{"http:\/\/dx.doi.org\/10.14288\/1.0080064":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Mathematics, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Tschirschwitz, Boris","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2009-12-21T20:38:13Z","type":"literal","lang":"en"},{"value":"2005","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Doctor of Philosophy - PhD","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"We compute the equivariant hypercohomology of the Koszul complex associated\r\nto an equivariant vector field. This hypercohomology is conjectured to be the\r\nequivariant cohomology ring of the moduli space M[sub 0,0] (Gr[sub k]C[sup n], 3) of stable maps of\r\ndegree 3 from genus zero prestable curves into Grassmannians.\r\nOur methods are based on the localization methods of E. Alkildis, M . Brion, J.B.\r\nCarrell and D.I. Lieberman.\r\nThis thesis is an extension of the work by K. Behrend and A. O'Halloran on stable\r\nmaps into projective spaces.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/16942?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"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 \u00a9 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 \/\u00bb[C] - \/3. The additional points are maps \/ : C \u2014 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 \u2014 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\"\u00bb 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, \u00a92) has to be replaced by the hypercohomology ring H\u00b0(X, Ky) of the Koszul complex Ky defined by the vector field V. H\u00b0{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\u00b0(X,KV) - H \u00b0 ( 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 \u2014 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) \u2022 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\u201ef*0(l), 7r*f* ^ (2), and n*\\*@{3) onM0p(P\u00b0\u00b0,3) (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*\u00a3l 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 (\u00a3) 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
\u2022 SfcC\u2122\/ ker<\/>, i.e. by (representatives of) bases of C\"\/kertp \u2022 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\u201eC\/P of the general linear group on C n by the parabolic subgroup \/ \u2022 ... \u2022 0 ... 0^ \u2022 \u2022 \u2022 0 \u2022 \u2022 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\\ A \u2022 \u2022 \u2022 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 \u2014 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 {\u00a3} be the subset of ( Z 2 ) n of numbers with exactly k ones. Then {\u00a3} 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 {\u00a3} and the set of Young diagrams with at most k rows and p columns. The vector a e {\u00a3} 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 {\u00a3}. 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\u00a3j J\u00a3| ^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' \u00bb2,3) v2,2 2^,3, 1^,3 \"1,4 \"2,3 \"2,4 17. We use {\u00a3} as index set for the following (\u00a3) coordinate charts on the Grass-mannian GrkQn: Let [7ff for a \u00a3 {\u00a3} 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\u00b0) e Gr{k,p)\\x\u00b0 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 \u00a3Gr{k,p) x 1 \u00a3 Matfc,pC (1) (2) and charts \u00a3 a : Ua \u2014 Gr(fc,p), y = (fi i>fc)r \u2022\u2014 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\u00b0 e Mat2,2} UL - t\/ ioio f 1 .X-^ ^ 0 JC-^ ^ [\\0 Xgj 1 x2,2> EGr {2,2) x 1 \u00a3 Mat 2,2c| 19Lemma. Let C\/0 be the chart UXk0? of Gr(fc,p) and let Q\u00b0 be its Schubert cell of codimension 0. Then u0 = n\u00b00. Proof. This just states that we implicitly chose the standard basis of C\". \u2022 20. We compute the coordinate change : UL -* t\/o.x1 = (jtf.) >-* x\u00b0 = (x\u00b0(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 \"\u2022fey By coordinates: v i , l r-, lx(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 '\u2022\u2022 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 \u2022 x A-0 Xoo = lim A \u2022 x A\u2014oo 18 for x \u00a3 X . 36. Let C be the connected component of X G M containing the fixed point F\u00a3 X G M . We call the set C + = {x \u00a3 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 {\u00a3} 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\u201e of Fa is given by Fa = Ua- U Uo'-a' 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 ,\u2022 - 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' 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 \u2022Pk-i,i,e o o 0 1 Pk,2,e \u2022\u2022\u2022 Pk,p,e) 67. We interpret the 3 points p^g e YQO found in proposition 66 \u2022 Poo,e: -u\\t( 0 0 Ufc-1 -Uk-U 0 0 0 1 r2,t \u2022\u2022\u2022 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 \u2014 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 (\u00a3) - 2 projections Y \u2014 P 1 are well-defined. Let p denote the projection pa o p : Y \u2014 P 1 onto the two homogeneous coordi-. . Then p is well-defined on x e Gr(fc,p) if detm0(x) ^ 0 or detm 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 ;\u2014-)i(dzeixe)dZe (=\\mite ze(xm) defines a relative vector field U e T(T x P\\,!J\"pi) with U{x() = i(, for \u00a3 = 1,2,3, and a parameter at X(. Furthermore, U i)u(- Yln.eun) n?\u00a3 n\u00b1e = {26 f +T]f)ue + ( r 2 i , - bKi - 6f)ge + E le.ngn-'Pe E rln,eUn + E ^ln,e