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Towards the equivariant cohomology ring of moduli spaces of stable maps into Grassmannians Tschirschwitz, Boris 2005

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Towards the Equivariant Cohomology Ring of Moduli Spaces of Stable Maps into Grassmannians by Boris Tschirschwitz Dipl.phys, Freie Universitat Berlin, 1998 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Mathematics THE UNIVERSITY OF BRITISH COLUMBIA April 2005 © Boris Tschirschwitz, 2005 Abstract We compute the equivariant hypercohomology of the Koszul complex associated to an equivariant vector field. This hypercohomology is conjectured to be the equivariant cohomology ring of the moduli space Mop (Gr^C 3) of stable maps of degree 3 from genus zero prestable curves into Grassmannians. Our methods are based on the localization methods of E. Alkildis, M . Brion, J.B. Carrell and D.I. Lieberman. This thesis is an extension of the work by K. Behrend and A. O'Halloran on stable maps into projective spaces. ii Contents Abstract ii Contents iii Acknowledgements iv Introduction 1 Computation of Cohomology 8 Grassmannians 8 Pliicker Coordinates on Grassmannians 10 Affine Coordinates on Grassmannians 10 Group Actions on Grassmannians 13 Gm-equivariant Vector Fields 17 Biah/nicki-Birula cells 18 The C* -equivariant Cohomology of Grassmannians 21 Equivariant Chern Classes 24 The Universal Quotient Bundle Over a Grassmannian 25 Computation of Chern classes a la Behrend-O'Halloran 25 Moduli Spaces of stable Maps 32 Cohomology of Stable Map Spaces 38 Chern Classes 45 Chern Classes in Terms of Coordinates on T 48 Conclusion 58 Outlook 59 Bibliography 61 iii Acknowledgements I thank Kai Behrend for suggesting the topic of this thesis to me and for his help in realizing it. I must also thank him for his patience. Thanks go to Bill Casselman, Jim Bryan, and Jim Carrell for educating me on the overall structure of a well-written thesis. I thank Greg Lomnes for opening the coffee shop "think!" just when I needed it. The thesis was in its entirety written there. I thank Wolfgang PriiEe for his financial support. Most importandy, I thank Susan Lazar for her moral support. iv ) s Introduction The goal of this thesis is a better understanding of moduli spaces of stable maps, in particular of stable maps of degree 3. It will be clear shortly why we prefer to concentrate on degree 3. In their paper [DM69] Deligne and Mumford introduce moduli spaces of stable curves of genus g with n marked points, denoted by Mg>n. This paper is also the birthplace of stacks. These spaces are fundamental to intersection theory of curves. As a generalization, stable map spaces are completions of spaces of smooth maps from algebraic curves into varieties. Let X be a smooth projective variety and let $ e H2X be the class of a curve in X. Then M & n (X, /3) is defined to have as generic member a smooth map / from a non-singular, connected curve C of genus g into X such that /»[C] - /3. The additional points are maps / : C — X from a nodal curve C of arithmetic genus d with n special points (nodes or marked points) such that every component of C contracted to a point by / has at least 3 special points. These stable map spaces were introduced by Kontsevich and Manin in [KM94] and have become fundamental in the study of Gromow-Witten theory. One recovers Mgtn as M^({pt.},0). The spaces of interest here are Mop (Gr f cC\ 3), where GrkC n is the Grassmannian of fc-dimensional quotient spaces of C n . We choose to study Grassmannians in this thesis because of the following property of moduli spaces of stable maps: 1 Let X be a complete non-singular variety with tangent bundle 3~x. Then Mo^tX, d) is a smooth projective stack if X is convex, i.e. if for every morphism ^ : P 1 — X Hl(P\n*Fx) = 0. The main examples of convex varieties are homogeneous spaces X = G/P. The simplest homogeneous spaces are the projective spaces studied in [BO03]. Now we study the next more general class of homogeneous spaces, Grassmannians. To gain a better understanding of the geometry of Mop (GrfcC", 3), we want to com-pute its cohomology ring and its equivariant cohomology ring. In number 60 we give a very short account of the concepts introduced in [BO03] and used here, but we must still ask the reader to study that article in detail. To ease the pain, we will now indicate the results of this thesis as we outline the article [BO03]. The main issue of the article [BO03] is to compute the cohomology ring of Mo,o(P"» d). The method to achieve this is inspired by the utility of C*-actions for studying in-tegrals over stable map spaces, and the existence of C*-equivariant vector fields V. In [BC04] Michel Brion and lim Carrell prove the following theorem (Theorem 1 on page 194): 1 Theorem. For a smooth projective regular variety X with an action of the group B of upper triangular 2x2 complex matrices of determinant 1, there is an isomor-phism of graded algebras p:H^X)-*n2,e2). Here T is the diagonal torus of the group B and 2 is the zero scheme of a C*-equivariant vectorfield on [/x A 1, where U is the big open cell of the C*-action X. The construction of the vector field is done in number 34 on page 18 and the grading comes from the fact that the coordinate ring of 2 is a direct sum of polynomial rings in one variable, see [BC04], page 190. This result depends crucially on the finiteness of the fixed point set 2 of the e n -action. We demonstrate the technique in proposition 44 and corollary 46 by computing the equivariant cohomology ring of the Grassmannian Gr{k,p). 2 Unfortunately, the zero scheme 2f has positive dimension in our case. Fortunately, for ordinary cohomology, the method in [BC04] was extended in [BO03] to stacks with zero schemes of dimension bigger than 0. We apply these general results to ]V%(GrfcCn,3) on pages 38 to 57. It turns out that, to compute the ordinary cohomology on an infinite zero scheme 2, the ring of global sections Y{2, ©2) has to be replaced by the hypercohomology ring H°(X, Ky) of the Koszul complex Ky defined by the vector field V. H°{X, Ky) depends on an open neighborhood of X in X, but surprisingly, at least for Mop(Pn, d), it is shown in [BO03] that the attention can be restricted to a single open subset U of X, even though U does not cover the fixed locus 2 completely. We generalize the construction of this open substack U to maps to Gr(k,p) and show in proposition 71 on page 38 that it is again the big Biafynicki-Birula cell. We assume that we will again be able to concentrate on this one open substack. This assumption is founded in the results of [BO03] noting that the projective spaces P f c and Pp form for each choice of k and p the extreme cases p - 1 and k = 1 of the sequence of Grassmannians Gr(k,p). Another indication is the fact that all Grassmannians of equal dimension have the same unique Schubert cells of codimensions 0 and 1, which constitute the arena for our study. This is true because the big Schubert cell of a Grassmannian is also its big BB-cell-we show this in proposition 48 on page 29. It is shown on page 404 of [BO03] that for a proper and smooth stack X endowed with a C*-action producing an equvivariant vector field V the morphism H{X,C) = H°(X,KV) - H ° ( U , K V ) = T{U,0z) is injective in all relevant degrees. We hope to be able to show this for Gr{k,p) in the near future. One of the main results in [BO03] is a description of the big Biafynicki-Birula cell of Mo,o(Pn, d) as a vector bundle over the moduli space MQJ modulo an action of the symmetric group Sa in proposition 3.8 on page 419. We restict our study to maps of degree 3, because MQ,3 is just a point. lust as in section 4.2 on page 430 of [BO03] the big Biafynicki-Birula cell U is then simply a vector space modulo an action of S 3 . 3 This outcome is not surprising in light part (ii) of theorem 2 in [FP97]. It states that the moduli space of stable maps into a convex space is locally the quotient of a non-singular variety by a finite group. In [BO03] the vector field V is found in proposition 4.5 on page 431, leading to a conjectural description of the cohomology ring of Mo,o(Pn,3) in Corollary 4.18. We find the corresponding vector field in theorem 76 on page 40. Unfortunately, we cannot yet make a very strong case for a conjectural description of the cohomology ring Mojo(GrfcC", 3). In [BO03] the conjecture was based on the knowledge of the Betti numbers of Mo,o(P",3), computed in the preprint [GP05] by Getzler and Pandharipande. Once we know the Betti numbers of Mop(GrfcC",3), we should be able to find that cohomology ring. In the limit n — oo the cohomology ring of Mop(P",rf) stabilizes, and the ring defined by this limit is called the cohomology ring of MopfP 0 0, d). The main result of the article is given for d = 3: H(M0x)(Poo,3),C) = C [ fo ,o- i ,p ,a 2 ,T ,CT 3 ]/((T 2 -pcr2) ,T(T3 ,po '3) • As a main result we were still able to compute the cohomology ring and also the C*-equivariant cohomology ring of the Kozul complex Kv in corollary 80 on page 44. In the projective case the generators can be expressed in terms of Chern classes of vector bundles jr„f*0(l), 7r*f* ^ (2), and n*\*@{3) onM0p(P°°,3) (similar to the diagram below). The degrees of b,o\ are 1, the degrees of p,az,T are 2 and the degree of 0 3 is 3. We have computed the Chern classes of the pullback-pushforward 8 - n*f*£l to Mop(GrfcC",3) of the universal quotient bundle over Gr{k,p) indicated in the diagram below in proposition 87 on page 48. 4 GrkC n Mofl(GrfcC",3) We think that the equivariant cohomology ring of Mop (GrfcC", 3) should be express-ible in terms of such pullback-pushforwards of symmetric powers of i2. 5 Organization of this thesis: We start with an introduction to pur notation for Grassmannians and their cell decomposition in numbers 2 to 9. Next we define two kinds of coordinate systems, the well known Pliicker embedding into projective space in number 10, and a covering by open subsets isomorphic to affine space that contains the big Biamicki-Birula cell as a chart in numbers 12 to 17. In lemma 19 we prove that the coordinate chart U0 coincides with the Schubert cell of codimension 0. In number 20 we compute coordinate changes between affine charts. Following these constructions on Grassmannians we introduce the group action on Gr(k,p) generating two equivariant vector fields V and V on G r ^ C m numbers 22 to 24 and give an explicit formula in equation 12 on page 16 for the action in any of the (£) charts. Before we can compute the equivariant cohomology ring of Gr[k,p) in proposi-tion 46 we need to prove that U0 is the big Biatynicki-Birula cell for this group action. We introduce Biatynicki-Birula cells in numbers 35 and 36 and compute the big BB-cell in corollary 43. Next we compute the C*-equivariant vector field on Gr[k,p) in proposition 44 and the C*-equivariant cohomology ring ofthe Grassmannian in terms of coordinates on U0 in proposition 46 to demonstrate the method in the simple case of only finitely many fixed points. We remind the reader of the tautological sequence over Grassmannians in num-ber 51. In number 53 we show how to compute the Chem classes of an equvariant vector bundle from an equivariant action of an equivariant vector field on the bundle. This was first shown in remark 1.10 in [BO03] on page 396. We compute the equivariant Chern classes of the universal quotient bundle over GrfcC" in terms of coordinates on U0 in proposition 54. It follows a short introduction to moduli spaces of stable maps in numbers 56 to 58. In number 59 we introduce the open subvariety Y of Gr{k,p). Stable maps into this 6 subvariety will form the big Biafynicki-Birula cell U of Mop (Gr f cC\ 3). In number 60 we give a concise summary of terms introduced in [BO03] and needed here. It is intended as a reminder after having studied that paper. Next we define the vector space T in number 62 parameterizing the stable maps in U up to an action of S3, the symmetric group in three letters. In numbers 64 to 70 we give geometric interpretations for the coordinates of T parametrizing stable maps in U c Mo,o(GrfcCn,3). In theorem 71 we prove that U is the big Bialynicki-Birula cell of Mop(GrfcCn,3). The computation of the equivariant hypercohomology ring of Kv starts in num-bers 72 to 73 with an outline of the method. In theorem 76 we compute the equivariant vector field Von the substack U. In corollary 80 we give the hypercohomology ring of the Kozul complex in terms of generators and relations. Here the generators are the coordinates on T and the relations describe the zero scheme of the Gm-action on Mop(GrfcCn,3) induced by the action on GrfcC". This is done with the help of the computer algebra system 'Maple' by inverting the isomorphisms in the diagram T{GrkC n,SrGrka) r T[V,9v) 1^  r(^,f*5-Gr f cC") Dn resulting in a homomorphism nGrkCn,g-GrkC*)-TV,9~T). We show the correctness of this map in number 78 and in proposition 79 in a way independent from any result using computers. Finally, we compute the Chern polynomial of the bundle 8 in number 83 to propo-sition 87 in the hope to gain in the future a representation of the cohomology ring independent of the chosen coordinates on the big cell, in terms of Chern classes of 8 and its symmetric powers. 7 Computation of Cohomology Grassmannians 2. Let V be a finite-dimensional vector space. For fc < dim V we denote the Grassmannian of fc-dimensional quotient spaces of V by Grk V. If the vector space is C" we also write Gr(k,p), where p = n-k. The dimension of GrfcCnis k(n-k) = kp. Put another way, it's the space of all epimorphisms Q n - L ^ E ^0, where dim E = k. We denote the Stiefel variety of sets of fc linearly independent vectors in V by Sfc V. We can always assume that V = C" and E c C n . We will represent a point <p e GrfcC" by • Cn/ker(/> • SfcC™/ ker</>, i.e. by (representatives of) bases of C"/kertp • By fc x n matrices of maximal rank, interpreting the rows as (representatives of) basis vectors of C n / ker (p. Since Gr^V = SfcV/GLfcV, two elements v, w e S^ V represent the same quotient space C"/ kercp if there is a G e GLfc V such that Gw = y. 8 We will also look at Gr^C71 as a homogeneous space, the quotient GL„C/P of the general linear group on C n by the parabolic subgroup / • ... • 0 ... 0^ • • • 0 • • 0 G GL n C In this representation it is obvious that dim Gr [k, p) = pk. Without loss of generality, we will assume that p > k, as this will simplify notation. 3 Example. G^C 4 = Gr(2,2) is the space of all epimorphisms from C 4 to planes through the origin in C 4. Its dimension is 2(4 - 2) = 4. 4. Fix a complete flag F : 0 c F i c . . . c F n = V of V. For every Young diagram1 A = (Ai,...,Afc) of at most k rows and p columns, we define a subvariety of the Grassmannian Gr(k,p): D.x = {x eGrkV | dim(xn F f c + ;-_ X j)> ;'(V1< j<p)}, called the A-th Schubert variety. The Schubert variety has codimension |A| = Xf=1 the number of boxes building the Young diagram. 5. Every Schubert variety Q\ represents a class S\ e H2WGrkV. The class is independent of the choice of flag. 6 Remark. The empty diagram A = () corresponds to the unique Schubert vari-ety Q() = GrkV and the diagram with one box A - • corresponds to the unique Schubert variety Qn S A** - 1 . 7. The Schubert cell of Gr{k, p) is defined by = {x| dim(xn F r) = j (Vfc + j-Xj<r< k + j - A ; + i , 0 < j <p}. The closure of the cell Q° is ~oJx = Q A , and Q° = A f c p - | A | . ^ee [Ful97] 9 Furthermore, the Schubert variety £l\ is the disjoint union of all Schubert cells of dimension at most |A|: nx = | J n°, A':|A'|<|A| 8 Remark. It follows that there are unique Schubert cells of codimensions 0 and I in every Gr{k,p). 9. We can use this cell decomposition to see all Gr{k,p) as subsets of Gr{k,oo): Gr(k, 0 ) c Gr(k, 1 ) c . . . c Gr{k,p) c . . . c Gr(fc,oo) Pliicker Coordinates on Grassmannians J 0. The Pliicker embedding p: GrkCn - P/ \ f c C \ i / — (v\,..., vk) •—• i>\ A • • • A vk is an isomorphism between Gr f cC n and a subvariety of P W _ 1 . II Example. On Gr 2 C 4 we find : Gr 2 C 4 — P 5 with , [V\,\ 1/1,2 t/1,3 1/1,4' 11/2,1 1/2,2 1/2,3 V2A) = (t/1,11/2,2 " V\t2 l / 2 , l , 1/1,11/2,3 - 1/1,3 ^ 2,1, ^1,1 ^2,4 ~ ^1,4 ^2,1. 1/1,2 1/2,3- f/1,3 1/2,2. 1/1,3 t/2,4 - 1/1,41/2,3) Affine Coordinates on Grassmannians 12. Let {£} be the subset of ( Z 2 ) n of numbers with exactly k ones. Then {£} has Q elements. The length \a\ of a is the number of inversions in a, i.e. the sum over the zeros in a of the number of ones in a after that zero2. We write a' > a if \a\' > \a\. 2This is the opposite of the convention in [KT03] 1 0 13Example. For G ^ C 4 we have {4} c ( Z 2 ) 4 (4l = {1100,1010,1001,0110,0101}, where the strings have lengths 0,1,2,2,3,4. 14. There is a bijection between {£} and the set of Young diagrams with at most k rows and p columns. The vector a e {£} is identified with the Young diagram having as many boxes in row i as the number of inversions in a caused by the i-th 1 from the right. We write A (cr) and cr(A). 15. Let v e SfcCn,<7 e {£}. We call the k x fc-minor build from the columns with index equal to the index of one of the k ones in o the c-th k x k minor of v, denoted bymCT(y). 16 Example. For G r 2 C 4 and v = (j£j J£| ^4) w e n n ( * : miiooC^) = miooi(y) = Tnoioi(y) = "id vh2 'V\,\ Vifi .^ 2,1 2^,4. 1^,2 V\A V2,2 2^,4 mioio(f) = mooii(f) = *>1,1 "1,3 V2,l' »2,3) v2,2 2^,3, 1^,3 "1,4 "2,3 "2,4 17. We use {£} as index set for the following (£) coordinate charts on the Grass-mannian GrkQn: Let [7ff for a £ {£} denote the open set of all points x representable by a matrix with ma(v) = Hfc. In the case a = l f c 0 p this is the matrix in reduced row echelon form for x. U0 = UlkQp = {x = (Hfc,x°) e Gr{k,p)\x° e Matfc,p} Ux = t/jfc-lQlOP-l 1,1 0 x 1,2 Dfc-l 0 x i j 1 x i 2 .1 \ -k,PJ £Gr{k,p) x 1 £ Matfc,pC (1) (2) and charts £ a : Ua — Gr(fc,p), y = (fi i>fc)r •— span{yi,..., vk}-11 18 Example. For Gr (2,2) we get the two open subsets U0 = l/noo = {* = (12. e Gr(2,2)|x° e Mat2,2} UL - t/ ioio f 1 .X-^  ^ 0 JC-^ ^ [\0 Xgj 1 x2,2> EGr {2,2) x 1 £ Mat 2,2c| 19Lemma. Let C/0 be the chart UXk0? of Gr(fc,p) and let Q° be its Schubert cell of codimension 0. Then u0 = n°0. Proof. This just states that we implicitly chose the standard basis of C". • 20. We compute the coordinate change : UL -* t/o.x1 = (jtf.) >-* x° = (x°(x1) i jy). Doing row operations we see that the point in U0 n Ux, represented in t/x by the matrix 1,1 0 x 1,2 lfc-1 0 Xk,l 1 Xfc,2 has the following representative in U0: c1 11 X k,pj l/*fel ll,2 ' *2,2 ' vfc-l,2 2,p fc-l,p "•fey By coordinates: v i , l r-, l<i<-fc- l , 7 = 1 ^1 x / " , i x f c j x fc, i — ^ x f c , i ' J — 1' * — fe i xr, = xr,- -r—S 2<j<p, l < i < f c - l x£;- = - ^ - , 2<j<p, i = k "•fel vfel 12 21 Example. On Gr (2,2) we find for the coordinate change £ 0 1 : UL -* U0, x 1 = (x^j) x° = (x°(x 1) ij). that the point in U0 n Ux, represented in U± by the matrix 1 x 1,1 1° 4 , i has the following representative in U0: l/x^j n,2 x22 1 _ X\,\X22 \ Xl,2 xi, Group Actions on Grassmannians 22. We denote the multiplicative group of invertible complex numbers by G m and the additive group of complex numbers by G a . 23. Let B be the Borel subgroup of lower triangular matrices in S L 2 C . This group contains representations of G m and G a on C 2 , such that B is isomorphic to the semi-direct product B S G a xi G m . The isomorphism is given by A:G m — T c B , f —A(f) = and (p: G a — U c B, u —• (p{u) -1 0 1 The subgroup T is the Cartan subgroup of SL 2 C. 24. On Gr f cC" we will use the right-action of G m induced by the representation of T on C™ with weight r -1 on the standard vector er: Let x e Gr(k,p). Then A e G m acts on x by right multiplication (A, x) • 1 0 I n - l (3) The representation of U on C " by right action v • u -* v • e"w" induces the following right action on Gr (k, p): («,x) — x-euN\ (4) 13 where Nn is the nilpotent matrix whose only non-vanishing elements are ones along the sub-diagonal. Combining the two actions we get a B-action O: {{u, A), x) ~ [u, A) = x -euN"Dn. Here we write Dr for the diagonal matrix (5) Dr = 1 ir-1 25 Example. The action on Gr (2,2) is <D: ((u, A),x) •— (u, A) = x • &uNnDn (6) with D 4 = A 2 A 3 and = \ + u (0 ro f° 1 0 u2 0 0 u3 0 0 + — + — 0 1 0 2 1 0 0 6 0 0 0 1° 0 1 1° 1 0 I 1 0 0 o ; 0 1 1 u u2/2 u u3/6 u2!2 0 0 0 0 1 0 u 1 26. Let x e Gr{k,p) be a point in U0, written as a k x n-matrix. Then there is a G e G LfcC and afcxp matrix x°, such that Gv is in reduced row echelon form (Hfc, x°). We compute the action of G m on the affine open U0: An x e Gr{k,p) has the form x = (G,Gx°) for some G e GL f cC when written as a k x fc-matrix followed by a k x p-matrix. The action of A e G m on (%, x°) is (%,x0)-X = (Dk,Xkx0Dp (7) 14 Therefore (%,x°)-A = (\,AkDl1x°Dp) The action on the coordinates is 1-1 A " f c + 1 0 i-fc+i (Xk 0 0 ^ o •. 0 0 0 A " _ 1 J (8) <Xhx° 1,1 [Xkx°h in-1 r o ( i f c v o i fc+l v o i n - l v o '\ /i/ l 2 1 * * * fc 1 (Ax° p A 2x£ p Xk,p ) (9) (10) 27Example. In Gr (fc, p) we get the action ^ "^ 2 1 ^ °^2 2 ^  28. Now we compute the action on a general affine open Ua for a e {"}. Denote the cr-th x fc-minor of x e Gr{k,p) by A and the rest by B. Then G m acts on A by right multiplication with (X°l A°2 1 A A 0 * - 1 A°k where oi , . . . , ok are the indices of the ones in a (we get the factor 1IX because we start counting indices at 1). Furthermore G m acts on B by right multiplication with 1 A XZ2 A z ?-i Xz» 15 where the indices of the zeros in a. Therefore the action of G m on the coordinates x° = A~lB of Ua is given by A " 0 1 A " 0 2 A" 0 " - 1 A" 0 * ] (A 2 1 A Z 2 Az? (11) A ^ ' - ^ x ^ A Z 2 - 0 l x f 2 A2,2 A z ? - 0 l x ^ A Z p - o 2 x a 2,P J Z l - O f c - l r ( T I Z 2 -Ofc- l y C I A ^ ' - ^ x ^ A xfc-l,p-l A Afc-1, A Z 2 - ° f c x^ 2 iZp- i -Oj i^ -a (12) Since < z 2 < • • • < zp and 01 < 02 < • • • < ok the exponents are strictly decreasing with increasing row index and stricdy increasing with increasing column index. The possible exponents are: l-(ra-fc+l) <zi - 01 <(n -p+ l ) - l p - (n-fc+1) <zp- -Ol <n- 1 or, simplified, 1-n <zi - o f c <(fc+l)-fc p - n )-Ofc <n -k -1 <Zl-Oi <zv-oi <k <n- 1 -(n+l) -k ^zi - ofc <Zp - Ofc <1 16 29 Lemma. Let G m act on Gr{k,p) as explained in number 28 on page 15. Then the induced Gm-action of A £ G m on the coordinates xa on the chart Ua, a e {£} is element-wise multiplication with powers of A. The number of negative exponents is equal to n- \o\. Proof. The exponents are of the form z; - oy where z,- is the position of the i-th zero in o and Oj is the position of the j'-th one. The length \o\ of o is the number of inversions. We have seen in number 12 on page 10 that this is the sum ofthe number of ones after every zero. But the j-th one comes after the i-th zero exactly if z,- - Oj < 0. • G m -equivariant Vector Fields 30. Let an algebraic group G act on the smooth algebraic stack X from the right and let 8 — X be a vector bundle over X. We say that a left G-action on 8 is a lift of an action on X if the following diagram commutes for all g e G: 8-^8 x ~ r x We call a group action on a vector bundle a geometric action if it is a lift of an action on the base space. The action on the vector bundle induces a left action on the corresponding sheaf (£ of sections such that the following diagram commutes for all g e G and all e e <B(U-g) = T{U-g,8): 8—^8 g-e e U—^U-g Note that the right action on Gr{k, p) induces a left action on sections of 8, because T{*,8) is a contravariant functor. 17 31. The above construction gives the following geometric action, Dg, on the tangent sheaf of X for any g £ G and ve3~x: (Dg)(g-v) = g*v 32. We call the fibre-wise G m -action of scalar multiplication the linear action on the bundle S — X. 33. We call a section e e T{X,S) of the vector bundle § with geometric Gm-action G | „ X ( ? ^ ^ (A, e) —• A • e equivariant if A • e = Ae, where the right-hand action is the linear one. 34. Taking the derivative of the Ga-action described in number 4 at u = 0 induces a Gm-equivariant vector field on Gr{k,p), (Xi,2 . . . Xhn 0\ Wx = du u=0 Q>x(u,l) = xNn = \Xk,2 Xk,n 0) Taking the derivative with respect to A at A = 0, we get another equivariant vector field on Gr(k,p), W'x = dA A=l 0^(0, A) = xA = x-(0 0 1 0 . . . 0 0 2 0 0 0 '•• 0 ^ 0 0 0 0 n-2 0 0 n-1) Bialynicki-Birula cells 35. If X is a scheme or smooth algebraic stack with a G m -action, then we denote by xo the limit and by XQQ the limit xo = lim A • x A-0 Xoo = lim A • x A—oo 18 for x £ X . 36. Let C be the connected component of X G M containing the fixed point F£ X G M . We call the set C + = {x £ X I x 0 e Q the plus cell of C , and the set C ~ = { X E X | X O O E C } the minus cell of C . 37Proposition. The fixed points of Gr(fc,p) under the given Gm-action and the strings in {£} are in a one-to-one correspondence a ^ Fa-The fixed point Fa is represented by those kxp matrices whose o-th k x fe-minor has maximal rank and whose kxp submatrix complementary to the minor is equal to the zero matrix. Furthermore, the plus cell F„ of Fa is given by Fa = Ua- U Uo'-a'<a Proof. Let's first look at U0 = UXk0P. From equations 1 on page 11 and 12 on page 16, we see that all coordinates are of the form A r x? , with positive exponent r. Therefore, each point x in U0 has as limit point, xo, the point with all coordinates equal to the zero matrix in U0. If G m acts with negative weight on a coordinate x?. on Ua, then it also acts with a negative exponent on xjj on Ua' with o' > o. This follows from Lemma 29 on page 17. On Ua the points x with xo = Fa are the points with coordinates xf. = 0, if z ,• - 0[ < 0. The other points do not approach a fixed point in Ua. In fact, the points of Ua that do not have Fa as limit point must have more negative weights. Thus they approach a limit point Fa1 for some o' <o. • 38 Corollary. If we arrange the weights of the G m - action in fe x p boxes according to their position in the coordinate matrix of v • A, for veUa, and reflect along the anti-diagonal, then the boxes with negative entries form a Young tableau whose diagram A is equal to A(<r). This Young tableau has strictly increasing entries horizontally and vertically. 19 39 Corollary. The fixed point Fa is represented by the k x n matrix in GL n C with the standard basis vectors €j1,...,ejk, as rows. Furthermore, j i < ••• < jk are the indices of the ones in a. 40. According to lemma 1 and proposition 1 on page 42 of [Car95], the cardinality of the fixed-point set Gr(fc,p)Gm is equal to the Euler characteristic %Gr{k,p) = (£) of the Grassmannian. Also note that Gr(fc,p)B = Gr(fc,p)Ga. 41. The only point of the Grassmannian fixed under the action of B is Fi*o?- We will also write o for Fi*o? • 42 Proposition. Let Fa be a fixed point of the Gm-action on Gr(k,p). Then r+ - o° Proof. By construction, pUa = Va and pFa = FC T P, with the cr-th fixed point pGr{k,p) in PW _ 1 denoted by FC T P. Now^to be concrete, let '>' denote a complete ordering of {^ }. We define If \a\ > \a'\ then o > a'. If \cr\ = \o'\, then a > cr' if ||a|| > ||a'||, where ||a|| = With this ordering, we index the homogeneous coordinates of P W - 1 by the index set {£} in increasing order and denote the standard opens by Va. Since the only points in Ua not in F a + lie in some Ua> with \a'\ < \o\, pGr(k,p) On P®'1 we know that (FC T P)+ = D°, thus p{Fa+) = Q° Since dim F„ = dim£}°, it is left to show that pQ.°a c Q°. pGr(fc,j>) This is certainly true for 0,°nk. Assume that the claim is true for all D°, with \o'\ < \a\ Then p(na—n°a) = p (J na-lff'l=lffl-l = (J g >KV)= U " a ' la'|=|a|-l |a'|=|a|-l = (oCT—n°) 1 7 Gr(fc,j>) Gr(fc,j>) 20 This proves the claim. • 43 Corollary. We see that for every Grassmannian Gr fcC" there are unique B-B+cells of codimension zero, H°, and one, Q°. Also, by example 8, Q° = A f c p and f2° = A f c p _ 1 . The C*-equivariant Cohomology of Grassmannians 44 Proposition. In coordinates on the big Bialnicki-Birula cell3 U0 of Gr (fc, p) the vector fields W and W' have the matrix representations (compare with [CL77]) W{x) = h,2 \Xk,2 "hp (o and xiP °J W'(x) = 0 x o A 1,1 Hfc-i i f c , i ; 1,1 „0 \xKi l,p (13) 'fcx^ ... (n-l)xf •0 \ V xfc.i (14) V ) in the standard basis d\,i,..., dk,\, d2,\,..., d^p of 3\jB. Proof. The action (14) of V follows immediately from (10) on page 15. Now we prove formula (13): For r > 1, let Nr denote the nilpotent matrix of dimension r defined in number 24. Then NnJN* ° C Npj' where C is the p x fc-matrix whose only non-zero element is a one in the upper right corner. For a fc x n-matrix (A, B) with A e M a t ^ and B e Mat^p we get (A,B)Nn=(ANk + BC,BNp). 3 see number 17 21 The matrix exponential function is formal for the exponential series, therefore ('=' denotes equality up to first order in u) ^,x°)euNn = {\x°)^ + uNn) = (1,x°) + u{Nk + x°C,x°Np) = {t + u{Nk + x°C),x0 + ux0Np} Since (i + u{Nk + x°C))~1 = (H-u(iVfc + x0C)) we conclude that (1,x°)e"W n represents up to terms in first order in u the same quotient space as (1 -u(Nk + x°C)) (11 + u(Nk + x°C), x° + ux°Np) =(1, (1 -u(Nk + x°Q) (x° + ux°Np)) =(H, x° + u(x°Np - Nkx° - x°Cx0)) The vector field W in the coordinate of U0 is now the second part of the derivative of this expression at u = 0: W{x) = -(JVfc + x°C)x° + x°Np = x0Np-Nkx°-x°Cx0. This is the equation we wanted to prove. The vector field on Gr{k,p) x C is then, using equation (8) Un Ws{x) = (x°Np - Nkx° - x°Cx°) + s(x°Ap - Afcx°) with Ap = k 0 0 fc+1 0 ... 0 \ 0 0 n-1 and A f c = (o 0 ... 0 "\ 0 1 : 0 0 0 fe-1 (15) (16) (17) • 45 Example. Over G r (2,2) we find W(x) = and K ,2 0\J0 X^fxl, X°2A o „0 \x°2 Oj U x2°J IV' (x) vX2,l X2,2 V A2,l '2,2 22 46 Corollary. We find the following presentation of the of the C*-equivariant cohomology ring: Hc*Gr(k,p) = T{Z,ffz) = C[xlps 11 < i < k, 1 < j < p] / R with the relations R expressed in the following matrix equation: (x° 1,2 V fe2 : 0 0 x o \ 1,1 Ufc-l X l , l +S (fcx°(1 ... (n-l)jc°^ „ o ^ xfc,l = 0 P J In coordinates and assuming that all undefined terms are zero: x°J+1 - x l h j - xllX°kj + s[k- i + j)xlj = 0 if 1 < i < k, 1 < j < p) 47Example. For the equivariant cohomology ring of Gr (2,2) we find HaGr(2,2) = T(Z,0z) = C[xlvxoh2,xi1,xl2,s} /R with the relations i? expressed in the following matrix equation: (0 xl,2 \x: 2,2 X\,l \ (X\,l X l , 2 1 X| ; 1 )\x2y X22 + S 2^x^  ^  3x^  2 2 j V X 2 , l 0 In coordinates: %2 2 - i^ 1 %2. 1 ^2 1 ^•^'2 1 — ^ (18) 4 , 1 A 2 , 1 2 ^ 2 1 ^2 2 ^^*^2 2 = ^ x^  ^ X2 2 3sx^  2 ~ 0 Some algebra shows (with c = x° 1): Hc*Gr(2,2) = T(Z.ffz) = C[c,s] / {0 = c(c + 6s2)(c-r-3s2)(c + 2s2)} According to number 55, c is the first Chern class of SI, the universal quotient bundle over Gr(2,2). 23 Equivariant Chern Classes 48. Let & be a vector bundle over a space X and let G be an algebraic group. Following the Borel construction, we take any contractible space E with a free action of G and define the fc-th G-equivariant Chern class, cG<f e H^X, of S as the k-th Chern class4 of the vector bundle 8G = {gxE)/G over the base space XG = (XxE)/G. 49. As shown in [BC04], the computation of the Chern classes reduces to the computation of a characteristic polynomial. In this computation we will use the elementary symmetric polynomials as defined for example in [Mac98]. The generating function of the elementary symmetric polynomials in the r variables £i,...,<f ris C£(r) = fl(^--f). Now the elementary symmetric polynomials Iii ( J C I , . . . , xr) in the r variables x\,..., xr are defined as the coefficients in the polynomial CfW = nrtfi,...,w + nr_i (fi.-, fr)( - t ) + ... + nitfi,.. .,£r)(-f)r"1 + (-flr We will mostly use the following values of these polynomials: 7 X ^ = 0,(1,2,...^). Clearly; they are only defined for j < r. 50 Example. If j = r we find nrKi,...,6) = flO a n d 7lr,r = r\. 4 which is of course independent ofthe choice of £ 24 If ; = 1: n1(tl,...,Sr) = 1£Si a n d ;'=i r(r + l) Jl\ r — • 2 The Universal Quotient Bundle Over a Grassmannian 51. Let V be an rc-dimensional vector space and let Gr^ V be the Grassmannian of fc-dimensional quotient spaces of V. There are three tautological bundles over the Grassmannian Gr{k,p): 1. The Product Bundle: This is just the trivial bundle Gr^ V x V —• V. 2. The Universal Subbundle: Its fibre at x e Grjt V is the (n-k)-dimensional subspace kerx c Gr^ V. 3. The Universal Quotient Bundle: Its fibre at x e Gr^ V is x itself. These three bundles form the tautological sequence (exact) (19) 52. A geometrically more meaningful presentation than in coordinates on U0 is one in terms of Chern classes of the universal quotient bundle £l over Gr(k,p) (see number 51). As proved in [BC04], lemma 1 on page 194, the equivariant Chern polynomial of SI can be computed as the characteristic polynomial of l/Vs. Computation of Chern classes a la Behrend-O'Halloran 53. Let e\,..., en be the standard basis of C". The bundle © n is then spanned by the constant sections o\,...,o„ e T@n with 0;(x) = (x, e{) for all x e Gr{k,p). 25 Now the restriction of the universal quotient bundle to the big cell U0 is trivialized by qv...,qk, where qt = r)Oi for all 1 < i < k and n is the projection in the tautological sequence (19). We can represent a point (x°, q) on Si \u0 by a k x (p + 1) matrix, where the addi-tional column contains the coordinates of e(x°) on the fibre @x° and in the basis <7i(x°) qk(x°). To find the action Ws of Ws on Si we can use the method described on page 398 of [BO03]: We first lift the B-action [u, A, x) ->• xeuNDx on Gr{k,p) to an invariant B-action on SI, for (x, q)eSl and b e B we find (x, q)b- (xb,q). Differentiating the G a -action induced by this B-action gives a vector field W on SI and differentiating the Gm-action on SI gives a vector field W. From these two vector fields (see remark 1.10 in [BO03]) W and W can be com-puted as Wq = (Dq)W and W can be computed as W'q = (Dq)W The derivative of the linear operator Dq(- from the tangent space of the Grassman-nian to the tangent space of SI is of course (Dc/,)x° = (x°,0), where we interpret x° as coordinates of the tangent vector in the standard basis {dtj 11 < i < k, 1 < j < p] of 3~v0. Then, using equation (15), we find (Dqt)W = (Dqf;)(x°ATp - Nkx° - x°Cx°) = (x°A/p - JV fcX 0 - x°Cx°, 0), and using equations (16,17), we find (Dqi)W'=(Dq.)ixoAp_AkXo] = (x°A p-A f cx°,0). -q*W, -q*W'. 26 For the second term we first compute the actions W and W on STg of the vector fields. The work is already done, leading to equations (15,16). The actions on the fc x (p + 1)-matrix (x°, q) are W(x°,q) = du u=0 (x°, q) • u and We find and = (x°Np - Nkx° - x°Cx°,-Nkq - x°Cq) W'(x°, q) = {x°W, q) = (x°Ap - A f cx°,-Akq) Wqi = (Dqi)W-q*W = {x°Np-Nkxo-x°Cxo,0) - (x°Np - Nkx° - x°Cx°,-Nke t - x0Cqt) = (0, (Nk + x°C)qi) W'qt = {Dqi)W'-q*WJ=(0,Akqi) In coordinates for the basis qly...,qk this linear operator is represented by the matrix (0 1 s 0 0 1 2s 0 W--0 0 0 42,1 43,1 0 1 (fc-2)s vfc-i,i (20) 0^ 0 1 x ^ + ffc-DsJ For the matrix A the characteristic polynomial is n det(A-r1) = £n ! (r i , . . . , r„)f i ' . Here r\,...,rn are the roots of the characteristic polynomial and n, is the i-th elementary symmetric function in n variables, see number 49. 27 54 Proposition. The C*-equivariant Chern p o l y n o m i a l 5 of the universal quo-tient bundle SI over Gr(k,p) is given i n coordinates of U0, and wi th = - 1 for simplicity, by 1 = 1 for even k, and for odd k by c f i2 = - x? + £ ( - l ) ' - 1 * ? ^ u -_ 2 ) s (f) - t{*-l {k_l)s(t). i=2 The r-th equivariant Chern class of Si is the coefficient of tk~r in the Chern polyno-mial . In this way we find the equivariant Chern classes: c f SI = -x° (21) fc-i c f & = £ i!jc? + 2s' (22) i=0 fc+l-r c ^ = ( - l ) r 52 * ? + r s ' * M _ 2 + r (23) j=0 r* „ (fc-2)(fc-3) „ ? ck~_1Sl = -x0k_l x^s + s ^ 2 , f c - i (24) c * 0 (fc-l)(fc-2) c^ , «»2 — "^ fc ^ s (25) Here nttT is the i - th elementary symmetric function on r variables evaluated at (1, . . . , i), see number 49. When s = 0, this is the same result as in [CL77]. Proof. We need to compute the determinant of the matrix in equation (20) minus t l To do this, we first notice that we can apply row operations to M - r l to bring it 5Note that the Chern polynomial is normally defined with the first Chern class being the coefficient of the term constant int,... 28 in the form with Next we use (0 fli 0 1 0 a2 M ' = o 1 0 0 0 1 0 O-k-2 0 1 bi = xf-({i-l)s-t)x°+1 for i<k bfc = jcg + ((fc-l)s-f) bk-i =x°k_1-((k-2)s-t)bk aj = -((j-l)s-t)(js-t). rfc/21 / - l detM'= }Z ("D'fei-i FI i=l ;=1 (26) The upper limit of this sum is the smallest integer at least as big as k/2. Now we get the same sum in terms of m for even k = 2m and odd k = 2 m -1. Then 29 expanding the a's and b's simplifies this expression nicely: m - l i-1 detM' = £ ( - D ' ' ( 4 / - i - « 2 i - 2 ) s - t)x°2i) (y's- t) i=\ j=l m-l + (- l) m (x£_ 1 -((fc-2)s-r)(x°+((fc- l)s-r))) n 02J-1 7 = 1 m - l 2;'-3 = -x°-tx23 + £ (-1) ' x 2 ° / _ 1 - ( (2 / -2)s - f )x 2 ° ; . r iO ' s - f ) 1=2 7=0 fc-3 + (-l)m(x^_! - ((fc-2)s- t)(x£ + ((fc- l ) s - f)))(-l)m n t j s - f ) 7=0 m - l 2i-3 = - x ? - f x 2 ° - Y 2 x ° . _ 1 - ( ( 2 i - 2 ) s - f ) x ° . r iO ' s - t ) i=2 j=0 fc-3 - - ((fc- 2)s - r)(x° + ((fc- i ) S - *•))) ]1 (y's -1) 7=0 fc 1-2 fc-1 = - x ° + y 2 x ? n o ' s - « + n o ' s - f ) i=2 j=0 j=0 To do the following substitution of a sum for the product only once, we set TT7l 0(./s-t) = 1 and x£ + 1 = - 1 . This gives "fc+l i-2 fc+l i-2 detM'=)z n o's - o = - E 4 f n o' s - « i=2 ;=0 i=2 ; = 1 fc+l = - X° - t X°Cs,..2,(;-2)S(f) !=2 Now we want to find the coefficients of the powers of t in these polynomials. By 49, C£(t) = nrtfi fr) + n r _i(ei , . . . ,fr)(-f) + ... + n i « i , . . . , f r ) ( - f ) r - 1 + ( - f ) r . We insert this expression in the polynomials above, leaving out the constant terms. fc+l fc+l i-2 - 1 E x ° C 2 , ( i - 2 ) s ( f ) = ~ F E X?E n i_ 2_ r(s,...,(i-2)s)(-r) r i=2 ' ' i=2 r=0 fc+l ;'-2 i=2 r=0 fc-1 fc+l = - t y > D V £ x ° 5 ' - 2 - ^ _ 2 _ r , _ 2 r=0 i=r+2 30 Now the i-th Chern class is the coefficient of tl 1 in this polynomial. We can calibrate the classes by demanding that the coefficent of tk be 1: We multiply the coefficients of detM' by ( - l ) f c + 1 (which is -1 for even k and 1 for odd k) to get the Chern classes. We get for the first Chern class the term we left out above: c f ^ = (-l) rx 1° The second Chern class, i.e. r = 0: * fc+1 cf & = £ xfsi-27lt-2,i-2 i=2 fc-1 i=0 The (fc- l)-st Chern class, i.e. r = fc-3: c * (fc-2)(fc-3) 2 c k - l ^ - ~ x k - l 2 xfcs + s Jt2,k-1 The fc-th Chern class, i.e. r = k- 2: C* o cfc -2 =xfc7r0,fc-2-S^l,fc-l _ 0 (fc-l)(fc-2) ~xk 2 S • 55 Example. We find the equivariant Chern classes of the universal quotient bundled over Gr(2,2): c f & = -x\ c2 °% — x2 They do not depend on the equivariant parameter s. 31 Moduli Spaces of stable Maps As reference for moduli spaces of stable maps see [FP97]. 56. Any smooth curve of genus 0 is isomorphic to P 1. If a curve is singular but has only nodal singularities6 then it is called a prestable curve. A prestable curve of genus 0 must then be isomorphic to a "tree" of P1 's. Figure 0.1: Some prestable curves of genus 0 57. Let X be a smooth projective variety. A genus 0 stable map to X without marked points of class /3 e Hz{X) is a morphism f from a genus 0 prestable curve C to X such that f * [C] = /3 and that each component of C, that is mapped to a point by f, contains at least three nodes. 58. We denote by Moo (GrfcC™, d) the moduli stack of stable maps of degree d into Gr f c C n . 59. LetYc Gr(k,p) be the open subvariety defined as (see number 15) Y={xe Gr(k,p)||m0(x)| # 0 v Im^x)! ± 0}. Then the map p : Y — P1, v — <|m0(x)|, Im^x)!) is well defined, it is the compo-sition of the Plucker embedding and the projection from p 0 _ 1 on the first two coordinates. P 1 (zo, Zi> We denote the fibre over oo = (0,1> by YQQ. A point x e Gr{k,p), lies in Yoo iff rankm0(x) < fc and rankmjx) = fc. Then there is a G e GL f cC such that Gm1(x) = Ifc. As mjx) and m0(x) share the first fc-1 columns, 6 a singularity analytically equivalent to the singularity of {xy = 0} in C 2 . 32 m0(x) must be of the form 1 x- \ 1 X 1  xk-l,l 0 ... 0 x ^ , We see that we need xkl=0, to have Im^x) | = 0. Therefore 5 A p f c _ 1 . Since the inverse images under p'of the two standard opens of P 1 are direct prod-ucts of the opens withA f c p -\ the map p defines a vector bundle of rank kp - 1 over P 1. 60 Fact (As seen in [BO03], section 3.3, p. 417ff). Let Tbe a complex vector space and let n: C = T xPl -~ Tbe the trivial bundle with sections xi,X2,X3 £ T{T, C). The section X£ then induces an effective Cartier divisor D( in T and the usual line bundle 0{D() associated to this divisor in C. We define the line bundles a)( = X*(OC/T and L(-u)y(. Then, as proved in proposition 3.3 on page 410 of [BO03], we can find morphisms hf.Lf^ JI*&{D(), which, through n*, are completely characterized by 1- Y-e^m M*m) = 0, where M*m) e T{T,(Om). 2. he{xe) = 1, where he{xe) £ T(T,(i>g®x*g©{De)) = T(C,@(De)). Given T( £ I[T,Le), we can interpret he(re) £ Y{T,n*0(De)) = T{C,©(D()) as a rational map C > P 1 using the pencil given by the global sections 1 and h({T() oW(Pt). The locus where this map h( is undefined is by definition Z( = Df n n~l {T = 0}. Let C denote the blow-up of C along Z = Y?g=i Z(- We then have regular maps M f f ) : C —P1. Given a regular function b e TIT,©?), define the map 3 (f)=b+Y^(pe-C-P\ (27) where (pi = heife):C-+P1. (28) 33 We write for £jt m: Vem = (Pe(xm) (29) By construction, cpp is a stable map of degree 3 to P , unramified over oo. The fibre (p~l{oo) is given by Let R is the ramification scheme of (p. As shown in the proof of proposition 3.8 on page 420 of [BO03], we can recover b through the formula We interpret b as the average of the ramification points. 61. After these general constructions, we will now define the vector space T. It parameterizes an open substack U of M(jn(GrfcCn,3) modulo S3, the stack of all maps of the form 27. 62. Let 3 0-1(c») = £ D , . (30) T = (A1) p x Li x Li x I 3 x (A3) (A3) with the following coordinates: • (A1)k p has coordinates b = • Lp has coordinate Tp for ( = 1,2,3 • (A3)fc 1 has coordinates u\tp,..., uk-i}p for £ = 1,2,3 • (A 3) p _ 1 has coordinates r2}p rp<p for £ = 1,2,3 34 For convenience we define the 3 matrices pp with components Pe U\,( l •(1 r2,p ... rpj) (31) ( u\t( uijr2,e u2,p u2,er^e Uk-\,e Wfc- i , / r 2,/ l r2,e u2,erv,e rv<( (32) 63 Remark. The vector space T, up to an action of S3, will parameterize the substack U c Mop (GrfcC", 3), consisting of the stable maps f: C — GrfcC" with the properties • f(C)cY, and is unramified over 00 = (0,1) e P . 64. Define the stable map f: C — Y, f = f u - h,i ffcl ••• fk,p' with U.j = bUj + Z Pi,j.e<Pe-e=\ 65. The following proposition shows that the coordinates r^p and Ujj determine the three points where the image of the stable map f intersects YQO. 66 Proposition. The composition f j o xp for £ = 1,2,3 defines morphisms (ffc,2- • • •, hp)0 x( '•T ~* Y o o - fKj °xe = rj,e and fl,l \ l f fc - i ,J All other components vanish. o xp : T — Yco, f/,i o xp = - uue 35 Proof. To compute the value of f (x^ ) we need to express f in coordinates on Y 0 By 20 on page 12 this gives ~t 0 f u " • f c 0 te-1 fc-1 ffc-1 ftel fl.lflt.2 h,p~ h.p~ h.iffep ^ ftel f2,1 f fc.2 ftel , f2,lffeP fk.p ffc-l,lffc,2 ffel • •• ffc-i,p-fk-l,lffe}> ftel fte2 ftel ftep ftel (33) 0 ... 0 1/ffc.i 1 At X(, and with pk,i,e = h this matrix is well-defined only if Pi,j,e = Pi,\,ePk,j,(-This gives p the form defined in (31 on the preceding page). Then the matrix (33) has the form - P i . U 0 0 Ufc-l •Pk-i,i,e o o 0 1 Pk,2,e ••• Pk,p,e) 67. We interpret the 3 points p^g e YQO found in proposition 66 • Poo,e: -u\t( 0 0 Ufc-1 -Uk-U 0 0 0 1 r2,t ••• rPie) (34) as the three points where the stable map f intersects the plane at infinity, i.e. YQO C GrkC n. 68. We have already used the projection px o p : Y — P 1, defining the plane at infinity, Y x , . We defined Y as the open subset of Gr{k,p) on which that projection is well-defined. Now we will investigate on which subsets of Y the other (£) - 2 projections Y — P 1 are well-defined. Let p denote the projection pa o p : Y — P 1 onto the two homogeneous coordi-. . Then p is well-defined on x e Gr(fc,p) if detm0(x) ^ 0 or detm<j(x) ^  0. 36 This condition is fulfilled for all points in U0. Now let x be a point in Gr{k,p) — U0, i.e. a matrix with detm0(x) = 0. A stable map f: C — Gr{k,p), represented by a point zeT, z = (bitj,Tf,uite,rj}( | 2 < j <p, 1 < i < k-l,£ = 1,2,3), intersects Yoo in exactly the three points j given in equation 34 on the previous page. The determinant of a fc x fc-minor of one of these points can only be non-zero if the minor contains fc - 2 of the first fc-1 columns, the fc-th column and one of the last p -1 columns. Then the determinant of the minor build from all but the j-th column of the first fc-1 columns and the (i + fc)-th column looks like (1 \ det after one row transposition and one column transposition7. Therefore the map Pi'-ioifc-'o^-1iOP-' °P°f 1< - C ^ - Y - t p O - i I Pa is well defined on all t e T with u^r^p ^ 0 for £ = 1,2,3. We summarize this in the following proposition. 69 Proposition. Fix 1 < j < p and 1 < i < fc. Then (.pi,j,i,pij,2,Pi,j^) ^ 0 on [71i-i0ik-ioj-iio''-J- The composition p1;-i01fc-/0;-i10P-j o f is unramified over oo = (0,...,0,1,0,...,0), where the '1' is at position l ^ O l ^ ' o ^ l O P ^ ' . Let Ritj c C be the ramification scheme of f j j . Then, by equation 30 on page 34, bi,j = \tiRL]n(hi\Ri,J). 7If 7 = 1 then we denote the 1 in column k+ 1 by r\ (. 37 70 Remark. We see that btj is the average of the ramification points, projected onto the (i, ;)-th line. 71 Theorem. The substack U of Mon(GrfcC",3) is the big Biatynicki-Birula cell of Mofl(GrfcC",3). Proof. We found the Gm-actions on the open subsets U0 and U± of Gr{k,p) in number 28. In induces the following Gm-action on z in T: Let z = {bij^Tp, uij, rjj | 2 < j < p, 1 < i < k- 1,( = 1,2,3). Then (z, A) — zA = {bX, rJ7(Xj-\u,-^Afc+1"1' , T ^ A ) , where i»A denotes the action of G m on an element of U0 defined in equation 10 on page 15. Now the action of A e G m on p^e has the form (see equation 34) (Poo,£> A) - > "A 0 -Aufc_i,^ 0 0 A - 1 Ar2,* 0 0 XprPie) We see that the fixed point is the triple cover of the P1 connecting the origin in U0 with the origin in Yco with ramification point x° = 0 in U0. Since the big Biatynicki-Birula cell is itself an affine open, and its fixed point is the curve described above, it must coincide with U. • Cohomology of Stable Map Spaces 72. We start this section with a description of the ideas behind the computation of the C*-equivariant cohomology rings over Gr{k,p) and Mofl(GrfcC",3). We use the method described in [BO03], based on the localization methods de-veloped in [CL73] and [AC87] for cohomology rings, and in [Car95] and [BC04] for G m -equivariant cohomology rings. First we build a Koszul complex: Let V be a G m -equivariant vector field on a smooth and proper Deligne-Mumford stack X, and let 2 c X be its zero scheme, i.e. the 38 fixed point scheme of the Gm-action that is part of the B-action inducing the vector field. The contraction with V then defines for all n e N maps between the sheaves of differential forms on X: ix • &x+1 ~* ^x with v\ - 0 because of the antisymmetry of the differential forms. We get a complex of sheaves A x Kx Kx Kx where A7 denotes the dimension of X, and Kxn = D.x for all n e N . We can extend this complex to the double complex Ap,cl of Kahler differentials of type {p, q) on X with the differential d and natural filtration F,- = dBp<i,q>oAp'q, leading to the total complex Kx with = 0 ( ? _p = r Ap,q and total differential As usual, such a double complex leads to the spectral sequence E~p'q = Hq{X,nx)=>Hq-pKx. (35) According to theorem 2.6 on page 267 in [CL77], all differentials in (35) vanish if V has at least one zero, thus E\ = Eoo and we get an isomorphism of graded C -algebras, grade by grade of the form Hp+q{X,nx) = &pHqKx. Here grHKx is the hypercohomology 0-QKx of the total complex Kx-73. We can now apply the method of number 72 to the spaces X and 2: H(X,nx) = HKx and mz,ci2) = HK2. We need only look at the 0-th hypercohomology object because all other objects vanish, as is also proved for Deligne-Mumford stacks lemma 1.4 on page 394 of [BO03]. As proved in theorem 2.1 of [CKP05], the inclusion 2 <—• X induces an isomorphism HKX = MK.Z, which leads to H*(x,nx) = H*(2,n*z). 39 For moduli spaces of stable maps to homogeneous spaces we know that for all p ^ q (see [BO03], remark 1.8 on page 395) Hp(X,D.qx) = 0 and that implies that Hnnx = 0 HP(X,nP) = HKjr = H°K2. P Here it is important to note that we use the algebraic grading on de Rham cohomol-ogy-74. Via the pullback-pushforward n*f* shown in the diagram ^ —*• GrkCn Mop(GrkCn,3), the B-action on Gr(k,p) induces a B-action on Moi0(GrfcCn,3) and Gm-equivariant vector fields V and V, ultimately defining a B-action and a Gm-equivariant vector field V + sV on Mo,o(GrfcCn,3) x C with coordinate s on C. The restriction of this vector field to T x C is computed in the following theorem, but we will first give an informal description of this vector field. 75. The ^-components of V are a deformation of the x°-components of W in proposition 44 by bilinear forms Q and Dg* on the coordinates M, and of the three points Po^e where the curve intersects Y ^ . Only the fo-components of V depend on the equvivariant parameter s. The ut-component of \/depends only on Kj_i, »,-, uk-\, bit\, and bkii. In particular, it does not depend on ry. Correspondingly, the ry-component of Vdepends only on r/+i, ry, r2, b^j, and bki\. In particular, it does not depend on Finally, the ^-component depends only on r2, rk-\, and b^y. 76 Theorem. The restriction to T x C of the vector field V= V + s V is given by fc-i p V = L Vi.idh] + L Uidu, + £ Rjdr} + QDqdq i,j i=l 7=2 40 and The coefficents are found to be for all 1 < i < fc -1 and all 2 < j <p Vij = biij+i - bt-ij - bi}ibkij - Q{Ui, r;) Vlj^Vc-i + jUbij-UiDwrp Ut = -u-i-i - bi,i 1 - Eu\ + Duu\ RJ = rj+l-bktJl-Ertj-Drrtj Q = r z - u k - 1 - 2 b k t l l - 4 0 m , where we use the following definitions: D{x,y) = xOyl fV1 2 )+20f ) D = 1 = E = E' = 3ni,2?72,i 3^1,3773,1 (1,1,1) E'-b^ (-12,1 -773,1 771,2 7?1,3 37?1,2772,1 lf + 2B? 3772,3773,2 3771,3773,1 ' 3772,3773,2 772,1 -771,2-773,2 772,3 773,1 773,2 -771,3- 772,3 J And (uk-i,i 0 0 1 0 Ufc-1,2 0 . 0 0 "fc-1,3; r r 2 , i 0 0 ' 0 r 2, 2 0 I 0 0 r 2, 3j 'qx 0 (T 0 q2 0 ,0 0 q3j (1™ 0 0 3(1) o e2" o l o o Of) 0m = (ef,df,ef) K 0 i (i) o cV™ o ^ o o dq3ef; Finally the definitions from [BO03]: r)f = -X\ne,m ef = E w 6f = E Ve.mVmJ We defined r\^m in equation 29. Proof. We only need to check these formulas on a dense open subset of T. We restrict to the subscheme where none of the X( vanish and then assume C = Cr = T x P1. Now it suffices to find a vertical vector field U for the projection T x P1 — T such that Df{V + U) - W. Then the vector field on U is given by U = V + U, over the open subscheme of T. • 77Example. We compute the vector field over Mo,o(Gr(2,3)2,:) The restriction to T x C of the vector field V= V + s V is given by 2,2 V= E Vijdbij + UidUl+R2dr2 + QDqdq i=i,]=i 42 and With coefficients: V= E Kjdhr i=i.j=i = bi,2 - 6i,i&2 ,i - 1 ) V\,2 = -bi,ib2,2-n{ui,r2) V2,\ = b2i2-bi,i -b2,\b2,i -Oil,I) v*2,2 = -bi,2-b2iib2i2-Qil,r2) V[il = 2(bili-u1De-lt) V1' i2 = 3(foi,2-MiD0-r2r) ^ ' > 1 = (6 2 li-lJDfl'l f) ^ ' , 2 = 2 ( ^ , 2 - 1 1 ) 8 ' ^ ) i?2 = - b2,2 \-Er^-DTr2l Q = r 2 - M 1 - 2 b 2 , i l - 4 0 ( 1 ) 78 Facf. By Lemma 4.6 of [BO03], the formula _ 3 z u = E l l f1 ;—-)i(dzeixe)dZe (=\mite ze(xm) defines a relative vector field U e T(T x P\,!J"pi) with U{x() = i(, for £ = 1,2,3, and a parameter at X(. Furthermore, U<pe = 26 f + rjf + 26 f (pe - (p2(, (36) and, for £ / m, <P(<Pm = i]m,e<Pe + Ve.mVm + ^ri£,mVm,e- (37) 79 Proposition. With the vector field on T defined in Theorem 76 on page 40 we have vfij + ufij = u,j+i - U-ij - hifnj+s(fc- i+j)hj Proof. This is a not very illuminating but very unpleasant exercise in using the definitions. • 43 80 Corollary. We find the C*-equivariant hypercohomology ring H°,(T,iCy) = C[biii,bk,i,bkiP,qe,r2,e,rpie,ui>e,uk-iie,s]IR where R contains the following conditions: -b\,ibk,p = rp) + s(n- l)(£>i,p - D 0/(ui, rp)) 0 = «fr (r2l{ - "fc-i/ - 2fofc,i - 40 m ) 3 b\,i = - ]T uiim{E-Du)m_( m=l 3 bk,p = ~ £ rp,m(E + Dr)m:( m=l Proof. We first eliminate the bjj for £{(1,1), (fc, 1), (k,p)} using the equations Vjj = 0. The first relation follows from Vk,\ = 0. We could also eliminate b\t\ and bk,p, but the recursion relations are surprisingly complicated. Using the equations Ui = 0 and Rj = 0 we can eliminate all M'S and r's but uk~\ and r2, using the following equations for 1 < i < fc and 2 < j < p: i-2 «fc_i = ( - l ) i + 1 «fc_i(£-D M ) ' _ 1 + £ ( - l ) ' ' + 1 fc f c _ i + J + 1 > 1 i . (£ - Du)i j'=o y -3 ( 3 8 ) r; = r2(E + Dr)J-2 + £ fc^-^KE + Dr)' Finally, relations 5 to 9 come from U\ - 0 and i?p = 0. • 81 Remark. Note that we can also eliminate b\t\ and fo^, but the recursion relations are very complicated. The coordinates u\ j and rk,p could be eliminated using equations (38). 82 Example. The C*-equivariant hypercohomology ring in the case of Gr (2,2) is H° * {T,KV) = C [fcu > fo2,i, 62,2, 4/> r2,e, ui,/, s] //? 44 where R contains the following conditions: -b\,\b2i2 = - Q ( t t i , r 2 ) + 3 s ( & i , 2 - D 0 ' ( H i , r 2 ) ) O = ^ ( r 2 > / - M u - 2 / > 2 l i - 4 0 m ) 3 h,i = - Y ui,m(E-Du)mie m=l 3 b2,2 = ~ E r2,m(E + Dr)m,e m=l Chern Classes of Mnn(Gr f cCB,3) 83. As indicated in the diagram Gr f c C n (39) J i . f ' S = S = Mop(£,3) >• M 0 0 (Gr f c C " ,3 ) we get the pullback-pushforward £ = 71*1*3. 84. All n sections are mapped to sections of 8 over T under TT* f * • For 1 < i < n: ei = n*f*qi. By the Riemann-Roch theorem, the rank of 8 is k+3. Over T we extend the linearly independent e i , . . . , e*; to a basis of 8 I by three sections8 gj, g2, g% e T(T, where ge = 2ZuM<Peei = (Peue-i=i 85Lemma. The sections ei,...)efc,gj,g2,g'3 form abasis of r(T,<f). (40) 8The ^ were denned in equation (28) on page 33. 45 (41) Proof. It's clear by construction that qx,..., qk are linearly independent. The gf can't be linear combinations of the e's because such combinations with regular functions as coefficients won't have the singularities that the g( have where the T'S vanish. In the same way we see that they are independent of each other, as they have different singularities. • 86 Proposition. The action of V on 8 on the basis ei,..., efc, gx, g2, g3 is Vet = (iVfc + fQe t for i = l,...,k {e,+i if 1 < i < fc-1 Zki=lbi,\ei + Z)=lge if i = k Vge = (26 f +7]f)ue + £ 3rie,nT}n,tun + (r 2 , / - few - 0™)g> + £ T7^ ,«g"„ Proof. The equations for n > i > fc +1 are clearly true. For ek we find: _ _ fc Vek=Vn.fqk=urqk = rWqk = f <7 f c + i = f* £ ">U <7« fc 3 fc 3 fc = E ( f o u e ; + E Pixe<P(ei) = E feMe/ + E E "W<Pf e,-j=l *=l i=l =^11=1 Now we use the definition of the g's in (40): fc 3 V e f c = £ & i . i e i + £ & i=i f=i Finally, we compute the action of V on the 3 additional basis vectors: vge = = (r/(/?^ ) ue+(peU ue 46 We know fc fc fc Uue = U{r £ Ui,eqt) = f W(£ ui,eQi) = it ui,ef*w°t) i=l i=\ i=l fc-1 _ _ fc 3 = L utjVet + uki(Vek = £ u,-_ue,- + b.,i + £ g„ 3 where we define u'. ( = Here we can use the fact that u'e + o.,i = -{Eu)e + {Duu)(. Then we find 3 Ofu, = ( (D M - .E ) i i ) ,+ £ g B . n=l For them's we get f / ^ = Vcp( + U<pe = (&<^ + Uipe = (r2,e - uk-u - 2bkA 1 - 46f)<pe + 26 f + Tff + 29fcpe - <p2 = (r2,e - uk-u - 2bk}l - 20f)ipe + 2df + iff - <p2( = X£ + (peye - <p2e. Now we put all the parts together 3 Vg£ = {U(pe)ue + (peUue = {xe + <peye-<p])ue + (pett(Du-E)u)e+ £ <pnun) n=\ = {x( + <p(ye -(p2e)ue + ip({{{Du -E)u)e + £ (pnun + (peu() = (xf + (peye)ue + <pe{({Du-E)u)e+ £ <pnun)-Next we use (pe(pn = r]n,e<Pe + ne,n(Pn + 3n/,n n,e Vge = {xe + (peye)ue + (pettDu -E)u)e + £ bln,e<Pe + rie,n<Pn + 3ri?,nrin,t)un = xeue + yege+Y, ne,ngn + (Pe((Du-E)u)e+ £ {-qn,e<Pe + 3,ne,n'nn,e)un-47 By xt = 2Bf+Tif ye = r2,e - " fc -u - 2bkii - 20f] and {{Du-E)u)e = {uk-.he + Of] + bk,i)ue- E i)nitun n?e we arrive at Vge = {2&f + rjf)ut + ir2,e ~ u k - u -2bkil -2df])g£ + E le.ngn-KPeKuic-w + O™ + bk>i)u(- Yln.eun) n?£ n±e = {26 f +T]f)ue + ( r 2 i , - bKi - 6f)ge + E le.ngn-'Pe E rln,eUn + E ^ln,e<Pe + ^Ve,n'nn,e)un njte n?( n±e = {26f + nf)ue+Y 3ve,nT]n,eun n?e + {T2,t - fc*-1 -Oe1])ge + E It.ngn-Chern Classes in Terms of Coordinates on T 87 Proposition. The first three Chern classes of 8 are 3 cx8 = bk + Hi (z) = -2bk + Y r2,e 3 c28 = fofc-i + n2(z) + Y i-De~lck,( e=i • c38 = ( - l ) ' " 1 ( c k - u - che{ Y zm + ef\ m?e 48 For Cfc +3_j and 4 < i < k - 1 , i.e. Chern classes C 4 , . . . , ck we find 3 ct+3-ig = bi-2 + £ Vu - ni(z)i>i_i + ll2(z)bi - bi+lU3{z) 3 3 The last three Chern classes are 3 Ck+iS = -rii(z)foi +U2(z)b2-b3n3(z)-b3 £ zeee e=i 3 + £ ( - l ) ' _ 1 ( c i , , + c3y izm + efzm + 3ee)-c2i({2Zzm + ef)) e=i m^e m^e 3 ck+28 = n2(«)foi - fo2n3(z) - b2 £ z ^ 3 + £ ( - l ) m _ 1 ( c 2 , m EI tee + 6%zt + 3em)-ci,m[Z ze + B™)) 3 ck+3S = £ ( - D m _ 1 c i , m n (z£ + 6g]z£ + 3em) 3 -&i(n 3(z) + £ z / e / ) . In these equations we use the following definitions: 1. z( = r2ig - bk,i - df 2. Yl{ (z) denotes the i-th elementary symmetric polynomial in three variables evaluated at (zi, z2, Z3) as defined in 49 3 . (2ef+r,f 3n i , 2 7?2 , i 3771,3773,1^ Ci = Ui 3771,2172,1 2 0 2 2 ) + 7 ? 2 2 ) 3772,3^3,2 V 3 n i , 3 r 7 3 , i 3r?2,3n3,2 ZOf+rif) 4. The other definitions are given in 76. 49 Proof. We want to compute the characteristic polynomial of the matrix M repre-senting the action V on 8. The matrix M-X has the following form: M-X--'-X 0 0 c u Cl,2 1 - A 0 0 1 '•• '•• 0 '•• - A bk-i 0 0 1 bk-X Cfc,l Cfc,2 Cfc,3 0 0 1 z\ - A "1,1 "2,1 0 0 1 "1,2 z2 - A V3,2 1° 0 1 "1,3 "2,3 2 3 - A y where we write ze = r2,<> - frfcj - Of, and Ci = Ui {lef+rif 3T7i, 2n 2,i 3 n i , 3 r / 3 , i ^ 3 n i , 2 n 2 , i 26)f+n 2 / C 2 ) 3772,3773,2 13171,3713,1 3772,3^73,2 2 0 f + 7j£ ,(2) and bi for fy,i for all £ = 1,2,3 and 1 < 1 < fc. To partition this matrix, we call the lower right 4 x 4-matrix F(X) and the lower right 3 x 3-matrix D(A). To compute its determinant, we start with the first k X's, getting (-A)fcdetD(A), and bk instead of the fc-th A we get (-A)fc_1fofcdetD(A). Next we get the terms coming from using all but the j-th A, expanding along the row (fr;,c;,i,Cj,2,Cj,3) instead. We write F;(A) for the 4 x 4-matrix resulting from replacing the first row of F(A) with the row {bi, c/,i, c J | 2, c,,3) and we write Det (A) for the 3 x 3-matrix resulting from replacing the ^-th row of D(A) with the row (Q , i , C J , 2 , c()3) and then bringing it to the top. It then follows that 3 detDf(A) = £ {-l)m-lcUmAetm{X), m=l where A^,m(A) is the determinant of the matrix consisting of the 2x2 minor of D(A) not containing row £ and column m. 50 Finally we get from using one to fc -1 of the first fc-1 ones on the subdiagonal 3 detFj(A) = fc/detD(A) - L d e t D f e=i All other terms vanish. Collecting terms, we have found so far: det(M-A) 3 fc—1 = (-l) f c(A f cdetD(A)- A f c _ 1b f cdetD(A) + A f c _ 1 £ detD£(A) - £ A i _ 1detF/(A)) (=1 i=l Now we compute the determinants Ap<m{X). To simplify notation, we assume that {a, b, c} = {1,2,3}. Then: Aa,a(A) = A 2 - X{zb + zc) + zbzc + ea &a,bW = -hlla,b + T]a,bZc + eb 3 detD(A) = - A 3 + ni(z)A 2 -n 2 (z)A + n 3(z) + £ zanbiCiic,b a=l detDf (A) = A2Cii£ + X( £ citmr)£im - cii( £ z m ) + cu{Y\ zm-ee)- 2~L ci,m(Zi<xr]fim + em) mjt( m±e Here n,(z) stands for the /-th elementary symmetric polynomial in the three vari-ables zi,Z2,Z3 and, as before, {a, b,c} = {1,2,3}. 51 Now we collect all terms (-l) f cdet(M-A) 3 fc—1 = AfcdetD(A) - A^/JfcdetDCA) + A f c _ 1 £ detD^(A) - £ detF,-(A) e=i i=i fc 3 3 = A f cdetD(A)-X;A ! '" 1(fo/detD(A)-£ £ ( - D ^ ^ ^ A ^ U ) ) = _A*+3 + n i ( z U f c + 2 _ n 2 (z)A f c + 1 + Afc(n3(z) + £ Z / e , ) /=i fc + 2ZUi+2bi-nl{z)biAi+l + n2(z)bixi-x^biinsiz 3 3 3 + X>'e«) + A ' _ 1 £ zZ{-l)m-lcitmA(tm{X)). e=l £=lm=l We continue, using 3 £ A , , m = A 2 - A( £ Z( + 0™) + + + 3em) / = 1 f^m /^m 52 and ea = t]bicT]c,b-(-l)k det(M - X) = - A f c + 3 + n!(z)A f c + 2 -n 2 ( z )A f c + 1 + A f c(n3(z) + E zeee) e=i fc 3 + Y\xi+2bi-Udz)bi\M +n 2 (z)M i ' -A ! ' - 1 fo / (n 3 (z)+ E we) + £ ( - l r - W A ' ^ - A ' t E z,+e<») m=l £^m + A 1 ' - 1 n (^ + 0 ^ + 3em))) = - A f c + 3 + n x (z)A f c + 2 - n 2 (z)A f e + 1 + A f c(n3(z) + £ ztet) + £fA i + 2 (fo ! ) + ( f : (- l) m - 1 C l - , M -n 1 (z)fc ! )A m / = 1 V m=l + (n2(z)i»,- £ ( - i r ' W L m=l <Vm + A i ' " 1 ( £ (-D^^/^fn (z^ + 0™z^ + 3em))-^(n 3(z) + | ; z ^ ) ) ) m=l ^ = 1 ' We change indices such that all powers of A in the sum over i are equal to i: (-l) f cdet(M-A) 3 = - A f c + 3 + ni(z)A f c + 2 - n 2 (z)A f c + 1 + (n3(z) + £ zee()Xk (=1 fc+2 fc+1 3 + £ fc,.2A' + E( E ( -D^Q- i .m -n iCzJ fc i -OA 1 1=3 j=2 m=l + £(n 2(z)fc,- E t - D ^ W E j=l m=l + E( E (- l) m _ 1 Ci + i ,m( EI ^ + 0 m ^ + 3em)) - ( n 3 ( z ) + £ z,e,)U t=0 m=\ e?m e=i We even out the limits: (-l) f cdet(M-A) = - A f c + 3 + ni(z)Afc+2 - n2(z)Afc+1 + (n3(z) + £ z/e/)A f c e=i fc-i + bkXk+2 + fofc_iAfc+1 + fofc_2Afc + £ bt-zX1 i=3 + (£ (-i)m-1cfem-n1(z)fofc)Afc+1 + ( £ (-i)m-1cfc_1,ra-n1(z)fofc_1)Afc m=l m=l + L(2Zt-»m~lci-l.rn-niWbi-l)l1^ i=3 m=l m=\ + (n2(z)&fc- £ {-Dm-lchm{2Z zt+e%))xk m=l (im fc-1, 3 + £(n2(z)fc /- £ (-i)m-1cl-,m( E z, + 0£)W i=l m=l t±m + (n2(z)b2- ti-ir-'czAL ze + 9m)U2 + (n2(z)fei- £ (-Dm _ 1ci,m(L ze + 0m)V fc-1 3 3 + E I ( - D m _ 1 c i + i , m J! (z£ + e%ze + 3em) - fc/+i(n3(z) + £ z,e,) A* ;=0 m=l /^m /=l 3 3 + (£ (-l)m_1c3,m fj (ze + em]ze + 3em)-b3{n3(z) + 2Zzeee))x2 m=l dm t=\ 3 3 + (£ (-l)m_1c2,m ]"[ (^  + 6>m)z^ + 3em)-fe2(n3(z) + £ ^ e / ) ) A m=l <ym /=1 3 3 L ( - D m _ 1 c i , m EI (^  + ^ )^ + 3em)-/Ji(n3(z)+£^e/) m=l *ym /=1 54 And sort by degree in A: = - A f c + 3 + 3 + \bk-i-n2(z) + ( £ {-\)m-lck,m-ni(z)fcfc) V m=l 3 3 A f c + 1 + ffofc_2 + (n3(z) + £ ztet) + ( £ (-l) m _ 1 c f c _i, m - n1(z)bk-l)Xk + (n2(z)fcfc- f ( - l r - 1 ^ ^ ^+ )^)Ufc m=l 'A fc-W 3 + £ b,-_2+ £ (-l)m-Vi,m-ni(z)fcf_i + n2(z;fei i=3 V m=l i=3V /n=l m = l / ^ m 3 + £ ( - l ) m 1 r] (z/ + 0« + 3em) - b i +i(n 3(z) + £ ztee) A' m = l / ^ m /=1 ' 3 3 + ( £ ( - D m _ 1 c 3 , m n (z, + 0£z , + 3 e m ) - f c ( n 3 ( z ) + £ z , e , ) m = l <Vm / = 1 + ( E ( - i ) m - 1 c 1 , f f l - n 1 ( * ) + ( n 2 ( z ) i ) r £ ( - i ) m - 1 c 2 , m ( ^ f + ^ 3 3 + ( £ ( - l ) w - 1 c 2 , m f | (z, + 0™z, + 3em) - fc2(n3(z) + £ z,e,) (n2(z)fo!- £ t-ir-'ciALzt+oW)))* m=l £±m (=\ 3 + m = l t?m 3 3 £ (-1)m~ lc h m f ] (z/ + 0&}z, + 3em) - bi (n3(z) + £ z,e,) m = l / ^ m /=1 lfc+1 We clean up: = -Xk+3 + (bk + Ili(z))Xk+2 3 + (fofc_x - n2(z) + £ ( - U ' - ^ A /=1 / 3 + bk-2 + n 3 (z) - rii (z)&fc_i + n 2 (z)bfc + £ ( - I ) ' " 1 k - w - che{ £ *m + e»)))]Afc fc-1 . 3 + L *>/-2 + E (-u '~Vu -n^ zjb/.x + n2(z)fe, - fc,-+in3(z) ;=3 /=1 3 3 + E(-l)/(cI-,/(E zm + e^-ci+u EI (zm +0j1}z/ + 3e*)) - £ z(ee i 3 + -ni(z)&i + n2(z)fo2 - ^ n3(z) - &3 £ z ^ e=i 3 /=1 m^/ m^/ r 3 + n2(z)/ji - fo2n3(z) - b2 £ z/e* /=i 3 + £ (- l ) m - x (c 2 , r a n (zt + 6%ze + 3em)-ci,m (E m=l 3 3 £ ( - D m _ 1 c i , m EI U / + 0m )z^ + 3em)-& 1(n 3(z)+E^) m=l /^m • /=1 Finally, we can read off the Chern classes: 3 I (=1 c\S = bk + ni (z) = -2/jfc + £ r2,t e=\ 3 c2^  = bte-i + n2(z)+ £(-r/ _ 1 c w e=\ c3<£ = {-l)e'Hck-U - cK({ E zm + Q™j\ For Cfc+3-; and 4 < i < fc-1, i.e. Chern classes c\,..., ck we find 3 ck+3-i<g = bt-2 + E ( - i / _ 1 c ( _ u - niCzjfc,-.! + n2(z)bi - fc,-+in3(z) 3 3 The last three Chern classes are 3 ck+i8 = - I i i (z)h + n2(z)b2 - b3Yl3(z) - fc3 E z ^ 3 + E(-1)/_1(ci/ + c3/ FI ^m + 0le1)zm + 3ee)-c2AL Zm + 0?)) (=\ m?e m£( 3 ck+28 = n 2(z)bi - b 2n 3(z) - b2 E zee( e=i 3 + ^ ( - l ) m - 1 ( c 2 , m II (^ + e ^ + 3em)-c I i m(E ^ + 0 m ) ) m = l < V m e±m 3 cfc+3<?= E ( - D m _ I c i , m EI (ze + 8^ze + 3em) m=l e±m 3 -fei(n 3(z) + E^ e/)-• 57 Conclusion In this thesis we realize a good part of the program laid out in [BO03]. We find representations of the equivariant cohomology ring of Gr^C" and the equivariant hypercohomology of the complex Kx-To accomplish this we find generalizations of the methods in [BO03] to moduli spaces over Grassmannians, to equivariant cohomology, and to equivariant Chern classes of vector bundles of rank bigger than one. We have good reason to conjecture that H°KT is the full C*-equivariant cohomology ringofM0 f l(GrfcC",3). Towards a more geometrical representation of this ring in terms of equivariant Chern classes of the vector bundle & and its symmetric powers we make good progress by computing the Chern classes in terms of the generators of our repre-sentation oflH 0KT. We also compute the equivariant Chern classes of the quotient bundle over GrfcC". Even though these classes are well known, they still give us valuable insight into the computations necessary to obtain the equivariant Chern classes of the bundle £ over H°Kx. This is the case because a big part of our representation of the ring over Kx is nearly identical to the representation we find for the ring over GrkCn. 58 Outlook We plan to continue this program. First, we want to find the representation of the C* -equivariant hypercohomology object in terms of equivariant Chern classes of § and its symmetric powers. After this is achieved, we can work on showing that these Chern classes generate the whole equvivariant cohomology ring of Mofl(GrfcC",3). For the standard cohomology ring ofMofl(P",d) {d = 2,3) this was possible in [BO03] because the Betti numbers are known. This is unfortunately not the case for M0p(GrfcC",3) in general. We believe that the key to the complete equivariant cohomology ring of the moduli space Mo,o(GrfcCn,3) might be found in [Opr04] as Dragos Oprea shows there that the cohomology ring is generated completely by tautological classes. We will have to study the tautological classes in more detail to be able to make this connection. The initial motivation for this work was to gain a better understanding of the cohomology rings of moduli spaces of stable maps to homogeneous spaces by examining more examples than just projective spaces. Only after we find the equivariant cohomology ring of Mojo(GrfcC",3) can we know if this will be possible. Other approaches to a better understanding of the equivariant cohomology ring of Mop(GrfcCn,3) would be the study of maps to homogeneous spaces or maps of degrees bigger than three. The problem in degrees d bigger than three is harder because one then has to work with a vector bundle over the moduli space of stable curves Mo^ instead of having to deal with only one vector space, T, a vector bundle over the point M03. It remains curious that so little is known about the cohomological structure of these 59 moduli spaces, even though they are studied intensively in Gromov-Witten theory and enumerative geometry. 60 Bibliography [AC87] E. Alkildiz and James B. Carrell. Cohomology of projective varieties with regular sl2 actions. Manuscriptae Mathematicae, 58:473-486, 1987. [BC04] Michel Brion and James B. Carrell. The equvivariant cohomology ring of regular varieties. Michigan Mathematical Journal, 52:189-203, 2004. [BO03] Kai Behrend and Anne O'Halloran. On the cohomology of stable map spaces. Inventiones Mathematicae, 154:385-450, 2003. [Car95] James B. Carrell. Deformation of the nilpotent zero scheme and the intersection ring of invariant subvarieties. Journal fiir die reine und angewandte Mathematik, 460:37-54,1995. [CKP05] Jim Carrell, Kiumars Kaveh, and Volker Puppe. Vector fields, torus actions and equivariant cohomology. arXiv:math.AG/0503004, March 2005. [CL73] James B. Carrell and D. I. Lieberman. Holomorphic vector fields and compact kaehler manifolds. Inventiones Mathematicae, 21:303-309,1973. [CL77] James B. Carrell and D. I. Lieberman. Vector fields, chern classes, and cohomology. In Jr. R. O. Wells, editor, Proceedings of Symposia in Pure Mathematics, Vol 30, pages 251-254, Providence, Rhode Island, USA, 1977. American Mathematical Society. [DM69] P. Deligne and D. Mumford. 61 The irreducibility of the space of curves of given genus. Institut des Hautes Etudes Scientifiques. Publications Mathematiques, (36):75-109, 1969. [FP97] William Fulton and Rahul Pandharipande. Notes on stable maps and quantum cohomology. arxiv:alg-geom/9608011 v2, May 1997. [Ful97] William Fulton. Young Tableaux, volume 35 of London Mathematical Society Student Texts. Cambridge University Press, The Edinburgh Building, Cambridge CB2 2RU, UK, reprinted 1999 edition, 1997. [GP05] E. Getzler and R. Pandharipande. The betti numbers of Mo,«(r, d). arXiv:math.AG/0502525, February 2005. [KM94] M. Kontsevich and Yu Manin. Gromov-witten classes, quantum cohomology, and enumerative geome-try. Communications in Mathematical Physics, 164:525-562,1994. [KT03] Allen Knutson and Terence Tao. Puzzles and (equivariant) cohomology of grassmannians. Duke Mathematical Journal, 119(2):221-260, 2003. [Mac98] I. G. MacDonald. Symmetric Functions and Orthogonal Polynomials, volume 12 of Univer-sity Lecture Series. American Mathematical Society, 1998. [Opr04] Dragos Oprea. The tautological rings of the moduli spaces of stable maps to flag varieties. math.AG/0404280, April 2004. 62 

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