic cycles i n Z and H P , P ( Z , 6 ) = H P , P(Z,C) fl H 2 P(Z , 6 ) . (0.3) equivalently: (the group generated by {C (v)|V i s a holomorphic vector bundle over z})<8> Q ., = H P' P(Z,Q), where C (V) i s the p t h Chern class p of V. The equivalence of these two statements can be found i n [9, Theorem Q]. We should remark that the o r i g i n a l statement of the Hodge conjecture involved i n t e g r a l c o e f f i c i e n t s (Hodge P , P(Z/20) for Kahler manifolds Z . Both formulations ((i) integer c o e f f i c i e n t s , ( i i ) Kahler manifolds) are now known to be f a l s e , and one can f i n d -a suitable counterexample f o r ( i i ) dn [30], and a comment on (i) i n [13], The main body of theorems required i n t h i s thesis w i l l be stated now. We f i r s t f i x an embedding i : Z C—>• Z of a smooth, ample d i v i s o r Z i n a p r o j e c t i v e manifold Z, and denote n = dim Z . Also we l e t [Z F C] € H 1 , 1(Z,7/.) be the fundamental class . o f ZFC , and r [Z T] = cup product (o>) of [ZFC] with i t s e l f r times. We introduce the following: (0.4) D e f i n i t i o n . Let mSn . The p r i m i t i v e m cohomology, Prim m(Z) i s defined to be the kernel of the cup product homomorphism , . ,n-m+l m. , T 2n-m+2 ' [Z t] : H (Z) y H (Z) . The following are true: (0.5) Theorem (weak Lefschetz) . i . : H (Z . 71) »• H {Z,7L) i s an * r t r isomorphism f o r r < n - 1 , and an epimorphism for r = n - 1 . r n—r n+r (0.6) Theorem (strong Lefschetz) . w [Zfc] : H (Z,£>) *• H (Z,©J i s an isomorphism f o r a l l 0 ^  r 5 n . (0.7) Theorem (primitive decomposition). r Z r-2-S H (Z,Q) ~ ffi [Z ] ^ P r i m (Z,£>) . femax{0,r-n} We remark that due to the bidegree property of \^ [Z ] , i . e . , \J [Z^_] : H P , q(Z,C) v H P + 1 , q + 1 ( Z , C ) , the p r i m i t i v e decomposition theorem holds f o r H P , q ( z , C ) , and also f o r H P , P(Z,Q). More p r e c i s e l y : (0.8) H P , q ( Z , C ) ^ © [z V^Prim P~' e' q'"' e(Z,C) femax{0,p+q-n} H P' P(Z ,6 ) © [Z ]* \^PrinP~L'P~*(Z,Q) . £>max{0,2p-n} I f one wants to prove Hodge P , P(Z,$) i n d u c t i v e l y f o r a l l p > 0 and a l l Z , then the following deductions can be made: (i) The weak and strong Lefschetz theorems imply that the conjecture need only be proven f o r dim Z = n = 2m and p = m. ( i i ) Using the f a c t that the composite: H^ m ^ ( Z ) — *• H^ m ^ ( Z ^ _ ) — - — H ^ m ( Z ) i s p r e c i s e l y the cup product with 1 2m [Z^] , the p r i m i t i v e d e c o m p o s i t i o n theorem i m p l i e s t h a t H. (Z) = P r i m 2 m ( Z ) © i # H 2 m 2 ( Z ^ ) . T h e r e f o r e i t s u f f i c e s t o p r o v e the f o l l o w i n g : (0.9) P r i m m , m ( Z , Q ) i s g e n e r a t e d over. 0 by a l g e b r a i c c o c y l e s . The b a s i c s e t t i n g o f t h i s t h e s i s i s d e p i c t e d by the f o l l o w i n g diagram: (0.10) k _ 1(U.) = Z° c Z .P" - E = u c-2_^  p 1 = complex, p r o j e c t i v e 1 space. where ( i ) k i s a morphism, dim Z = 2m ( i i ) E = s i n g u l a r s e t o f k ( i . e . , the f i n i t e s e t o f p o i n t s i n P ..where k i s n o t smooth), ( i i i ) the cohomology o f Z d i f f e r s from t h a t o f Z by the cohomology o f a~ v a r i e t y o f d i m e n s i o n "" n-2 . We remark t h a t k i s smooth and p r o p e r o v e r U . Now i t i s known t h a t the f i b e r s o f k a r e a l l t o p o l o g i c a l l y CO e q u i v a l e n t ( i n t h e C sense) so t h a t the L e r a y cohomology s h e a f 2iti — 1 R k^c o v e r u a s s o c i a t e d t o the p r e s h e a f : V c U open f ^ H 2 m x(k 1(V),C) , i s l o c a l l y c o n s t a n t w i t h s t a l k s Urn H ° ( . V , R 2 m - 1 k A C ) = H ^ t k - ^ t ) ^ ) . V open t<cVCU In the s i t u a t i o n o f (0.10);, t h e r e i s the L e r a y s p e c t r a l sequence f o r k which abuts t o H (z\<C) , and whose* terms a r e : E P , q ( k ) = H P.(P 1,R qk^C). Under c e r t a i n c o n d i t i o n s t h i s s p e c t r a l sequence degenerates at the E^ terms, i . e . , E P , q ( k ) > H P + q ( Z , C ) , and t h i s f o r example occurs when (0,10) arises from a Lefschetz p e n c i l {Z } of hyperplane sections of Z . The d e f i n i t i o n of a Lefschetz t t€P" p e n c i l i s given i n (1.1) and the existence of such p e n c i l s i s proven i n [6, expose XVII]. Due to the nature of the techniques i n t h i s paper, i t becomes necessary to u t i l i z e the sheaf of i n v a r i a n t cycles defined by j^R k^K where K i s the integers, r a t i o n a l s , or complex numbers. The cohomology of main i n t e r e s t i n t h i s thesis i s E^' 2™ ^(k) = H^"(p\R^m "'"k^ O which can be compared with the more n a t u r a l l y a r i s i n g 1 >V 2m- L." cohomology H (p ,j*R Jc^C), by means of the l o c a l cycle i n v a r i a n t property which' states the following: (0.11) Theorem ([31, (15.12)]). The r e s t r i c t i o n homomorphism 2m-1.- ^ . 2m-1, - . R k^C *• 3^R k^C i s an epimorphism. We a c t u a l l y prove that the above i s an isomorphism for our p a r t i c u l a r morphism k , however i t should be remarked that the kernel of the above epimorphism has zero dimensional support, hence H V ', R ^ X c ) ^ H V , j , R 2 m _ 1 k ^ ) . (0.12) Focusing on the a n a l y t i c aspects of (0.10), there i s a holomorphic vector bundle over U with corresponding l o c a l l y free sheaf F = 0 <8> R k C and integrable connection V = 3 ® 1 which u can also be derived a l g e b r a i c a l l y ([19]) v i a the r e l a t i o n : R^m "hc..^  -° . ~ 6 ® R 2 m "'"k^ C , where R i s the r i g h t derived * Z /u U * ''"We define 0^ to be the sheaf of germs of holomorphic functions on U. hypercohomology coming from the s p e c t r a l sequence of a s u i t a b l e bicomplex ([12, Chapter 0 ] ) . The complex of r e l a t i v e d i f f e r e n t i a l s Q " ° / t t i s defined i n the following way: z / u a (0.13) Q- _ G i s the cokernel sheaf i n the short exact sequence: Z /U 0 »• fi"^ — »• fijc >• fi_c 0 and fir„ = A ^ 1 , There i s Z /u /U Z /U a d i f f e r e n t i a l 9 : .•+1 fi_0 induced from the d i f f e r e n t i a l on z /u z /u fi _ 0 , and one checks that d i s 9 l i n e a r , making fi e i n t o a U Z /U complex. There i s a f i l t r a t i o n on fi 0 defined by: z /u z /.u fi^0 i f q > p Z /U-i f q < p (0.14) This defines a f i l t r a t i o n F = F ° 3 F 1 ^ F 2 3 ... F 2 m _ 1 o 0 2m-1, by Hodge subbundles, where F P = R m k F ^ f i * 0 Z°/U (0.15) The i n f i n i t e s i m a l period r e l a t i o n holds: VF P c fi1 <8> F P _ 1 f o r U a l l p > 0 ( [8]) . (0.16) There i s a b i l i n e a r form Q(-,-) defined on Hr(Z,<£) i n the following way: Let £, n € H r(Z,0) and from (0.7) express 5,1 as H = © [ z ^ ^ ^ , n = © [ Z t l ^ •0 £ Define Q(£,TI) = „, .. (r(r+l) /2) +1 t . 1n-r+2-g^ . Given the Hodge femax{0,r-n} Z decomposition H r(Z,C) = © HP,q(Z,<C) , Define P r P , q : H r(Z,C) — p+q=r H P , q(Z,C) to be the canonical p r o j e c t i o n . Then the Weil operator C : H r(Z,C) -> H E(Z,C) i s defined as: C = E i ^ P r 1 ^ . p+q=r I t i s a standard f a c t ([11]) that the b i l i n e a r form <£,n> Q (5^ : G r i) i s p o s i t i v e d e f i n i t e Hermitian. (0.17) Let L be a r e l a t i v e l y f l a t ample d i v i s o r i n Z , i . e . induces an ample d i v i s o r on the f i b e r s k "'"(t) over P 1. Then CO.6) and (0.7) generalize to. m T O N . - r r • „ 2 m - l - r n „ . 2m-1+r, (0.18) \J L : j^R k^G—*- j^R k^C i s an isomorphism r r—2$ j.R k^e ~ © L w( R . kJ2 . * * i, r~ T ..J* Prim *^ -femax{0,r-n} In p a r t i c u l a r (0.18) holds over U , and the constructions i n (.0.16) generalize here. 2 (0.19) The Hodge f i l t r a t i o n i n (0.14) i s i s o t r o p i c , that i s (FP)"^~ = F 2 m - P , where (Fp)"^~ = {u€F |p_ (u, Fp) = 0}. This n a t u r a l l y i d e n t i f i e s the dual bundle FP'* with F/F 2 m~ P (see (2.12)). def 1 n (0.20) The holomorphic bundles •tfP'2m-1-P ^= F P/F P + 1 are not holomorphic subbundles of F , however i f we l e t be the sheaf of C functions on u , and define e P = e ® F P , £ P ' 2 m ^ P = u e u e ® Hp>2m 1 P then there i s a';C d i r e c t sum decomposition: u 2 We are s t i l l working over U. e = e° = © e P , q . We remark that Q (.-,-) i s defined on e , and p+q=2m-l p,2m-l-p p n 2m-p-l p _ p+1 X : = e (I e = (1 (- means complex conjugation) (0.21) There are the Riemann-Hodge b i l i n e a r r e l a t i o n s associated to the bundles e p , q and Q(-,-), which are mentioned i n [8, p. 133], According to the r e s u l t s i n [8], the data (0.12) ->- (0.21) define a p o l a r i z a b l e v a r i a t i o n of Hodge structure over u . This terminology w i l l be useful i n Chapter 4. I t should be remarked that not a l l v a r i a t i o n s of Hodge structures necessarily come from the s i t u a t i o n (0.10) above. Those that do are s a i d to a r i s e from a geometric s i t u a t i o n . For more on t h i s see [8]. In Chapter 2 we discuss the quasi-canonical extensions of the sheaves {F P} over 0 to l o c a l l y free sheaves {F p} over P • p^0 The Gauss-Manin connection also extends to an operator V on the F P . There i s the r e g u l a r i t y theorem ([20]) associated to the F P and V , which implies the following r e s u l t : (Q-. 22) Theorem; The underlying^vector bundles associated to the sheaves {F P} - • are algebraic. p>0 A basic c o r o l l a r y to (0.22) i s that we may also view the sheaves' {' }^ _>Q a s algebraic sheaves; and by the r e s u l t s of Serre ([23]), there i s no d i s t i n c t i o n between these points of view i n terms of t h e i r cohomology. Thus we w i l l think of { r}p>g p r i m a r i l y as a n a l y t i c sheaves, and whenever convenient, as algebraic sheaves. 14 There are a few technical points one should know before reading t h i s t h e s i s , i n p a r t i c u l a r Chapter 3. (0.23) Let S be a p r o j e c t i v e , algebraic manifold of dimension r . A d i v i s o r D c s i s sa i d to have normal crossings i f f o r every € D, r there i s a polydisk A c s centered at z = (0,.;. ;,0). with coordinates o {Z^,...,Z^}, such that D IT A r ='{z = ,(z ,...,z ) d A r s a t i s f y i n g the r e l a t i o n Z .,.Z = 0} f o r some 1 S t S r . An example of such a . . . 2 d i v i s o r D i s i n the case S = P and D = node. (0.24) We define the log deRham complex tt (log D) (only when D i s a d i v i s o r with normal crossings) i n the following way: Let z^ € S and A r as above with A r fl D = {z = {z^ , z p s a t i s f y i n g the r e l a t i o n Z 1...Z t = 0} . Define Putlog D) to be the l o c a l l y free sheaf which i s dZ n dZ^ generated over A by {—— , . , — — , d z , ,...,dZ }, and set Z 1 Zfc t+1 r . . • ftP(log D) = A P f i ^ ( l o g D). There i s the standard d i f f e r e n t i a l • • +1 • a : £2_(log D) »- Q, (log D) making 0, (log D) into a complex. There i s a bicomplex associated to the complex, with corresponding hypercohomology • • • spectral sequence abutting to • H (S, tt (log D)) = • H (S-D,C). We also have a Hodge f i l t r a t i o n {F tt (log D)} of tt (log D) S p=0 s which defines a Hodge f i l t r a t i o n F ^ " (S-D) ,C) = H* (S^tt' (log D)) on to H*(S-D,C). (0.25) We can a r r i v e at F^H (S-D,C) i n an e n t i r e l y d i f f e r e n t way as follows: There i s the complex of.meromorphic d i f f e r e n t i a l s • fig(*D) with poles of a r b i t r a r y ( f i n i t e ) order, and an order of pole p * f i l t r a t i o n G Q, (*D) on t h i s complex, which i s defined i n Chapter 3. We obtain the r e l a t i o n s h i p : F PH*(S-D,C) =.H'(S,F Pfi'Uog D ) ) = H * ( S , G P n * ( * D ) ) . I t should be remarked that f o r ^ G ( * D ) , the d i v i s o r D need not have normal crossings or be nonsingular. Further information on t h i s can be found i n [11]. (0.26) Given S as above with D C S any smooth d i v i s o r , there i s the Gysin sequence: -v H R _ 2 ( D ) v H R ( S ) • H R ( S - D ) H R _ 1 ( D ) - v H R + 1 ( S ) • . . . . The homomorphism R i s c a l l e d the residue homomorphism. More generally i f D i s a smooth subvariety i n S of codimension I , then the corresponding Gysin sequence i s the following: ... ^ H ' ( S ) — y H' ( S - D ) R > H ~ 2 ^ + 1 ( D ) — » • H * + 1 ( S ) — v . . . . The homomorphism R i s c a l l e d the generalized residue homomorphism. A more e x p l i c i t d e s c r i p t i o n of R can be found i n [17] and [7]. As a conclusion to t h i s section we outline two proofs of the Hodge conjecture f o r surfaces Z (m = 1) . We i n f a c t prove ):: ; Hodge"*"'^{Z,7JL) , the integer ' c o e f f i c i e n t version of the conjecture. (0.27) The f i r s t proof r e a d i l y adapts i t s e l f to version (0.3) of the conjecture, and we proceed as follows: One checks that the exponential exp * short exact sequence: 0—>- ~7L *• 9 * >- 6 >- 0 y i e l d s the z z following diagram: 16 (0.28) c ~ 1st o — » • H 1 ( z , e ^ ) — • H 1 ( z , e * ) — y E2(Z,7/.) y H 2(z,-e_) Z Z ^ Chem class H^" (Z,7!L) homomorphi sm || (def'n) || (def'n) || I Dolbeault isomorphism 0,2 0 v P i c " ( z ) y P i c ( z ) y H"(Z,7Z) — y H"'"(Z,C) o c l 2 P r 0 2 Picard group: of Z 0 2 I t i s now obvious from (0.28) that c 1 ( P i c ( Z ) ) = ker'.P r-' = H2(Z,7/1) n H 1 , : L(Z,C) = H 1 # 1 ( Z , ^ ) . (0.29) For the second approach, we give a sketch of Lefschetz's proof of the conjecture f o r surfaces. Version (0.2) of the Hodge conjecture i s more useful here. Lefschetz's proof consists mainly of two steps: Step I. Construct a p e n c i l (Lefschetz pencil) of hyperplane sections { z } of Z , with corresponding family of (generalized) t f C P 1 ' Jacobians ' U J(Z ).• Lefschetz proves that every class T t t€'P Y € Prim^'^(z,75£.) i s the cohomology class ( in some natural way) of a holomorphic cross section v : P 1—>• U J(Z ). These holomorphic teP 1 - t cross sections are c a l l e d i n our modern day language, normal functions. Step I I . By a process of in t e g r a t i o n , there i s the Abel-Jacobi morphism $ : (Zfc) y J(Z ) which i s b i r a t i o n a l when Z 17 i s smooth (Jacobi inversion theorem), where g i s the genus of Z and S^ g' (Z^ _) i s the g ^ symmetric product of . Now l e t Y € Prim"'"'''' (Z,7K) with corresponding normal function v . For t^P 1, v(t) defines a d i v i s o r on Z^_ v i a the Jacobi i n v e r s i o n theorem, and as t varies i n P^, the v(t) trace out a d i v i s o r D ' C Z. One checks that the fundamental c l a s s of D i s p r e c i s e l y y . We want to state which parts of Lefschetz's techniques above generalize to higher dimensions. (0.30) F i r s t of a l l , Step I completely generalizes and two proofs can be found i n [28] and [31]. (0.31) In Step II we can s t i l l obtain the Abel-Jacobi morphism: Let cm(Z^_) be the group of codimension m algebraic cycles i n Z^ which are homologous to 0 . Then there i s a homomorphism $ : C™(Z^_) y J(Z^_) defined by a process of i n t e g r a t i o n . We remark that no analogue of the Jacobi inversion theorem ex i s t s ([7, §13]) which i s the e s s e n t i a l d i f f i c u l t y i n adapting Lefschetz's proof to higher dimensions. (0.32) I t i s easy to show how an algebraic c l a s s y ^ Prim m , m(Z,7d) gives r i s e to a normal function v : P"*"—>- U J(Z ) . Given such a y , 1 t t€P i t follows from the d e f i n i t i o n of p r i m i t i v e cohomology and the weak Lefschetz theorem, that y f\ Z i s homologous to 0 i n Z^. Thus there e x i s t s £ t a r e a l 2m-l cycle i n Z^_ with boundary, 18 9 £ t = y (1 , and by int e g r a t i n g % with a basis of forms i n (Z^,C) (precisely the Abel-Jacobi morphism), we are able to define a holomorphic cross section v : P^ "—>• d, J(Z ). t€P7 fc (0.33) F i n a l l y we wish to introduce some useful terminology. (0.34) D e f i n i t i o n . (i) The i n v e r t i b l e part of J ( z t ) denoted by J ( Z f c ) i n v , i s defined to be <5(c m(Z t)) , $ being defined i n (0.31) as the Abel-Jacobi morphism. ( i i ) A normal function v : P1—>- U, J(Z ) i s sa i d to be i n v e r t i b l e i f v(t) € J(Z ). f o r qeneric t€P^ . t mv (0.35) D e f i n i t i o n . Given the va r i e t y Z with diagram (0.10), we say that there i s an inversion theorem f or the group G of normal functions v : 'P1 >• U, J(Z = k 1(t)) i f there e x i s t s a subgroup t€P X , * G ^ n v °f G s a t i s f y i n g the following two properties; (i) Every v € G. i s i n v e r t i b l e inv ( i i ) The group generated by the cohomology classes of v , as v ranges i n G^^, i s p r e c i s e l y Prim m , m(Z,2£.) . I t i s hoped that such an inversion theorem holds f o r our s p e c i a l v a r i e t i e s Z = XxC, and i n some cases we can be more s p e c i f i c ((4.44)). We now introduce some notation which w i l l be assumed throughout t h i s t h e s i s . (i) X, C are smooth, p r o j e c t i v e v a r i e t i e s of dimensions 2m-l, 1 r e s p e c t i v e l y . We w i l l denote Z = XxC. ( i i ) A l l i n t e g r a l cohomology i s intended modulo t o r s i o n , ( i i i ) 7/_ i s the r i n g of integers, 0 i s the r a t i o n a l numbers, C i s the complex numbers, and P N denotes complex, pr o j e c t i v e N space. 1 2m— 1 *•* Chapter 1. The Hodge Structure on E 2 ' (k). A p e n c i l of subvarieties covering Z i s equivalent to a non-constant r a t i o n a l map k : Z—*• P"^  , and the purpose of t h i s section i s to show that for a p a r t i c u l a r morphism k , the important part of Prim 2 m(Z) i s contained i n E^' 2 1" ^"(k), where by d e f i n i t i o n l,2m-l - 1, 1 2m-l- . „, , n l,2m-l - . E ' (k) = H (P ,R k..C) . The module E ' (k) i s e a s i l y seen to be the E^ term of the Leray s p e c t r a l sequence f o r k , abutting 2 m to H (Z,C). I t also contains the image of the normal functions defined i n Chapter 2, which j u s t i f i e s the importance of t h i s section. (1.1) D e f i n i t i o n . A Lefschetz p e n c i l of hyperplane sections {X^J^^pl of X i s a p e n c i l s a t i s f y i n g three conditions: (i) f o r generic t^P 1, X i s smooth ( i i ) a singular section has only a s i n g l e ordinary double point ( i i i ) the base locus D of the p e n c i l i s smooth. Matters can be arranged so that X and X are both smooth, and we o . 0 0 w i l l assume t h i s throughout the r e s t of t h i s paper. Given a p e n c i l as above, we define X = B D ( X ) = blow up of X along D . There i s a morphism f : X—•••P"'" with f i b e r s f ^(t) = X^, and a commutative diagram: (1.2) X - f 1(Z) = x°« » X f f p - I = u ^ —3 » p x where E = singular set of f, i . e . , P 1-! i s the set f o r which f i s smooth (and proper). 21 To a r r i v e at a degeneration r e s u l t f o r the Leray s p e c t r a l sequence f o r k , we f i r s t prove: (1.3) Proposition. For a l l integers q , the r e s t r i c t i o n homomorphism q— q R f #C—*• D^R f*C i s an isomorphism. Proof. This i s (6.3.1) of theorem 6.3 i n [6, p. 319], One way to see t h i s i s f i r s t to note that f o r q ^ 2m-l, 2m-2 the proposition i s cl e a r [6, p. 195], and consider the following argument: Localize over 1 * a disk A c p with A - {0} = A c u, A D E = 0, X q = singular f i b e r , * and X F C = smooth f i b e r (t € A ). There i s a l o c a l vanishing cycle 6 i n H2m-2^ Xt'^ a n C ^ a "*"on9 e x a c t sequence ([6, p. 196]): (1.4) 0 — H 2 M - 2 ( X , C ) — > H 2 M - 2 ( X -C) ( ' 6 ) > C - > H 2 m - 1 ( X ,C) o t o - « 2 m - L ( X t f C ) — K ) Now the r e s u l t s i n [6, p. 196] imply that ( ,6) i s an epimorphism so that H (X ,C) = H ~ (X^,C) and furthermore by the Picard-o t Lefschetz formula, the i n v a r i a n t subspace of H 2 M 2 ( X t , C ) i s a sub-space of codimension 1. Therefore 1 R 2 m 2 f * C ^ 3 * R 2 m 2 f * l C a n c ^ R 2 m "*~f*C ~ J * R 2 m ^ f * c ' hence the above assertion i s proven. Re c a l l that-Z = XxC. NoW define Z = XxC. There i s a morphism k:Z—>• with f i b e r s k '^"(t) = X xC,'and a diagram analogous to-(1.2) above: X°xC = Z ° ^ > Z (1.5) |k |k '1 P -L - U<- - 7> P From (1.3) we obtain the following useful lemma: 1 '2 iti"" 2 " The sta l k s of j'^ R f ^ l consists of those coeyles which are in v a r i a n t (via deformation) under the Picard-Lefschetz transformations (see (2.10)"). Also H ° ( A , R 2 m - 2 i ^ ) c H 2 m - 2 ( X 0 , C ) . (1.6) Lemma. For a l l integers q. , the r e s t r i c t i o n homomorphism q — q R k^C —^-j^R k^C i s an isomorphism. Proof. Use (1.3) applied to the Kunneth formula decomposition of H q(X txC). (1.7) C o r o l l a r y . The Leray s p e c t r a l sequences for f and k degenerate at the terms. Proof. Let L be a very ample d i v i s o r class which induces on the fi b e r s of f (resp. ' k) a very ample d i v i s o r . For a l l 1 * 0 , there i s a commutative diagram f o r f ( r e s p . k ) : r •• . _2m— 2—•£ „ \-/[L] : j R f.C (1.3) (1.8) (resp . ~ (1.6)) (strong Lefschetz theorem) . 2m-2+^ (1.3) (resp. (1.6)) 2m-2~£- „ f * c 2m-2+-e-(induced isomorph-ism) where the bottom row induced isomorphism i s the unique one making the diagram commutative. We now apply Deligne's c r i t e r i o n ([5]) . We now set out to i d e n t i f y the p r i m i t i v e cohomology of Z . More p r e c i s e l y we have the following: (1.9) Proposition. Prim 2 m(Z) = P r i m 2 m - 1 ( X ) ® H 1 (C) • © ( [X xC] - [ XxC ])-^Prim 2 m~ 2(X) ® H°(C) t t where C^ i s a hyperplane section of C , and [ ] denotes the Poincare dual. 23 Proof. V i a the Segre embedding; ( 1 . 1 0 ) , the pullback of the hyperplane c l a s s , * * i * 0 N ( D i s equal to P^ 6 ( 1 ) <8> ^  P 2 0 (.1 ) , which corresponds to the P 2 very ample d i v i s o r X x C + XxC . Denote L.. = X xC, L„ = XxC, L = L n+L„ t t 1 t 2 t 1 2 2m (6„(L) a* i * G M ( D ) . Now v^L acts on H (Z) i n the following way: Z pN v_/L = : H 2 m~ 2(.X) ® H2(.C) »• H 2 m(X) ® H 2 (C) v-z-L =v^L : H 2 m - 1 ( X ) ® H^C) > H 2 m + 1 ( X ) ® H l C ) = UC^+L ) : H 2 m(X) ® H°(C) »• H 2™* 2 (.X) ® H°(C) © H 2 m(X) ® H 2 (C) Now l e t [ y ^ € H 2 m - 2(X)., [y 2] € H2m(.X) . Then L V ^ ^ ] ® [C f c]) = [ Y , 0 X . ] <8> [C J and L ^ ( [ Y o ] ® [C]) = I Y ~ 0 X . ] ® [C] © [ Y ~ ] 3 [C.] -I t t 2 2 t 2 t Now L ^ ( [ Y - j J ® [C 1) = L ^ C t Y 2 ] ® [C]) i f and only i f [ Y 2 1 = [ Y - ^ ^ and [ Y 2 1 ^ [ X ] = 0 , thus [ y ^ * Prim (X). I t i s easy to check that the above argument implies ( 1 . 9 ) . Remark. Since the cohomology of main i n t e r e s t i n t h i s thesis w i l l be H 2 m _ 1 ( X ) ® H 1(C), there i s no loss i n generality i f we assume Prim 2 m(Z) = P r i m 2 m - 1 ( X ) ® H 1(C). .To a r r i v e at-the main r e s u l t of..this section, the following lemmas w i l l be needed. -Throughout t h i s paper we w i l l denote Z^ = X^xC. We f i r s t introduce the following i n c l u s i o n morphisms: (i) i : Z 0* 1—y Z , j : X ° c — X o Jo ( i i ) i : Z <=—-> Z , j - . \ c — ( i i i ) i 2 : D x C ^ Zfc , J 2 : D-*» X ( i v ) ± 3 : Zt*_+ z , J 3 : X t V X (1.11) Lemma. -2 (i) H (X) ~ H (X) © H (D) ( i i ) there i s a commutative diagram: H 2 m ( I ) 1 m 2 * ^ ) H 2 m(Z) © H 2 m- 2(DXC) ( i i i ) the Gysin homomorphism i : H 2 m 2 ( Z ) *-H 2 m ( z ) becomes: 1,* t H 2 m - 2 ( Z t ) ± 3 ' * 0 ( " ± 2 ) >H 2 m(Z) © H 2 m - 2 ( D X C ) Proof. (i) follows from [6, p. 272]. By applying the Runneth formula to H (Z = XxC), the assertions ( i i ) and ( i i i ) are a t r i v i a l consequence of proposition 5.1.1 i n [6, p. 279]. 1 ?m— 1 — * ?m — ?m * (1.12) Lemma. ET' (k) - ker i , : H (Z) -4i (Z ) / ker i : 2 1 t o n2m(z)—m2m(z°). Proof. This i s (3.6) i n [28, p. 194], Note that t h i s induces a Hodge structure on E 1 ' 2 m "*~(k). An i n t r i n s i c d e f i n i t i o n of the Hodge structure on E 1 ' 2 m ^ (k) i s given i n [31]. We have been leading up to our main r e s u l t , namely: (1.13) Theorem. E 2 m 1(k) ~ Prim 2 m(Z) © H 2 m 2(DxC) where 2 V ..2m—2 , „ „. H (DxC) = ker l .2m-2. .2m, n * ' H (DxC) HH (.Z. ) (t € U) Proof. \—' = [Z t] can be described as a composite of the following homomorphisms (p = Poincare d u a l i t y ) : * i . ( i . i 4 ) ^ m i n ^ j j L , Applying the Kunneth formula, the following commutative diagram i s obtained: H 2 m(Z) = H 2 m ( X ) ® H ° ( C ) © H 2 m 1 ( X ) O H 1 ( C ) © H 2 m 2 ( X ) ® H 2 (C) v 12m. ,2m. i ^ (weak Lefschetz) H""(Z t) = H"""(X t)«>H 0(C) © H 2 m 1 ( X t ) ® H 1 (C) © H 2 ™ 2 ( X ) ® H 2 (C) H 2 a - 2 ( Z t ) = H2m-4 (V® H2 ( C ) @ H2m-3 (V ® 1 ( C ) ® H2m-2 (V^ Ho ( C ) H "3,* i # (weak Lef.) 2m-2 (Z) = H2 m - 4 ( X ) < g , H 2 ( C ) 8 H 2m-3 i ^(weak ' Lef.) [X) ® H (C) © H i (weak J ' Lef.) 2m-2 (X)®H (C) o Note that H " (X t) = J 3H " (X) © ker j 3 *([28, p. 200]),so that * * i _ ^ i s i n j e c t i v e on the image of i . Therefore ker i i n (1.14) i s equal to ker W x, = © 2 H 2 m ~ q ( X ) ® H q (C) , where H* (X) = k e r V h : ± o o q=o •+2 H (X) >H (X), h being a hyperplane class of X . Therefore (1.15) ker i * = H 2 m ( X ) ® H ° (C) © Prim 2 m(Z) . 3 o 26 There i s a commutative diagram (t € U) H 2 m - 2 ( Z t ) - H 2 m - 2 ( X t ) ® H ° ( C ) © H 2 m - 3 ( X t ) ® H 1 ( C ) © H2**"4 (X^ © R 2 (C) (1.16) (weak Lef.) '3,* "3,: H 2 m(Z) ~ H 2 m ( X ) ® H ° ( C ) © H 2 m ~ 1 ( X ) « > H 1 (C) © H 2 ™ 2(X)<8>H2 (C) I f we denote X q = singular section (o € I ) , then from [6, p. 196] we have H q ( X j ~ H q(X ) f o r a l l q f 2m-2. Also H 2 m ~ 2 ( X j ^ t o t j!H 2 m" 2(X) © H 2 m _ 2 ( X J , where H 2 m ~ 2 ( X ) = ker j , t , which i s the 3 t v t v 3,* 2m~* 2 ^ 2m~" 2 subgroup of H (Xfc) generated by the vanishing cocycles (jgH (X) corresponds to the f i x e d part of a v a r i a t i o n of Hodge structure r e l a t e d to the Hodge bundle over U with f i b e r s H 2 m 2 ( X ^ ) ) . From the above discussion and (1.16) i t i s c l e a r that the image of R 2 m 2k^C (via , . 2m. , x )^ i n H (Z) i s a constant system. -3 / There i s the analogous diagram to (1.16) above: H 2 m - 2 ( Z t ) ^ H 2 m - 2 ( X J ® H ° ( C ) (1.17) H 2 m- 2(DXC) 2m-2 H 2 m 3 ( X ) ® H 1 ( C ) © H 2 m _ 4 ( X )®H 2(C) (D)®H (C) © H (weak Lef.) 2m-3 (weak Lef.) (D)®^ 1 (C) © H 2 m 4(D)®H 2(C) * 2m-2 1 To show that i 2 H (Zfc) i s i n v a r i a n t i n t € P , i . e . , the image of R2m ( v£ a j_ j ^ s a c o n s t a n t system i n H 2 m 2 (DxC) , i t s u f f i c e s to 27 c o n s i d e r the image o f j 2 : H 2 m 2 ( X f c ) H i 2 m " 2 ( D ) u s i n g a s i m i l a r r e a s o n i n g as above. There i s a commutative diagram: P .(1.18) j * 33,* H 2 m 2(D) ~ H .(D) 2m-4 n D ^ 2 m - 2 ( X ) ^ 2 m ( X ) P wh ^ ! > H 2 m - 4 ( V ^ H 2 m - 4 ( X ) ^ ( X ) (weak L e f . ) (weak P L e f . By a s i m p l e diagram chase, i t i s easy t o see t h a t J 2 H ^ X t v = From (1.11) p a r t ( i i i ) we co n c l u d e t h a t the i m a g e ( v i a i , .) o f 1,* R 2 m 2k^C i n H 2 m ( Z ) i s a c o n s t a n t system so t h a t v i a the G y s i n sequence we have: (1.19) k e r i * : H 2 m ( i ) —>H 2 m(i°) = Im i , 4 : H 2 m _ 2 ( Z ) — H 2 m ( Z ) o 1, * t f o r any t € u . There i s a commutative diagram: (1.20) ~ 0 [hyperplane c l a s s ] H 2 m - 2 ( x t > < : H 2 m - , 4 ( x j (weak Lef.) _ „ W [hyperplane c l a s s ] H 2 m _ 2 ( D ) < H 2 m ~ 4 ( D ) ~ ( s t r o n g L e f . ] and i t i s easy t o see from t h i s t h a t j * : H 2 m 2 ( X f c ) ^ H 2 m 2(D) i s s u r j e c t i v e . Also H 2 m 3(D) ^ j * H 2 m - 3 ( x J © H 2 m~ 3(D) , therefore from 2 t v (1.17) we get (-i*) H 2 m " 2 ( Z t ) fl H 2 m _ 2 ( D x C ) v = 0. There i s an analogous diagram to (1.20): ^ [ h y p e r p l a n e class] t T m /(X ) < : H , ( X j j (weak Lef.) t j 3,* [hyperplane class] H (X) H Z(X) (strong Lef.) so that j _ . : H 2 m 4 ( X ) ^H 2 m 2(X) i s s u r j e c t i v e . Note also that 3 , t 2 in 1 2 m 3 2itt 1 H (X) ~ j . H (X ) © Prim (X). Combining these r e s u l t s with 3, * t (1.11), (1.16) and (1.17), i t i s easy to check that the expression i n (1.12) becomes: (1.22) E i'. 2 m- 1<k) c Prim 2 m(Z) © H 2 m _ 2(DxC) . 2 v (1.23) C o r o l l a r y . H m , m ( Z ) fl ' 2 m _ 1 (it) ^ Prim 1"' 1"" 1 (X)<8> H ° ' 1 (C) © Prim m" 1' m(X)^»H 1' 0(C) © H m" 1' I t" 1(DxC) v Proof. Apply (1.9) to (1.13) Chapter 2. The Intermediate Jacobian J( Z . ( J We w i l l define J ( z t ) a n c ^ show that i t admits a n o n - t r i v i a l Abel-Jacobi morphism. The image of the Abel-Jacobi morphism w i l l be c a l l e d the i n v e r t i b l e part of J ( z t ) • As the terminology suggests, we w i l l be i n t e r e s t e d i n i n v e r t i n g normal functions so that the cohomology classes i n Prim m' m(Z,Q) w i l l come from normal functions whose cohomology classes are r a t i o n a l multiples of algebraic cocycles. The-ultimate attempt i s to f i n d an-inversion theorem (;(0.35) ),-which i s only known i n c e r t a i n cases (for example, see (4.44)). 2 i t i — 1 There i s a Hodge f i l t r a t i o n on H (Z^) defined by: (2.1) F PH 2 m- 1(Z t,C) = H 2 1" - 1' 0^) ©....© H P ' 2 m _ 1 " P ( Z t ) where p £ 0 and t € u . (2.2) Define F^'* H2m_"'* (Z^_,C) = H 2 m _ 1 (Z^_,C) / F 2 m _ P H 2 m _ 1 (Z^ _ ,C) . m * 2m—1 For J ( Z ^ ) , we are mainly inter e s t e d i n F ' H (Z t,C) = H m _ 1 , m ( Z t ) © ... © H ° ' 2 m " 1 ( Z t ) . Via p r o j e c t i o n , H 2 m - 1 (Zfc,7£) embeds m * 2m—1 i t s e l f as a l a t t i c e i n F ' H (Z^_,C) . We now have the following: (2.3) D e f i n i t i o n . For t € U, the intermediate jacobian J(Z^) i s defined to be: j ( z ) = F™'* H 2 m _ 1 ( S t , C ) / H 2 m - 1 ( z , ID . Remarks: (i) t h i s d e f i n i t i o n appears elsewhere (['11, p. 331] K.-( i i ) J ^ z t ) ^ s n o t general an abelian v a r i e t y ([27]). The Kunneth formula provides us with an i n c l u s i o n H 2 m ^(X^)® H (C) c m ( Z J . Let {y} € H n .(X^.,"^), where y i s a 2m-2 t 2m-2 t chain. Then y defines a cy l i n d e r homomorphism p ^ : H^(C,72-) • H 2 m - l ^ Z t ' ^ obtained from the correspondence c € Ch—* yxc i n x t x C -The graph of the c y l i n d e r map i s a r e a l 2m cycle T c CxZ^ and v i a 2ro Poincare* d u a l i t y we obtain [T] € H (CxZ^,7Z.) . V i a the Kunneth 1 2lQ— 1 formula, the pro j e c t i o n of [T] i n H (C,7£) ® H (Z ,7J.) defines 7C t a homomorphism: (* denotes dual space) (2.4) H. (C,7Z) £ H 1(C,^.)* >H 2 i n~ 1(Z .X) £ H. , ^ , 3 1 ) which 1 t 2m-1 t recovers the cylinder homomorphism . Thinking of [T] € H 2 m(CxZ t,C) and p r o j e c t i n g [T] i n t o H 1 , 0 ( O ® ( H m ~ 1 , m ( Z ) © ... © H ° ' 2 m " 1 ( Z t ) ) = H 1 ' 0 ( C ) ® F m ' * H 2 m " 1 ( Z t , C ) we obtain a complex homomorphism: ( denotes conjugate) (2.5) p C : H 0 , 1 ( C ) — ^ F m ' * H 2 m _ 1 ( Z ,C) which i s induced by p o p . y t Y Therefore p defines a morphism $ : J ( C ) — M ( Z ) which i s a y y t transcendental analogue to the Abel-Jacobi morphism defined i n [27]. (Note: J(C) i s the usual Jacobian •of the curve C.) We now prove: (2.6) Theorem. (i) Every cl a s s i n H 2 ™ " 2 (X f c,C)®H 1 (C,C) fl J(Z f c) (t€u) l i e s i n the subgroup generated by $^(J(C)) where y ranges through H 2 m - 2 ( X t ' 7 ^ ' 31 Moreover ( i i ) i f y i s an algebraic cycle, then $^ i s the usual Abel-Jacobi morphism"''. ( i i i ) The image generated by $ (J(C)') as y T Im-l,m-l. _.. „2m-2. . r, „m-l,m-l, . ranges i n H ( X ,71.) = H ( X ,7D f l H ( X ,C) i s t t t p r e c i s e l y H m _ 1 ' m - 1 ( X ^ ,71) <S>H0,1(C) f l J(Z ) . t 7L t (2.7) C o r o l l a r y . I f the Hodge (m-l,m-l) conjecture i s true f o r X ^ , then the image of the (algebraic) Abel-Jacobi morphism contains Hm-l,m-l ® H 0' 1^) f l J(Z„). t 7Z. t Proof .of (2.6) . (i) Let [ x ] _ € J ( Z j f l H 2 m " 2 (X. ^ J ^ H 1 (C) . Then 1 " J t t [x] comes from [x] € H 2 m ~ 2 (Z .C)® H 1 (C,C) . Since H 2 m - 1 ( Z .£) ~ J t t H 2 m ^ (Z ,7D C§ c, we may assume that [x] = E N r ^ [y^ ] <8> [C.l where i = l r ± € C, [y±] € H 2 m 2 ( X t , - * ) , [c±] € H 1(C,7Z). Now each y defines a c y l i n d e r homomorphism p . : H(C,7£) ., (Z .7£) , and hence an yi 1 2m-l t Abel-Jacobi morphism $ . : J(C) KJ(Z-). Furthermore, p ,(r.) = yi t yi I y^  <8> . There i s a commutative diagram: H.(C,/fc) ^ v H 2 m - 1 ( Z 71) t (2.8) H 0' 1^) ^ , F m ' V ^ V ,C) J (mult, by r j (mult, by r j H ^ 1 * (C) ^ -> F ^ V ^ V ,C) 1"Usual" r e f e r r i n g to the d e f i n i t i o n of the Abel-Jacobi morphism, i n v o l v i n g .algebraic .cycles .(.s_e.e__(A._l)J .. ' . • 32 which simply says that p i s a complex homomorphism. I t i s now <C obvious from (2.8) that p (r^[£j)= r ^ [y^] ® so that N E $ . (J(C)) contains [x] . F i n a l l y ( i i ) and ( i i i ) are obvious. i = l Y 1 J 2m 2 Remark. I f H (X^jg) i s generated by algebraic cocycles, then i t i s expected that the following statement holds: (2.9) Statement. Every holomorphic cross section (normal function) of the family U J(Z ) i s i n the image of the (algebraic) Abel-Jacobi P 1 mapping, hence i t i s i n v e r t i b l e . Therefore the Hodge conjecture holds f o r Prim m , m(Z,$>) . (This statement i s v e r i f i e d i n Chapter 4) . In order to discuss normal functions, we need to rigorously define J(Z^) f o r t € E . This i s done i n terms of the nearby f i b e r s of i n (1.5). The d e f i n i t i o n of J ( z f c ) w i l l involve the extension of some l o c a l l y free sheaf over U to one over , therefore we w i l l now s h e a f i f y everything. Over U we have the Leray cohomology sheaves F = 6 T T®_R 2 m _ 1k.C ~ R 2 m _ 1 k . f i * with Hodge f i l t r a t i o n subbundles U C * * -o . z /u F P = R 2 m " V F V , where fi i s the complex of r e l a t i v e i°/u z°/u d i f f e r e n t i a l forms, F P i s the p Hodge f i l t r a t i o n , and R denotes the r i g h t derived hypercohomology. By the monodromy theorem, the l o c a l monodromy (Picard-Lefschetz) transformations T s a t i s f y the i d e n t i t y 3 3 (T N - I ) q + 1 = 0 for N, q > 0 ( i . e . , T i s quasi-unipotent). We can be more precise though: ( 2 . 1 0 ) Proposition. The l o c a l monodromy transformations T s a t i s f y 2 the i d e n t i t y T = 1 . Proof. We l o c a l i z e the family i n ( 1 . 5 ) over a disk A c p 1 with A (1 E = 0 € A . Applying the Kunneth formula to H 2 ™ " 1 ^ ) (t € A * ) , i t s u f f i c e s to consider the l o c a l monodromy transformation T associated to H 2 m 2(X f c) about O C A . Let y € H 2 m ~ 2 ( X t ) . From [ 6 , p. 1 9 6 ] we have the Picard-Lefschetz formula: ( 2 . 1 1 ) T(y) = Y + ( - D m - 1 ( ( e ( T ) - l ) / 2 ) (-y,6) 6 where e(T) = - 1 , * e : I I ^ ( A ) — ^ - { - 1 , 1 } being a character, and 5 = l o c a l vanishing cocycle. Therefore T 2 ( y ) = T ( Y ) + ( - 1 ) m _ 1 ( ( e ( T ) ~ 1 ) / 2 ) ( Y , 6 ) T ( 6 ) . I t i s easy to compute T ( 6 ) = 6 + ( - 1 ) m _ 1 ( (e ( T ) - 1 ) /2 ) . (6 ,<5) 6 where ( 6 , 6 ) = ( - l ) M _ 1 2 , i . e . , T(6) = 6 + 2((e(T) - 1 )/2)6 = e(T)6. Therefore T 2y = T(y) - ( - 1 ) M - 1 ( ( e ( T ) - l ) / 2 ) ( Y , 6 ) 6 =y, i . e . , T 2 ( Y ) = Y -( 2 . 1 2 ) We now define F P ' = F/F2m P . There i s a. short exact sequence: ( 2 . 1 3 ) 0 — ^ R 2 m - 1 k ^ 7 /— ^ F m ' * vj K) over U, where the cokernel sheaf J can be interpreted as the sheaf of holomorphic cross sections of the family U J(Z ) of jacobians over U . We wish to extend t€u ( 2 . 1 3 ) over P 1 , and we choose the so c a l l e d "quasi-canonical" extension. This extension i s a c t u a l l y a l o c a l procedure, so we l o c a l i z e over a disk A c p 1 with A fl E = 0 (: A . This procedure of extension i s discussed i n :[31, §6], however since- the monodromy i s so simple, a s i m p l i f i e d approach w i l l s u f f i c e . We f i r s t extend F to P 1 . Via it the Kunneth formula i t s u f f i c e s to consider the Leray cohomology sheaves F^ . with Hodge f i l t r a t i o n subsheaves F P associated to OTYI— O (1.2), where Vc = 9 ® R f j c . t u c * * 2m-2 Let t € A , and choose {v^,... v^} € H ( X t , ( t ^ s u c h that (v^,6) = 0 for a l l i = 1,...,N (6 = l o c a l vanishing cocycle), and so 2m— 2 that {v^,...,v ,6} i s a basis of H (X^,€). Now F f defines a * f l a t bundle over A , so by p a r a l l e l t r a n s l a t i o n we can extend * {v ,...,vN,6} to h o r i z o n t a l multivalued sections of F^ over A (horizontal with respect to an integrable connection - the Gauss-Manin connection, defined i n (2.14)). By the Picard-Lefschetz formula, the {V^,...,Vn} are single valued, and 6 i s the "eigenvector" section associated to the eigenvalue -1 of the l o c a l monodromy transformation T, i . e . , T6 = -6([6, p. 196]). Define v(t) = exp((log(t))/2)6 . Then * v(t) i s a si n g l e valued section of F^ over A . The quasi-canonical extension of F^ over A i s defined to be the sheaf generated by {v ,...,v N»v(t)}. One checks that t h i s sheaf i s i n fa c t free over A , and that v(t) vanishes at the o r i g i n . By 2 * * considering the double covering s = t : A > A , i t i s easy to see that the l o c a l monodromy transformation associated to s i s * t r i v i a l , a f o r t i o r i .unipotent. Therefore, s F extends t r i v i a l l y to a free sheaf over A and t h i s i s known as the canonical extension of * s F^, which i s the extension associated to unipotent l o c a l monodromy transformations (see [3, p. 91] for more information). (2.14) There i s the natural integrable "Gauss-Manin" connection V 2m 2 defined on F^ = 6 ® R f C , which i s of the form d ® 1. Over r U P 1, F^ extends to F- , and V extends to V . Now l o c a l l y over the disk A mentioned above, Vv(t) = (v ( t ) / 2 ) d l o g ( t ) , so that VF- £ 9. (logE)0 F ~ . The extensions F- of F^ are defined as: P - P _P - 2 2 F- = j^F". fl F- . Recall the morphism s = t defined above. Then f * f f the following i s true: (2.15) Proposition. H°(A,F-) = {a £ H°(A,s*F-) | a(t) = a(-t) for} a l l t € A o Z * 2 Proof. Let a € H ( A , F - ) . Then s a = a(t ) which l i e s i n the RHS of (2.15). Conversely i f a i s a member of the RHS then a(/s) i s well defined f o r a l l /s~ € A . We need to prove that a(/s) £ H°(A,F-). Now o(i/s) = £ N g ^ / s j v . + g (/s)6, where i = l {v^,...,vN,6} i s the ho r i z o n t a l multivalued basis of H°(A,F-). Since {v^,...,v } i s a si n g l e valued set of hori z o n t a l independent sections, c l e a r l y g^(/s) must be holomorphic i n s i n A for i = 1,...,N , and since each g. i s bounded on A , g. ( / i ) must be holomorphic i n s over A for i = 1,...,N . Since g o(/s) i s a holomorphic function times v ( s ) , i t i s obvious that the above assertion holds. 2 A further discussion of t h i s appears i n (A.2) 36 ~P (2.16) C o r o l l a r y . (i) r- i s completely characterized by i t s growth, of periods near E . ( i i ) F^ i s uniquely characterized by the property ~ ~p that r- / F- i s l o c a l l y free. Proof. This i s a r e s u l t of the above discussion, (2.15) and the properties of the canonical extension ([28, p. 190], also [3, p. 91]). — p i (2.17) The i n f i n i t e s i m a l period r e l a t i o n holds: VF- c fi ^(logE) P ® FT 1 . f Everything c a r r i e s over to the F p associated to (1.5) . The short exact sequence of (2.13) extends over P 1 to: (2.17) 0 " X " * — * P • • ^ J ^ ° ' with coboundary homomorphisms 6 : H°(P1,J)—>-H1(P'1', j ^ R 2 m "*"k^,"2i), and commutative diagram: (2.18) H°-(P\J) &- - ^ 1 ( P 1 , j # R 2 m - \ t Z ) (composite) E ^ ' ^ - ^ k ) I t should be remarked that ker § /ker <S = T , where T i s a t o r s i o n sub C o o group of H°(P 1, J)/ker 6 . This follows from the f a c t that H^P 1, j . R 2 m hi. 71) "may have t o r s i o n , i . e . v i a 6, T embeds i n * * o H"'"(P''",. j ^ R 2 m 7/) so that i f we. divide out the t o r s i o n i n H^CP1, j^.R 2 m hfi'-yc.) , then T i s also removed.- For the purpose of . o s i m p l i f y i n g the statements of the r e s u l t s i n t h i s t h e s i s , we assume H^P 1, j.R 2™ "Sc.TZ) i s t o r s i o n l e s s , so that T = 0 thus ker <5 = ker 6 * * o <E __ A precise i n t e r p r e t a t i o n of t h i s statement can be found i n (A.2). 37 We introduce the following basic d e f i n i t i o n : (2.19) D e f i n i t i o n . A normal function i s an element of H°(P"'",J) . The Gauss-Manin connection defines a homomorphism: (2.20) V : F m ' * = F / F m (logZ) <8> F / F m _ 1 , and since V k i l l s P j ^ R 2 m ±ki!~Z£, i t defines a homomorphism: (2.21) V : J • J}1 (logE) ® F / F 1 " " 1 . P (2.22) D e f i n i t i o n . The h o r i z o n t a l normal functions, written H°(P 1,J), , are defined to be the kernel of V : H°(P 1,J) *• h H°(P 1 , n 1 (logZ) ® F / F m _ 1 ) . P Remark. I t w i l l l a t e r be shown (Chapter 3) that H°(P1,J) = H°(P 1,J). We now prove the following: (2.23) Proposition. (i) j ^ R 2 m - 1 k A K ^ © 2 J A R 2 m _ 1 ~ q f ^ K ® H q(C,K) q=o where K = 7£ , <g , or C . ( i i ) F m ' * c F ? ' * ® H°(C) © Vf1'* ® H 0' 1^) f f . . . . . rm-1,* r m - l / r m as,- r m#* ( i n ) F - - F - / F - © F - i f a n d o n l y i f t h e cup product p a i r i n g defined i n (0.20) induces a holomorphic cup product p a i r i n g : F ^ " 1 ' * x F m _ 1 ' * — > 6 -f f " U " " Proof, (i) , ( i i ) . a n d (==>) of ( i i i ) are obvious. Therefore we prove only ( i i i ) (<£=. ) . There i s a short exact sequence: 38 0 >• Fm > F- — y Fm y 0, and. an i n c l u s i o n of sheaves: F- c >• F-  c y ¥- , so that F- /F- i s a holomorphic subsheaf of F? 3"' We now l o c a l i z e over a disk A . Let 9 ' be the l o c a l i z a t i o n A,t of 0^ at t € A , and l e t {w^ tw^} be a basis of H°(A, Fm 1 ' ), _m-l * -m-1 * which we can assume generates the stalks F-^  ' = F- ' ® 0, as f , t f A,t an 9. ^  module. Note that N = dim F m _ : LH 2 i n~ 2 (X ,C) for t € A fl U. A , t C t Let r = dim H m 1 , m 1 ( X ,C). There e x i s t s a basis {s n,...,s ) of C t 1 r T To,. 7-m-l , r m K , • •, . , ,, ,T=m-l . T^m. H (A,r- / r-) which generates the stal k s ( r- / r - ) ^ = F-| 1 / Fm ® 0^ ^  for a l l t € A . There i s no loss of generality i n assuming that s. = w.,...,s = w . A t y p i c a l a € H (A, F- ' ) 1 1 r r f N o i s of the form a = E a.w, where a. € H (A, 0,) . Also av_>w. = j = 1 i i : E a.w. v_> w. . Define c. . = w . w . € H (A,F- ^ F- ) . 3 3 1 i ] i ] f f 2m~~2 2in~"2 Then c.. i s a holomorphic section of the bundle U H ' (X ,C), which i s i n f a c t a t r i v i a l holomorphic l i n e bundle over A . Therefore c.. can be i d e n t i f i e d as an element of H°(A,0„). Define I J A T = {a C H°(A , F m 1 ' ) I E N a .c . . = 0 for a l l i = 1,...,r}. There i s a f j=l 3 3 1 short exact sequence: 39 (2.24) 0 — T — F m _ 1 ' * — 1 — 0 ^ — > 0 , with res ty : ff1 / F™ r ~m * y 0^ i n j e c t i v e . Note also that T £ F-' . Let K be the sheaf of meromorphic functions on A . ' Tensoring (2.24) with - ® Ry we see that the • " • • - • • ' : - V morphism res ty : F™ 1 / F m ® K -> K r i s an isomorphism, and also T ® K = F m' ® K . Let M be the sheaf which makes the following a short exact sequence: (2.25) 0 • F j _ 1 / F m ^ > 0^—»- M »•• 0 . M i s i n f a c t t o r s i o n l e s s . Then tensoring (2.25) with K and looking at the corresponding short exact sequence, we obtain:-0 y F| - 1 / F| ® K *• K r v M ® K »• 0 • Therefore M ® K = 0, hence M = 0 so that ty i n (2.25) i s an isomorphism. This immediately implies that F m ^' = T © F™ ^ / F m , so that v i a the pr o j e c t i o n r- > r- , T = r-(2.26) Co r o l l a r y . (i) E ^ ' 2 m _ 1 ( k ) ~ © 2 H 1(p 1, j ^ R 2 m ~ 1 _ q f (p) ® H q(C) q=0 ( i i ) H^P 1, j ^ ^ k ^ / ) - © 2 H^P 1, j . R 2 " 1 " 1 ^ ^ / ) q=0 ®H q(c f-7^) ( i i i ) Fm'* r f ' * ® H°(C) © ~ff* ® H X(C) © ff1'* ® H 2(C) © F m _ 1 / F| ® H 0 , 1(C) , provided that —m-1 * the s p l i t t i n g of F ' i n (2.23) ( i i i ) holds. f 40 Proof. Obvious. We now define J to be the image of F m ' <8> H^CC) i n J , v i a the composite r- ® H (C) c F »- J , whenever F = F_ / F_ © F_ . f f f f We now prove the following which b a s i c a l l y says that we can d i s c a r d a l l h o r i z o n t a l normal functions which take t h e i r values i n a p a r t i c u l a r Hodge l e v e l i n the family of intermediate Jacobians. (2.27) Proposition. H°(P 1, J ) h c kerfi . o 1 - ' Proof. Let cr € H -(P , J ) . Then a i s equivalent to the data a = {a , u } , a - aD € H ° ( u '= u fl u Q , j t R 2 l I h l k 4 7 / ) , a' a a 3 a3 a 3 J * *"-''• a€l o ~"m * 1 _ a € H (u , F-' ) ® H (C) and Va =0. We can write a a f a a = E 2 g. y ® v. where {v , ...,v } i s a basis of H1 (@ ,7£) , g 0« . _ Ci . ~i _L «^ CT :=i - D J being the genus of C . Now (Sa) = a - a = E 2 g ( Y - Y Q ) ® v. , where n a p . a. p. ] a3 :=1 3 3 Y - Y 0 € H°(U a , j . R 2 m ~ 2 f . 7 1 ) . Now define a 1 = y ® v, . Then a. 3. ap * * a a, 1 3 3 1 a 1 = {a \ u } s a t i s f i e s the properties a 1 - a 1 € H°(U „, j^R 2™ 1kJ..'^ .) , a a ,,. a 3 ap * * a 1 6 H°(U , F™/*) ® H 1(C), and Va 1 = 0 . That i s a 1 defines a a a f h o r i z o n t a l normal function. From [31, §9], 6a 1 i s of type (m,m). 41 Now i t can be checked that no c l a s s of type (m,m) can be represented i n the form y <8> v^ unless i t i s zero. Such an exercise w i l l be l e f t to the reader, so that the above (2.27) i s proven. Remark. The theorem we used i n the reference mentioned above, [31, §9] i s the optimal version of the theorem-on-normal functions which states the following: (2.28) Theorem. A l l the i n t e g r a l classes of type (m,m) i n El,2m coincide with the cohomology classes of the h o r i z o n t a l normal functions. In the next section we prove our main r e s u l t on the normal functions, namely that a l l the normal functions are h o r i z o n t a l . Knowing t h i s enables us to prove a stronger r e s u l t than (2.27), 0 1 _ i namely H (P , J ) c ker6^ .- - In f a c t we can do much better by e x p l i c i t l y describing kerdi^ (3.58). 42 Chapter 3. A Vanishing Theorem arid i t s C o r o l l a r i e s . In t h i s chapter an e x p l i c i t d e s c r i p t i o n of ker 6 : H°(P"'',J) y H 1 ( P \ j * R 2 m ^k^T^) i s given, and as a c o r o l l a r y to the general machinery developed we deduce our main theorem on the normal functions. The d e s c r i p t i o n of ker 6 comes from a vanishing theorem, and at t h i s point we f i r s t e s t a b l i s h the machinery to provide the correct s e t t i n g fo r the statement of the theorem. We f i r s t prove a few lemmas. , -> -, v r T 2 m , „ . . . „2m-2,„ . „ . 2m. „. (3.1) Lemma. H (Z) / I H (Z ) ~ Prim (Z) . 3,* t Proof. There i s a commutative diagram: (3.2) H 2 m _ 2 ( Z ) ^ H 2 m - 4 ( X J ® H 2 ( C ) © H 2 m ~ 3 (X ) ® H I C ) © H 2 m _ 2 ( X J -2> H° (C) I ' - i t t H 2 m ( Z t ) ® V C ) fflH2m-l(Xt) 0 H 1 ( C ) ® H 2 m - 2 ( V ® H 2 ( C ) 3,* "3,* "3,* (weak Lef.) 3,* H2m ( Z ) '" H2m ( X ) ® V C ) ® H 2 m - l ( X ) " V F * ®H 2m-2(X) ® V C ) ,2m 2m-2 2m-1 2m, H (Z) ~-H (X) ® H (C) © H (X) O H (C) © H (X) ® H (C) for which H (X) / j H (X ) o* Prim (X) by the p r i m i t i v e •J , t decomposition theorem, and there i s a commutative diagram: (3.3) 2m-4, . H (Xt) 2m-2, , -> H, ( X J H 2m-2 J 3 / ( X ) -j ^(weak L e f . ) - H 2 m ( X ) ( s t r o n g L e f . ) where h i s a h y p e r p l a n e c l a s s which i n d u c e s a h y p e r p l a n e c l a s s on X^ From (3.2) and (3.3) we see t h a t H 2 m ( Z ) / i ^ H 2 m ~ 2 ( Z ) ~ P r i m 2 m 1 ( X ) 3 , t ® H 1 ( C ) , which i s d e f i n e d t o be P r i m 2 m ( Z ) (see the remark on p. 23). (3.4) Lemma. H 2 m _ 1 ( Z j ~ H 2 m - 1 ( Z j © i * H 2 m _ 1 ( Z ) . t t v 3 P r o o f . There i s a commutative diagram: H 2 m - . 1 < Z ) - e , H 2 m - 1 ( X ) ® H ° ( C ) © H 2 m _ 2 ( X ) ® H 1 ( C ) © H 2 m " 3 ( X ) ® H 2 ( C ) (3.5) 2m-l (weak Le f . ) (weak Lef.) (zt) 3,' ^ H ? m 1 ( X t ) ® H°(C) © H 2 m 2 ( X J ® H 1(C) © H 2 m 3 ( X j ® H 2 (C) (weak Le f . ) L3,' (weak Lef.) "3,* "3,* H 2 m + 1 ( Z ) ^ H 2 m + 1 ( X ) ® H ° ( C ) © H 2 m ( X J ' ® H 1 { C ) © H 2 m _ 1 ( X ) ® R 2 (C) A l s o H 2 m 2(X f c) ~ j * H 2 m _ 2 ( X ) © H 2 m ~ 2 ( X t ) v . T h e r e f o r e the * m i d d l e column o f (3.5) s a t i s f i e s image i fl k e r i ^ = 0 . F o r the 3 3, o t h e r two columns we use the f o l l o w i n g commutative diagram: t j , 4 (weak Lef.) '3/ H ^ c x ) ~ U . H 2 m + 1 ( X ) which implies that i „. : H 2 m 3 ( X ) *> H 2(c)—»- H 2 m -"-(X) 8> H 2(C) i s 3,* t * i n j e c t i v e , and therefore image i fl ker i ^ = 0 f o r a l l three 3 3, , 2m-l,„ . r T2m-l,„ . „ . *2m-l / r 7, columns, i . e . , H (Zfc) H ( Z t ) v @ i 3 H (Z) . Remark. I t should be noted that i n general i 3 : H 2 m ^(Z) >• H 2 m 1(.Z ) i s not i n j e c t i v e . An easy example of t h i s i s when X i s a hypersurface. In t h i s case H 2 m - 1 ( X t ) = 0 , so that i * : H 2 m _ 1 ( X ) ® H°(C) • H 2 m "^(X ) ® H°(C) i s not i n j e c t i v e . Referring to (3.5) we see that i ^ : H 2 m •''(Z) »• H 2 m ^"( z t) i s obviously not i n j e c t i v e . From t h i s i t i s also c l e a r that Z i s not ample i n Z f o r generic X . From (3.1), the Gysin sequence f o r the p a i r ( z> z t) provides us with the short exact sequence: (3.7) 0 • Prim 2 m(Z) •• H 2 m ( Z - Z ) »• H 2 m _ 1 ( Z ) - »• 0 . (3.8) The complements'!". There i s a commutative diagram: This discussion evolves from the ideas i n [28, i 4] . 45 (3.9) where C - ZxP - z h _ 1 ( t ) = Z -pr i s the pro j e c t i o n The short exact sequence (3.7) s h e a f i f i e s to: (3.10) 0 • Prim 2 m(Z) * R 2 l \ c *• R 2 m "Sc^ • 0 over U 2 (where 2 m Prim (Z) i s the constant sheaf over U). Tensoring with 8 <8> - we obtain: (3.11) 0 v 8 T T ® Prim 2 m(Z) *• 9 T T ® R 2 i nh .C *• 9„ ® R 2 1"" 1^ v 0 U U * U * v From (3.4) there i s a natural s p l i t t i n g of F (over U) into F c* F v © i * H 2 m _ 1 ( z ) — — £ Om 1 3 y , which extends to F ~ © i 3 H (Z) ® 6 P There i s a Hodge f i l t r a t i o n defined on K = 0 ® R t i . C ~ ,2m R " L p r * ^ * Z x U / U ( l o g ° ° V e r U ' i - e - = R 2 l I l p r* F P^ZxU/U d o g Z ° ) , which provides us with the following short exact sequence: /-, -i-ix „ v « ^ m + l ^ • 2m. „. ^ m + l residue rm (3.12) 0 y 6„ ® F Prim (Z) »- £ — »• F »• 0 . U v (3.13) Lemma. In terms of the quasi-canonical extensions, the short exact sequence (3.12) extends to: (3.14) 0' ® F Prim (Z) >• £ > F y 0 over P ,1 v "h,k are the r e s t r i c t i o n s of h,k over V.. Proof. I t should be noted that the remarks concerning the quasi-canonical extension of F apply s i m i l a r l y to £ . In p a r t i c u l a r (2.16) holds f o r £ . We f i r s t l o c a l i z e over a disk A with A* c u. ™~ 2m ~ Now over A we have £ ~ 0, <8> Prim (Z) © F I claim that A v t, ~ 0^ ® F Prim (Z) © F^ . To prove t h i s we j u s t simply note that £/(9. ® F m + 1 P r i m 2 m ( Z ) © Fm) ~ 0 A ® F m'*Prim 2 m(Z) © F / F™ A v A v v which i s free. We now invoke (2.16) to conclude the above claim. I t i s now c l e a r that (.3.12) extends l o c a l l y , hence g l o b a l l y as w e l l . Tensoring (3.14) with J^" ® - we obtain: P ,-, ir-N r> ^1 o „m+l . 2m.„. _1 „ m^+1 _1 „ 7-m (3.15) 0 fi ® F Prim (Z) fi , <8> £ >" , ® r •> 0 P P P hence the long exact sequence: (3.16) • H 1 ( P 1 , fi11 ® Em+1) • H 1(P 1, fi11 O Fm) v 0 . P P V As a preliminary to s t a t i n g the main vanishing theorem, we prove the following: (3.17) Proposition; (i) H 2 m ( Z - Z ) ^ H 2 m - 1 ( x - X ) <8>H1(C) .... „m+12m,„ „ , m 2m-1, , ,. ' 1 , 0 , X ( i l ) F H (Z - Zfc) - F H (X - X ) « H (C) @ F m + l H 2 m - l ( x _ ^ 0 H O f - l ( c ) 47 where FPH2m'(.Z - Z ) = H2m(.Z, F P f i (log Z )) and where H• denotes t z t hypercohomology. Proof. (i) Z - Z = XxC - XfcxC = {X - X^xC. Therefore v i a the Kunneth formula H 2 m ( Z - z 1 o- H 2 m ( X - x ) ® H°(C) © H 2 m _ 1 ( X - X ) ® H 1(C) © H 2 m _ 2 ( X - X ) ® H2CC) . Now H 2 m(X - X ) = H 2 m(X, h" (*X ) ) , the hypercohomology of the t X t complex of meromorphic d i f f e r e n t i a l s with poles of a r b i t r a r y ( f i n i t e ) order. The terms of t h i s hypercohomology s p e c t r a l sequence abutting to H 2 m ( X - X ) are of the form E = H 2 m~ P(X, ((p+1)X )) t 1 X t for p > 0 . By Serre d u a l i t y , E = HP~1(X, fi2m_1~P (- (p+1) X^ _) ) which JL X t i s zero by Nakano's generalization of Kodaira's vanishing theorem. 2m Therefore H (X - X^) = 0 , and one can argue from t h i s r e s u l t that 2m~* 2 H (X - X^) i s zero also. ( i i ) i s a t r i v i a l consequence of (i) and i t s proof w i l l be omitted. Remark. (3.17) can also be proven from the following Gysin sequences: 48 (3.18) H 2 m - 2 ( X t ) - ^ H 2 m ( X ) (weak Lef.) H 2 mCX - X t) (weak Lef.) which implies that H 2 m(X - Xfc) = 0 . (3.19) H 2 m - 4 ( X J - i ^ H 2 m- 2(X) — * H 2 m" 2CX - X ) H ^ V ) - ^ H ^ C X ) t t Now a quick inspection of (3.3) w i l l reveal that i ' i s s u r j e c t i v e , hence r i s i n j e c t i v e . S i m i l a r l y (3.6) implies that i " i s i n j e c t i v e . A combination of these two facts immediately implies that H (X - X^ _) = 0. There i s a diagram analogous to (3.9) (3.20) X c > XxP <r g where C- = XxP - X g - 1 ( t ) = X - Xx 2m-2 As before, we define Kf = ^ R g^C over U , with Hodge f i l t r a t i o n subsheaves £ P over U , and extend to over P ] From (3.17) part ( i i ) we obtain: (3.21) C o r o l l a r y . f " 1 c* Z-f ® H 1'°(C) © Zf1 ® H 0' 1 (C) We now state our main vanishing theorem: 1 1 1 —r-> (3.22) Theorem. H (P , fi ± = 0 f o r a l l p > m+1 P The proof of-the above theorem w i l l be obtained through a series of lemmas, propositions, and other theorems. This treatment i s the same as i n [28, § 4] so that only an out l i n e w i l l , be given. The only r e a l difference between t h i s treatment and the one i n [28] i s that the hyperplane sections X of X are even dimensional. Now over U , £ f — "Spr • (log X°) * XXU/U ,2m-l , ^.-o. c- R pr . (*X ) XxU/U The order of pole f i l t r a t i o n on the complex fi* , (*X°) ^ XxU/U i s defined as: ( 3 ' 2 3 ) ( G P ^ X x U / U ( * X ° ) ) q = ^ X x U / U ( ( q - p + 1 ) x 0 ) l f q - P 0 i f q < p Over u we have: (3.24) v,p _2m-l D n . ,.-o. f ~ p r * G f l XXU/U (* X > 50 Now there i s an obvious extension of R 2 m 1 P r * G ^ > ^ ' X x r j / u ( * x ° ) over U to P 1 , namely; (3.25) ^ P = R 2 m V * G P f i - - . (*X) XxP /P -'P -P We w i l l l a t e r compare to but for now we w i l l prove the following: (3.26) Proposition. H1(P"L, Q1 <8> R 2 m 1pr J tG PfT ± (*X)) =0 P XxP /P Proof. We w i l l proceed i n three steps. Step I; Lemma. (i) X c XxP 1 i s a hyperplane section under the Segre embedding of p'^xP1 i n P 2 N 1 (X c p N) therefore ( i i ) X i s l i n e a r l y equivalent i n XxP 1 to the d i v i s o r X xP 1 + Xx{°°} . OO Proof. This i s proven i n (4.43) i n [28]. Now denote Y = x^xP 1 + Xx{°°} . Then i t i s easy to prove: (3.27) Proposition. H 1(P 1, ft1., ® R 2 m~ 1pr J.G Pfi' n ,(*Y)) = 0 . P XxP /P Proof. This i s (4.44) of [28], however since the proof i s easy we w i l l provide i t here. Let Px : X x P 1 — X be the canonical p r o j e c t i o n . Using algebraic d i f f e r e n t i a l s , the complex G Pfi* ^i*^) takes the form: XxP /P 51 0 >- Px tPiX.) ®pr*0.. 1(l) >• Px ftP+1(2X ) ® pr*9 (2) X 0 0 1 X ° ° 1 >r1' p- x p-This complex c l e a r l y contains the subcomplex L* : 0 y Px* Sp(X ) ® pr*9 . (1) v Px ftP+1(2X ) ® pr*9 , (1) • ••• X °° 1 X 0 0 1 p p Now i t i s obvious that R 2 m _ 1 p r ^ L ' = H 2 m - 1 ( X , G P tt' ( * X J ) 0 8 • (1) P so that H 1(P 1, tt1 ® R 2 M 1pr.L*) = 0 . I t i s easy to check that the P l cokernel of the i n c l u s i o n ft1, ® R 2 M 1 p r J L i ' c ft1, ® R 2 M 1 p r J G P 1 -f * 1 * P P ft" .. , (*Y) has support over pr 1(°°), therefore since H 1(*) i s XxP /P r i g h t exact on the above i n c l u s i o n of sheaves, we must have H^P , ft1 <8> R ^ ' V ^ f t " (*¥•)) =0. P XxP /P Step I I : We would l i k e to replace Y by X i n (3.27). For t h i s we need the following version of the semicontinuity theorem: (3.28) Theorem. Let f : V *• S be a proper, f l a t morphism of Noetherian schemes, and l e t K* be a complex of sheaves on V , whose terms K 1 are coherent 9^ - modules that are f l a t over S , and whose d i f f e r e n t i a l s are l i n e a r over f * 9 g • For each p > 0 , the function: si >• d i m ^ ^ H P ( , K * ) i s upper semicontinuous on S Proof. This i s (.4.45) of [28]. The r e l a t i o n f o r l i n e a r equivalence of the two d i v i s o r s X "and- Y i n the above lemma ( i i ) i s given by a d i v i s o r N c XxP^xP"^ where N q = b 1~ 1(0) = X and N ^ = b 1~ 1(°°) = Y,'b , b 2 being the canonical projections i n : (3.29) N -» XxP 1xP 1 Now introduce the complex K* = G P fl- - (*N), We need the " XxP xP /P xP following: (3.30) Lemma. R 2 mb K " = 0 2, * Proof. This i s (4.46) of [28] (3.31) Co r o l l a r y . For a l l s i n P 1, R 2 mpr #K^ = 0 (3.32) C o r o l l a r y . H 1(P 1, Q,1 _2m—1 ., ..2m 1 * 1 ~ R p r / * ) = tt ( X X P \ pr fi"\ ® K ) P 1 S P 1 S Proof. Identify the two E terms f o r H ^ X x P 1 , pr**!1, ® K ' ) and 2 P 1 3 use (3.31). 53 2m 1 1 Step I I I : Now define d(.s) = dim H (XxP , pr*fi ^ ®-K*) . I t i s easy to see that d(°°) = 0 ((3.27)).T so that by (3.28), d(s) = 0 i n a Z a r i s k i neighbourhood of 0 0 . Therefore (3.26) i s a consequence of the following: (3.33) Lemma. d(s) i s closed under s p e c i a l i z a t i o n . Proof. This follows from (4.49) i n [28]. To a r r i v e at a proof of the vanishing theorem (3.22), i t w i l l be necessary to give a desc r i p t i o n of the l o c a l dual and vanishing cycles associated to the Lefschetz p e n c i l of X . A thorough treatment of t h i s can be found i n [7, §15] .- We l o c a l i z e the family i n (1.2) over a disk A with A (1 E = 0 € A . Let X be the o corresponding singular f i b e r with singular point Z Q € X q . We consider an open polydisk (radius c) neighbourhood i n X centered at z , so that a l o c a l d e s c r i p t i o n of X, near z i n sui t a b l e o t o coordinates z = ( z ^ , . . . 'Z2m-j} -"-s P r e c i s e l y : {z = (z. ,...,z_ n) I z*z = £ 2 m 1 z 2 = t and | z | 2 5. c, where t € A}. 1 2m-1 ' . , 1 1 1 U=l We can.write z = x + i y where x = (x,,...,x_ ,) and y = (y,,...,y„ 1 2m-1 1 2m-3 S p e c i a l i z a t i o n i n t h i s context r e f e r s to the notion of continuity, i . e . , d(s) = 0 for a l l s 6 P 1 . 54 are r e a l . Now define f o r c > e > 0,, the following c y c l e : (3.34) 6 = {z = x + i y I x-x = e , y = 0} c x . Also we w i l l denote U £ = X £ fl above polydisk. Note that f o r a l l z € u , x f 0 otherwise z«z = e < 0. Therefore U r e t r a c t s onto e e 6 ^ v i a the r e t r a c t i o n map z = x + i y i > (e/x #x) x. Note also that 6 ^ = 2m-2 sphere. Therefore we obtain the following: r J/_ f o r q = 2m-2, 0 (3.35) H g(U e , 7 Z ) = ^ 0 otherwise I t i s not hard to check that 6 ^ i s the l o c a l vanishing cycle i n H (X ,7L) associated to z € X 2m-2 E o o The r e l a t i v e dual c y c l e : From (3.35) and Poincare-Lefschetz d u a l i t y , we obtain: (3.36) H (U , SU , 7Z) = 0 f o r q ? 2m-2, 4m-4; and H _ (U , 3U , 7L) C£ £ £ ^ i l t l £ £ £ ~7l. • We e x h i b i t the generator of H _ 9 ( U , 3U , ~//_) as the r e l a t i v e dual cycle 9^ . Define 0 to be the 2m-2 c e l l i n U defined by: E £ 55 (3.37) 0 e = 1 Y = ( Y 1 y 2 m - 2 ' 0 ) x = (0, ,+/E + y y ) y y 5 Cc - e)/2 Then <5 fl 6 = ( 0 , ,+ /£) and 96 i s a 2m-3 sphere i n BU e e e e There i s a short exact sequence: (3.38) 0 — H 2 m _ 2 ( U £ , VI) — H 2 m _ 2 ( X £ , 7L)—> H 2 m _ 2 ( X £ , U £, 7D — 0 which by d u a l i t y defines a short exact sequence: (3.39) 0 v H_ _(X - U , 7Z) y H „(X , ID • H„ (U , SU ,1L) 0 2m-2 £ £ 2m-2 £ 2m-2 £ e where r(y) = (Y, <5 )6 £ £ From (3.38) and (3.20), there i s a commutative diagram (we denote U to be a polydisk i n XxP"'" centered at Z q and inducing the polydisk described above i n X): (3.40) 0 T ° — * H 2 m - 2 ( U £ ' 7 ^ --1. H (X ,71) *• H ,(X ,U ,S) ^0 2m-2 £ 2m-2 E £ ,--1, H 2 m - l ( U f 1 g U ) ' ^ ^ ^ H 2 m - l ( g ( e ) ' X ) H 2 m - l ( g " 1 ( £ ) ' UH g 1 ( E ) ,7Z) ° = H 2 m - l ( U n p r ' 1 ( £ ) , 7 L ) * H 2 m - l ( X ' 7 / 1 ) * H 2 m - 1 ( X , U 0 p r 1 ( e ) ' ^ ) - > 0 2m-1 Note that the homomorphisms i \ are Inverse to the "tube over cycle" homomorphism.4 A simple diagram chase w i l l reveal that i ' i s i n j e c t i v e and r' i s s u r j e c t i v e . Note also that r ^ i s an isomorphism. Therefore combining (3.39) with (3.40) we obtain the following: ( t ^ = tube over cycle homomorphisms) ° — * H2m-2 ( X £ " V 7 £ ) — H 2 m _ 2 ( X £ , 7 ^ H 2 m _ 2 C U £ , 3U £, 7/) (3.41) ,—1. ° — * H 2 m - l ( g ^ > " ^ > — ^ H ^ ^ F ' . C e ) ,7/) — H ^ . t i ^ U } 'ft U a {g 1 ( E ) n u } , 71.)—* o — 1 — 1 Note that the image of H„ ,(g (e) - U, C) i n H • .(g (e) , C) i s 2m-1 2m-1 a codimension 1 subspace. We make the following remark before f i n i s h i n g o f f the proof of (3.22): (3.42) Remark. As i n [28], we assume that the to r s i o n of £ f - ''P (whichican only be supported on Z) i s divided out so that i s to r s i o n l e s s . This w i l l i n no way a f f e c t the vanishing theorem arguments to follow. We need the following useful r e s u l t : A discussion of the "tube over cycle" homomorphism appears i n [7, §3], (3.43) Proposition. (i) X_P <= f o r p > m+1. f f cj'm-k+l ^m-k+1 (11) K c X (.ki) f o r 1 S k 5 m+1 Proof. The proof i s almost the same as i n [28], except that i n t h i s case n = 2m-l i s odd. Therefore whenever necessary, some estimates have to be checked. Using Dolbeault resolutions to compute the -•p hypercohomology and l o c a l i z i n g over A , a l o c a l section of X_ i s f CO — — a r e l a t i v e l y closed C 2m-l form cp on XxA - X with a pole along X of order at most v = 2m-p. We determine the period growth of cp = r e s t r i c t i o n of cp to X - X • , around O C A , and t h i s i s done by •integrating over the i n t e g r a l cycles coming from H. . (g 1(t),Z^) t 2m—1 (t € A*). Note that the f i b e r s , g 1 ( t ) - U are t o p o l o g i c a l l y the same fo r a l l t € A , so that cycles i n H2m-1^ ^~ ^ t) ~ u ' 7/J c a n fc»e kept uniformly away from X as t varies i n A . Therefore using (3.41) i t i s c l e a r that -/cp i s uniformly bounded over cycles i n H_ 1 ( . g ~ 1 ( t ) , 721) which come from H.„. 1 ( i " 1 ( t ) - U, 7L) . Therefore 2m-1 2m-1 i t s u f f i c e s to check ftp over the tube of the dual cycle described by the homomorphism i n (3.41) . Since taking residues i s dual to taking tubes, i t s u f f i c e s to evaluate / res cp , where 6 i s the e t t r e l a t i v e p o r t i o n of the dual cycle. There i s no loss of generality i n assuming t i s r e a l and p o s i t i v e . The e x p l i c i t d e s c r i p t i o n of 8 i s given i n (3.37). We u t i l i z e the following r e s u l t s from [28]: (3.44) Lemma. res cp f fJ- , 2.p-2m+*s 2m-3n A J t t + r ) r r dr 0 where A i s some constant. Now the expression on the RHS of (3.44) i s bounded above by (3.45) • l - t _ m _ 1 A / (t + r ) P m dr. Therefore f o r p > m+1, (3.44) i s 0 obviously bounded as t 0 . This proves (i) of (3.43) . We now make a change of va r i a b l e s by s e t t i n g w = r / / t . Then dr = dw and (3.45) becomes: (3.46) (A St) t?-™-1 " 1 } 2 (1 + w 2 ) ^ 1 " " ^ Now assume 0 5 p 5 m . Then i t i s clear that (.1 + w 2 ) P m 1 2 -1 (1 + w ) , so that the following holds: 59 (t -i)h _ _ (t  1-i)h , (3.47) / (1+w ) P m ±dw <• / (1+w ) _ 1dw = tan i ( ( t -1) 2) 0 0. which i s bounded at t >• 0 . Therefore (3.46) i s bounded by ( B / t ) / t m + 1 P where B i s some constant. By s e t t i n g p = m-k+1 (for 1 5 k 5 m+1) we obtain part ( i i ) of (3.43). (3.48) Co r o l l a r y . Let 1 < k < m+1. Then H 1(P 1, ft1 ® ^ m - k - 1 (kE)') = o . P f Proof. There i s a short exact sequence: (3.49) 0 • fi^.® K_P *• n1± ® £ P (kl) • • L' r 0 where p = m-k+1 and 1 has 0 dimensional support. Since H^"(p\ ft1 o> £_ P) = 0 P f H 1(P 1, L ), the assertion i s obvious. We now have the following: (3.50) Main Vanishing Theorem. H 1(P 1, ft1 ® £p) = o f o r a l l p > m+1. P f Proof. There i s a short exact sequence: (3.51) 0 » tt1 ® Z'f >- Q,1 <8> K? »• L" • 0 where l " has 0 P f P f dimensional support. We now proceed as i n (3.48). This concludes the proof of our f i r s t main r e s u l t . The r e s t of t h i s section w i l l be devoted to gi v i n g c o r o l l a r i e s to (3.50), i n p a r t i c u l a r our main theorem on the normal functions. 60 (3.52) Co r o l l a r y . H^p 1, fi1 ® £ m + 1 ) ~ H1(.P1, fi1 ® (£™ / £ m + 1 ) ) c3>H1,0 (C). P P f f Proof. This follows from (.3.21) , (3.50) and the following short exact sequence: (3.53) 0 — Km+1 — Zm • Zm / Zm+± • 0 . f f (3.54) Co r o l l a r y . H°(p 1, F™'*) ~ H°(P 1, F™ 1 / F m ) ® H 0 , 1 ( C ) f,v f,v Proof. There i s a commutative diagram: (3.55) 0 - n 1 ® Zm+1 ® H 1' 0^) v fi1 ® £ m ® H 1'°(C)* P f P 1 f 0 • o 1 ® F m P f ,v P I f f H 1 , 0 ( C ) + 0 H ^ V ) • fi\ ® F m _ 1 ® H 1' 0^) P f ,v fi ®.{ (F_ /F_ ) P f,v f,v ® H 1 , 0 ( C ) } r-»- 0 Now one e a s i l y checks that (3.50) implies that H1(.P1, fi1(8>£™+1 ® H 1 , 0 ( C ) ) P f -1 1 1 £m 1.0 '.; = H (P ., fi ., ® F <8> H ' (e) ) = 0 . Since F i s madeuup of T . 1 ^ V • P f,v ~f% and F1 -^ 1 , we can combine (3.55) , (3.52) and (3.16) to obtain the f,v f,v following commutative diagram: HV1, fi1 ® f* 1)-^— E'CP1, fi1. e ^  / e^ 1!® H1'0^) p p f f (3.56) HV1, fi\ ® Fm) P 1 V -HV1, fi\® F m _ 1 / F m ) OH^V) P f,v f,v Now the bundle F™ 1 / F m i s s e l f dual, so that Serre d u a l i t y f,v f.,v provides us with the desired r e s u l t . A good consequence of the above r e s u l t i s that we can give a more e x p l i c i t d e s c r i p t i o n of the ker <5 , defined i n (2.18). More p r e c i s e l y , we obtain: (3.57) C o r o l l a r y . The ker <5 i s equal to the image of H ° ( P \ F m _ 1 / F m ) S H ^ O C ) © J 2 m _ 1 ( Z ) i n H ° ( P 1 , J ) , where f,v f,v ,2m-l,„. . , j . . , _2m-l,_. m,* 2m-1,_. / T T2m-l, J (Z) i s defined as: J (Z) = F H (Z)/H (Z,7L). Proof. Use (3.55), the s p l i t t i n g defined just preceeding (3.12) and the short exact sequence (2.17). (3.58) c o r o l l a r y . 5 H°(p 1, J') l i e s i n the image of J 2 m - 1 ( Z ) h (3.59) Co r o l l a r y . H (P , Q <8> F P ) = 0 for a l l p > m+1. P V Proof. The two decompositions: F p ^ F f ® H ° ' 1 C C } © F P _ 1 ® H 1 ' 0 ^ ) f,v f,v ZP+1^ ZP+1®n0'\c) © ^ H ^ t C ) are r e l a t e d i n an obvious way v i a the residue homomorphism. We now apply (3.50), using an analogous sequence to (3.16). (3.60) C o r o l l a r y . H°(P"L, ttl± <8> F P ' * ) = 0 f o r a l l p > m+1 . P Proof. We f i r s t note that F P ~ F P © M where M = i * F P H 2 m - 1 ( Z ) so that F P ' * ~ F P ' * © M. Now c l e a r l y H°(P 1,.p 1 ® M) = 0 , and since r- ' P Q1± = 0 (-2), i t i s obvious that ^ ( P 1 , ft1 ® F P ' * ) • H°(P 1, F P ' * ) P P P V V i s i n j e c t i v e . But H°(P 1, F P ' * ) ~ H 1(P 1, ft1 ® F P ) which i s 0 by v 1 v P (3.59). Therefore we have: H°(P 1, fi1 ® F P ' * ) ~ H°(P 1, fl1 ® F P ' * ) © H°(P 1, Q 1 ® M) =0. P P P We now a r r i v e at our main r e s u l t f o r the normal functions: 5 The proof of (3.58) can be found i n (A.3) (3.61), Theorem. H°(.P1, J) = H°(P 1, J) . . Remark. (3.58) becomes: H°(.p\ J') l i e s i n the image of J 2 m (Z) Proof„of (3.61) . We f i r s t l o c a l i z e F over a disk A with A 0 E = 0 € A. Let N = T-I, where T i s the l o c a l monodromy transformation. Then N acts on the sheaf F so as to obtain a subsheaf W = ker N of F , which defines a f l a t subbundle of dimension b„ ,-2g (where 2m-1 we assume for s i m p l i c i t y that H 2 m (.z ) = H 2 m 2 ( X t ) ® H^C) , b„ = dim H 2 m 1 ( Z ^ ) , and g = genus of C). We l e t 0, be the 2m-l t A,o l o c a l i z a t i o n of 0. at O C A . Define F (respectively W ) to be A o o the s t a l k of F (respectively W) over O C A , i . e . F = F ® 0, o A,o Let : v F ^ ' be the morphism obtained from the composite W  c y F v (F / F m ) = F m ' * . We remark that a h o r i z o n t a l basis o o o o o f o r W over A i s given by { v ^ , ; . i , v } <8> {basis of H 1(c)}, where the {v^} are orthogonal (under the cup product pairing) to the l o c a l vanishing cocycle <5 , hence W i s n a t u r a l l y s e l f dual under the cup product p a i r i n g . Also i f 6 € H m 1 , m 1(X t,Q) v i a h o r i z o n t a l d i s -* placement for generic t € A , then i n f a c t the i r r e d u c i b i l i t y of the II Cu)'/ action on the vanishing cohomology associated to the Lefschetz 64 p e n c i l (due to the Picard Lefschetz formula, and the f a c t that ir^(U) acts t r a n s i t i v e l y on the vanishing cocycles) implies that H (xt»'2)v £ H ' (Xt,CQ) for a l l t e U. Now l e t v be any normal function. Then from the e x p l i c i t d e s c r i p t i o n of V i n (2.14), and the i n f i n i t e s i m a l period r e l a t i o n (2.20), i t i s c l e a r that V v = 0 i n t h i s case. Therefore i t s u f f i c e s to assume that 6 i s not of type (m-l,m-l) for generic t e A * . We f i r s t j u s t i f y the claim: F™ c W_ implies that H 2 M ~ 2 (X , J ) ) v £ H* 1" 1'* 1 - 1 (X for t e U, f,o f,o where F™ i s the l o c a l i z a t i o n of F™ at 0 e A(resp. W_ ) . f,o f f,o ' Any c l a s s cr i s represented i n the form a = E g.v. f,o i = l 1 1 + g Q v ( t ) where v(t) = 6 / t and 9^"^ } a r e holomorphic functions around 0. Now the r e l a t i o n F™ c W implies that for a l l such a above, N a= 0, f,o f ,o I.e. 0 = -2g Qv(t) which implies g Q = 0. This immediately implies that ^ * — ^ p m f j 2 m 2 ( X t ) = 0 f o r t i n a neighbourhood of 0 e A . Therefore 6 remains of type--\(m-1 ,m-l) under h o r i z o n t a l displacement i n U, thus j u s t i f y i n g the above claim from the properties of the IT (U) action on H 2 m- 2(X t,CQ). Now u t i l i z i n g the above claim and the above assumption that 6 i s not of type (m-1,m-1) f o r generic t e A , we deduce that f_ n W_ f,o • "f ,o c F™ , and therefore applying Nakayama's lemma to the unique maximal jt f ,o r 77- 1 , •r' T B I . ^ m ide a l m c 0 , we obtain: i W_ n F } /m F_ c F /m F_ A , 0 f,o f,o £ , 0 ? f,o f,o Now taking into account the d e s c r i p t i o n of F™ in terms of 1 and o -f,o F_' , we can deduce (using the symmetry of any given basis of H"1" (C)) that dirn {F™ n W } /m F = h ,/2 C o o o 2m-l/ 2g. Since dim W A,o b - 2g and dim„ F 2m-l ^ 0. o A,o a commutative diagram: = b2m-l / /' 2' w e c o n c x u < 3 e that there e x i s t s -> F™ -> F mod m o -> F V2»-l / 2 o ->'0 °. ( b2m-l/ 2 ) " 2 q 2m-l * b2m-l/ 2 7-U 0 > W n F™ o o .3 b 2 m - r 2 q b2m-l/ 2 •> W /{ W n F1"} ? o O ' ' o o f o r which i t i s c l e a r by a su i t a b l e diagram chase (and Nakayama's lemma) that ty ( W ) = F™'* . r o o — —m * (3.62) The upshot of the r e s u l t ty ( W ) = F ' i s that given a section o o a e H°(A, F™' ), a can be represented by a section cF e H°( A, F ) with holomorphic d e r i v a t i v e with respect to the Gauss-Manin connection. That i s V a e H° ( A, JJ 1. —m+1 * F ). Therefore i f v i s any normal — o 1 1 —m+1 * function, we have V v e H (P , Q, ® F ').We now invoke (3.60) to P conclude the proof of the theorem. Remark. As a by-product of the r e s u l t s of t h i s section, the following i s easy to prove and w i l l be l e f t to the reader: (3.63) Proposition. H w(p", F F' ) = i * F P ' * H 2 m - 1 (Z,C) f o r a l l p > m+1. This concludes chapter 3. 66 Chapter 4. A Summary on the Normal Functions. The purpose of t h i s chapter i s twofold, namely: (i) Prove the meromorphicity of the Abel-Jacobi mapping and apply i t to the cases where an inversion theorem f o r the group of normal functions i s known to hold (cf (0.35)). ( i i ) Discuss a somewhat deeper aspect of the normal functions not already mentioned i n chapter 3. In regard to ( i ) , we remark that the meromorphicity i s a s i g n i f i c a n t l y stronger r e s u l t than that attainable i n the case of a Lefschetz p e n c i l ([28,(4.58)]), f o r i n [28] the meromorphicity i s guaranteed provided the family of intermediate Jacobians l o c a l l y embeds (over P 1) i n a Kahler manifold which i s proper over P 1 ([30, -p.202]). This c r i t e r i o n has been v e r i f i e d i n a few geometric cases ( [30]). The reason f o r the stronger assertion i n (i) as opposed to that i n [28] 2 i s e n t i r e l y due to the differences i n the l o c a l monodromy (T =1 rather 2 than (T - I) = 0). We remark that the purpose of proving the meromor-p h i c i t y of the Abel-Jacobi mapping i s to be able to construct algebraic cycles from normal functions which are known to be i n v e r t i b l e . I t i s an e a s i l y v e r i f i a b l e f a c t that the knowledge of (i) implies the equiv-alence of the following two statements: (A) Prim m' m(Z,{)) i s generated by algebraic cocycles. (B) Every c l a s s Y e P r i m m , m ( Z , 1 L ) i s the cohomology cla s s of an i n v e r t i b l e normal function v The d i f f i c u l t y i n proving (B) i s the lack of knowledge of the image of the Abel-Jacobi morphism i n J(Z ) (for t e U). 67 In chapter 2, we i d e n t i f i e d a subgroup of J ( Z t ) (denoted by J A(Z^) below) which from inductive assumptions i s contained i n the image of the Abel-Jacobi morphism. In t h i s chapter we i d e n t i f y the part of 1 2 m.—1 E 2 ' (k) that l i e s i n the image of the normal functions which take t h e i r values i n u J ( z ) ((4.30) and (4.32)) to be the classes of type (m,m) i n the cohomology of a c e r t a i n sheaf, namely H (P ,L ) where L = k k L f®^H 1(C , "2) and L f i s defined i n (4.9) part (i) . The sheaf describing these normal functions i s defined i n (4.9) part ( i v ) , and i s denoted by I. We then deduce from (i) and inductive assumptions that those classes i n E 1 ' 2 m _ 1 ( k ) n 6H°(P 1,I) are i n f a c t algebraic ((4.43)). To discuss ( i i ) , we remark that from the above discussion i t i c l e a r that the n o n - t r i v i a l part of proving (B) would be to study those normal functions v which are not elements of H°(P 1,I). We indicate a d i r e c t i o n f o r studying such v by introducing the terms (p) i n the remark following (4.29). The basic philosophy i s the following: The coh-m 1 2m—1 1 omology F E 2 ' (k) projects into E2(m-1) ® H (C), and those normal functions which induce the zero cohomology cla s s i n E2(m-1) ® H"*" (C) are p r e c i s e l y those normal functions which are elements of H°{P^,1) + kerfi . Now the terms E 2(p) are expressed p a r t l y i n terms of the Gauss-Manin connection and the general idea i s that those normal functions v ^ H°(P 1 f o r which <$v jt 0 i n E 2 ' (k) w i l l have a n o n - t r i v i a l cohomology c l a s s representation i n E2(m-1) ® H 1 ( C ) written i n terms of V , thus r e f l e c t i n g on the i n f i n i t e s i m a l properties of v . i t i s believed that the knowledge (B) l i e s i n a f u l l understanding of the i n f i n i t e s i m a l properties of the normal functions. 68 In the s i t u a t i o n of our Lefschetz p e n c i l { X^ } t £ p l / i t c a n be shown that for "general enough" X, the sheaf L^, and hence I are constant over P 1 (see (4.10) p a r t ( i i i ) ) . This would imply that 6 H°(P 1,1) = 0 i n 1 2m"~l E ' (k) . To-get an o v e r a l l p i c ture of what i s happening, we have the following cases that can occur f o r a given Z: Case 1 E2(m-1) = 0. This i s implied"'" by the cohomological condition FmH2m-2 = 0 for t e U, and i n t h i s case, H ^ P ^ L ® C) = E ^ ' 2 m ~ 1 ( k ) . We prove that Sa°(p1,1) i n E 2 ' 2 m 1 (k) i s equal to a l l the 71 classes of -j^  2ltl~" 1 type (m,m) i n E ' (k) ((4.30)). The r e s u l t s i n t h i s chapter are the strongest i n t h i s case, f o r i f we assume that the Hodge (m-1,m-1) conj-ecture holds for X (for generic t e U) , which i s equivalent to: ".', H 2 m 2 ( X t , $ ) v i s generated by algebraic cocycles, when F m H 2 m 2 ( X f c ) v = 0, then i t i s proven that the Hodge (m,m) conjecture holds f o r Z. Case 2 E2(m-1) j- 0 and 6H°(P" L,J) fi 0 i n E 2 ' 2 m - 1 ( k ) . We remark that i n t h i s case F 1 1^ 2 1 1 1 2 ( X ) ? ' 0, and as a consequence, H"L(P1,L ) = 0 and ^ ( P 1 , ! ) = 0 i n E 2 ' 2 m - 1 ( k ) . This i s the most d i f f i c u l t case to handle and the techniques i n t h i s chapter are i n s u f f i c i e n t i n proving the Hodge conjecture f o r such Z. Case 3 E2(m-1) f 0 and SH 0^ 1,!) = 0 i n E 2 ' 2 m _ 1 ( k ) . We also remark here that F m H 2 m 2 ( X ) f 0, and consequently H 1(P 1,L ) = 0. Now t h i s case occurs t V .K p r e c i s e l y when Prim ' (Z,$) 6 H ' (DxC,?) = 0(as suming E 0 (m-1) ^ 0 ) , 2m and examples of Z f o r which t h i s occurs are when X c P i s a generic hypersurface of degree >_ 2 + 3/(m-1) , where m>_ 3 ([7, §13]). I t would be nice to be able to detect from the knowledge of the normal functions """It can be shown that i f E 2 ' 2 m 1 (k) ^ 0, then E (m-1) = 0 i f f F m H 2 m ~ 2 ( X t ) v =0; see (A.5 ) f o r the proof. 69 when such a s i t u a t i o n o c c u r s f o r a g i v e n Z = XxC, and we g i v e some comments i n t h i s d i r e c t i o n ((4.47) and (4.48)). We now e x h i b i t some n o n - t r i v i a l examples o f c a s e s 1 and 2 so as t o v e r i f y t h a t a l l 3 cases a r e not vacuous. We a l s o work i n t h e case m = 2 so t h a t X i s a body ( t h r e e f o l d ) . The e a s i e s t c l a s s o f b o d i e s X s t u d i e d were i n t r o d u c e d by Fano and s a t i s f y t h e embracing c o n d i t i o n s : P i c ( X ) =7L and H ° ' 3 ( X ) = 0. From t h e s e c o n d i t i o n s we see t h a t e v e r y l i n e bundle o f p o s i t i v e degree on X i s ample, and t h a t t h e c a n o n i c a l d i v i s o r K has A n e g a t i v e degree. C o n s e q u e n t l y some i n t e g r a l m u l t i p l e N o f t h e p o s i t i v e g e n e r a t o r o f P i c ( X ) i s e q u i v a l e n t t o 6 (-K ), and t h i s N i s c a l l e d t h e A X i n d e x o f t h e Fano body. I f we choose our L e f s c h e t z p e n c i l ^ x t ^ t e p l s o t h a t 0 (X ) g e n e r a t e s P i c ( X ) , t h e n t h e case N> 2 p r o v i d e s examples x t — f o r which c a s e 1 h o l d s . The more i n t e r e s t i n g examples f o r which c a s e 1 h o l d s o c c u r when N = 2, and such X a r e of the f o l l o w i n g 3 forms ([27, 4 p.43]): (1) a c u b i c h y p e r s u r f a c e i n P (2) t h e i n t e r s e c t i o n o f 2 q u a d r i c s i n P^ 4 (3) t h e i n t e r s e c t i o n o f t h e Grassmannian o f l i n e s i n P w i t h 9 3 h y p e r p l a n e s i n P . The example o f t h e c u b i c X i s the most w e l l s t u d i e d , and i n t h i s case i t i s known t h a t : 3 1 1 1 1 2 -(1) dim^H (X) = 10, t h e r e f o r e H (P ,L O <E) = H (P ,R f*(C) = H 3(X) f 0. (2) I f we denote F(X) t o be t h e v a r i e t y o f l i n e s i n X (as a 4 s u b v a r i e t y o f the Grassmannian o f l i n e s i n P ) , t h e n i t i s p r o v e n i n 2 2 [27,p.17] t h a t t h e r e e x i s t s a cu r v e C C F ( X ) f o r which H ' (XxC) n 70 H 3 ( X , $ ) 8 H ^ C g ) f 0. (3) C o n s e q u e n t l y ( s i n c e J A ( Z f c ) = J ( Z f c ) ) t h e r e e x i s t s normal f u n c t i o n s v t a k i n g t h e i r v a l u e s i n u J- (Z ) f o r which 5v ^ 0 i n YT' (k) . t 6 P X A Z 2 „ l , 2 m - l t T A l t h o u g h t h e Hodge c o n j e c t u r e i s a l r e a d y known t o h o l d f o r Z = XxC, f o r b o d i e s X w i t h N > 2 ( f o r example see [ 3 0 ] ) , t h e t e c h n i q u e s o f p r o o f r e q u i r e some knowledge o f t h e geometry o f X ( e x i s t e n c e o f a c o v e r i n g f a m i l y o f l i n e s ) , whereas from t h e t e c h n i q u e s o f t h i s t h e s i s , o n l y a knowledge o f t h e geometry of. . X ^ . i s r e q u i r e d . I n p a r t i c u l a r , we need o n l y check t h a t the g e o m e t r i c genus o f X^_ i s z e r o . Some examples of Fano b o d i e s X s a t i s f y i n g c ase 2 a r e t h o s e X where N = 1, X a complete i n t e r s e c t i o n , and where X embeds as a h y p e r p l a n e s e c t i o n i n a smooth f o u r f o l d . The most w e l l known example o f such X i s 4 t h e q u a r t i c h y p e r s u r f a c e i n P . I t f o l l o w s from [27,p.42] t h a t t h e r e e x i s t s 2 2 3 1 a c u r v e C f o r which H ' (XxC) fl {H (X,$)® H (C,$)} ? 0, and hence t h e r e e x i s t s a normal f u n c t i o n v which does n o t t a k e i t s v a l u e s i n u , J (Z ) 1 - t £ P ( i . e . v $ H (P , 1 ) ) , and f o r which <5v ^ 0. I t i s a l s o a s t a n d a r d f a c t t h a t t h e Hodge c o n j e c t u r e h o l d s f o r Z = XxC where C i s any c u r v e and X i s any complete i n t e r s e c t i o n Fano body o f i n d e x 1 s a t i s f y i n g t h e above embedding c o n d i t i o n . F i n a l l y , one o f t h e i n t e n t i o n s i n the p r o c e s s o f w r i t i n g up c h a p t e r 4 i s t o g i v e somewhat more g e n e r a l p r o o f s (whenever p o s s i b l e ) o f t h e r e s u l t s i n t h i s c h a p t e r t h a n needed f o r the c a s e ^ x t ^ " t e p l a L e f s c h e t z p e n c i l , w i t h the hope o f w r i t i n g f u t u r e papers stemming from t h e s e r e s u l t s . Due to the product structure on Z^ , the Jacobian J ( z t ) i s reducible as a p r i n c i p a l torus. (.[27, p. 10]) . More e x p l i c i t l y : (4.1) J(Z t) ^  ( F m ' * H 2 m 1 ( X t ) ) ®tt°CC) / H 2 m' , : L(X t,^) |H°(C,7/J Fm,*(H2m 2 ^ HV)) / H 2 M 2(x,7/} WccT/J t C t 7C ( Fm-l,* H2m 3 } % 2 ( c ) H2m 3 ( ^ ^ ^ - ^ t C t 7C (4.2) We define J = F m ' * ( H 2 m ~ 2 ( X J ® H1 (C)) / H 2 m 2 ( X .7/1) ® H 1 ( C , ^ . ) t C t Since the main i n t e r e s t i s Prim 2 m(Z) ~ P r i m 2 m "^(X) ® H^CC) , we need only focus on J . In view of (2.23), we may assume the following: (4.3) (I): f™'* F m - 1'* 9 H 0' 1^) © F1"'* ® H 1' 0^) f f ( i i ) j ^ R 2 m " \ VI- = j * R 2 m 2 f ^ 7Z ® H 1(C,7/.) , i . e . , H 2 m 1 ( Z t , 7 / ) = H 2 M ~ 2 ( X t , 7 £ ) ® H 1 ( C , 7 ^ ) ( i i i ) and J modified accordingly. A l l constructions i n Chapters 1, 2, and 3 carry over to t h i s case. Assuming only the Hodge (m-1,m-1) conjecture f o r X^, we see that only a r e s t r i c t e d part of J i s known to be i n v e r t i b l e . In fa c t we can describe t h i s part very p r e c i s e l y : (4.4) Lemma. (i) H m 1 , m " 1 C X ®R 1(.C,7Z) i s a l a t t i c e i n Hm-l,m-l z ) 0 H O f l } t 7/ ( i i ) j c z . ) = H™ 1 ' - m ' 1 ( x . ®H 0 , 1-(.C,C) / A t t (Hm-l,m l ( x % HV,^)) t 7i-i s an abelian v a r i e t y . Proof. C D i s obvious.; ( i i ) — the induced Hermitian inner product on J from J has the required properties f o r J to be an « A abelian v a r i e t y ( v e r i f i c a t i o n l e f t to reader). There are a few preliminary remarks that we w i l l need before proving the main r e s u l t s of t h i s section, which w i l l be stated now. ~"m * r-m * (4.5) Let F ' , be the h o r i z o n t a l subsheaf of F ' (with h respect to the Gauss-Manin connection). There i s a short exact sequence: ( 4 . 6 ) o * ( F m _ 1 / F m ) ® H^CC) — F m ' * — v F m ' * ® H^ C) — v 0 i f f Also the i n c l u s i o n F c > F ' defines a homomorphism: h 7-m * (4.7) F"1' *• F ' ® H^CC) . We w i l l use t h i s homomorphism l a t e r i n h f Chapter 4. We consider the embeddings: , „ ~v • 2m-2_ , -m-1,* (4.8) j^R f*.7^c »- F_ f rm-1 . _ .-m-1,* F / F c — - v f In t h i s context define the following sheaves over P"*" (4.9) (i) L = j ^ R 2 m  2f^L n ( F _ ) /ft f f ( i i ) L, = L ® H^C, 7/.) K f 7C-( i i i ) F = L ® H 0 , 1(C,C) ® 9 n 1 t „ T , l C P (iv) I = (4.10) Remarks. (i) Via the stalks over U, the stalk of over t i s p r e c i s e l y equal to {v £ H m  1 , M 1 (X^,~7/-) |v remains of type (m-1,m-1) under l o c a l h o r i z o n t a l displacement i n u} ( i i ) I should be interpreted as the sheaf of normal functions taking t h e i r values i n U J (Z ) . p i A t ( i n ) For our Lefschetz p e n c i l {x } , , t , 1 t€P L^ ~ H m - 1 ' m _ 1 ( X , 7 Z ) 2 i s the constant sheaf i f H m ~ 1 _ r ' m ~ 1 + r ( x ) ^ 0 f t v for generic t and some r > 1. This follows from the f a c t that II. (U) acts i r r e d u c i b l y on H (X ) 1 t v (iv) L^ i s l o c a l l y constant over U . There i s a commutative diagram of short exact sequences: (4.11) n . • „2m-l, r m , : 0 >• k ^ : + F -» I v 0 -> J y 0 y i e l d i n g the following commutative diagram of long exact sequences: (4.12) H°(P 1,I) ^ H 1(p 1,L k) -> H 1(P 1,F) • 0 > ker 6 *-.H°(p\j) v H°(P 1,j 4R 2 m " V 7 t ) : • H^P 1,?"'*) • Now suppose that some i n t e g r a l class v of type (m,m) i n E l , 2 m - l ( - ) ^ H 1 ( p 1 j j ^ R 2 m - l k ^ 7 / : ) g c c o m e s f r o m H 1 ( P 1 , L k ) , and l e t a be any normal function i n H°(P 1,J) f o r which 60 = v (this i s possible 2 U m 1' , m !* 1(x,7£) i s interpreted as the constant sheaf on p \ and' the isomorphism's obtained v i a J 3 75 by (2.28)). I f we prove the i n j e c t i v i t y of i ' , then a diagram chase w i l l reveal the existence of a € H P (P^ ,J) coming from H ° (P^ " ,1) , such that a - a d ker 6 . I t w i l l l a t e r be proven that a i s i n v e r t i b l e so that <5a = v i s algebraic over g . We now set out to v e r i f y the remaining d e t a i l s . (4.13) Lemma, i ' i s i n j e c t i v e . Proof. The short exact sequence: (4.14) 0 y F ' >- F y V ' y 0 , y i e l d s the long exact sequence: (4.15) ••• y H 1(p 1 , F M ) • H 1 ( P 1 , F ) — y H 1 ^ 1 , ? 1 " ' * ) y • • • The cup product induces natural isomorphisms: (via p r i m i t i v e decomposition) (4.16) (i) F* ~ F ( i i ) F* ~ F There i s a composite of morphisms: - i n c l i „ ? r* i n c l * . - * - , . . . n . . . ^ E F f F r * F F , which implies that r i s s p l i t with d i r e c t summand F. Therefore i n c l ^ : H 1(P 1,F) ->- H ^ ( P \ F ) 76 i s i n j e c t i v e . Now there i s a commutative diagram: (4.17) H^P^F) H W . F ) H^P 1,^'*) and since H 1(P 1,F) fl H 1(P 1,F m) = 0 i n H1(.P1,F) , i t i s obvious by (4.15) that i ' i s i n j e c t i v e . We now introduce some further notation: (4.18) Denote (i) V = cokernel sheaf making the following a short 2iu~" 2 exact sequence: 0 — L „ »- j R f 7L V 0 f -* * * f ( i i ) V = cokernel sheaf making the following a short exact sequence: 0 »- >• j^R ~"k^ ~#- >• Vfc »• 0 . Note that V ^ V. ® H 1(C,^) . JC r 7i_ ( i i i ) L = ® K, L, = L, ® K, where K = Q, on: C (iv) = V. ® K, vf = V, ® K f f a: k k 7c We consider the sheaves , , , over U and state the following: O 0 0 0 (4.19) Theorem. (i) 1Q:• , L~ , V* , VT define underlying p o l a r i z a b l e f k f k v a r i a t i o n s of Hodge structures over U . ( i i ) The short exact sequence diagrams: (a) 0 >• L, -> R 2 m - V c > — • 0 (b) T,2m-1, „ R k C * are exact sequences of va r i a t i o n s of Hodge structures over U . Proof. One proves t h i s by going through the requirements f o r a v a r i a t i o n of Hodge structure defined i n [8], and t h i s w i l l be l e f t to the reader. Note that the i n f i n i t e s i m a l period r e l a t i o n i s t r i v i a l l y C C s a t i s f i e d f o r f i l t r a t i o n s of L <8> 0 , L. ® 9 . , hence also f o r f U k U v f ® eu ' vk 9 eu • (4.20) C o r o l l a r y . There are i n t r i n s i c a l l y defined Hodge structures on H P^1,^ ), HW,^), R 1 ( P 1 f j i R 2 n , " 2 ^ C ) f A v \ ^ R 2 m - \ c ) r 1 1 C 1 1 G H (P ,V ), H (P ,V ) . Furthermore, the following long exact sequences I K . are exact sequences of Hodge structures: (a) • H 1^ 1,!^) • H 1CP 1,j*R 2 m \ c ) • H 1CP 1,V^); y ( B ) • H1(P1,LS y H 1(p 1,j J TR 2 m - 2f^c)—- H1(.P1,V^ ) y Proof. This i s a d i r e c t consequence of (4.19) and the r e s u l t s i n [31, §7]. We remark that from the morphism defined i n (4.7) and the in c l u s i o n j ..R2m "^k.'^ - < 1 y F ™ ' , we get a homomorphism: * * h (4.21) v : j * R 2 m - 1 k A 7 ^ ^ F m ' * ® H 1 ( C ) . f I t i s t r i v i a l to v e r i f y the following: (4.22) Proposition; (i) ker v = hence ( i i ) lm v = V, k We also have the following which w i l l a i d i n computing a 1 1 C p a r t i c u l a r Hodge l e v e l of H (P , V ) : K. (4.23) Proposition. The Gauss-Manin connection defines a short exact . f m , * f m , * V :a _ fm+1,* sequence: 0 y F_ y F • 0, <S> F_ —y 0 f,h f P f Proof. This follows from a s i m i l a r argument to the proof of (3.62) , combined with (3.12) of [28]. 1 1 -m * 1 1 -m * (4.24) C o r o l l a r y . (i) H (P ,F_' ) H (P ,F ' ) i s i n j e c t i v e , f ,h f hence ( i i ) there i s a commutative diagram; (4.25) H 1(p 1,V k) •H 1(P 1,F m/*) (Sl f ^ C ) f C i n j e c t i o n f ,h C Proof. (i) From (3.60) we have H°(P 1, ft1 <8> F m + 1'*)= 0, therefore P f (i) follows from (4.23). ( i i ) i s obvious from (.i) There i s a commutative diagram: (Note: J , J are by L V d e f i n i t i o n cokernel sheaves) 3 See (A.4) f o r more d e t a i l s . 80 (4.26) 0 i ^ . ^m-L, 1 -v ( F m _ 1 / Fm) ^ H 0 ' 1 ^ pin,* f H 1(C) 0 1 J -> J I -> 0 0 -> 0 Remark. Since V , V define v a r i a t i o n s of Hodge structures of weights 2m-2, 2m-l res p e c t i v e l y , i t follows from [31, §7] that H 1(P 1,V^) and H 1(P 1,V^) ^ H 1(P 1,V^) ® H 1 ( C , C ) have Hodge structures of weights 2m-l, 2m res p e c t i v e l y , as i s r e a d i l y v e r i f i a b l e i n our case of the Lefschetz p e n c i l {x } ^ P 1 We now u t i l i z e some r e s u l t s from [31, §9] (4.27) Theorem. (i) H 1 ( P 1 , F P ) £ F P E 1 , 2 m 2 ' * f,h, (f) f o r a l l p 5 2m-2, s i m i l a r l y ( i i ) F ^ 1 , * E ] ' 2 m " 2 (f) = H 1(P 1,F P'*) f,h ( i i i ) H^P 1,^'*) = F ^ ' V t P 1 , ^ ) = P ^ ^ V ' 2 " - 2 ^ , , f,h 81" . l,2m-2,_. T T1 / T,1 . T,2m-2_ „, where E (f) = H. (P. ,-j^R f^C) Proof. Using the notation of [31], (i) and ( l i ) follow from the f a c t that the complex F- >• F- • V » £ c ft1 ( i o g I) <8> F- i s f i l t e r e d f ,h f ^o p l f quasi-isomorphic to the complex ^ ( 2 ) (t'1^-s ^ s (^-D i n [31]). Since the above Hodge f i l t r a t i o n s i n (i) and ( i i ) come from H (P' ) and l i p * . H (P ,F^' Q' . ) , using the above quasi-isomorphism and computing the E^ terms we see that (i) and ( i i ) are obvious. Now ( i i i ) follows from a s i m i l a r reasoning as above plus the following f a c t : The Hodge C <E f i l t r a t i o n on 6 -j^ CV^ ) = canonical extension of 0 y ® V f over U to P C P"*" , denoted by FP(V^1) i s the unique one induced from F P * F p(vp . f ~iu * ~rn. * C Since F_' = F ' (V ), the r e s u l t follows. f We need one more r e s u l t to prove our next theorem i n t h i s section, namely: (4.28) Proposition. H ^ P 1 , j . R ^ f . x ) n HV1, F™"1) = H^ P1,^ ) f,h Proof. There i s a short exact sequence: 82 (4.29) 0 • L f j ^ 2 m _ 2 f A 7 / . © F M _ 1 - ^ T C F ? J H f,h . 2 i H " ~ 2 where d(x,y) = x-y i n F- , , for a l l x € j.R f. 7£ , and r, n * , 7m-1 . • , . 2m-2 _. pm-1 ^ y € r . Now i t i s easy to see that j^R f*'^ - + ' _ = ' » f,h f,h therefore modulo possible t o r s i o n , H^(p\T)—>• H^(p\ F- ,) i s t, n i n j e c t i v e . The i n t e r s e c t i o n defined i n (4.28) i s a c t u a l l y an i n t e r -section i n H^(p\ F_ ) , and taking cohomology we get: f,h H 1 ( P 1 f L . ) ^ H 1 ( P 1 , j # R 2 n b - 2 f t 7 B : ) © H 1 t P 1 f F ^ ^ H ^ P 1 , F- ) . t f,h 1 , n Now i t i s easy to check that the ker d^ i s the LHS of (4.28). One checks that t h i s i s p r e c i s e l y the statement of (4.28). We now make the following remark: Remark. F P E ^ ' 2 m " 2 (f) /H^P1, F P ) =. E (p) , where E (p) i s the f,h 2 second E^ term of the s p e c t r a l sequence mentioned i n the proof of (4.27). Moreover, one checks (using the r e s u l t s i n [31, §9]) that E_(p) = H°(P1, ( ( j J . ( f i 1 ® F P - 1 ) ) fl V F-)/V F P) , which i s an expression 2 * U f f £ inv o l v i n g the i n f i n i t e s i m a l behaviour of F P . We also remark that i n f our case of the Lefschetz p e n c i l {X } 1 , either H (P , ) = 0 or f , h E^ (p) = 0 (for p=m-l, see the proof i n (A.5X - . T s T h e general case i s s i m i l a r ) . I t i s easy to prove the following: 83 (4.30) Theorem. ( i ) The i n t e g r a l c l a s s e s o f type (m,m). i n H^CP1, F™ 1) <8> (C) coincide with those of type (m,m) i n f, h H 1 ( p 1 , L k ) , moreover, i f E 2(m-1) = 0 then ( i i ) t h e i n t e g r a l c l a s s e s 1 2m— 1 — of type (m,m) i n E ' (k) c o i n c i d e w i t h those o f typ e (m,m) i n H-'-CP1,!^ ) P r o o f . Ci) F o l l o w s from ( 4 . 2 8 ) ; ( i i ) f o l l o w s from ( i ) and the above d i s c u s s i o n . Remark. We can now complete o ur s t o r y c o n c e r n i n g the Hodge s t r u c t u r e s d e f i n e d i n (4.27) and the diagram (.4.26). I f we look a t t h e cohomology o f (4.26) and a p p l y our known r e s u l t s t o i t , we o b t a i n the f o l l o w i n g summarizing diagram: (4.31) Q I 0 1 •4' 0 — v E m " " ( k , 7 Z ) fl{H (P , F m - 1 ) - , H i ( p i , L ) • E m - ± ' m ( f ) ® H U ' X ( C ) f , h k H1 CO } o • E m' m(k , 7 ^ ) o — ^ E 2 ( k , 7 / ) n {(E m' m - : Lcv^ E m - 1 ' m ( V ^ ) ) 9 HV)} .1,-1 „ ,r „m+l . l,2m-l,-->• E 2(.k,Z/) y F , * E 2 ' (k) -y ircp^vj y ( F m + 1 ' * E ] : ' 2 l n " 2 ( f ) HV) 84 where (i) E ' (k,^L) = i n t e g r a l classes of type (m,m) i n _ ^ 2 l T l — 1 — ( i i ) E2(k,7Z-) = i n t e g r a l classes i n E 2 ' (k) ( i i i ) E P ' q ( f ) = Hodge (p,q) l e v e l of E 2 ' 2 m _ 2 ( . f ) (iv) E P , q ( V ^ ) = Hodge (p,q) l e v e l of H^p 1,^) As a c o r o l l a r y to the r e s u l t s i n t h i s section so f a r , we obtain the following: (4.32) Co r o l l a r y . Let a € H°(P 1, J) and assume <5a = 0 i n H1(P"'",V1 ) . Then a decomposes i n t o a = a + a, , where 5a, = 0 k r o 1 1 and a (t) C J (Z ) f o r a l l t . O A t (4.33) One of the tasks involved i n i n v e r t i n g normal functions i s to prove that the Abel-Jacobi mapping i s meromorphic. Before doing so, we make a base change so as to t r i v i a l i z e the Picard-Lefschetz transformations. We f i r s t check things out l o c a l l y . Let A , A be 2 small open disks with E fl A = 0 , and l e t s = t : A 1 > A be onto. There i s a pullback diagram: 85 (4.34). A l 1 -> X = X r e s t r i c t e d to A A A with respective monodromy transformations T = I and T with T =1. Note that X^ i s singular, otherwise the l o c a l cycle i n -2 variant property would f a i l to hold (also note that s = t i s not smooth at 0) . To gl o b a l i z e (4.34) consider the h y p e r e l l i p t i c Riemann sur-2 face M obtained from the a f f i n e plane curve i n C defined by: 2 2 (4.35) {(w,z) € C | w - II (z - x) = 0}, by f i r s t completing and x<E£ then d e s i n g u l a r i z i n g . We obtain the corresponding global diagram f o r Z: (4.36) M -M x -Z- = Z p 1 M Z = U e-M M •+ M (deg 2) » P By a desi n g u l a r i z a t i o n process, we may assume that Z i s smooth. M There i s a corresponding diagram to (1.5): 86 (4.37) Note that a l l the l o c a l monodromy transformations associated to (.4.37) over u are a l l t r i v i a l . In a rather obvious fashion the family of M intermediate jacobians U JlZ ) extends over M with compact f i b e r s over the singular set E, M Let W be the family of r e l a t i v e codimension m algebraic cycles i n Z over P 1 , which are homologous to zero fiberwise. Then W p u l l s back to W = W x , M over M . A'normal function M p l a € H°(P 1, J) w i l l p u l l back to ir*a : M • UJ(.Z ) . I f the cohomology M class of T r*a, 6 M(ir*a) = qfalgebraic cocycle (for q € ©) y then 6a = (q/2) • TT^ (algebraic <cocycle).In view of the r e s u l t s i n [30], we need only check that the Abel-Jacobi morphism defined over extends to a meromorphic map : $ : W. M M > UJ(Z ) over M M t M (4.38) Now we w i l l show that i n f a c t $ extends continuously across M the f i b e r s of W„ over E„ , therefore we can a c t u a l l y deduce that $ M M x M i s a n a l y t i c . Assume that $„ i s meromorphic and l e t a € H°(.P. , J) have the property that cr(t) € i n v e r t i b l e part of J-(Z ) f o r generic t € P^ . There i s a diagram: where V = graph of $ M and Y i s some pr o j e c t i v e v a r i e t y dominating T (Y e x i s t s since V i s a Moishezon space [11]) . Now ir*a (M) i s a curve i n U J ( Z ) which p u l l s back to I 1 Tr*a(M) i n Y . Choose a M t curve D c I 1 ir*a(M) which maps onto M v i a Pr°£ . D also maps int o W and determines an algebraic cycle D on Z.. . I t i s easy M M M to check that (l/2deg (Pr°^ | )) TT^ [D ] i s the algebraic cocycle corresponding to a . (4.40) We now v e r i f y (4.38): Since we are dealing with a family over M with t r i v i a l Picard-Lefschetz transformations, the extended Leray cohomology sheaf F over M i s described p r e c i s e l y as (4.39) Y M F v = M 6. R 2m-1. M M,* } n F. Z /U M M M Since by d e f i n i t i o n F P = j / f l F , we must have M M,* M M (4.41) F P = {j .R^ 'V fi* } fl F . M M, * M, * Z ° /TJ There i s the l o c a l cycle i n v a r i a n t property: (4.42) The r e s t r i c t i o n homomorphism res: R 2 m ^k. ..C »- j„ .R 2 m "he .C M, * M,* M,* i s an epimorphism. Therefore we have 6 ® R 2 m "he X * F 3 Fm . Now l o c a l i z e over M M, * M M a disk A C M with A fl Y.^ = 0 € A . By the above discussion, a l o c a l o 7m 0 0 section v € H (A, F ) can be represented as a C form over k "*"(A), hence i t i s obvious that the Abel-Jacobi map extends across the M . . s i ngular f i b e r s of Ww over E„ . M M Summary. Looking over the above discussion, we are able to deduce the following: (4.43) Theorem: Given Z = XxC where (i) X i s smooth, p r o j e c t i v e of dimension 2m-l, and ( i i ) C i s a smooth curve. Assume there e x i s t s a Lefschetz p e n c i l {x } 1 of hyperplane t€P sections of X such that the Hodge (m-1,m-1) conjecture holds f or the generic X . Then i f a normal function v : > U ,J(Z ) P t 89 s a t i s f i e s the property that 6v = 0 in H 1(P 1,V ), then up to a r a t i o n a l s c a l a r , 6v i s the fundamental cla s s of an algebraic cycle i n Z. (4.44) We remark that i n view of (4.30) p a r t ( i i ) , the above theorem has i t s greatest strength when E^dn-l) = 0, as mentioned i n case 1 of the i n t r o -duction to t h i s chapter. In case 3, the above theorem becomes t r i v i a l and consequently a l l the p r i m i t i v e 2m algebraic cocycles of ..such Z = XxC must/lie, :in H m i m ( x , e ) ®H°(C,£) ©. H™" 1 , m ~ 1 (X,£) ® H 2 ( C , $ ) (cf (1.9)). O I T We now put together our facts on the normal functions H (P , J ) . Now J was defined i n chapter 2 to be the image of F™' ® H 1(C) in J , provided we have the s p l i t t i n g r _ = r / r © r _ , and we f f _ i f f have JT defined i n (4.26), J T = image of (F™- / F™ )® H 0' (C) i n J . f f Our summary on the normal functions i s given by the following theorem: (4.45) Theorem. (i) A l l the normal functions are h o r i z o n t a l . .... „o ,„1 T I . _ ,.2m-l ,„. . ( i i ) H (P ,J ) 5_ J (Z) (we are assuming for s i m p l i c i t y that ker 6 = ker <5 , see the discussion following (2.18)). ( i i i ) I f a e H° (P1,3 ) and the (m-l,m-l) .conjecture-r;-holds f or the generic. x ,. then &a i s - a - r a t i o n a l multiple of. an algebraic cocycle. (iv) ker 6 = { H 0(P 1,"F^" 1/ F^ )® H 0' 1^) / f,v f,v H°(P 1,L^ ) ®H1(C,7Z.)} f ,v 2m-l, . © J (Z) . Moreover, i f H'L ( P ^ i L ) = 0, e.g. L i s the constant sheaf over P 1, such as i n cases 2 and 3 of the introduction to t h i s chapter, then: (V) H0<p\7T) = H'V/F*-1/ P*) « H0'1^ ) / H'V.L. L f f k 2m (4.46) Recall the s i t u a t i o n of a generic hypersurface X c P of degree >_ 2 + 3/(m-1) , (m >_ 3) , where we have Prim m , m(Z,$) ® m-* 1 m—1 H ' (DxC,$))v = 0. Then the following i s true: (4.47) Cor o l l a r y . Given Z = XxC where X i s defined i n (4.46). Then H°(P1,T) = ker 6 = H°(p 1 ,F I t l ~ 1 / T™ ). ® H0'1^ ) / H°(L ) f,v f,v k , V e j ^ - ^ z ) . One could make the following guess: (4.48) Conjecture. (4.47) holds f o r a l l (smooth) hypersurfaces X c p ^ m ( m > 3) of degree >_ 2 + 3/(m-1), i . e . , a l l the normal functions a decompose into a = a + a' where o_ e H°(P ,^J ) and a-e J 2 m _ 1 ( z ) . 91 More s p e c i f i c a l l y we can conjecture the following: (4.49) Prim m , m(Z,©J = 0 for a l l hypersurfaces X c p 2 m (m >. 2) of degree • > 2 + 3/(m-l). Now (4.49) would imply the Hodge conjecture t r i v i a l l y f o r such, products Z = X x C, and hence f o r a l l Fermat fourfol d s , since Fermat v a r i e t i e s are r a t i o n a l l y dominated by v a r i e t i e s of the form Z = X x C ([25]). We remark that given a Fermat v a r i e t y Z c p 2 m + 1 f there e x i s t s a dominating morphism^- B'-('XxC) -—» z", where X c p m i s a Fermat v a r i e t y , c i s a Fermat curve, and E i s a Fermat subvariety of Z = XxC. I f one can prove i n t h i s case that Prim m' m(.Z,pJ = 0 , then i t i s easy to see that a l l the cocycles i n Hm'm(z',Q) come from blowups of cycles from E, X g x C where X^ i s a Fermat subvariety of X . This adds weight to the following conjecture (see [24, §4J). (4.50) Conjecture. Let Z }' be a Fermat v a r i e t y of dimension 2m . Then Hm,m(z",Q) i s generated by algebraic cocycles a r i s i n g from Fermat surfaces E and products of Fermat curves C x C . B (XxC) i s defined to be the blow up of XxC along E 92 Bibliography W. Barth and A. Van de Ven, "Fano-varieties of l i n e s on hyper-surfaces", Arch. Math. Vol. 31, 1978. A. Conte,,J.P. Murre, "The Hodge conjecture f o r fourfolds admitting a covering by r a t i o n a l curves", Math. Ann. 238, 79-88 (1978). P. Deligne; "Equations d i f f e r e n t i e l l e s a points s i n g u l i e r s r e g u l i e r s " Lecture Notes i n Mathematics 163, Berlin-Heidelberg-- New York: Springer 1970. P. Deligne, "Theorie de Hodge I I " , Publ. Math. I.H.E.S. 40, 5-57 (1972) . P. -Deligne, "Theoreme de Lefschetz et c r i t e r e s de degenerescence de sui t e s spectrales", Publ. Math. I.H.E.S. 35 (1969), 107-126. P. Deligne-and N. Katz, "Groupes de monodromie en qeometrie al g l b r i q u e " , (SGA 7 I I ) , Lecture Notes i n Math. 340, B e r l i n -Heidelberg •— New York: Springer ;i973. P. G r i f f i t h s , "On the periods of c e r t a i n r a t i o n a l i n t e g r a l s : I and I I " Annals of Math. 90 (3), 460-541 (1976). P. G r i f f i t h s , "Periods of i n t e g r a l s on algebraic manifolds, I I I " , Publ. Math. I.H.E.S. 38, 125-180 (1970). P. G r i f f i t h s and J . Adams, "Topics i n algebraic and a n a l y t i c geometry" Math. Notes (13), Princeton University Press: Princeton, New Jersey, 1974. P. G r i f f i t h s , "A theorem concerning the d i f f e r e n t i a l equations s a t i s f i e d by normal functions associated to algebraic cycles", Amer. J . Math. 94-131 (1979). 93 [11 ] P. G r i f f i t h s and J . Ha r r i s , " P r i n c i p l e s of algebraic geometry", Wiley, New York, 1978. [12] A. Grothendieck, "Elements de ge"ometrie algebrique III (premiere p a r t i e ) " , Publ. Math. I.H.E.S. 11 (.1961). [13] A. Grothendieck, "Hodge's general conjecture i s f a l s e f o r t r i v i a l reasons", Topology 8, 299-303. Pergamon Press 1969. [14] R.C. Gunning, "Lectures on Riemann surfaces", Math. Notes (2), Princeton University Press: Princeton, New Jersey, 1966. [15] R.C. Gunning, "Lectures on Riemann surfaces: Jacobi v a r i e t i e s " , Math. Notes (12), Princeton University Press: Princeton New Jersey, 1972. [16] R. Hartshorne, ^Algebraic geometry", Graduate Texts i n Mathematics 52, Springer-Verlag, New York, Heidelberg, B e r l i n , 1977. [17] R. Hartshorne, "Residues and duality,;", Lecture Notes i n Mathematics 20, Berlin-Heidelberg — New York : Springer 1966. [18] R. Hartshorne, "Equivalence r e l a t i o n s on algebraic cycles and subvarieties of small codimension", Proceedings of Symposia i n Pure Mathematics, Vol. 29, 1975 -, A.M.S. [19] N. Katz and T. Oda, "On the d i f f e r e n t i a t i o n of DeRham cohomology classes with respect to parameters", J . Math. Kyoto Univ. 8-2 (1968), 199-213. [20] N. Katz, "The r e g u l a r i t y theorem i n algebraic geometry", Actes, Congres Intern. Math., 1970. Tome 1, 437-443. [21] J . Lewis, "The Hodge conjecture f o r q u i n t i c f o u r f o l d s " , manuscript, 1980. 94 [22] W. Schmid, "Variation of Hodge structure: The s i n g u l a r i t i e s of the period mapping", Inventions math. 22, 211-319 (1973) [23] J.-.P.. Serre, "Geometries'algebrique et geometrie analytique", Annales l ' I n s t . Fourier 6, 1-42 (1955-56). [24] T. Shioda, "The Hodge conjecture and the Tate conjecture for Fermat v a r i e t i e s " , Proc. Japan Acad., 55, Ser. A. (1979). [25] T. Shioda : and T. Katsura, "On Fermat v a r i e t i e s " , Tohoku Math. Journ. 31 (1979), 97-115. [26] J . Steenbrink, "Limits of Hodge structures", Inventiones math. 31, 229-257 (1976). [27] A. N. Tyurin, "Five lectures on three-dimensional v a r i e t i e s " , Russian Math. Surveys 27 (1972), 3-50. [28] S. Zucker, "Generalized intermediate jacobians and the theorem on normal functions", Inventiones math. 33, 185-222 (1976). [29] S. Zucker, "Intermediate jacobians and the Hodge conjecture for cubic f o u r f o l d s " , Proceedings of Symposia i n Pure Mathematics, Vol. 30, 1977. [30] S. Zucker, "The Hodge conjecture for cubic f o u r f o l d s " , Compositio Mathematica, Vol. 34, Fasc. 2, 1977, 199-209. [31]. S. Zucker, "Hodge theory with degenerating c o e f f i c i e n t s : L 2 cohomology i n the Poincare" metric", Annals of Mathematics, 109 (1979), 415-476. 95 Appendix (A.l) The Abel-Jacobi morphism. We give a b r i e f outline of the construction of the Abel-Jacobi morphism, and r e f e r the reader to [8, p. 165] f o r further d e t a i l s . We l e t (i) Y be a smooth, p r o j e c t i v e v a r i e t y of dimension 2m-l ( i i ) W be the group of codimension m algebraic cycles on Y (modulo r a t i o n a l equivalence) which are homologous to zero in 2in—1 ( i i i ) {w, ,.. . ,w } be a basis of F H (Y,<£) . 1 n The Abel-Jacobi morphism i s a map 0 : W —>-J(Y) defined i n the following way: Let D e W. Then by d e f i n i t i o n of W, there e x i s t s a r e a l dimensional 2m-l cycle y with 8y = D. We define $(D) = ( /wlf...,/w ) Y Y as a c l a s s i n J(Y). One checks that $ i s well defined and i s indeed a morphism from the H i l b e r t scheme W to J(Y). (A.2) Some remarks on the sheaves {FP} . Associated to the p>_ o diagram (1.5) i n chapter 1 i s the period mapping (defined i n [8, p. 156]) which associates to every point t e U the corresponding Hodge f i l t r a t i o n 2rci—1 {F^H ^ Z t ' C ^ p > o a s a P°^- n t i n a c l a s s i f y i n g space E of Hodge structures (modulo the action of a given monodromy group on E). Now by the construction of E, there i s a natural embedding of E i n a product of Grassmannians, so that the bundles {Fp} , are pullbacks of universal p>_o bundles by c e r t a i n maps (which are i n f a c t a l g e b r a i c ) . These maps extend to P 1 (since P 1 i s a curve), giving the bundles {P3} p>o Now the sheaf j^F over P"*" can be described as the sheaf of 96 holomorphic sections of F over U with a r b i t r a r y growth near E . One can e a s i l y check from our sheaves { F p } defined above that p>o * (as an i n t e r s e c t i o n i n j ^ F ), and from t h i s i t i s easy to deduce part ( i i ) i n (2.16). I f we l o c a l i z e the sheaves over a disk A c with A n E = 0 e E, and assume the notation on page 34, then from prop. 5.2 i n [3,p. 91] ((i) and ( i i ) of prop. 5.2 are equivalent i n our case since we are extending a Hodge bundle which i s s e l f dual under ^ ) the canonical extension s * F i s completely characterized by the growth of sections about 0 with respect to any given h o r i z o n t a l multivalued basis ( c o e f f i c i e n t s grow l i k e powers of log ft) ). Consequently we obtain part (i) of (2.16) from (2.15). (A.3) We wish to v e r i f y (3.58) which says that H°(P 1,J') c J 2 m - 1 ( z ) . Let a e H°(P"'',J'). Then a i s equivalent to the following data: {a , U } where (i) a e H ( U , F ' ) ® H (C) a a a el a -( i i ) U d i l = n u n U Q c u ( m ) a a - c 3 e H (U a 3,R f ^ n F _ ' ) ® ^(0,7/0 where R 2 m 2f^7^and F m ' are viewed as subsheaves of F m f f Now such a give r i s e to i n t e g r a l classes v e F m H 2 m 2(X^) © m * 2m—2 F ' H (X^ _) which remain i n t h i s Hodge l e v e l under l o c a l horizontal displacement around t i n U . I t i s easy to check that such a v w i l l remain i n the above Hodge l e v e l under hor i z o n t a l displacement i n U . We 2 in - 2 use the i r r e d u c i b i l i t y of the TT^ (U) a c t i o n on H (xt/<£>)v to conclude that there are two cases to consider: case (i) a - aD </i*H (Z) f o r some a,3 e i . Then M p 3 97 H 2 m 2 ( X f c ) v c F m H 2 m 2 ( X t ) © F m ' V m 2 ( X t ) and one checks that (3.58) i s obvious from (3.57). case ( i i ) &a - a g e i * H 2 m - 1 ( Z ) f o r a l l a,.6 e I. Then (3.58) follows from a simple argument using (3.63). (A.4) Some'remarks • on;. (4. 23) . To prove (4.23), we argue i n a s i m i l a r fashion as (3.61) that 7™ 1 n W_ c F™ 1 and then deduce f,o f,o ^ f,o s i m i l a r l y that F™' = VJ_ n F™ 1} . The s u r j e c t i v i t y of V f,o f,o f,o f,o i n (4.23) follows from the f a c t that ft1 ® F c_ V F_ ( analogous P f ~ f argument to [28,(3.12)]). 1 2ltl—2' — (A.5) We prove that i f E 2 ' (f) jt 0, then E (m-1) = 0 i f f F m H 2 m ~ 2 ( X t ) v = 0 (Recall the d e f i n i t i o n of E 1 , 2 m ~ 2 ( f ) i n (4.27)). Proof. If F V m " 2 ( X j = 0, then j * ( P.1 ® F m _ 2) n V F" ' / V F™ - 1 t v U f - -1_ T*tn*~'2 * -— — j * ( A ® r. • ) n V (-_ / v r _ =.0, so that E (m-1) = 0 (we don't f f 1 2m—2 need the added hypothesis E ' (f) / 0 i n t h i s case); conversely, suppose that E (m-1) = 0. Then F m ~ 1 E 1 , 2 m ~ 2 ( f ) = H 1(P 1, F™ - 1). 1 claim f ,h that i f F m H 2 m _ 2 ( X ) ft 0, then F™"1 = i * F n i' 1H 2 i n' 1 (X) i s the constant f,h ' sheaf over P 1. ~m—1 To see t h i s , we simply note that a l o c a l section a of F f ,h always extends to a global multivalued section a of F™ 1 . Now the f,h hypothesis F m H 2 m 2 ( X f c ) v f 0 implies that f o r generic t e u , the vanishing 2m- 2 cocycles {6 1,. . . N } of H ^ X t ^ v a r e n o t o f t y p e (m-1,m-1). Now i f we consider the horizontal section a near a singular point i n £ 98 corresponding to 8-^ say (where we may assume WLOG that o ^  ^ 0) , then for a to be a multivalued section of F™ 1 , i t must induce f ,h classes i n F m ^~tt2m 2 (X ) f o r a l l t e U, and hence by the Picard-Lefschetz formula, o + [(-l)™ \^ 8^] 8^ must also induce classes i n ^"H2m 2 (X ) ( a l l t e U). Since 6 ^ i s i n t e g r a l , i t must be of Hodge type (m-1,m-1) over U, and hence so are the { 6 . ^ . . . , 8^} ,-..contradicting the above statement: f o r generic t e U, { 6 ^ . . . , 8^} are not of type (m-1,m-1). This j u s t i f i e s the above claim. Now using the claim, i t i s easy to see that given E^dn-l) = 0, then either F m H 2 m ~ 2 ( X t ) v = 0 or else F m H 2 m " 2 ( X f c ) v fi 0 and F ^ E 1 ' 2 m _ 2 (f) = H^P 1, Tf" 1) = 0. Since E * ' 2 m _ 2 ( f ) ft o, we must have F ^ 2 ™ " 2 (X ) = 0. f ,h V (A.6) Referring to the construction of X^ i n (4.34), i t should be noted that the s i n g u l a r i t y of X A i s p r e c i s e l y a single ordinary double point ( v e r i f i c a t i o n l e f t to the reader). (A.7) A remark on the proof of (4.13): From the s.e.s. 0 —> "F™ > F > "F"1' the L.E.S.: H^P 1, ? > - * H ° ( P 1 , T'*) — ^ ( p 1 , T) — H ^ P 1 , F ) — ^ ( P 1 , T'*) Now I claim that H° (P 1, F ) >H°(P 1, "f™' ) i s s u r j e c t i v e . I t follows from (3.54) that i t s u f f i c e s to prove the following: -> 0 , we obtain 99 ^ ( p h r 1 ) f ,v ^ T T o , 1 r±m-l , —m , . . . ->H (P f r _ / F_ ) i s s u r j e c t i v e . f , v f , v „ ^ -pm r=m-l From t h e s . e . s . 0 * F > F f , v f , v -> 1 f t 1 / ft -f , v f , v ->0 p l u s (3.60) and S e r r e d u a l i t y , i t i s c l e a r t h a t t h e above c l a i m i s j u s t i f i e d . U s i n g t h e above c l a i m , i t i s n o t h a r d t o check t h a t t h e r e e x i s t s an isomorphism: (P^, F ) W^p1, F ™ ) ® H 1 ( P 1 , p™'*) such t h a t t h e f o l l o w i n g diagram commutes: H1(P1, F ) i n c l * H ^ P ^ F ) - ^ ( P 1 , F") • ^ ( P 1 , p™'*) 0 © i n c l . p r o : -^ H^ P1, r1'*) Now we s h o u l d a l s o remark t h a t u t i l i z i n g t h e p r o p e r t i e s o f our L e f s c h e t z p e n c i l { X } t e p l , one can g i v e a c a s e by case argument ((4.10) p a r t ( i i i ) ) t o v e r i f y t h e i n j e c t i v i t y o f i ' . 

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