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On the Local Donaldson-Thomas theory of curves Song, Yinan 2006

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On the Local Donaldson-Thomas theory of curves by Yinan Song B.Sc , Harvey Mudd College, 2000 M.Sc , California Institute of Technology, 2002 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF Doctor of Philosophy i n The Faculty of Graduate Studies . (Mathematics) ; . The University Of British Columbia June 2006 © Yinan Song 2006 11 Abstract In this thesis, we study the Donaldson-Thomas theory of local curves. The motivation is the Gromov-Witten/Donaldson-Thomas correspondence. First, we review the gauge theory motivation of the original construction and the history of the Donaldson-Thomas theory. Then we review the construc-tion of Gieseker-Maruyama-Simpson moduli spaces and their relation with Hilbert schemes of threefolds. We also review the concept of a perfect ob-struction theory and its relation with the virtual fundamental classes. Then we describe the Gromov-Witten/Donaldson-Thomas correspondence and the equivariant generalization. We study the equivariant Donaldson-Thomas theory of two types of three-folds. First, we consider the total space of P2-bundles over smooth curves of genus g with (C*)3 action along the fiber, and we consider the curve class of a section plus the class of a line in the fiber. We compute the Donaldson-Thomas partition function up to the first order and we find that it agrees with the Gromov-Witten prediction. The second case we consider is the local curve case. The threefold is the total space of a direct sum of two line bundles of opposite degree with an anti-diagonal action along the fiber. Based on the calculation from Gromov-Witten theory, the only non-trivial contributions to the Donaldons-Thomas partition function come from components that parametrize subschemes of pure dimension. We compute these contributions and find that they agree with the Gromov-Witten prediction. We are also able to verify the vanishing of contributions from the so-called product-type components. i i i Contents Abst rac t i i Contents i i i L is t of Figures v Acknowledgments v i 1 In t roduct ion 1 2 Donaldson-Thomas theory 7 2.1 The moduli space 7 2.2 The perfect obstruction theory 9 2.3 The Gromov-Witten/Donaldson-Thomas Correspondence . . . 11 2.4 Generalization ; ; ' : 14 3 Donaldson-Thomas Theory of P 2 -bundles over a curve . . . 17 3.1 Results . 17 3.2 Preliminaries on P 2 -bundles over curves 18 3.3 Computing Extg( J , J ) . 20 3.4 Proof of the Proposition 3.1.3 21 4 Donaldson-Thomas Theory of Loca l Curves 27 4.1 Results 27 4.2 Degree zero Donaldson-Thomas parti t ion functions 29 4.3 Partitions 30 4.4 Determination of Fixed Locus 31 4.5 Equivariant Computation wi th K-classes 36 4.6 Loc i of subschemes of pure dimension 39 4.7 Proof of Theorem 4.1.2 42 4.8 Product-type components 44 4.9 Proof of Theorem 4.1.3 Bib l iography V List of Figures 4.1 A n example of 2D partitions 30 4.2 A n example of 3D partitions 31 4.3 A n example of singular fixed loci: first component 35 4.4 A n example of singular fixed loci: second component 35 vi Acknowledgments I would like to thank my supervisor Jim Bryan, who teaches me how to do mathematics. Without his support, this thesis would not be possible. I would also like to thank Dr. Kai Behrend, who answered my questions on stacks and virtual fundamental classes. I would like to thank my fellow graduate students, especially Amin, Yun-feng, Dagan, Maciej, and Boris, for many stimulating discussions over the last four years. I would like to thank Edwin, from whom I learnt much during my first two years at U B C I would like to thank NSERC, U B C and Webster Foundation for financial support during the last four years. Chapter 1 i Introduction Donaldson-Thomas invariants were first conceived as the complex analogue of gauge theoretic invariants for three dimensional real manifolds [7] [29]. Gauge theory was first developed by Atiyah, Donaldson and others in the study of four-dimensional real orientable manifolds. In the complex setting, the orientability condition is replaced by the Calabi-Yau condition. We say that an n-dimensional complex manifold X is Calabi-Yau if its canonical bundle is trivial. It then implies that it has a trivializing holomorphic n-form, which is the analogue of the real volume form of a real orientable manifold. Here is the formal picture of the complex analogue of the gauge theory restricting to a Calabi-Yau threefold X. Let E —> X be a C°°-bundle and B be the space of gauge-equivalence classes of connections on E. We can define the complex analogue of the Chern-Simons functional, CS on B. The critical points of this Chern-Simons functional are those connections that induce a holomorphic structure on E. They are called Hermitian-Yang-Mills connections. It is a standard fact from Morse theory that the number of critical points, counted with proper signs, equals the Euler characteristic of the space B. This count should be the holomorphic Casson invariant, the analogue of the Casson invariant of three-dimensional real manifolds. In practice, this formal picture works well for three dimensional real manifolds. Unfortunately, there are some analytical difficulties in the case of complex threefolds to make this formal picture rigorous. Consequently, Thomas decided to work within the algebraic category. We have the Kobayashi-Hitchin correspondence between Hermitian-Yang-Mills connections and stable holomorphic bundles. Mumford had constructed the moduli spaces of stable bundles on curves and Gieseker had constructed moduli spaces of torsion-free sheaves on surfaces. Maruyama generalized Gieseker's results to smooth varieties of all dimensions. Simpson generalized their constructions to coherent sheaves of pure dimension. These coherent sheaves of pure dimension can have support of lower dimension than the am-2 bient space, i.e. they can be purely torsion. If their support has the same dimension as the ambient space, then they have to be torsion-free. Gieseker-Maruyama-Simpson moduli spaces are in general singular and of excess di-mension. Therefore, we have the same problem as in the analytic category. However, motivated by the Gromov-Witten theory, Li-Tian and Behrend-Fantechi developed a standard technique to take care of singularities and excess dimension at the same time, i.e. the so-called perfect obstruction theory and virtual fundamental class [23] [2]. Once Thomas showed that cer-tain Gieseker-Maruyama-Simpson moduli spaces have a perfect obstruction theory with expected dimension equal to zero, it followed from Li-Tian and Behrend-Fantechi's result that these moduli spaces have an invariant which is invariant under the deformation of the complex structures of the underly-ing complex manifold. Thomas called these invariants holomorphic Casson invariants. We will follow his terminology and use Donaldson-Thomas invari-ants in the sense of [24]. In other words, the Donaldson-Thomas invariants are a class of holomorphic Casson invariants. When Thomas defined the holomorphic Casson invariants, Calabi-Yau threefolds had been the subject of intensive study in mathematics for a while, partly because Calabi-Yau threefolds were the central objects in string theory in theoretical physics. For example, Gromov-Witten theory on Calabi-Yau threefolds was developed as the mathematical equivalent of Type IIA topo-logical string theory. Let X be a Calabi-Yau threefold and (3 E H2(X,Z). Let Mg(X,P) be the moduli space of degree (3 and genus g stable maps to X. It has a well-defined perfect obstruction theory and a virtual fundamen-tal class, [M5(X,/?)]v i r in A0(Mg(X, /?), Q), where A*(Mg(X, (3), Q) are the Chow groups of Mg(X,f3). Let be the degree /3, genus g Gromov-Witten invariant of X, which is defined to be the degree of [Mgtp(X)]vlv . Intuitively, Nfffi is the virtual count of the -number of stable maps from genus g curves to X, whose image is of class j3. In 1998, two physicists, Gopakumar and Vafa considered another way to calculate the Gromov-Witten potential functions. In particular, they con-jectured that Gromov-Witten potential functions can be computed from so-called "BPS counts" [11]. Conjecture 1.0.1. Let X be a Calabi-Yau threefold and let (X) be the genus g degree (3 Gromov-Witten invariant of X. Then there exist integers n9JX) with two conditions. For a fixed /3, ni(X) = 0 for g » 0. These 3 n9p(X) satisfy the following equation: 0^0 G>0 0^0 G>0 K>0 In string theory, these integers,5 rip are supposed to count the number of B P S M2-branes. There are sti l l no rigorous definitions of B P S counts in mathematics. One proposal is to regard B P S M2-branes as bundles sup-ported on embedded curves in the Calabi -Yau threefolds, and B P S counts come from the cohomology of the moduli spaces of such bundles [16]. Re-cently, K a t z proposed to use another class of the holomorphic Casson invari-ants to define rip for g = 0 [18]. Coming back to the holomorphic Casson invariants, Thomas computed some examples following his construction and found that some of them were related to Gromov-Wit ten invariants [29]. In November 2003, Maul ik , Nekrasov, Okounkov and Pandharipande studied a type of holomorphic Cas-son invariants, where the moduli spaces could be identified wi th Hilbert schemes of one dimensional subschemes of the Calabi -Yau threefolds [24].' They called these invariants Donaldson-Thomas invariants. Maul ik , Nekrasov, Okounkov and Pandharipande applied vir tual localization and computed Donaldson-Thomas invariants for toric Calabi -Yau threefolds and discovered a correspondence wi th Gromov-Wit ten theory. They then made a series of conjectures on Donaldson-Thomas invariants and their correspondence wi th Gromov-Wit ten invariants for arbitrary nonsingular threefolds [24] [25]. (see section 2.3) Soon after, Bryan and Pandharipande generalized this conjec-tural correspondence to the equivariant setting [6]. The G W / D T correspondence looks similar to the Gopakumar-Vafa con-jecture. It also states that two theories are equivalent. Another feature of these two conjectures is that both Donaldson-Thomas invariants and B P S counts are a priori integers while Gromov-Wit ten invariants are rational numbers. We can compute Donaldson-Thomas invariants in terms of Gromov-Wi t t en invariants and vice versa. Therefore, certain linear combinations of Gromov-Wit ten invariants are integers, and this is known as the integrality property of Gromov-Wit ten invariants. This thesis is motivated by the conjectural equivariant G W / D T corre-spondence. First , let us review what has been done in the last two years. There are three techniques so far in computing Donaldson-Thomas invariants. The first is vir tual localization. It was first applied by Maul ik , Nekrasov, 4 Okounkov and Pandharipande to toric Calabi -Yau threefolds. It is also the technique used in this thesis. The second technique is degeneration, which allows one to reduce the computation to simpler spaces. Often degeneration is used together wi th vir tual localization. The foundational papers on degen-eration and relative Donaldson-Thomas theory, however, have not yet been written [25]. Finally, the thi rd technique uses weighted Euler characteristics to compute Donaldson-Thomas invariants by stratification, also reducing the computation to simpler spaces [3] [5] [4]. The first two techniques are well-known and they have been applied suc-cessfully in Gromov-Wit ten theory for the last ten years. The last technique of weighted Euler characteristic, however, is a relatively new technique devel-oped by Behrend to study Donaldson-Thomas theory of Calabi -Yau three-folds. In this case, the perfect obstruction theory on the Donaldson-Thomas moduli has certain symmetric properties as described in [3]. Heuristically, these symmetries reflect the fact that this moduli space is the set of cri t i -cal points of a holomorphic functional (the Chern-Simons functional). Using this symmetry, Behrend is able to show that Donaldson-Thomas invariants are given by the topological Euler characteristic, weighted by a constructible function. A s mentioned before, the correspondence for toric Calab i -Yau threefolds was first verified by Maul ik , Nekrasov, Okounkov and Pandharipande in November, 2003, using vir tual localization. Soon after, in A p r i l 2004, they verified the degree zero Donaldson-Thomas conjecture for toric threefolds, using relative Donaldson-Thomas invariants [25]. Based on their results on toric threefolds, Jun L i gave a proof of the degree zero Donaldson-Thomas conjecture for arbitrary threefolds in March, 2005 [22]. In November, 2005, Behrend and Fantechi gave a new proof, using the thi rd technique of weighted Euler characteristics [5]. In November, 2005, Okounkov and Pandharipande solved the equivariant Donaldson-Thomas theory for local curves and proved the correspondence using the degeneration technique [28]. In December, 2005, Bryan and Behrend gave a proof of the correspondence for Calabi -Yau quintic in degree one and two [4]. This is the first proof of the correspondence for compact Calabi -Yau threefolds, since the toric Calabi -Yau threefolds are non-compact.There was also work done on products of surfaces and curves in [19] [8]. Recently, Levine and Pandaripande gave a new proof of the degree zero conjecture using algebraic cobordism [21]. In this thesis, we compute parts of the equivariant Donaldson-Thomas parti t ion functions in two cases and we find that they al l agree wi th the 5 Gromov-Wit ten predict ions/The first case is the curve class of a section plus a fiber in a P 2 -bundle over a curve. We compute the equivariant Donaldson-Thomas parti t ion functions up to first order, Theorem 3.1.1. The second case is the case of local curves, where we specialize to line bundles of oppo-site degrees and anti-diagonal action. We compute the contribution to the equivariant Donaldson-Thomas partition functions from two types of smooth fixed loci, see Theorem 4.1.2 and Theorem 4.1.3. In chapter 2, we introduce Gieseker-Maruyama-Simpson moduli spaces and show that some of them are isomorphic to Hilbert schemes. Then, we introduce the perfect obstruction theory and define the vir tual fundamental class. We also introduce the equivariant and local version of Donaldson-Thomas theory. Finally, we wi l l introduce the conjectural correspondence in its various forms. In chapter 3, we look at the global case of P 2 -bundles over a smooth curve and consider the class of a section plus a fiber and prove that the correspondence holds up to the first order. We can compute the equivari-ant Donaldson-Thomas invariants explicitly in this case. The moduli space involved can be described explicitly as a disjoint union of two copies of the given smooth curve. The perfect obstruction theory, and hence the vir tual class, can also be written down explicitly in terms of known classes of the smooth curves. Therefore, we get the invariants by direct evaluation of the vir tual class over the moduli space. In chapter 4, we look at the local case of the total space of L i ©L2, a direct sum of two line bundles of degree fci and k<i over a curve, wi th a torus action along the fiber. We consider the equivariant residue theory of local curves. In their recent work [28], Okounkov and Pandharipande have completely solved the correspondence. W i t h degeneration, they reduce from the case of curves of arbitrary genus to P 1 . Then they use additional C*-action on P 1 to cut down the fixed loci again. We have adopted a different approach here. Our approach to evaluating the invariants is by direct computation. Our technique does not rely on degeneration methods, which have not been fully developed in the literature. We specialize our action to the anti-diagonal C * action and specialize our line bundle to have opposite degrees. In this case, the degree zero parti t ion function becomes 1, and the conjectural correspondence takes a simpler form. In fact, if the curve is elliptic, i.e. g = 1, then we also have the so-called "equivariant Calabi -Yau cas". The Gromov-Wit ten parti t ion function was also obtained by Vafa in a physics calculation, (cf. page 8 of [30].) In general, 6 we have an equivariant Calabi-Yau case if ki + k^ = 2g — 2 with anti-diagonal action. For the case ki — —k2, it is conjectured that only fixed loci parametriz-ing subschemes of pure dimension contribute to the partition function. We evaluate these contributions, which agree with the predictions. It is left to show that all other fixed loci contribute zero to the partition function and we verify this in some cases as well. Similar to chapter 3, the first step of our calculation is to describe the fixed locus under the anti-diagonal ac-tion. We show that the fixed loci are described by one-leg 3D partitions. They embed into product of various symmetric products of the curve. The products of symmetric products of the curve C are varieties of the form (j(ni) x Q(n2) x . . . x C^n'\ for some natural number I, where denotes the n-th symmetric product of C, which is isomorphic to the Hilbert scheme of n points in C. This product of symmetric products of the curve is determined by the 3D partition. Here we run into the difficulty that some of the fixed loci are not smooth. Therefore, we have to restrict ourselves to cases where the fixed loci are products of C, i.e. C x C x . . . x C. In particular, they are all smooth. Again we are able to describe the perfect obstruction theory as classes on the fixed loci. The virtual localization theorem implies that the virtual class is given by the Euler class of the obstruction bundle. The final step involves combinatorics with partitions and we will show that the equivariant Euler class of the obstruction bundle is trivial. Therefore, the equivariant virtual fundamental class is zero on these fixed loci, and their contribution to the partition function is zero. 7 Chapter 2 Donaldson-Thomas theory 2.1 The moduli space The Donaldson-Thomas invariants as used in [24] are in fact a subclass of the invariants defined by Thomas in his original paper [29]. The original motivation was to define invariants on moduli spaces of stable bundles over threefolds. A good reference for moduli spaces of sheaves is [17]. Def ini t ion Let (X,L) be a polarized complex projective algebraic variety and E a coherent sheaf on X, the Hilbert polynomial of E with respect to L is defined by: PE{™) =x(E®L®m). Remark The Hilbert polynomial depends on the polarization L. We will always fix a polarization. If E has support of dimension r, then Hilbert polynomial of E is a poly-nomial of degree r [17]. Def ini t ion Let (X, L) be a polarized complex projective algebraic variety and E a coherent sheaf whose support has dimension r. Suppose its Hilbert polynomial is Pe(m), then its reduced Hilbert polynomial is ( N pE(.m) pE(m) = , Ctf where ar is the leading coefficient of Pe(m). Remark We only consider torsion-free sheaves in this thesis, i.e. the support is X. R e m a r k Even if we are originally only interested in bundles, we must include torsion-free sheaves if we want to have a compact moduli space. In other words, torsion-free sheaves can appear as limits as bundles degener-ate. 8 Now we can define Gieseker stability for torsion-free coherent sheaves. D e f i n i t i o n Given (X, L), a polarized complex projective threefold, a torsion-free coherent sheaf E on X is (semi)stable if for any subsheaf F of E, PF(<) < PE-We use the lexicographic ordering on polynomials. Equivalently, p < (<)q if and only if p(m) < (<)q(m) for m » 0. Given a polynomial P, we can consider the following moduli problem, M.(P), as a functor from the category of schemes to the category of sets of equivalence classes of flat families of coherent sheaves, whose Hilbert poly-nomial is P: B^M(P)(B) where M(P)(B) is the set of equivalence classes of flat 5-families of semistable coherent sheaves on X wi th Hilbert polynomial P wi th the following equiv-alence relation. If S and £ ' are two flat families on B x X ^* B, they are equivalent if there exists a l i n e bundle L on B such as 8 is isomorphic to S'®p\{L). \ Maruyama constructed a coarse moduli space for this moduli problem. T h e o r e m The moduli functor M(P) has a projective coarse moduli space M(P) [17]. R e m a r k We wi l l also fix the determinant of coherent sheaves to be t r ivial . More precisely, If S is a flat 5-family, we require d e t £ to be isomorphic to OBXX- Then we get a submoduli problem and the resulting space is a closed subschemes of M(P). R e m a r k Once the moduli space contains strictly semistable sheaves, the moduli space can never be a fine moduli space, (cf. Lemma 4.1.2. [17].) O n the other hand, if the coefficients of the Hilbert polynomials satisfy some numerical conditions, the moduli space may not contain any strictly semistable sheaves. Even in this case, the moduli space may st i l l not be a fine moduli space. In this paper, al l moduli spaces wi l l be isomorphic to Hilbert schemes. Therefore, they are al l fine moduli spaces Hilbert schemes are moduli spaces of ideal sheaves.' Let Px to denote PoXi the Hilbert polynomial of the structure sheaf of X. Let (3 be a class 9 in H2(X, Z) and d = f5.C\(L) the degree of (5 wi th respect to L. Let Y be a 1-dimensional subscheme, whose associated 1-cycle is of class (5. The Hilbert polynomial of Oy is given by PoY (m) = dm + n, for some n G Z. The Hilbert polynomial of the ideal sheaf of Y, Iy, is given by Px-PoY. Proposition 2.1.1. W i t h the notation as above, The Hilbert scheme parametriz-ing 1-dimensional subschemes of X wi th Hilbert polynomial PjY is canoni-cally isomorphic to the Gieseker-Maruyama-Simpson moduli parametrizing coherent sheaves wi th Hilbert polynomial, PjY and t r iv ia l determinant. P R O O F : see [26]. • Finally, there exists a canonical map from the H i l b ( X , PjY) to H2(X, Z). This map is the composition of the Hilbert-Chow morphism and the cycle map from the Chow group to the homology group. Therefore, by restricting the homology class of the associated 1-cycle to (5, we get a closed subscheme. Definition Given f3 G H2(X, Z) and n G Z, we define In(X,j3) to be the Hilbert scheme parametrizing 1-dimensional subschemes, whose asso-ciated 1-cycles have homology class (5 and whose holomorphic Euler charac-teristic is n. 2.2 The perfect obstruction theory In this section, let (X, L) be a polarized smooth projective threefold and P be a Hilbert polynomial. Let M(P) denote the Gieseker-Maruyama-Simpson moduli space parametrizing stable sheaves wi th Hilbert polynomial P. We wi l l only consider fine Gieseker-Maruyama-Simpson moduli spaces, i.e. the universal sheaves exist. One way to get invariants that are invariant under deformation is to construct vir tual fundamental classes as developed by L i and T ian , [23] or by Behrend and Fantechi [2]. We wi l l use Behrend and Fantechi's construction in this thesis, though Thomas's original paper uses L i and Tian's construction. B y Behrend and Fantechi, we have the following, concept of a perfect obstruction theory 10 Definition Let Y be a proper Deligne-Mumford stack and Db(Oy), the bounded derived category of coherent sheaves on Y. A perfect obstruction theory is a morphism cf> in Db(Oy) from E* to Ly, the cotangent complex of Y. The complex, E" is perfect of perfect amplitude contained in the interval [—1,0], and 4> induces isomorphism on h° and epimorphism on h~l [2]. E* is perfect of perfect amplitude means that its homology is only non-trivial within [—1,0]. And locally, we can think of E* as a complex of vector bundles [E~l —> E0]. The rank of E' is well-defined and can be expressed as rk£* = rk£° -rkE'1. The main results of [2] is that a perfect obstruction theory, E' —• Ly, de-fines a virtual fundamental class [Y]v ir € A-k£«(Y). In [2], the authors need to assume that E* is globally a complex of vector bundles in order to obtain the existence of the virtual fundamental class, due to a lack of intersection theory of Artin stacks. In [20], Kresch constructed an intersection theory of Artin stacks and removed this assumption. Under suitable conditions, Thomas showed that's moduli spaces carry a perfect obstruction theory. Let £ be the universal sheaf on M(P) x l A M(P). And let Rp*?iomo(£, — ® ui) be the derived functor of p*Hom0(£,— <g> a;), where subscript 0 of 7iom0 means that we are taking the traceless Tiom and ui is the relative dualizing sheaf of p. Theorem With the notation as above and assuming that Extl{E,E)=0 for all E £ M(P), then there exists a morphism from Rp*Hom0(£, £ <g)u;)[2] to LM{P) that makes Rp*Homo(£, £ ®w)[2] into a perfect obstruction theory, where [2] represents a shift of degree 2 in the derived category. [29] Once we have a perfect obstruction theory, we can apply Behrend and Fantechi's machinery and obtain a virtual fundamental class. Remark In practice, we will use the tangent-obstruction complex, which is the dual of the perfect obstruction theory. Recall that there exist natural trace maps from Extl(E,E) to Hl(Ox), which are surjective with 11 kernels Ext0(E,E). (cf. p l02 of [17].) We think that the tangent sheaf has fiber ~E,xtl(E,E) at the point [E] and the obstruction sheaf has fiber Extl(E,E) at the point [E]. Note that the vir tual dimension is given by d i m E x t J ( £ , E) - d i m E x t 2 , ^ , E). Moreover, H o m 0 ( £ , E) = 0 by stability and Extg(-E, E) = 0 by the hypothesis, so the vir tual dimension is also given by 3 ^ ( d i m E x t ^ , £ ) - dlmWiOx)) = X(E,E) - X{Ox). i=0 R e m a r k If X is Calabi-Yau, then Kx and hence u> is t r iv ia l . B y Serre duality, E x t J ( ^ , E) and Exil(E, E) are dual to each other, so the vir tual dimension is zero. 2.3 The Gromov-Witten/Donaldson-Thomas Correspondence In this section, we wi l l assume X is a smooth projective Calabi -Yau threefold. We wi l l use In(X, (5) to denote the moduli space of ideal sheaves parametriz-ing subscheme Y of X, such as that Y is one dimensional, and the associated 1-cycle of Y is of class (3 G H2(X, Z), and X{@Y) = n. From previous dis-cussion, we know that In(X,(3) carries a perfect obstruction theory wi th the expected dimension equal to zero. R e m a r k 2 .3 .0 .15. If Y is a 1-dimensional subscheme, from primary decom-position, Y have a canonically determined component which is of pure dimen-sion one. This 1-dimensional component determines the associated 1-cycle of Y and sometimes we refer this component as the associated 1-dimensional component of Y. O n the other hand, the 1-dimensional component of a flat family is not flat. D e f i n i t i o n The Donaldson-Thomas invariants of the class (3 and holomorphic Euler characteristic n is ,N°T(X,P)= f , 1 J[in(XMvir 12 Then we can form the degree P Donaldson-Thomas partition function: ZDT(X,P) = J2^T(X,p)qn n e Z When P = 0, In(X,P) is in fact H i l b n ( X ) , the Hilbert scheme of n points in X, and we have the following result, which was first conjectured in [24] and see Remark for a discussion on the history of the proof. Theorem Z o r ( X , 0 ) = M(-g)*<*> where x(X) is the topological Euler characteristic of X and n > l V H ' is the McMahon function. R e m a r k This theorem can be generalized to an arbitrary smooth projective threefold, X. Then we have ZDT(X,0) = M(-q)^c^Tx®Kx). This theorem is proved for toric threefolds in [25] assuming the existence of the relative theory. Jun L i announced a proof for arbitrary smooth projec-tive threefolds assuming that it is true for toric threefolds in March 2005 at the Workshop on Donaldson-Thomas Invariants at the University of Illinois at Urbana-Champaign [22]. In December 2005, in [5], Behrend and Fan-techi gave a proof for the Calabi-Yau case using the technology developed in [3]. Recently, there was a new proof of this generalization by Levine and Pandaripande using algebraic cobordism [21]. We define the reduced degree P Donaldson-Thomas partition functions to be A priori, Z'DT(X, P) is a Laurent series in q. Conjecture 2.3.1. [24] The reduced partition function Z'DT(X,P) is ratio-nal function of q symmetric under the transformation q —> 1/q. 13 Before we introduce the Gromov-Witten/Donaldson-Thomas correspon-dence, we wi l l review the Gromov-Wit ten theory briefly. See [15] for an intro-duction. St i l l fix a nonsingular projective Calabi -Yau threefold X and a class j3 G H2(X,Z), there exists a Deligne-Mumford stack MG(X,P) parametriz-ing stable maps / : C —* X from a connected curve C of genus g to X, such that /*([C]) = p. This stack has a perfect obstruction theory wi th vir tual dimension zero [2] [1]. Therefore, we get an invariant: D e f i n i t i o n 2 .3 .0 .19. We define the degree /?, genus g, Gromov-Wit ten in-variant of X to be N°W(X,f3)= ( 1 J[Mg(X,P))ViT R e m a r k 2.3 .0 .20. Where Donaldson-Thomas moduli spaces are schemes and their invariants take values in Z , the moduli spaces of stable maps are Deligne-Mumford stacks and their invariants may take values in Q . Enu-merative invariants, by definition, must be integers since they are the counts of objects in some finite sets. Therefore, how to get enumerative invariants from Gromov-Wit ten invariants has always been an important question. A s usual, we put these invariants into parti t ion functions D e f i n i t i o n 2 .3 .0 .21 . We define the reduced Gromov-Wit ten potential of X to be FGW(X,u,t;) = £ £ N ™ ( X , V . P*0 g<0 Here, we call this potential reduced because we remove the degree zero con-tribution, i.e. constant maps. Exponentiating the reduced potential gives us the reduced parti t ion func-tion: Z'GW (X, it, v) = exp F'GW (X, u,v). We can think Z ' G W as the generating function of Gromov-Wit ten invari-ants where the domains of the stable maps are allowed to be disconnected and the maps are required to be non-constant on each component. We can write Z ' G W as: z'GW(x, u,v) = i + Y; Z'GW(X, p)(uy. P^o 14 We call Z'GW(X, (5) the reduced Gromov-Witten, partition function of degree 0 • ••' Now we can state the Gromov-Witten/Donaldson-Thomas correspon-dence. Conjecture 2.3.2. After a change of variables, em = —q, we have Z'GW(X,P)(u) = Z'DT(X,0)(-e™), 2.4 Generalization In [6], Bryan and Pandharipande have generalized the original Gromov-Witten/Donaldson-Thomas correspondence. In this section, let X be a non-singular projective threefold, not necessarily Calabi-Yau. The virtual dimen-sions of In(X, 0) and Mg(X, 0) are no longer zero , but equal to D = —Kx.p. We need a torus action to define invariants equivariantly. Let T be an al-gebraic torus, i.e. (C*)n and suppose that T acts on X. Identify H^(pt) with Q[t\, £2, • • • > tn]. Sometimes, we say that £1, . . . , £ n are the equivariant parameters. Then this torus action induces natural actions on the moduli spaces In(X,P) and Mg(X,P). Then we define the equivariant Donaldson-Thomas invariants of (X, T) to be N°T(X,P)= [ 1 Here, [In(X, P)]mr is the equivariant virtual class of degree D and the integral is taken to be equivariant pushforward to a point, so that the definition makes sense for negative virtual dimension and N^T(X, 0) is a homogeneous polynomial in t\, £2,..., tn of degree —D. If X is quasiprojective, but not projective, then we need to include an ex-tra condition in the definition of In(X, 0), to require In(X, 0) to parametrize proper subschemes of X in the class P and with holomorphic Euler char-acteristic n. Alternatively, we can take a compactification of X, X. We define In(X,0) to be the open subscheme of In(X,P), whose point [Y], as a subscheme of X, lies entirely in X. For quasiprojective X,'In(X,P) and Mg(X,0) might not be compact, so the integrals are not well-defined. If the fixed loci are compact, however, we 15 can st i l l define residue equivariant invariants, using vir tual localization. 1 N™(Xt0)= [ J\i [in(x,p)T]™ e ( N o r m m r ) Here, N o r m m r is the vir tual normal bundle and e ( N o r m m r ) is the equivari-ant Euler class of the vir tual normal bundle. Here, because we are tak-ing residues, the definition make sense eyen for positive vi r tual dimension, D. Then N^T(X, (3) are homogeneous rational functions of degree — D in ti, t2, • • •, tn. R e m a r k 2.4 .0 .22. In the non-Calabi-Yau case, we need to verify that Ext 3 , ( I , / ) = 0 for al l the ideal sheaves in the moduli space to have a well-defined. This is done in [26] for al l threefolds, but we wi l l verify it in our cases in Lemma 3.3.1. In both cases, we can form Gromov-Wit ten and Donaldson-Thomas par-t i t ion functions as before. Here is the formulation of their correspondence in the equivariant (residue) setting. Again , the M c M a h o n function determines the structure of the Donaldson-Thomas degree zero parti t ion function as proved recently by Levin and Pand-haripande [21]. T h e o r e m 2.4 .0 .23. The degree zero Donalds on-Thomas partition function is where the integral is taken to be equivariant pushforward to a point if X is projective. If X is quasi-projective, we take the residue, i.e. / c3(Tx®Kx)= / — — r - , Jx JXT e{NXT/X) where e(Nxr/x) is the Euler class of the normal bundle of XT in X. [21] Similarly, we formally divide the Donaldson-Thomas parti t ion function by the degree zero parti t ion function to get the reduced Donaldson-Thomas parti t ion function. 16 C o n j e c t u r e 2 .4 .1 . The reduced Donaldson-Thomas parti t ion function is a rational function in q and it is symmetric under the transformation q —> 1/q. Finally, we have the equivariant version of the Gromov-Wit ten/Donaldson-Thomas correspondence. C o n j e c t u r e 2.4.2. After a change of variables elu — —q, (-iu)fec^Z'GW(X,0) = {-q)-y^T^Z'DT{X,(5). R e m a r k It is worth to emphasize a crucial difference between the equivariant theory and the equivariant residue theory. In the equivari-ant Donaldson-Thomas theory, the vir tual dimension of the moduli space is always non-positive to get non-trivial invariants and the invariants take val-ues in homogeneous polynomials in the equivariant parameters. O n the other hand, in the equivariant residue Donaldson-Thomas theory, the moduli space can have positive vir tual dimensions to have non-trivial invariants. Hence the invariants take values in homogeneous rational functions in the equivariant parameters. Similar statements apply to the Gromov-Wit ten theory as well. 17 Chapter 3 Donaldson-Thomas Theory of P2-bundles over a curve 3.1 Results In this section, we consider the equivariant GW/DT correspondence of a class of threefolds: P2-bundle over a smooth curve of genus g. We verify the Conjecture 2.4.2 up to the first order. Let C be a smooth and connected curve of genus g. Let L\ and L2 be line bundles of degrees fci and k2, and let X be the total space of the P2-bundle P(0 © L\ © L2) over C. The three dimensional torus, (C*) 3 acts diagonally on the fibers of X. We denote by £o> *i a n d the equivariant parameters of this action along the first, second and third summand, respectively. We define Po to be the curve class of the section given by the locus ( 1 : 0 : 0 ) , and / to be the line class of a fibre in X. Then we prove the following result: Theorem 3.1.1. GW/DT Conjecture 2.4-2 holds for the leading term in q, when P = 0o and P = Po + f> and hi, k2 < 0. Remark This is the first example of the GW/DT correspondence of a compact threefold. Subsequently, Bryan and Behrend has verified the full GW/DT correspondence for the quintic threefold in degrees one and two [4]. Remark We will write out the proof of the theorem for P — Po + f and ki,k2 < —1. The other cases are similar and in fact easier. The theorem is proved by direct computation. The Gromov-Witten computation is a consequence of the work of Gholampour [9] and the result is presented in proposition 3.1.2. This result has been written up in a joint paper with Amin Gholampour and will appear in Mathematical Research Letters [10]. 18 P r o p o s i t i o n 3.1.2. [10] [9] Let k\, k2 < —I, then the parti t ion function for the degree (5 = Po + / equivariant Gromov-Wit ten invariants of X is given by Z0(g | fci, k2) = (t0 - h)9-kl-Ht0 - t2)9-k2-3 • ((4g - 4 - h - k2)t0 - ( 2 g - 2 - k2)h - (2g - 2 - h)t2)) (2 sin | ) f e l + f c 2 + 3 . Let IN(X,P) be the Donaldson-Thomas moduli spaces denned in Sec-t ion 2.3. Let 7r be the projection from X to C . Unlike the computation in Gromov-Wit ten theory, we can only compute the first term of the parti-t ion function in Donaldson-Thomas theory on the right hand side of Con-jecture 2.4.2. The power of q of the first term is the holomorphic Euler characteristic of the structure sheaf of a purely 1-dimensional subscheme, Y, in the class /?, which is equal to X(@Y) = 1 ~~ 9- From now on we fix n = 1 - g. B y definition, the coefficient of the first term of the right hand side of Conjecture 2.4.2 is N°T(X,P)= [ 1 = [ 1 , (3.1) J[/„(x,/3)]«- J[in{x,pY\viT e(JNorm j where IN(X,(3)T is the fixed locus of the induced torus action on IN(X,P), and N o r m O T r is its vir tual normal bundle, and e denotes equivariant Euler class. Now we express the result of this section: P r o p o s i t i o n 3.1.3. Let X, n, and ki, k2 < —1 be as above. Then the equivariant Donaldson-Thomas invariant of class P = Po + f and holomorphic Euler characteristic n = 1 — g is given by: N°T(X, p) =(*0 - t ! ) « - f c i - 3 ( t 0 - t2y-k>-3 • ((4g - 4 - h - k2)t0 - (2g - 2 - k2)tx - {2g - 2 - fc^)). 3.2 Preliminaries on P 2-bundles over curves The geometry of X is easy to understand. The projective bundle X can be constructed in three ways, as P(0 © LX © L2), and P ( L ? © O © L2L\), 1 19 and P(I/2 ® LiL*2@0). The tautological bundles change depending on the construction. We use the following notation for the hyperplane divisors: H0 = P ( L i © L2) C X = P(0 © L1 © L 2 ) #x = P ( L J © L 2 L * ) CX = P ( L * © £> © L2L\) H2 = P ( L * © LiL*2) C l = P ( L * © L X L ; © O) The corresponding line bundles are Ox(l), 7r*Li <8> Ox(l) and 7r*L 2 <S> ^ x ( l ) . The fixed points of the torus action lie in the three sections coming from the pairwise intersection of the above hyperplane divisors. The homology class of these three sections are 0, 0 — kif, 0 — k2f respectively. Next, we collect some facts on some cohomology groups of X. L e m m a 3.2.1. Let C be a genus g curve and X = P(0®L1® L2) !>C, and L\ and L2 are line bundles on C with degree ki and k2 respectively. Then, (a) Kx = n*tKcLlL*2)®Ox(-3). (b) H°(X,Ox) = H°(C,Oc) H\X,Ox) = H\C,Oc) H2(X,Ox) = Q H3(X,Ox) = Q (c) II°(X, Kx) = 0 Hl(X.Kx) = 0 H2(X,KX) = H°(C,KC) P R O O F : (a)The relative Euler sequence, 0 -» nn T T * ( L ; © L\ © Oc) <S> Ox(-l) -> O x -> 0, 20 implies • ^ = Ox{-2>)®L\Ll. Then the exact sequence, 0 -> 7 T * ^ C ttx -> nw 0, gives the result. (b) Apply Leray's spectral sequence to the fibration X-+C. R°7r*(C?x) = and higher images are zero. The results follow. (c) Again use Leray's spectral sequence, so we need to compute Hi(C,Rp7v*(Kx)), which is Hq{C, KCL\L*2 ® R p 7r*(O x ( -3))) . The fiber is P 2 . From the coho-mology of (9p2(—3), we have H V O x ( - 3 ) = 0 R 1 7 r , O x ( - 3 ) = 0 R2n*Ox(-3) = LXL2 The last equality uses relative duality and the 7r-relative dualizing sheaf is Ox(-3) ® L\L*2. And the lemma follows. • 3.3 Computing Exto(/,7) The moduli space In(X, (3) has a perfect obstruction theory if Exto(/, / ) = 0. The moduli spaces considered in this paper satisfy this condition by the following lemma. L e m m a 3.3.1. Let I be an ideal sheaf in In(X,(3). Then, Ext 3 ( / , /) - 0. 21 P R O O F : It suffices to show that E x t 3 ( / , J ) = 0. B y Serre's duality, E x t 3 ( / , i " ) is dual to Hom(J , / <g> Kx)- A p p l y Hom(J , *) to the following short exact sequence 0 -> / ® Kx Kx KX\Y 0. The corresponding long exact sequence is 0 -»• Hom(J , J ® Kx) -»• Hom(I , K x ) . We claim that Hom(J , Kx) is zero. This implies our lemma. To see the claim, Hom(J , Kx) is dual to H3(X,I) by Serre's duality again. From the short exact sequence 0 -y I - » e> x C y -»• 0, we have the long exact sequence: H2(X,OY) = 0 because F has dimension one and H3(X,Ox) = 0 by lemma 3.2.1. Therefore, H3(X,I) = 0. • 3.4 Proof of the Proposition 3.1.3 Before giving the proof, we first describe the fixed locus explicitly: L e m m a 3 .4 .1 . IN(X,/3)T is isomorphic to the union of two disjoint copies of C, and moreover the universal subscheme in IN(X, (3)T x X is a complete intersection. P R O O F : The closed points in IN(X,P)T are T-invariant subschemes in IN(X,p). The T-invariant curves in X are the three canonical sections and the three canonical lines in the fibers. Since they must be of class (3 and wi th holomorphic Euler characteristic n = 1—g, they must be the reducible curves wi th two components, the section @0 and a line in the fiber intersecting (30. The lines can move in two different families along the curve. There cannot be any embedded points on these subschemes, since they would increase the holomorphic Euler characteristic. 22 First , we construct these two families. Let M = C\ \\C2 where C\ and C2 are two copies of C. We wi l l consider M as H i l b ^ M ) and show that M is isomorphic to In(X, (3)T by showing that they represent isomorphic functors of points. In other words, we wi l l show how to construct a family of one-point in M from a family of fixed subschemes and vice versa. First we construct an M-fami ly of fixed subschemes: where pr\ andpr 2 denote projections from MxX onto M and X respectively. We define A to be the diagonal divisor in C x C. We use Ai and A 2 to distinguish between the two diagonals in M x Q. We use id X7T denote from the map from M x X to M x C. Recall the definition of the three canonical divisors, Ho, Hx and H3 and their corresponding line bundels. We let D\ = pr2Hx and D2 = pr2H2 U (id XTT)*AI and D[ = pr\H2 and D'2 = pr*2Hx U (id X T T ) * A 2 . The family y is a disjoint union of two components and y2. One, component is the intersection of D\ and D2 and the other is the intersection of D[ and D'2. This family is fixed under the T-action. This family gives a map from M to In(X,f3)T. From the analysis at the beginning, we know that this map is bijective on the set of closed point. To show that it is an isomorphism of schemes, it suffice to show that this map has injective differential, (cf. p 179, [13]) The tangent space to In(X,(5)T at closed points is the fixed part of the tangent space of In(X,/3) at those closed points, (cf. p 6, [12]) To show that this map is an isomorphism, we need to analyze the T-fixed sections of the normal bundle of a subscheme yp parametrized by p E M. These subschemes are complete intersections, so their normal bundles are direct sum of line bundles. Therefore sections of the normal bundles are specified by sections of the component line bundles. The subschemes are reducible curves, whose irreducible components are the section /30 and a line in the fiber. Then to specify a section of a line bundle on a reducible curve is equivalent to specify sections on the irreducible components and a gluing of these sections over the node. (cf. p 249, [14]) The normal bundles to the sectional curve have no section that is invariants under the T action and the normal bundle to the line in the fiber splits into a t r iv ia l line bundle, and the normal bundle of the line in the plane. The normal bundle of the line the plane has no T-fixed sections, but the t r iv ia l line bundle has one 2 3 dimensional T-fixed sections, which can be identified wi th the tangent space of the sectional curve at the nodal point p. The above argument shows that H°(Y, NY/X)T is one-dimensional and can be identified wi th the image of Kodaira-Spencer map from TPM to H°(yp, Ny /x)- Therefore, M is isomor-phic toIn(X,P)T. • P R O O F O F P R O P O S I T I O N 3 . 1 . 3 : Consider the first component constructed in the previous lemma. For convenience, we wi l l suppress the subscript in C\ and J ^ i from now on. Recall that 7r and id X7r are the projections from X and C x X to C and C x C respectively, that pr\ and pr2 are projections from C x X onto C and X respectively, and that p\ and p2 are projections from C x C onto the first and the second factor respectively. Recall that y is a zero section of rank two vector bundle on C x X, which is the direct sum of the following two line bundles: Ocxx(D1)=pr*(Tr*L1®Ox(l)) Oc*x(D2) = pr*2(7T*L2 <g> Ox(l)) <8> (id X 7 r ) * 0 C x C ( A ) . The K-theory class of the perfect obstruction theory on C, considered as a connected component of In(X,P)T, is given by R*prh*(Hom(l,l) - 0 C X X ) , where X is the universal ideal sheaf (cf. [24] page 1 9 ) . According to [12], in terms of equivariant K-theory classes, we can regard R > i , , ( W o m ( I , I ) - Ocxx) = - D e f m - Def 7 + Q b m + Qbf, where the superscripts, / , and m, mean the fixed, and the moving parts under the induced action of the torus, respectively. Since the fixed locus is smooth, we wi l l get J\Ir e ( O b / ) e ( O b m ) e ( D e f m ) ' We wi l l show later that Ob-^ is zero, so we can write N°T(X,P) = [e(K'prUnom(X,l) - 0Cxx))m. ( 3 . 2 ) Jc 24 Since y is the complete intersection of two divisors, D\ and D2, we have the standard Kozsul resolution of the ideal sheaf of y: 0 Oc^xi-D, - D2) -> OcM-DJ © 0CxX(-D2) - + 2 ^ 0 . Therefore, the equivariant K-theoretic class of X is Ocxx{-Dx) + Ocxx(-D2)-0Cxx(-Vi ~ D2). Then a formal calculation of equivariant K-theoretic classes gives us R>i ,* (Hom{J, 1) -Ocxx) = - R > i , . ( 0 C x x ( £ i ) ) - *L'priAOcxx(D2)) - R'pn^Ocxxi-Di)) - R>i ,*(e>cxx(-£> 2 ) + R > i , * ( C c x x ( £ > i - D2) + 2R> 1 , * (0 C xx) (3.3) + R>i,*(Ocxx(^ 2 - Dx)). The map pr\ can be factored through C x I i d 4 f C x C ^ C , where p\ is the projection onto the first factor, and therefore = R*Pi,* ° R*(id X7r)*. We will use the following facts in the rest of the proof and we summarize them here in the following lemma: Lemma 3.4.2. With the same notation as in Proposition 3.1.3, we have the following identities. R*(id xn)*(0CXX) = 0 C X C , R-(id xnUpr*Ox(l)) = p*2L*0 ©p\L\ ®p*2L*2, R'(idx7r)*(pr*0x(-l)) = O, R'piM(L0L{LL2)) = (l-g + jki + lk2)Cit0+jtl+lt3, R'Pl^{p*2{L0L{LL2) <8> C C x C ( - A ) - (1 - g + jh + lk2)Cito+jtl+it2 - LQL{L12, R*PIAP*2(L0L{L12) ® Ocxc(A)) = (l-g + jh + lk2)Qito+jtl+lt2 + L0L{LL2TC, where Cs denotes the trivial line bundle on C with weight —s, and LQ denotes the trivial line bundle on C with weight —to. 25 P R O O F : The first three are elementary. The fourth follows from Riemann-Roch. The last two are proved by applying R'pi,* to the divisor sequence of A twisted by p\(L§L\l}2) and pl(Ll0L[L2) <8> Ocxc(A) respectively. • Now by the first and fourth equalities in Lemma 3.4.2, we have R>i,*(C>cxx) = (l-<?)Co. The second and fourth equalities in Lemma 3.4.2, implies that R>i,.(pr 2*(O x(l) ® T T ' L O ) = R'(PM(P1LZ®P1L1®P'2LZ)<8>PZL1) = (fci + l - ^ ) C t l _ t 0 + ( l - y ) C 0 + - k2 +1 - g ) c t l _ t 2 , and similarly, using the second and the sixth equalities in Lemma 3.4.2, we have R>1 , * (^ (0 X (1 ) <g>7T*L2) <g> (id X7T)*(C>C><C(A))) -= R'pi,*((p2*L5 ®P*2L\ ®p*2L*2) ®p*2L2 ® Ocxc(A)) = R'Pl,*(p*2(L*L2)(A) @p*2{L\L2)(A) © 0 C x C ( A ) ) = (1 - g + k2)Ct2-t0 + L*0L2TC + (1 - <?)C0 + (1 - 3 - fci + A; 2 )C_ i l + t 2 + L;L 2 7b + T c . Finally, we can write R > V ( ( i d xnY(p*(LlL2)(A))) = R'(p1),(pJ(LIL2)(A)) = (1 - g - h + A: 2 )C_ t l + t 2 + L\L2TC) and R>i,.((id X7r)^(L1L5)(-A))) = R - ^ O ^ L i ^ X - A ) ) = (1 - ^ + fcx - fc2)Ctl_t2 - L i L 2 , where for the first one, we used first and the sixth equalities in Lemma 3.4.2, and for the second one, we used the first and fifth equalities in Lemma 3.4.2. So far, we have computed all the terms in the right hand side of (3.3), so we can write e ( R > i , „ ( H o m ( l , 1) - 0Cxx)T fo-*i)-*+g-1fo-^-,»+g-1 ^ ((to-t2) + (k2 + 2-2g)\p})((t2-t1) + (k1-k2)\p}y {°- ) 26 (3.5) where [p] is the class of a point in C: " Similar computations for the other component of IN(X,fi)T, yield e(R'pr^(Hom(l,l) - 0Cxx))m = ( * 0 - t i ) - f c l + g - 1 f t o - * 2 ) - f c a + g - 1 ((t0 - t i ) + (fci + 2 - 2g)\p})((t1 - t2) + (k2 - h)\p})" Equation (3.3) also shows that the fixed part of R'pr^(Hom(l,l) - 0CXX) is just Tc- Since we have already shown that the fixed locus is smooth, we can conclude that O t / is zero. Therefore, by (3.2), NPT(X,P) is equal to the sum of the integrals of (3.4) and (3.5) over C. To do the integral, we expand the fraction in terms of [p] and integrate over C. This proves the proposition. • 27 Chapter 4 Donaldson-Thomas Theory of Local Curves 4.1 Results In this section, we wi l l study the Donaldson-Thomas theory of local curves. The motivation is Conjecture 2.4.2. The local curve case is another non-toric case. In this case, the threefold under consideration is not compact and the moduli spaces are not compact either. Therefore, we have to use the equivariant residue theory. Let fix some notation. Let C be a non-singular curve of genus g and L\ and L2 are two line bundles on C wi th degrees k\ and k2. Let X = LX@L2^C be the total space of the direct sum of these two line bundles. Let T = C* x C* and C [ s i , s2] be the T-equivariant cohomology ring of a point. A n d let T0 = C* «->T (t,t~l) be the anti-diagonal subgroup of T and C[s] be the T 0-equivariant cohomol-ogy ring of a point. R e m a r k 4 .1 .0 .27. We can take the compactification of X to be X = P ( C x © L\ © L2). Then X is in fact the space X from last section. The equivariant parameters can also be identified as Si = t\ —10 and s2 — t2 —10. A s in Section 2.4, we have two moduli spaces, In(X,P) and Mn(X,P). They are not compact. There is the canonical T-action on X by acting along the fiber, wi th induced anti-diagonal T 0 -act ion. There are corresponding 28 actions on the moduli spaces as well. We will show that these two actions have the same fixed loci, which are compact. Then the assumption of [6] is satisfied. Therefore, we have well-defined equivariant residue theories and Conjecture 2.4.2 applies to the space X with T 0 action. In this section, we also specialize to the case where L\ and L2 have op-posite degree, i.e. k = k\ = — k2. We require that h°(Li ® L^) = 0. In this setting, it is easy to see from Lemma 4.2.1 that ZE)T(X)Q = 1. Therefore the reduced and the non-reduced partition functions are the same. From the calculation of the partition function of the Gromov-Witten residue invariants, the authors in [6] make the following conjecture on the partition function of Donaldson-Thomas residue invariants: Conjecture 4.1.1. Suppose that k = k\ = —k2 with the anti-diagonal T 0 action, so that s — si = — s2. And let (5 be the curve class of the zero section of X. The partition function of Donaldson-Thomas residue invariants of the class dp takes the following form: Z D T ( X ) D 0 = Sd ^- 2 )(- l )^- 1 + f c ) ^ ( - j ! L ) 2 « - 2 ( - g ) d ( l - » ) + * C ( A ) ) \\-d d l m A where A h d means that A is a partition of d and C(A) is the total content of the partition A. See Section 4.3 for all the facts about partitions used in this paper. The To-fixed locus consists of disjoint components, so we can evaluate the integral on each component seperately. Our first result is Theorem 4.1.2. The components of the fixed locus parametrizing subschemes of pure dimension are all isolated smooth points. They are characterized by a partition A h d. Let Y\ be the subscheme corresponding to A, then X ( 0 Y X ) = d(l -g) + C(X)k and f 1 = d(2g-2)r_^_\2g-2(_1\d(9-l+k)+kC(\) J [ l Y x ] V i r e(Normvir) . Kdim\J K J , Remark The exponent of —1 in this theorem is different from conjecture 4.1.1, because of the contribution from —q. Thus we have proved that the contributions from components parametriz-ing subschemes of pure dimension have accounted for all the terms in the par-tition function. This suggests that the contributions from all other compo-nents should be zero. We are able to verify this on product-type components 29 which are smooth, (see Definition for the definition of components of product-type) Theorem 4.1.3. Let S C In(X,dj3)T° be a product-type component, then f — J r s i v i r e(Norr fo-.(Norm-)-0 4.2 Degree zero Donaldson-Thomas partition functions The following lemma has also appeared on page 11 of [28]. L e m m a 4.2.1. Suppose that X is the total space of L\®L2 where L\ and L2 are line bundles on C with degree ki and k2 respectively. Tx is the tangent bundle and Kx is the canonical bundle of X. Jx JXT e{NxT/x) sis2 P R O O F The fixed locus, XT, is the zero section of the vector bundle and NXT/X — L i © L 2 , so e(NxT/x) = ( s i + c 1 ( L 1 ) ) ( s 2 + C i ( L 2 ) ) . K x = -K*{KCL\L*2) and Tx <8> KX is ir*{KcLl ® KCL*2® LIL*2). Therefore, cz(Tx <8» Kx) = c3(n*(KcLl © KCL*2 © L\L\)) = ( - S l - C i ( L i ) + ctiKp)) x ( - s 2 - c i ( L 2 ) + ci(iT c )) x ( - s i - s 2 - C i ( L i ) - C i ( L 2 ) ) . Then we wi l l have c3(Tx®Kx) XT e(NXT/X) ( - S l - C i ( L i ) + c i ( J C c ) ) ( - a 2 - C i ( L 2 ) Hr C i ( i C C ) ) ( - 5 i - s 2 - C i ( L i ) - d ( L 2 ) ) ' W l + . ^ ) ( 1 + ^ ) 30 To push forward equivariantly, expand the integrand in C\(L\) and c i ( L 2 ) and integrate out the appropriate terms and the result is -(h + k 2 ) + { S l + 5 2 ) 2 (2 (7-2) . 5 i S 2 • 4.3 Partitions This part introduces some notation and facts about parti t ion A of d. Given a parti t ion A of d of length n, we wi l l define two sets: Ax = {(1, A i + 1)} U {(i, Xi + l)\Xt < Xi-u n > i > 1} U {(n + 1,1)} and Bx = {(i, Xi)\Xi > A i + i , i < n} U {(n, Xn)}. Here is an example of A and B for the parti t ion (3, 2, 2,1,1) . We use Ferrar's diagram to represent 2D partitions. A3 ,2 ,2 , i ,D = {(1,4), (2, 3), (4, 2), (6,1)}. A n d B(3,2,2 , i , i) = {(1,3), (3, 2), (5,1)}. R A R R A A Figure 4.1: A n example of 2D partitions If / is an ideal of C[x, y] generated by {xm~1yn~1}(m,n)eAx, then C[x, y]/I is a vector space over C wi th basis elements given by {xl~ly^~l)^j)€\. 31 Given a box £ A, we define the hook length hij = Aj + X'j - i - j + 1, where A' is the conjugate parti t ion obtained by reflecting the Ferrar's diagram of A about the i — j line. We define the content of A, C ( A ) , to be 2~2(i,j)ex^~3-We wi l l also need to deal with one-leg 3D partitions, which are always denoted by 7r. Let use z for the axis wi th one infinite leg. Then for z large enough, the (x, y) cross section wi l l stablize to a 2D partit ion. We use A to describe this 2D partition, but sometimes we use Xn to emphasize that the 2D parti t ion A is determined by the 3D partit ion IT. We can associate each block in the 3D partit ion wi th a weight, which is the (x,y) coordinates. Then for each weight let T^J) be the number of blocks contained in the 3D partit ion wi th that weight. T(jj) can be zero. It can also be infinity, which means € A^. Let r denote the set of finite, non-zero T(JJ). Let lT denote the cardinality of the set r and | r | denote the sum of r^j) in r . A n y one-leg 3D partit ion determines uniquely the set of data A and r, though not al l pairs of A and r can give us a one-leg 3D partition. We wi l l use A and r to describe one-leg 3D partitions. For example, A = (1) andr(i ) 2 ) = 2,T(i ) 3) = l ,T(2 ,i) = 1>T(2,2)) = 1 describe the following 3D partition: Figure 4.2: A n example of 3D partitions 4.4 Determination of Fixed Locus We use two group actions in this paper. T acts canonically along the fiber on X and T 0 is the induced anti-diagonal action. They both act on In(X, (5) and let In(X,P) T and In(X,{3)T° denote the fixed loci under T-act ion and Trj-action respectively. They are both subschemes of In(X,/3). It is easy to 3 2 see that the fixed loci of the T-action correspond to ideal sheaves that are locally generated by "monomials". More precisely, let C = U Spec 4* and X = \jSpecAa[xa,ya] be affine open covers of C and local tr ivialization X respectively. Then T-invariant ideal sheaves are those whose restriction to Specy4.Q[x a, ya] are generated by monomials in xa and ya wi th coefficients in Aa. Lemma 4 . 4 . 1 . With the notation as above, In(X,(3)To = In(X,f3)T P R O O F : Consider the anti-diagonal action, given by t —> (£, We claim that the invariant ideal sheaves are sti l l locally monomial ideals. Under the anti-diagonal action, the fixed locus is s t i l l supported on the zero section. In other words, on each piece of the affine cover, the fixed ideals are sub-ideals of the ideal (xa,ya), so we can localize to the local ring Aa[xa,ya](xa,ya)-Suppose is a generator of the fixed ideal. If it is a monomial, then there is nothing to prove, so assume that it is not. The anti-diagonal action maps / is to itself multiplied by a unit, i.e. This forces i — j equal to a constant for al l Let io be the minimum value of i. Since i — j is constant, i — i' = j — j' for two different sets of The associated value jo is in fact the minimum value of j. Then we can write / as: / = (*V°)( E ^ V ^ + aiojo), In the local ring Aa[xa,ya], E aijixy)1'10 + aiodo is invertible. Therefore we can'take the generator to be xloyio. 33 Now we know that any fixed ideal is generated by monomials in the local ring A a [ x Q , ya](Xa,ya), it must be generated by the same generators in an affine neighborhood of the generic point (xa,ya). Therefore we have shown that invariant ideal sheaves are locally monomial ideals. • Next, we will show that the T°-fixed loci can be described by one-leg 3D partitions. Recall that C is a smooth curve of genus g, Li and L2 are line bundles of degree ki and k2 respectively. X is the total space of L\ © L 2 . Let 7T be the projection map: X —> C. We can also consider X as Spec(Sym*(Z/JffiL2)). In other words, n*(Ox) is isomorphic to Sym*(L^©L2) as Cc-modules. Then subschemes of X can be regarded as certain types of quotients of Sym*(L* © L\). Remark Sym*(Z/j; @ L2) is not a coherent (9x-module, because it is not finitely generated. On the other hand, we would like to describe our fixed loci as quotients of a fixed coherent C^-module. There are two ways to solve this problem. First, under the full torus action, each weight space is a coherent O^-module, or more precisely, a line bundle. Secondly for a given In(X,dPo)T, all fixed subschemes are supported on the zero section, since they are proper. Thus, we can fatten up the zero section enough, or pick a non-reduced curve, S, whose support is the zeroi section and it has a multiplicity high enough, such that any subschemes in In(X, d(50)T is a subschemes of S. We consider them as quotients of 7r*(C?s), which is a finite Ox module. Now, fix In(X, d(3o)T and let 7 .6 In(X., d/?o)7' denote a subscheme in X in the fixed locus and let Z denote the unique one dimensional component of Y. Let m be the difference of n and xiPz)- Recall that a one-leg 3D partition IT can be described by a 2D partition A and a set of weighted natural numbers r , who add up to |r|. L e m m a 4.4.2. With the notation as above, a component of In(X,d(3o)T can be described by a one-leg 3D partition n, so let's call it In. Also, In is a subscheme of CT = C^'h)) x C^a-fc)) x . . . x C{T(i'r^\ where C ( T<^> denote the T(ij)th symmetric product of C, which is also H i l b r ( i ' j ) C. Let i : In —> CT denote the embedding. P R O O F : We will show that a family of subschemes in In(X,d(30)T cor-responds to a families of points in CT. Consider the following diagram over B x X where B is the base scheme of a family. 34 0 -> IB - Q B x X - OyB - 0. Pushforward this exact sequence to B x C by id XTT, we get another short exact sequence: 0 (id X 7 r ) , ( / f l ) (id X 7 r ) * ( C W ) (id X7r)„(CV f l) 0-Under the T-action, this exact sequence splits into different weight com-ponents. For a particular weight (x,y), three things can happen. First, (id X7r )*(0y B ) has no weight (x,y) component, which is case for all but finitely many (x,yfs. Again, this is because that those subschemes are proper. Secondly, (id X7r)*( / s ) has no weight (x, y) component, this means the one dimensional components will contain a (x,y) component. Finally, we have non-zero but torsion quotient of weight (x,y). This quotient of a fixed line bundle on curve parametrizes points on the curve. Therefore, this B family of fixed subschemes is equivalent to a B-family of points on C. • Remark Intuitively, we should think those subschemes, Y's in the fixed locus in the following way. Let z-axis be the direction of the curve and x and y axis's be the fiber directions. Then there are rows of infinite length of blocks along the 2-axis, representing the one dimensional component of Y. The shape of rows is determined by the partition A: the cross section of these rows is the Ferrar's diagram of A. Then there are m embedded points that will move along the fat curve. The 3D partition 7r in fact describe a subscheme in the fixed locus where all the embedded points are supported at a single closed point. Because of the torus action, we can associate each point with a unique weight. In the 3D partition, this point is represented by a single block moving along those infinite rows. If we look at a (x, y) cross section, this block will always have its weight as its (x, y) coordinates. There can be more than one points that have the same weight and these points are indistinguishable. Therefore, the set r describe these embedded points. In addition, every block has,to be supported both from below and from the left. This imposes constraints, so-the fixed locus is a closed subscheme of C^T\ From this description of fixed loci, we know that subschemes of pure dimension are isolated points in the fixed loci. Coro l l a ry 4.4.3. If I € In(X,df3)T° and I corresponds a subscheme with pure dimension, then I is an isolated point, of In(X,df3)To and such isolated points I are in 1-1 correspondence with partitions A of d. 35 P R O O F : If Oy is the structure sheaf of the corresponding subscheme of pure dimension, there can't be any torsion in the quotient of n*(Gx)-Therefore, TT.CV = Y, Lr I + 1L 2- J + 1. In Lemma 4.6.1, we verify that the tangent spaces to these points have no fixed components. Therefore, they are isolated smooth fixed points. • If /„• is a component parametrizing subschemes wi th embedded points, then it is hard to analyze them in general. We wi l l consider a special case where T^J) are all equal to one and are in A\n. In this case, we wi l l show in section 4.8 that such In is isomorphic to a product of lT copies of the curve C. For a general one-leg 3D partition n, the component 1^ can be singular. For example, let TT be the one-leg 3D partit ion described in the last section. There are two generic configurations of a subscheme in this components as shown in the following two diagrams: / A A A / A A A A / / / / A / / / Figure 4.3: A n example of singular fixed loci: first component / / / / / / / / / / / / / / A / Figure 4.4: A n example of singular fixed loci: second component Each of these two diagram represents a subschemes wi th two embedded points. We think them as two 3D partitions glued together. Each 3D parti-t ion represents an embedded point. 36 These two generic configurations can both degenerate to the configura-tion given by the 3D partition. Therefore, this component is reducible and singular. 4.5 Equivariant Computation with K-classes The calculation in this section is based on the virtual localization developed in [12]. A key component of the calculation is the equivariant Euler class of the virtual normal bundles. Since we are interested in the equivariant Euler class, we will work in the category of equivariant K-theory. The equivariant K-theory class of the virtual normal bundle is part of the equivariant K -theory class of R-p*(Hom(l,l)-0MXX-). Here p is from the universal diagram: where M denotes a component of the fixed locus, and X-* MxX is the universal ideal sheaf. In fact, Hom(I,X) - 0 M X X - , is supported on the zero section of X. Therefore we can first restrict to MxX and then push forward: L e m m a 4.5.1. As equivariant K-theory classes, R'p*{Hom{l,l) - 0 M x x ) = R'p*{Hom(l,l) - 0MXX). P R O O F : Since R 'p* is the derived functor of p*, it suffices to check equal-ity for p*. We have the following short exact sequence: 0 -»• Hom0(l,l) -»• Hom(l,l) -»• 0 M X X -* 0 Hence, the K-theory class of Hom(l,2) - 0 M X X -37 is represented by 7iomo(X,X), which is supported entirely wi thin X. It fol-lows directly from the definition that Pt,(Hom0(l,l)) = p*(Hom0(X, T)\X). • Roughly speaking, R'p*(Hom(l,l) - C W ) is an equivariant K-theory class on M whose fibers are E x t 2 ( / , I ) - E x t J ( I , I ) , since that E x t ° ( / , / ) = 0 by stability and Exto( / , / ) = 0 by lemma 3.3.1. It is in fact the K-theory class of the perfect obstruction theory on M. Therefore, its moving part, i.e. the part wi th non-trivial weight, is the vir tual normal bundle. The fixed part, i.e. the weight zero part, is a perfect obstruction theory for the fixed locus. In the following sections, we wi l l compute the equivariant K-theory class of R'p*(Hom(l,l) -QMxX). Furthermore, MxX^M can be decomposed as MxX^MxC^M. B y an abuse of notation, we use TT to denote both X^C and M x X -> M x C. We can decompose R'p* as R*P* = R*Pi* O R*7T*. If E is a locally free sheaf on M x C, then by projection formula: 38 R " 7 r , ( 7 T * ( £ ) ) = E ® R ' T T , ( C W ) . Since X —> C is a vector bundle, OMXX is equivariantly isomorphic to Sym*(L* © L2) as an OMXC algebra. Therefore, oo i=0,j=0 In the two cases that we study in this paper, the universal ideal sheaves J al l admit a locally free resolution in terms of pullbacks of bundles on M x C. This means the K-theory class of R V * ( W o m ( J , I ) -OMXX) can be expressed as sum of bundles o n M x C using the projection formula. Then we can use Grothendieck-Riemann-Roch to evaluate R*p i* . Let introduce more notation. Let x and y denote the pullback t o M x C from C of LT1 and L 2 L respectively. B y an abuse of notation, we use them to denote their pullbacks to M x X as well. In the case of components of product-type, i.e. M = C x . . . x C, we use Zi to denote OMXC0(~^i) a n d its pullback to MxX, where Co is another copy of C and Aj is the big diagonal of the i t h copy of C in M wi th CQ. In summary, x = n\L7x) y = **(L-21) Zi = OMXCO(~^I)-Definition It wi l l be shown that the equivariant K-theoretic class of J can be expressed in terms of 7 r * L i , 7T*L 2 , and OMXC0(—AJ). W i t h the notation above, we can express X as a polynomial Q in x, y, Z{. We wi l l call Q the equivariant K-theory class of X. This polynomial Q w i l l be a polynomial in x and y in the case of sub-schemes of pure dimension, and a polynomial in x, y, and Zi in the case of product-type. Since x, y and Zi are al l pullbacks by 7r, the push-forwards of Q by 7r is the same as multiplication by Y2i=o,j=o X%V^ • Formally, this multiplication by infinite sum is the same as multiplication,by ( 1_ g^ 1_ y) • Let 39 Then R'n*(Hom(l,l) — OMXX) QQ-1 ~(l-x)(l-j) =P(P - x~xP - 1) + x-xy~xP(P - xP - 1), where Q(x,y) — Q(x^1,y~1) and P(x,y) = P(x~1,y~1). In the following calculation, it is easy to figure out the K-theory class of R * / T * ( 0 y ) from the geometry of the subschemes parametrized by M for various components M. The above formula will then allow us to compute the K-theory class of R"n.(Hom(I,T) - OMXX). 4.6 Loci of subschemes of pure dimension In this section, we will apply the technique discussed in the last section to compute the contribution from components parametrizing subschemes of pure dimension. First, we show that their tangent spaces have no fixed components. Lemma 4.6.1. If I parametrizes a subscheme of pure dimension, then Ext0(I, J ) T ° = 0. P R O O F : By definition, we have the following exact sequence: 0 ExtJ(I, I) Ext1 (I; I) H\Ox) ^ 0 Since HX(X,OY) = HL(C,Oc) has weight zero, it suffices to show that the weight zero part of Ext 1(7, /) has dimension no greater than h}(C, Oc)-To get the dimension of weight zero part of Ext 1 ( J , / ) , we use the long exact sequence from the short exact sequence: 0 -> I -> O Y -> Oy -> 0. We have 40 -> H o m ( / , O y ) -> E x t 1 (I , / ) - * E x t 1 (I , Ox) E x t 1 ^ , O y ) . B y the following two lemmas, Hom(J , O y ) has no weight zero part and E x t 1 ( J , Ox) has weight zero part wi th dimension equal to h}(C, Oc)- There-fore, Ex to( / , I) has no weight zero subrepresentation. • L e m m a 4 .6 .2 . / / / is an ideal sheaf defining a subscheme with pure dimen-sion, then Hom(J , O y ) contains no weight zero TQ-representation. P R O O F : Suppose that a is a weight zero non-trivial global section of the sheaf Tiom(I, Oy). The localization of a has to be non-zero somewhere. It suffices to restrict to p G Y. Ox,P — R[x,y](XiV), where R is a discrete valuation ring. Ip is generated by {xmyn}(rn,n)€Ax where A corresponds to the pure subscheme Y. Order {xmyn} in the descending order of the x exponents. Suppose that xm° is the first generator, then in order to have weight zero, a maps xm° to zero. Let x m i y n i be the next generator, so that mi < mo and ni > 0. suppose that a does not map xmiyni to zero, then in order to have weight zero, a(xmiyni) = xmi-Syni-s f o r s o m e positive integer <5. Then consider xm°yni. a(xmoyni) = ynia(xmo) = yni x 0 = 0. O n the other hand, a{xmoyni) = xm°-mia(xmiyni) = x ^ - ^ x ^ - s y ^ - s = x m ° - 5 y n i - 5 . A n d xm°~syni~s £ Iv. This is because n\ is smallest non-zero exponent of y in the generators. Therefore a(xm°yni) ^ 0, a contradiction. Therefore, a(xmiyni) = 0. Continuing this argument, we can show that a is in fact a zero map. • L e m m a 4 .6 .3 . The weight zero subrepresentation o / E x t ^ J , O^) has dimen-sion equal to hl(X, Ox). 41 P R O O F : Ext1 (I, Ox) is dual to Ext2(Ox, Kx®I) = H2(X, KY ® I) by Serre's duality. A p p l y long exact sequence to the short exact sequence 0 -»• KT ® / -+ KT -* KX\Y 0, we get - H\Kx) -> H\Kx\Y) -> H2{Kx®I) - J ^ i t f y ) - # 2 ( ^ | y ) -Hl(Kx) = 0 as stated before and i f 2 ( i £ x | Y ) = 0 because Y's dimension is one. The above exact sequence is in fact a short exact sequence: 0 H\Kx\Y) -> H2(Kx ®I)^ H2(Kx) -> 0 B y Lemma 4.6.4, ^(K-jAY) has no weight zero part, so the dimension of the weight zero part of H2(K-x® I) is the same as H2(Kx) which is dual to H\X,Ox). • Lemma 4.6.4. Hl(K^\Y) has no weight zero part. P R O O F : Use Leray spectral sequence wi th respect to the projection n : Y —> C. Since the dimension of the fiber is zero, IV-K^^XIY) = 0. Moreover, Y lies in X, so Kx\Y = KX\Y. A n d Kx = n*(KcL\Li). Therefore, R\*{Kx\Y) =KcL\L*2-K^OY) =KCL\L\ ® ( £ L'rL'p). (».j')eA A p p l y this to the Leray spectral sequence, we have H\X,Kx\Y) =H\C,KcL\Ll®{Y, LlrL\-j)) ' ' (iJ)6A =H1{C:KC®{Y, L?L2j)). 42 By Serre's duality, H\C,KC®{ Y £rV)) ( i j ) 6 A is dual to H°(C, ^ e A L\L32). Since we are considering the anti-diagonal action, L\L32 has weight zero global section if and only if i = j. Since we assume L\ and L2 are generic with opposite degree, (Z/iL2)1 is generic of degree 0. They have no global sections. Therefore, HX{K^\Y) has no weight zero part. • 4.7 Proof of Theorem 4.1.2 We have seen that these loci are all isolated smooth points. Then we need to calculate x(CV A ) to determine to which term in the partition function these components contribute to. Recall that X —> C ^ pt and that ( i j ) € A Using the Grothendieck spectral sequence, R'(Plon),(0Yx):. =R>,,(R-7r*(CV A)) =R'Pl,.( Y {(LiY-'il*)1-') ( i , i ) e A = Y R * P i , * ( ( L i ) 1 - i ( ^ ) w ) -( i J ) € A Since X(OYX) equals to the alternating sum of R'(pion),(0Yx), we conclude that x(Oyx) = Y xdLx)1-*^)^). ( J J ) e A 43 We can apply Riemann-Roch to compute x((L1)l-\L2)1-i) = l - g + (i-j)k, where L\ and L2 have opposite degree, k and —k respectively. Summing up, we obtain d(l — g) + C(X)k as expected. Next, we compute the K-theory class of R'%*{Hom(l,l) -0Mxx)-B y the argument from Section 4.5, this is the same as computing QQ-1 (l-*)(l-y) = P{P - x~lP - 1) + x-ly~1P(P - x P - 1) withP = E ( u ) 6 A ^ - V - 1 . Luckily, this has been computed before in another context. We formulate the following lemma, which appears in [27]: Lemma 4.7.1. Given " (l-a;)(l-y) then Then we use Riemann-Roch to evaluate R'(pi)*(L\L{) wi th k = kx = -k2, we get Y ( g - 1 + A-(/iii))C_/,.., + (<? - 1 + K-hti))^ (4.1) where ChijS is a one dimensional representation of C* wi th weight hij. Recall that the hook length of a element G A is defined by 4.3. This is an equivariant K-theory class on the moduli space, which is a point. Note that there is no weight zero term and a priori the moduli is smooth, so this is actually the K-theory class of N o r m v i r . Taking the equivariant Euler class, we get 44 = s (2 S -2 )d^ J J ( / l i j . ) 2 s - 2 ^ _ 1 ^ ( f l - l ) + E ( i , i ) € x f c ' H j (ij')6A _ „ ( 2 g - 2 ) d / rf! \2g-2/ -i Nd(fl-l+fc)+fcC(A) M i m A j 1 } and it is easy to see that htj = d + C(A) mod 2. (<j')eA • 4.8 Product-type components In this section, we consider'components in the fixed loci that parametrize subschemes with certain types of embedded points. Recall that components in the fixed loci correspond to one-leg 3D partitions. A n d each one-leg 3D parti t ion determines a 2D partit ion A and a set of weighted natural numbers r. We wi l l only consider the case where T^J) are al l equal to one and are al l in A\. Definition A component of the fixed locus, corresponding to a one-leg 3D parti t ion IT is of product-type if n satisfies the following two conditions. A l l elements in r equal to one and al l the weights are in A\n, where A„- is the associated 2D partition. A generic point in this component parametrizes a subscheme wi th distinct embedded points. Each embedded point can be assigned a distinct pair ( m r , n r ) € A\. The ideal at these embedded points can be described as ( Z ^ y - 1 W m r i n ? 4 „ r U ( x — y - l Z ) x m r y n r - ^ xmr-lynry Here, the local ring can be described as R[x,y](Xiy) where R is the local ring of the curve at closed point wi th z as the local parameter. Lemma 4.8.1. If M is a product-type components, corresponding to a one-leg 3D partition TT with lT embedded points, then M = C 1 t , the product of lT copies of C. 45 P R O O F : First , as in Lemma 4.4.2, we have a map from M to CLR. We can also construct a C ' T - fami ly of subschemes lying in the product-type components, so there exists a morphism from C1T to M. This morphism is bijective. Next lemma shows that M has tangent space of dimension I everywhere. Therefore, they are isomorphic as schemes. • L e m m a 4 .8 .2 . / / / is an ideal sheaf in a product-type component parametriz-ing a subscheme with I embedded points, then dim(Extl(I, I)T°) = I-P R O O F : B y the same reasoning as in Lemma 4.6.1, it suffices to show that E x t 1 (7, / ) has weight zero part wi th dimension no greater than hL(C, Oc) + I. Starting from the short exact sequence of coherent sheaves: 0 - » I -> Ox -> Oy -> 0, we have the following long exact sequence in Ext*(I, •): -> Hom(I , Oy) -> E x t 1 (I , / ) Ext1 (I, Ox) -> Ext1 (I, OY) - » . B y Lemma 4.8.3, H o m (7, Oy) has at most I dimensional weight zero part. A n d the same argument from Lemma 4.6.3 can also show that E x t 1 (7, Ox) has no weight zero representation, with Lemma 4.6.4 replaced by Lemma 4.8.4. • L e m m a 4 .8 .3 . Horn(7, Oy) has at most l-dimensional weight zero subrep-resentation. P R O O F : If a G Hom(J , Oy) — T(X,Hom(I, Oy)) is a nonzero weight zero global section of Hom(I, Oy), then there exist some point p in X, such that locally ap in H o m ( / p , OytP) is not zero. If p is not on Y, then H o m ( / p , Oy:P) is zero. If p is on y, but not one of these embedded points, then by the argument from Lemma 4.6.2, H o m ( / p , OytP) is zero as well. Therefore, it suffices to show that the sum of d im H o m ( / p , CV, P ) is I, where the sum is over al l points that support embedded points. Now suppose that p is point that supports an embedded point of length q. Formally, if Y0 is the associated 1-dimensional 46 component of Y, then IY0,P/IY,P is of length q. In other words, the ideal is generated locally in by the following generators: {s m - 1 y B - 1 }(m,n) ? t (m P ,n P ) U { x ^ y ^ z } U {x^'1} U {^I,"'}. Here, r = 1, . . . ,q and (m, n) G where A corresponds to the maximum one dimensional subscheme. . ( m r , n r ) r = i . . . g is a subset of ^4A- A n d note that sometimes a term of the form xmrynr~1 or xmr~1ynr can be absorbed as a multiple of a generator from the first group, i.e. xm~lyn~l. We wi l l argue that the dimension of Hom(7 p , Oy:P) at this point is q and a basis is pro-vided by maps that map generators xmr~lynr~lz to xmr~xynr~x and a l l other generators to zero. Now order these generators by decreasing order of the x exponents. The first generator, xM must be mapped to zero by ap if the weight of ap is zero. The next generator may or may not contain 2 as a factor. In the first case, the generator is of the form xm~1yn~1z. Suppose that it gets mapped to xm-i-Syn-i-sz-y^ w h e r e 5 > 0 and 7 can be 1 or 0. Then O p C z V 1 * ) = (yn~1~Sz)ap(xM) = 0 on one hand. O n the other hand, av{xMyn-xz) = {xM-m+l)ap{xm-lyn-lz) = x M " V - 1"^ 7-If 5 = 0, then xM-8yn-x~6z1 = 0 since it is in the ideal. If 5 > 0, then xM-&yn-i-sz~i Q s j n c e jt i s n o t j n the ideal, so we have a contradiction. Therefore 5 = 0, and to get a nonzero map, 7 = 0. If the next generator contain no z, then the same argument from Lemma 4.6.2 shows that this generator is mapped to zero by ap. We can thus argue inductively on the generators. , The only difference is when we. are at the induc t ive , steps among the following generators: x m r y n r - 1 , x m r - 1 y n r - 1 z , x m r - l y n r . Assuming xmrynr~l is mapped to 0, the same argument wi l l show that x m r - \ y n T - \ z j g m a p p e d to xmr~lynr~l if the map is a non-zero map. In the next step, we need to show that xmT~lymr is mapped to zero whether x m r - i y n r - i z i s m a p p e d to zero or not. The same argument w i l l show that xmr-iynr c a n D e m a p p e c i to xmT~2ynr~xz, but by examining the image of x m r ~ 1 y n r z , we wi l l have a contradiction in this case. • 47 L e m m a 4.8.4. H1(KX-\Y) has no weight zero part. P R O O F : We will first argue that HL{KX\Y) = HL{KX\YQ), where Y0 is the associated one dimensional subscheme of Y. The lemma will follow because H 1{KX-\YQ) has no weight zero part by Lemma 4.6.4. We know that Y0 is a closed subscheme of Y and they agree on an open dense subset of Y. They only differ at finitely many closed points. Therefore, if we use Cech cohomology to compute the first cohomology, we can pick an open cover such that all double intersections miss these points. Then, Kj^\Y and K^YQ will have the same one cycles and hence the same cohomology. • 4.9 Proof of Theorem 4.1.3 We have shown that the product-type components are smooth, so their vir-tual class is given by the Euler class of the fixed part of the obstruction bundle. The next lemma will show that the K-theory class of the fixed part of the obstruction bundle contains a degree zero line bundle with trivial T° action. Therefore, its Euler class is zero and its contribution to the residue invariants is zero. Let M to denote the product-type component, Cl. Recall the universal diagram: M xX ±*M. with X and y as the universal ideal sheaf and the structure sheaf of the universal subscheme respectively. Denote the projection to the rth copy of C as 7 r r . L e m m a 4.9.1. The weight zero part of the equivariant K-theory class of R'p*(Hom(l,l) -0Mxx) consists of ^TM = — (Xlt=i 7 R * ( ^ C ' ) ) and I line bundles of degree zero. P R O O F : Consider the following commutative diagram: 48 M xX M x C-—— M = Cl r c Recall that x and y are used to denote respectively TTQLV1 and TTQL^1 on M xC and by abuse of notation pullbacks by 1 to M x X as well. zr denotes OMXC{~Ar) and its pullback to M x I . A r is the big diagonal in M x C between the rth copy of C in M and the second factor of M x C. In this notation, The K-theory class of R*7r*(C?;y) is given by: P = R \ ( O y ) = ( 1 y ) = F + E ^ V ^ d -With F = . j e A xl~ly^~l and (m r ,n r ) 6 4 - A is the partition associated to the canonical maximal 1-dimensional component of y. Q denotes the equivariant K-theory class of I. Compare this expression with the one from the pure dimensional case, there are extra terms of the form i J 2 x m r - l y n r - l { l _ Z r ) r = l They appear because over the big diagonal that parametrizes the embedded points there are more regular functions, namely xmr~lynr~l. They glue to-gether to form a line on the big diagonals, L\~MR LL2~NT ® 0&r. The K-theory class of the structure sheaf of these big diagonals can be represented by l — zr. Recall that the K-theory class of R'ir*{Hom(l,l) -OMXX) is given by QQ ~ 1 r>/"r> „ - l T > i \ , - „ - l „ . - l ~ (l-x)(l-y) = P(P - x~lP - 1) + x^y^PiP - x P - 1). (4.2) 49 It consists of terms of the forms: xly3zr, xlyJzr 1 , xly3'zsj'zr, and xly°. Recall that R'p*(Hom(l, 1) - 0MxX) = RmPl.(Rmir,(Hom(I,T) ~ C W ) ) -The following lemma wi l l describe what these terms contribute after applying R * P i * L e m m a 4.9.2. We have the following: (a) R " ( p i ) » ( L i ^ O M x c ( - A r ) ) ) - (1 - g + ikx + jk2)C{i-j)s -K(L\Li). (b) R'(pML\LiOMxC(Ar)) = (l-g + ih+ jk2)C(i_j)s + K*r(L\LiTc). (c) R * ( P l ) * ( L i L J 2 0 M x C ( A r - A s ) ) = (l-g + ih1 + jk2)C{i_j)a + TT;(L\4TC) - o ( A „ ) 7 r ; ( L r V)-(ri) R'(Pl)*(L\Li) = (l-g + ik1+ jk2)C(i_j)s. Here C^-j)s represents the equivariant bundle on M with pure weight (i—j)s. PROOF: (d) is implied by Grothendieck-Riemann-Roch. To see (a), con-sider the following exact sequence: 0 - H - 0 M x C { - A r ) ® L\4 - oMxC ® 0Ar ® L\L{ -> 0. We have R'(pML\LiOMxC(-Ar)) = R-(pML\L{) - R ' ^ O ^ L i L ^ A , . ) . We can apply (d) to the first term to get the pure weight terms. To see the second term, consider the following commutative diagram: M = C x . . . x C x . . . C M = Cx ... 50 Then, R'(PlUL\Li\AR) = A*(L\LJ2) = K(L\Li). To prove (c), use the following exact sequence: 0 - L\LJ2 - 0 M X C ( A R ) ® L\L{ -> 0AR(AR) ® L i L 2 ' -> 0, which implies that R-(Pl)*(L\LiOMXC(AR)) = R'(Pl)>(L\L{) + R"(p i ) . (0 (A r ) <g> L i L J 2 | A r ) . Use (d) again for the pure weight terms. For the second term, use the commutative diagram above and the fact that A*(OMXC{At) = n*(Tc). To prove (c), we use the following exact sequence: 0 - 0 M X C { A R - AA) <g> L\L{ - 0 M X C ( A R ) ® L\L{ C A 3 ( A r ) ® L \ L ^ 0. Then the result follows from part (b) and the fact that A * ( C A s ( A r ) ) = C>M(ARS). • Using this lemma, the pure weight contribution can be obtained by setting zr equal to 1 in Equation 4.2. Then the weights we get are ±hijS as in the Expression 4.1 which are never zero. This takes care of al l the contribution of the form C(j__j)s. Next we need to find al l the non-pure-weight and weight zero K-theory classes. The above lemma shows that the weight zero K-theory classes come from terms where x and y have the same exponents, we look only for these terms then. Recall that the expression on the right hand side of Equation 4.2 consist of four types of monomials: x1yJzr, xly]z~1, xly3zs/zr, xlyJ. Those terms, xlyJ, only contribute pure weight classes, so we wi l l examine the other three types of terms one by one. The terms contains zr/zs are of the form: zr x y (i „ - i _ i . _ i _ i > , zs xmayT ( l - x - ' - y - ' + x^y-1) Without loss of generality, we can let mr > ms and nr < ns. It is easy to see that the above expression contains no terms wi th the same x and y exponents. Therefore, zr/zs terms contribute 0. 51 Next we analyze terms with zr . They are -z~l((F - x~lF - 1 + x'ly~lF - y-1F)x-mry-rlr i +(E x m , J / n ; > _ m r y _ n r ( i - x'1 - y'1 + x~ly-1)). First, consider F — xF — yF + xyF — 1: F - xF - yF + xyF - 1 = E ^-y-1 - E xiyj~l - E + E xV - 1 ( J J ) e A ( i j ' ) G A (i.j ')GA (t j ' )GA A i N A i AT = E ^ 1 - E * V - E ? / + E xVjyj+l - 1 j=i j=i j=i j=i =i - E z m ~ y _ 1 + E ^ - y - 1 - 1 = - E z m - y _ i + E z m - y - 1 -(m,n)€Ax (m,n)eBx Note that (mr, nr) G A^. The only way for a term in x-mry-(m,n)&Ax to have the same x and y exponents is when m = mr and n = nr and that term is — x~1y~1. And i " m r y - n r ( E ^M_1?/N"1) (m,n)eBx doesn't have any terms with the same x and y exponents. Therefore, (F - xF - yF + xyF - l)x~mry-nr contribute — r e - 1 ? / - 1 . The term (1 - x~l - y 1 + £ ~ y ^(XlLi x~msy~ns)xmTynr contributes 1 and Together, the contribution from z~l terms is —z~l. 52 Similarly, the zT term contribution wi l l be —zr(xy)~1 by the symmetry between terms containing zr and z~l: zrxxyi <-> z~lx~l~ly~^~l. To summa-rize, the contribution from terms wi th zr and z~l are — zr(xy)~1 — z~l. Their pushforwards under R*pi* wi l l contribute K-theory class L1L2 and —ir*Tc on the moduli space M by Lemma 4.9.2 and LiL2 has degree zero, and weight zero. ' • 53 B i b l i o g r a p h y [1] K . Behrend. Gromov-Witten invariants in algebraic geometry. Invent. Math., 127(3):601-617, 1997. [2] K . Behrend and B. Fantechi. The intrinsic normal cone. Invent. Math., 128(l):45-88, 1997. [3] Kai Behrend. Donaldson-Thomas invariants via microlocal geometry. [4] Kai Behrend and Jim Bryan. Super-rigid Donaldson-Thomas invariants. [5] Kai Behrend and Barbara Fantechi. Symmetric obstruction theories and Hilbert schemes of points on threefolds. [6] Jim Bryan and Rahul Pandharipande. The local Gromov-Witten theory of curves. [7] S. K . Donaldson and R. P. Thomas. Gauge theory in higher dimensions. In The geometric universe (Oxford, 1996), pages 31-47. Oxford Univ. Press, Oxford, 1998. [8] Dan Edidin and Zhenbo Qin. The Gromov-Witten and Donaldson-Thomas correspondence for trivial elliptic fibrations. [9] Amin Gholampour. On the equivariant Gromov-Witten Theory of P2-bundles over curves. [10] Amin Gholampour and Yinan Song. Evidence for the Gromov-Witten /Donaldson-Thomas Correspondence. [11] R. Gopakumar and C. Vafa. M-theory and topological strings-II. [12] T. Graber and R. Pandharipande. Localization of virtual classes. Invent. Math., 135(2) :487-518, 1999. 54 Joe Harris. Algebraic geometry, volume 133 of Graduate Texts in Math-ematics. Springer-Verlag, New York, 1995. A first course, Corrected reprint of the 1992 original. Joe Harris and Ian Morrison. Moduli of curves, volume 187 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1998. Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow. Mirror symmetry, volume 1 of Clay Mathematics Monographs. American Math-ematical Society, Providence, RI, 2003. With a preface by Vafa. Shinobu Hosono, Masa-Hiko Saito, and Atsushi Takahashi. Relative Lefschetz action and BPS state counting. Internat. Math. Res. Notices, (15):783-816, 2001. Daniel Huybrechts and Manfred Lehn. The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braun-schweig, 1997. Sheldon Katz. Genus zero Gopakumar-Vafa invariants of contractible curves. Sheldon Katz, Wei-Ping L i , and Zhenbo Qin. On certain moduli spaces of ideal sheaves and Donaldson-Thomas invariants. Andrew Kresch. Cycle groups for Artin stacks. Invent. Math., 138(3):495-536, 1999. M . Levine and R. Pandharipande. Algebraic cobordism revisited. Jun L i . Zero dimensional Donaldson-Thomas invariants of threefolds. Jun L i and Gang Tian. Virtual moduli cycles and Gromov-Witten in-variants of algebraic varieties. J. Amer. Math. Soc, 11(1):119—174, 1998. D. Maulik, N . Nekrasov, A . Okounkov, , and R. Pandharipande. Gromov-Witten theory and Donaldson-Thomas theory, I. ITEP-TH-61/03, IHES/M/03/67. D. Maulik, N . Nekrasov, A . Okounkov, , and R. Pandharipande. Gromov-Witten theory and Donaldson-Thomas theory, II. 55 [26] Maulik N and R. Pandharinpande. Foundations of Donaldson-Thomas Theory, in preparation. [27] H. Nakajima. Lectures on Hilbert Schemes of Points on Surfaces. Amer-ican Mathematical Society, Providence, Rhode Island, 1998. [28] A . Okounkov and R. Pandharipande. The local Donaldson-Thomas the-ory of curves. [29] R. P. Thomas. A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations. J. Differential Geom., 54(2):367-438, 2000. [30] C. Vafa. Two Dimensional Yang-Mills, Black Holes and Topological Strings. 


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