Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten Theories by Artan Sheshmani*

Home , Dominic Joyce, Eric Zaslow, Nikita Nekrasov, Sheldon Katz

Abstract. This article provides the summary of 1. Introduction [GSY17a] and [GSY17b] where the authors studied the In recent years, there has been extensive math- enumerative geometry of “nested Hilbert schemes” ematical progress in enumerative geometry of sur- of points and curves on algebraic surfaces and faces deeply related to physical structures, e.g. their connections to threefold theories, and in around Gopakumar-Vafa invariants (GV); Gromov- particular relevant Donaldson-Thomas, Vafa-Witten Witten (GW), Donaldson-Thomas (DT), as well as and Seiberg-Witten theories. Pandharipande-Thomas (PT) invariants of surfaces; and their “motivic lifts”. There is also a tremen- dous energy in the study of mirror symmetry of sur- Contents faces from the mathematics side, especially Homo- logical Mirror Symmetry. On the other hand, phys- 1 Introduction ...... 25 ical dualities in Gauge and String theory, such as Acknowledgment ...... 26 Montonen-Olive duality and heterotic/Type II du- 2 Nested Hilbert Schemes on Surfaces ...... 26 ality have also been a rich source of spectacular 2.1 Special Cases ...... 27 predictions about enumerative geometry of moduli 2.2 Nested Hilbert Scheme of Points . . . . . 27 spaces on surfaces. For instance an extensive re- 3 Nested Hilbert Schemes and DT Theory of search activity carried out during the past years was Local Surfaces ...... 28 to prove the modularity properties of GW or DT in- References ...... 30 variants as suggested by the heterotic/Type II duality. The first prediction of this type, the Yau-Zaslow conjecture [YZ96], was proven by Klemm-Maulik- Pandharipande-Scheidegger [KMPS10]. Further recent developments in this particularly fruitful direction * Center for Mathematical Sciences and Applications, Har- include: Pandharipande-Thomas proof [PT16] of the vard University, Department of Mathematics, 20 Garden Street, Room 207, Cambridge, MA, 02139 Katz-Klemm-Vafa conjecture [KKV99] for K3 sur- Centre for Quantum Geometry of Moduli Spaces, Aarhus faces, Maulik-Pandharipande proof of modularity of University, Department of Mathematics, Ny Munkegade 118, GW invariants for K3 fibered threefolds [MP13], building 1530, 319, 8000 Aarhus C, Denmark Gholampour-Sheshmani-Toda proof of modularity of National Research University Higher School of Economics, PT and Gholampour-Sheshmani proof of modular- Russian Federation, Laboratory of Mirror Symmetry, NRU HSE, 6 Usacheva str., Moscow, Russia, 119048 ity of DT invariants of stable sheaves on K3 fibra- E-mail: [email protected] tions [GST17, GS18] (moreover, the generalizations

DECEMBER 2019 NOTICES OF THE ICCM 25 of the latter in [BCDDQS16]), and finally Gholampur- Their main exciting realization was when L = KS, the Sheshmani-Thomas [GS14] proof of modular prop- canonical bundle of S. In this case the above DT in- erty of the counting of curves on surfaces deforming variants of X recover some well known invariants freely (in a fixed linear system) in ambient CY three- in physics, the Vafa-Witten (VW) invariants [VW94], folds. which are known to have modularity property, fol- SU(2)-Seiberg-Witten (SW) / DT correspondence. lowing the work of Vafa-Witten [VW94] (and verified The introduction of other versions of gauge theo- by Tanaka-Thomas [TT17), TT18(2)]). On the other ries in dimensions 6 started numerous exciting de- hand the authors also showed that in some cases the velopments in physical mathematics and mathemat- nested Hilbert scheme invariants also have modular ical physics involving enumerative geometry, mirror property [GSY17a, Theorem 7]. Therefore, we were symmetry, and related physics of 4 dimensional man- able to show that in some cases, the SW invariants ifolds, realized as a complex two dimensional sub- can be described in terms of modular forms. variety which could exhibit deformations inside ambient higher dimensional target varieties, such as CY threefolds. The SW/DT correspondence is one of Acknowledgment such platforms to find interesting structural sym- The author was partially supported by NSF metries between theory of surfaces and theory of DMS-1607871, NSF DMS-1306313 and Laboratory of threefolds: having fixed a spin structure on a com- Mirror Symmetry NRU HSE, RF Government grant, ag. plex surface S the SW invariant of S roughly counts No 14.641.31.0001. The author would like to further with sign the number of points in the parametriz- sincerely thank the Center for Mathematical Sciences ing space (the moduli space) of solutions to SW and Applications at Harvard University, the center equations defined on S [D96, Section 1]. The fo- for Quantum Geometry of Moduli Spaces at Aarhus cus of SW theory in physics is the study of mod- University, and the Laboratory of Mirror Symmetry in uli space of vacua in N = 2, D = 4 super Yang-Mills Higher School of Economics, Russian federation, for theory, and in particular certain dualities of the the great help and support. theory such as electric-magnetic duality (Montonen- Olive duality). In [GLSY17] Gukov-Liu-Sheshmani-Yau conjectured a relation between SW invariants (asso- 2. Nested Hilbert Schemes on ciate to SU(2) gauge invariant theory) of a projec- Surfaces tive surface S and DT invariants of a noncompact Hilbert schemes of points and curves on a non- threefold X obtained by total space of a line bun- singular surface S have been vastly studied. Their dle L on S. In a later work [GSY17a, GSY17b], which rich geometric structures have proved to have many will be elaborated below, Gholampour-Sheshmani- applications in mathematics and physics (see [N99] Yau proved the conjecture in [GLSY17, Equation for a survey). The current article is a survey of the 56]. articles [GSY17a] and [GSY17b], where the authors The key object which bridges the geometry of studied the enumerative geometry of “nested Hilbert complex surface to the ambient noncompact com- schemes” of points and curves on algebraic surfaces. plex threefold is the “nested Hilbert scheme of sur- Given the sequence face”. This space parametrizes a nested chain of configurations of curves and points in the surface. n := n1,n2,...,nr,r ≥ 1 Let us assume that S is a projective simply con- nected complex surface and let L be a line bundle of nonnegative integers, and β := β1,...,βr−1, a se- on S. In [GSY17a], Gholampour-Sheshmani-Yau an- 2 quence of classes in H (S,Z) such that βi ≥ 0, we de- alyzed the moduli space of stable compactly sup- note the corresponding “nested” Hilbert scheme by ported sheaves of modules on the noncompact three- [n] [n] Sβ . A closed point of Sβ corresponds to fold X. In physics terms these are D4-D2-D0 branes wrapping the zero section of X (i.e. sheaves are sup- (Z1,Z2,...,Zr), (C1,...,Cr−1) ported on S or a fat neighborhood of S in X). The authors showed in [GSY17a] that the DT invariants of where Zi ⊂ S is a 0-dimensional subscheme of length X satisfy an equation in terms of SW invariants of S ni, and Ci ⊂ S is a divisor with [Ci] = βi, and Zi+1 is a and certain correction terms governed by invariants subscheme of Zi ∪Ci for any i < r, or equivalently of nested Hilbert schemes: (1) IZi (−Ci) ⊂ IZi+1 . DT invariants of X = (SW invariants of S) · (Combinatorial coefficients) + Invariants of nested In [GSY17a] and [GSY17b], in order to define in- Hilbert scheme of S variants for the nested Hilbert schemes (see [GSY17a,

26 NOTICES OF THE ICCM VOLUME 7,NUMBER 2 [n] Definitions 2.13, 2.14]), the authors constructed a vir- where ωS is the rank n tautological vector bundle [n] vir [n] 1 tual fundamental class [Sβ ] and then integrate ap- over S associated to the canonical bundle ωS of S. propriate cohomology classes against it. More pre- 3. If β = 0 and n = n2 = n1 − 1, then it is known that [n+1,n] ∼ [n] [n] cisely, they constructed a natural perfect obstruction Sβ=0 = P(I ) is nonsingular, where I is the uni- [n] [n] theory over Sβ . This is done by studying the defor- versal ideal sheaf over S × S [L99, Section 1.2]. mation/obstruction theory of the maps of coherent Then, sheaves given by the natural inclusions (1) following [S[n+1,n]]vir = −[S[n+1,n]] ∩ c (O (1) ). Illusie. It turned out that this construction in partic- β=0 β=0 1 P ωS ular provides a uniform way of studying all known 4. If n = n = 0 and β 6= 0, then S[0,0] is the Hilbert obstruction theories of the Hilbert schemes of points 1 2 β [0,0] vir and curves, as well as the stable pair moduli spaces scheme of divisors in class β, and [Sβ ] coincides on S. The first main result of [GSY17a] is as fol- with virtual cycle used to define Poincaré invariants lows (see also [GSY17a, Proposition 2.5 and Corollary in [DKO07]. [0,n2] 2.6]): 5. If n1 = 0 and β 6= 0, then Sβ is the relative Hilbert scheme of points on the universal divisor over S[0,0], Theorem 1 ([GSY17a, Theorem 1]). Let S be a nonsin- β which as shown in [PT10], is a moduli space of sta- gular projective surface over C and ωS be its canonical [0,n ] [n] ble pairs and [S 2 ]vir is the same as the virtual fun- bundle. The nested Hilbert scheme S with r ≥ 2 car- β β damental class constructed in [KT14] in the context ries a natural virtual fundamental class of stable pair theory. If Pg(S) = 0 this class was used r−1 in [KT14] to define stable pair invariants. [n] vir [n] 1 [Sβ ] ∈ Ad(Sβ ), d = n1 + nr + ∑ βi · (βi − c1(ωS)). 2 i=1 In certain cases, the authors constructed a re- [n1,n2] duced virtual fundamental class for Sβ by reducing the perfect obstruction theory leading to Theorem 1 2.1 Special Cases ([GSY17a, Propositions 2.10, 2.12]): In the simplest special case, i.e. when r = 1, we Theorem 3 ([GSY17a, Theorem 3]). Let S be a nonsin- [n] [n1] have Sβ = S is the Hilbert scheme of n1 points on gular projective surface with pg(S) > 0, and the class β S which is nonsingular of dimension 2n1, and hence it be such that the natural map [n1] [n1] has a well-defined fundamental class [S ] ∈ A2n1 (S ). [n] 1 ∗∪β 2 [n] H (TS) −−→ H (OS) is surjective, For r > 1 and βi = 0, S := S(0,...,0) is the nested Hilbert scheme of points on S parameterizing the flag of then, [S[n1,n2]]vir = 0. In this case the nested Hilbert 0-dimensional subschemes β [n1,n2] scheme Sβ carries a reduced virtual fundamental Zr ⊂ ··· ⊂ Z2 ⊂ Z1 ⊂ S, class

[n1,n2] vir [n1,n2] 1 [S ] ∈ Ad(S ), d = n1 +n2 + β ·(β −KS)+ pg. which is in general singular of actual dimension 2n1. β red β 2 The authors were specifically interested in the [0,0] vir [n] [n1,n2] 2 The reduced virtual fundamental classes [S ] and case r = 2, that is: Sβ = Sβ for some β ∈ H (S,Z). β red [0,n2] vir Interestingly, in the following cases the invariants of [Sβ ]red match with the reduced virtual cycles con- nested Hilbert schemes coincide with the Poincaré structed in [DKO07, KT14] in cases 3 and 4 of Theorem and the stable pair invariants of S that were pre- [0,n2] vir 2. [Sβ ]red was used in [KT14] to define the stable pair viously studied in the context of algebraic Seiberg- invariants of S in this case. Witten invariants and curve counting problems. The following theorem is proven in [GSY17a, Section 3]. 2.2 Nested Hilbert Scheme of Points Theorem 2 ([GSY17a, Proposition 3.1]). The virtual One interesting specialization of the nested fundamental class of Theorem 1 recovers the follow- Hilbert schemes is the case where β = 0 and n = n1 ≥ n2. ing known cases: In [GSY17a] the authors studied the nested Hilbert [n,n] ∼ [n] [n,n] vir schemes of points 1. If β = 0 and n1 = n2 = n then Sβ=0 = S and [Sβ=0] = [n ,n ] [S[n]] is the fundamental class of the Hilbert scheme [n1≥n2] 1 2 S := Sβ=0 of n points. [n,0] ∼ [n] in much more details. Let ι : S[n1≥n2] ,→ S[n1] × S[n2] be 2. If β = 0 and n2 = 0, then Sβ=0 = S and the natural inclusion. For the case where S is toric

[n,0] vir n [n] [n] 1 [Sβ=0] = (−1) [S ] ∩ cn(ωS ), We were notified about this identity by Richard Thomas.

DECEMBER 2019 NOTICES OF THE ICCM 27 with the torus T and the fixed set ST, the authors The operators provided a purely combinatorial formula for comput- Z n ,n [n1≥n2] vir 1 2 ing [S ] by torus localization along the lines of − ∪ cn1+n2 (EM ) S[n1]×S[n2] [MNOP06]: were studied by Carlsson-Okounkov in [CO12]. Here Theorem 4. For a toric nonsingular surface S the n1,n2 M ∈ Pic(S), and E is the rank n1 + n2 virtual vector T-fixed set of S[n1,n2] is isolated and given by tuple of M bundle over S[n1] × S[n2] obtained by taking the alter- nested partitions of n ,n : 2 1 nating sum of all the relative extensions of I[n1] and T I[n2] M defined as follows: (µ2,P ⊆ µ1,P)P | P ∈ S , µi,P ` ni . n1,n2 Definition 2.1. For any line bundles M on S, let EM ∈ Moreover, the T-character of the virtual tangent bun- [n ] [n ] K(S 1 × S 2 ) be the element of virtual rank n1 + n2 de- dle T of S[n1≥n2] at the fixed point Q = ( ⊆ ) is µ2,P µ1,P P fined by given by n1,n2 0 ∗ h [n ] [n ] ∗ i E := Rπ p M − RHom 0 (I 1 ,I 2 ⊗ p M) , tr vir (t1,t2) = VP, M ∗ π TQ ∑ P∈ST where p and π0 are respectively the projections from where t ,t are the torus characters and V is a Lau- 1 2 P S × S[n1] × S[n2] to the first and the product of last two rent polynomial in t ,t that is completely determined 1 2 factors. Let i be the inclusion of the closed point (I ,I ) ∈ by the µ , µ and is given by the right hand side of 1 2 2,P 1,P S[n1] × S[n2], then, we define the following formula (2) En1,n2 ∗ 0 ∗ M |(I1,I2) := Li Rπ∗ p M Z2 (1 −t1)(1 −t2) ∗ [n1] [n2] ∗ trT vir = Z1 + + Z1 · Z2 − Z1 · Z1 − Z2 · Z2 . − Li RHomπ0 (I ,I ⊗ p M) I1⊆I2 t1t2 t1t2 ∈ K(Spec(C)). It turns out that when S is toric and Fano, by torus localization, one can express [S[n1≥n2]]vir in terms of the If M = O, we sometimes drop it from the notation. We fundamental class of the product of Hilbert schemes also define the following generating series [n ] [n ] S 1 × S 2 : Z n1 n2 n1,n2 n1,n2 Zprod(S,M) := ∑ q1 q2 c(E ) ∪ c(EM ). Theorem 5. If S is a nonsingular projective toric Fano S[n1]×S[n2] n1≥n2≥0 surface, then, Note that by equation (5) in [CO12], we know [n1≥n2] vir [n1] [n2] En1,n2 n1,n2 ι∗[S ] = [S × S ] ∩ cn1+n2 ( ), ci(EM ) = 0 for i > n1 + n2, and so the integrand in the definition of Z (S,M) can be replaced by n ,n [n ] prod where E 1 2 is the rank n1 + n2 vector bundle on S 1 × S[n2] obtained by the first relative extension sheaf of the n1,n2 n1,n2 cn1+n2 (E ) ∪ cn1+n2 (EM ). universal ideal sheaves I[n1] and I[n2]. Carlsson and Okounkov were able to express 2 1 1 Theorem 5 holds in particular for S = P , P × P , these operators in terms of explicit vertex opera- which are the generators of the cobordism ring of tors. As an application of Theorem 6 and the re- nonsingular projective surfaces. The authors used sult of [CO12], the following explicit formula was this fact together with a degeneration formula devel- proven: oped for [S[n1≥n2]]vir to prove: Theorem 7. Let S be a nonsingular projective surface, Theorem 6. If S is a nonsingular projective surface, ωS be its canonical bundle, and KS = c1(ωS). Then, and α is a cohomology class in Hn1+n2 (S[n1] × S[n2]) with Z the following properties: (−1)n1+n2 ι∗c(En1,n2 )qn1 qn2 ∑ [n ≥n ] vir M 1 2 n ≥n ≥0 [S 1 2 ] • α is universally defined for any pair of a non- 1 2 n−1 nhKS,KS−Mi n n hKS−M,Mi−e(S) singular projective surface and a line bundle on = ∏ 1 − q2 q1 (1 − q1q2) , it, n>0 • α is well-behaved under good degenerations of S, n1,n2 where h−,−i is the Poincaré paring on S and EM is then as in Theorem 5. Z Z ∗ n1,n2 ι α = α ∪ cn1+n2 (E ), [S[n1≥n2]]vir S[n1]×S[n2] 3. Nested Hilbert Schemes and DT

n ,n Theory of Local Surfaces and E 1 2 is the rank n1 + n2 virtual vector bundle on S[n1] × S[n2] obtained by taking the alternating sum (in In this section we give an overview of the results the K-group) of all the relative extension sheaves of of [GSY17b]. Let (S,h) be a nonsingular simply con- [n ] [n ] I 1 and I 2 . nected projective surface with h = c1(OS(1)). Let ωS be

28 NOTICES OF THE ICCM VOLUME 7,NUMBER 2 the canonical bundle of S with the projection map q perfect obstruction theory over Mh(v). The restriction ∗ to S, and X be the total space of ω 2. X is a noncom- ωS C vir [n] S of [Mh (v) ]red to a type II component Sβ is identified pact Calabi-Yau threefold and one can define the DT [n] vir ∗ ∗ with [Sβ ] . invariants of X by using C -localization, where C -acts on X by scaling the fibers of ωS. More precisely, let When r = 2, then types I and II are the only compo- ∗ ωS C nent types of Mh (v) . This leads us to the following 2 2i v = (r,γ,m) ∈ ⊕i=0H (S,Q) result:

ωS Theorem 9 ([GSY17b, Theorem 3]). Suppose that v = be a Chern character vector, and Mh (v) be the moduli space of compactly supported 2-dimensional sta- (2,γ,m). Then, ble sheaves E on X such that ( E) = v. Here stabil- ch q∗ ωS ωS ωS DT (v;α) = DT (v;α)I + DT (v;α) [n ,n ] , h h ∑ h ,S 1 2 ity is defined by means of the slope of q∗ E with re- II β n1,n2,β spect to the polarization h. In [GSY17b] the authors ωS vir vir [n1,n2] ωS DT (v) = χ (Mh(v)) + χ (S ), provided Mh (v) with a perfect obstruction theory h ∑ β by reducing the natural perfect obstruction theory n1,n2,β ∗ given by [T98]. The fixed locus MωS (v)C of the moduli h where the sum is over all n1,n2,β (depending on v) for ∗ space is compact and the reduced obstruction theory [n1,n2] ωS C which Sβ is a type II component of Mh (v) , and gives a virtual fundamental class over it, denoted by the indices I and II indicate the contributions of type I ω ∗ S C vir ω [Mh (v) ]red. They defined two types of DT invariants: S and II components to the invariant DTh (v;α). Z 1 DTωS (v; ) = ∈ [s,s−1], The stability of sheaves imposes a strong condi- h α ω ∗ Q S C vir vir [Mh (v) ]red e(Nor ) tion on n1,n2,β appearing in the summation in the the- ∗ ∗ ωS ∈ H ∗ (M (v)C ) orem above. For example, if S is a generic complete in- α C h s ω ω ∗ tersection in a projective space, then for any n1,n2,β S vir S C ∗ DTh (v) = χ (Mh (v) ) ∈ Z, [n1,n2] ωS C for which Sβ is a type II component of Mh (v) , vir the condition in Theorem 3 (leading to the vanishing where Nor is the virtual normal bundle of [n1,n2] vir ∗ ωS C ωS vir [Sβ ] = 0) is not satisfied. Mh (v) ⊂ Mh (v), χ (−) is the virtual Euler charac- vir [n1,n2] ωS The invariants χ (S ) and DT (v;α) [n ,n ] teristic [FG10], and s is the equivariant parameter. β h ,S 1 2 II β If α = 1 then the authors were able to show that (for a suitable choice of class α e.g. α = 1) appearing in

ωS −pg the theorem above are special types of the invariants DTh (v;1) = s VWh(v),

NS(n1,n2,β;−) where VWh(−) is the Vafa-Witten invariant (which were also mathematically studied in detail by Tanaka which are defined as follows: and Thomas in [TT17), TT18(2)]) and is expected to have modular properties based on S-duality conjec- Definition 3.1 ([GSY17a, Definition 2.13]). Let M ∈ [n1,n2] ture (see [VW94]). Pic(S). Define the following elements in K(Sβ ) of vir- ∗ ∗ ωS C tual ranks respectively n +n and −β ·β D/2+β ·c (M): The C -fixed locus Mh (v) consists of sheaves 1 2 1 supported on S (the zero section of ωS) and its thick- h i ω ∗ n1,n2 [n1] [n2] S C K := Rπ∗M(Zβ ) − RHomπ (I ,I ⊗ M) , enings. One can write Mh (v) as a disjoint union β;M β of several types of components, where each type is h i G := Rπ M(Z )| . indexed by a partition of r. Out of these component β;M ∗ β Zβ types, there are two types of particular importance; If β = 0 we will instead use the notation K[n1≥n2] := Kn1,n2 One of them (we call it type I) is identified with Mh(v), M 0;M the moduli space of rank r torsion free stable sheaves (see [GSY17a, Definition 5.4]). We also define the rank on S. The other type (we call it type II) can be identi- 2ni twisted tangent bundles fied with the nested Hilbert scheme S[n] for a suitable h i β M [ni] [ni] T [n ] := [Rπ∗M] − RHomπ (I ,I ⊗ M) choice of n,β depending on v. The reason that types S i h i I and II are more interesting, is the following result 1 [ni] [ni] = Extπ I ,I ⊗ M . proven in [GSY17b]: 0 M Note that if M = O then T = T [n ] . Theorem 8 ([GSY17b, Theorem 2]). The restriction of S[ni] S i ωS ∗ vir Let P := P(M,β,n1,n2) be a polynomial in the Chern [M (v)C ] to the type I component Mh(v) is identi- h red n1,n2 vir classes of K , Gβ;M, T [n ] , and T [n ] , then, we can de- fied with [Mh(v)]0 induced by the natural trace free β;M S 1 S 2 fine the invariant 2 In [GSY17b], we consider a more general case in which X Z is the total space of an arbitrary line bundle L with H0(L ⊗ −1 NS(n1,n2,β;P) := [n ,n ] P. ω ) 6= 0. [S 1 2 ]vir S β

DECEMBER 2019 NOTICES OF THE ICCM 29 Moreover under the condition in Theorem 3, the au- [D96] Simon Donaldson. “The Seiberg-Witten equa- thors defined the reduced invariants tions and 4-manifold topology.” Bulletin of American Mathematical Society (N.S.), 33(1) Z (1996): 45–70. red NS (n1,n2,β;P) := [n ,n ] P. [DKO07] Markus Dürr, Alexandre Kabanov, and Chris- [S 1 2 ]vir β red tian Okonek. “Poincaré invariants.” Topology, 46 (2007): 225–294. Mochizuki in [M02] expresses certain integrals [ES87] Geir Ellingsrud and Stein Arild Strømme. “On against the virtual cycle of Mh(v) in terms of Seiberg- the homology of the Hilbert scheme of points in the plane.” Inventiones Mathematicae, 87 Witten invariants and integrals A(γ1,γ2,v;−) over the (1987): 343–352. product of Hilbert scheme of points on S. Using this [EGL99] Geir Ellingsrud, Lothar Göttsche, and Manfred result the authors were able to prove the following: Lehn. “On the cobordism class of the Hilbert scheme of a surface.” Journal of Algebraic Ge- Theorem 10 ([GSY17b, Theorem 4]). Suppose that ometry, 10 (2001): 81–100. pg(S) > 0, and v = (2,γ,m) is such that γ · h > 2KS · h and [FG10] Barbara Fantechi and Lothar Göttsche. R “Riemann-Roch theorems and elliptic genus χ(v) := S v ·tdS ≥ 1. Then, for virtually smooth schemes.” Geometry & Topology, 14 (2010): 83–115. ωS (v ) = − ( ) · 2−χ(v) · A( , ,v P ) DTh ;1 ∑ SW γ1 2 γ1 γ2 ; 1 [F13] William Fulton. “Intersection theory.” Vol. 2. γ +γ =γ 1 2 Springer Science & Business Media (2013). γ1·h<γ2·h [G90] Lothar Göttsche. “The Betti numbers of the + ∑ NS(n1,n2,β;P1). Hilbert scheme of points on a smooth pro- n1,n2,β jective surface.” Mathematische Annalen, 286 ωS 2−χ(v) (1990): 193–207. DT (v) = − SW(γ1) · 2 · A(γ1,γ2,v;P2) h ∑ [GP99] Tom Graber and Rahul Pandharipande. “Local- γ1+γ2=γ γ1·h<γ2·h ization of virtual classes.” Inventiones Mathe- maticae, 135 (1999): 487–518. + N (n ,n ,β;P ). ∑ S 1 2 2 [GS16] Amin Gholampour and Artan Sheshmani. “In- n ,n , 1 2 β tersection numbers on the relative Hilbert schemes of points on surfaces.” Asian Here SW(−) is the Seiberg-Witten invariant of S, Pi and Journal of Mathematics, 21(3) (2016): 531– Pi are certain universally defined explicit integrands, 542. and the second sum in the formulas is over all n1,n2,β [GS18] Amin Gholampour and Artan Shesh- [n1,n2] mani. “Donaldson-Thomas Invariants of (depending on v) for which Sβ is a type II component ∗ 2-Dimensional sheaves inside threefolds and ωS C of Mh (v) . modular forms.” Advances in Mathematics, If S is a K3 surface or S is isomorphic to one of the 326(21) (2018): 79–107. five types of generic complete intersections [GS14] Amin Gholampour, Artan Sheshmani, and Richard P. Thomas. “Counting curves on sur- 3 4 4 5 6 faces in calabi-yau 3-folds.” Mathematische (5)⊂P , (3,3)⊂P , (4,2)⊂P , (3,2,2)⊂P , (2,2,2,2)⊂P , Annalen, 360 (2014): 67–78. [GST17] Amin Gholampour, Artan Sheshmani, and ωS ωS the DT invariants DTh (v;1) and DTh (v) can be com- Yukinobu Toda. “Stable pairs on nodal k3 fi- pletely expressed as the sum of integrals over the prod- brations.” International Mathematics Research Notices, 2017(00) (2017): 1–50. uct of the Hilbert schemes of points on S. [GSY17a] Amin Gholampour, Artan Sheshmani, and Shing-Tung Yau. “Nested Hilbert schemes on surfaces: virtual fundamental class.” References arXiv:1701.08899. [GSY17b] Amin Gholampour, Artan Sheshmani and [BF97] Kai Behrend and Barbara Fantechi. “The intrin- Shing-Tung Yau. “Localized Donaldson- sic normal cone.” Inventiones Mathematicae, Thomas theory of surfaces.” American Journal 128 (1997): 45–88. of Mathematics (To Appear), arXiv:1701.08902 [BCDDQS16] Vincent Bouchard, Thomas Creutzig, Duiliu- (Preprint). Emanuel Diaconescu, Charles Doran, Cal- [GLSY17] Sergei Gukov, Artan Sheshmani, Chiu-Chu lum Quigley, and Artan Sheshmani. “Vertical Melissa Liu, and Shing-Tung Yau. “Topologi- d4-d2-d0 bound states on k3 fibrations and cal approach to local theory of surfaces in modularity.” Communication in Mathematical Calabi-Yau threefolds.” Advances in Theoret- Physics, 350(3) (2016): 1069–1121. ical and Mathematical Physics, 21(7) (2017): [BFl03] Ragnar-Olaf Buchweitz and Hubert Flenner. 1679–1728. “A semiregularity map for modules and appli- [HL10] Daniel Huybrechts and Manfred Lehn. “The ge- cations to deformations.” Compositio Mathe- ometry of moduli spaces of sheaves.” Cam- matica, 137 (2003): 135–210. bridge University Press (2010). [CO12] Erik Carlsson and Andrei Okounkov. “Exts and [HT10] Daniel Huybrechts and Richard P. Thomas. vertex operators.” Duke Mathematical Journal, “Deformation-obstruction theory for com- 161 (2012): 1797–1815. plexes via Atiyah and Kodaira-Spencer [C98] Jan Cheah. “Cellular decompositions for classes.” Mathematische Annalen, 346 (2010): nested Hilbert schemes of points.” Pacific 545–569. Journal of Mathematics, 183 (1998): 39–90.

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