We construct relative moduli spaces of semistable pairs on a family of projective Deligne-Mumford stacks. We define moduli stacks of stable orbifold Pandharipande-Thomas pairs on stacks of expanded degenerations and pairs, and then show they are separated and proper Deligne-Mumford stacks of finite type. As an application, we present the degeneration formula for the absolute and relative orbifold Pandharipande-Thomas invariants.
Maulik and Ranganathan have recently introduced moduli spaces of logarithmic stable pairs. We examine the theory in the case of toric surfaces, and recast the theory in this case using three ingredients: Gelfand, Kapranov and Zelevinsky secondary polytopes, Hilbert schemes of points, and tautological vector bundles. In particular logarithmic stable pairs spaces are expressed as the zero set of an explicit section of a vector bundle on a logarithmically smooth space, thus providing an explicit description of their virtual fundamental class. A key feature of our construction is that moduli space…
We generalize the notion of expanded degenerations and pairs for a simple degeneration or smooth pair to the case of smooth Deligne-Mumford stacks. We then define stable quotients on the classifying stacks of expanded degenerations and pairs and prove the properness of their moduli's. On 3-dimensional smooth projective DM stacks this leads to a definition of relative Donaldson-Thomas invariants and the associated degeneration formula.
Using a homotopy approach, we prove in this paper a conjecture of Maulik, Nekrasov, Okounkov and Pandharipande on the dimension zero Donaldson-Thomas invariants of all smooth complex threefolds.
In this paper, we prove Thomae's formula for a triple covering of $\bold P^1$ with arbitrary index. This formula gives a relation between theta constants, determinants of period integrals and the difference products of branch points. To specify a symplectic basis of the curve, we use the combinatorics of binary trees on $\bold P^1$. This symplectic basis behaves so well for degenerations that we obtain the absolute constant in this formula and reduce it to a special case treated in [Bershadsky-Radul], [Nakayashiki].
We relate Pandharipande-Thomas stable pair invariants on Calabi-Yau 3-folds containing the projective plane with those on the derived equivalent orbifolds via wall-crossing method. The difference is described by generalized Donaldson-Thomas invariants counting semistable sheaves on the local projective plane, whose generating series form theta type series for indefinite lattices. Our result also derives non-trivial constraints among stable pair invariants on such Calabi-Yau 3-folds caused by Seidel-Thomas twist.
Let $Y$ be a smooth projective threefold and let $f:Y\to X$ be a birational map with $Rf_*\mathcal{O}_Y=\mathcal{O}_X$. When $Y$ is Calabi-Yau, Bryan-Steinberg defined enumerative invariants associated to such maps called $f$-relative stable (or Bryan-Steinberg) invariants. When $X$ has Gorenstein singularities and $f$ has relative dimension one, they compared these invariants to the Donaldson-Thomas, or equivalently the Pandharipande-Thomas invariants of $Y$. We define Bryan-Steinberg invariants for maps $f$ as above without assuming that $Y$ is Calabi-Yau. For $X$ with Gorenstein and rationa…
A degeneration formula of Donaldson-Thomas theory on local Calabi-Yau 4-folds
Institute of Mathematics & State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematics and Statistics, University of Melbourne（墨尔本大学数学与统计学院）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
Relative BPS state counts for log Calabi-Yau surface pairs were introduced by Gross-Pandharipande-Siebert. We describe how in the case of del Pezzo surfaces they are linearly related to local BPS state counts by means of generalized Donaldson-Thomas invariants of loop quivers.
The Donaldson-Thomas invariant is a curve counting invariant on Calabi-Yau 3-folds via ideal sheaves. Another counting invariant via stable pairs is introduced by Pandharipande and Thomas, which counts pairs of curves and divisors on them. These two theories are conjecturally equivalent via generating functions, called DT/PT correspondence. In this paper, we show the Euler characteristic version of DT/PT correspondence, using the notion of weak stability conditions and the wall-crossing formula.
The generalized Donaldson-Thomas invariants counting one dimensional semistable sheaves on Calabi-Yau 3-folds are conjectured to satisfy a certain multiple cover formula. This conjecture is equivalent to Pandharipande-Thomas's strong rationality conjecture on the generating series of stable pair invariants, and its local version is enough to prove. In this paper, using Jacobian localizations and parabolic stable pair invariants introduced in the previous paper, we reduce the conjectural multiple cover formula for local curves with at worst nodal singularities to the case of local trees of smoo…
We introduce marked relative Pandharipande-Thomas (PT) invariants for a pair $(X,D)$ of a smooth projective threefold and a smooth divisor. These invariants are defined by integration over the moduli space of $r$-marked stable pairs on $(X,D)$, and appear naturally when degenerating diagonal insertions via the Li-Wu degeneration formula. We propose a Gromov-Witten (GW) / PT correspondence for marked relative invariants. We show compatibility of the conjecture with the degeneration formula and a splitting formula for relative diagonals. The results provide new tools to prove GW/PT correspondenc…
Using the degeneration formula for Donaldson-Thomas invariants, we proved a formula for the change of Donaldson-Thomas invariants of local surfaces under blowing up along points.
We consider the moduli space of the McKay quiver representations associated to the binary polyhedral groups G < SU(2) < SU(3). The derived category of such representations is equivalent to the derived category of coherent sheaves on the corresponding ADE resolution Y=G-Hilb(C^3). Following the ideas of Nagao and Nakajima, by making particular choices of parameters in the space of stability conditions on the equivalent derived categories above, we recover Donaldson-Thomas (DT), Pandharipande-Thomas (PT) and Szendroi (NCDT) moduli spaces. We also compute the Gromov-Witten (GW) partition function…
We introduce the notion of a polynomial stability condition, generalizing Bridgeland stability conditions on triangulated categories. We construct and study a family of polynomial stability conditions for any normal projective variety. This family includes both Simpson-stability, and large volume limits of Bridgeland stability conditions. We show that the PT/DT-correspondence relating stable pairs to Donaldson-Thomas invariants (conjectured by Pandharipande and Thomas) can be understood as a wall-crossing in our family of polynomial stability conditions. Similarly, we show that the relation be…
We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative Donaldson-Thomas theory.…
The ADHM construction establishes a one-to-one correspondence between framed torsion free sheaves on the projective plane and stable framed representations of a quiver with relations in the category of complex vector spaces. This paper studies the geometry of moduli spaces of representations of the same quiver with relations in the abelian category of coherent sheaves on a smooth complex projective curve $X$. In particular it is proven that this moduli space is virtually smooth and related byrelative Beilinson spectral sequence to the curve counting construction via stable pairs of Pandharipan…
We construct a moduli space of stable pairs over a smooth projective variety, parametrizing morphisms from a fixed coherent sheaf to a varying sheaf of fixed topological type, subject to a stability condition. This generalizes the notion used by Pandharipande and Thomas, following Le Potier, where the fixed sheaf is the structure sheaf of the variety. We then describe the relevant deformation and obstruction theories. We also show the existence of the virtual fundamental class in special cases.
We show the existence of semiorthogonal decompositions (SOD) of Pandharipande-Thomas (PT) stable pair moduli spaces on Calabi-Yau 3-folds with irreducible curve classes, assuming relevant moduli spaces are non-singular. The above result is motivated by categorifications of wall-crossing formula of PT invariants in the derived category, and also a d-critical analogue of Bondal-Orlov, Kawamata's D/K equivalence conjecture. We also give SOD of stable pair moduli spaces on K3 surfaces, which categorifies Kawai-Yoshioka's formula proving Katz-Klemm-Vafa formula for PT invariants on K3 surfaces with…
Using the degeneration formula and absolute/relative correspondence, one studied the change of Gromov-Witten invariants under blow-up for six dimensional symplectic manifolds and obtained closed blow-up formulae for high genus Gromov-Witten invariants. Our formulae also imply some relations among generalized BPS numbers introduced by Pandharipande.
In this paper, we propose a definition of the moduli stack of stable relative ideal sheaves, and prove that it is a separated and proper Deligne-Mumford stack. It is the first part of the project of relative Donaldson-Thomas theory of ideal sheaves on projective threefolds, which in the end provides a degeneration formula of Donaldson-Thomas invariants of threefolds.
We introduce a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold $X$. More precisely, we develop a moduli theory for frozen triples given by the data $O^r(-n)\rightarrow F$ where $F$ is a sheaf of pure dimension 1. The moduli space of such objects does not naturally determine an enumerative theory: that is, it does not naturally possess a perfect symmetric obstruction theory. Instead, we build a zero-dimensional virtual fundamental class by hand, by truncating a deformation-obstruction theory coming from the moduli of objects in the derived categor…
We use degeneration formula to study the change of stable pair invariants of 3-folds under blow-ups and obtain some closed blow-up formulae. Related results on Donaldson-Thomas invariants are also discussed. Our results give positive evidence for GW/DT/P correspondence, and also give partial correspondence for varieties not necessarily toric or complete intersections.
We study Pandharipande-Thomas's stable pair theory on $K3$ fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai-Yoshioka's formula for the Euler characteristics of moduli spaces of stable pairs on $K3$ surfaces and Noether-Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the $K3$ fibration. In the case that the $K3$ fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniq…
We identify Le Potier's moduli spaces of limit stable pairs $(F,s)$, where $F$ is a 2-dimensional sheaf on a nonsingular projective 4-fold $X$ and $s \in H^0(F)$, with the moduli spaces of polynomial stable 2-term complexes in derived category. These stable pairs are 2-dimensional analogs of Pandharipande-Thomas' stable pairs defined for 3-folds. We establish categorical correspondences involving these stable pairs, ideal sheaves of 2-dimensional subschemes of $X$, and 1-dimensional sheaves on $X$. Under some conditions on the Chern character, these lead to Hall algebra correspondences. The ge…
This article is an attempt to briefly introduce some of the results from arXiv:1011.6342 on development of a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold X. More precisely, we develop a moduli theory for highly frozen triples given by the data O^r-->F for r>1 where F is a sheaf of pure dimension 1. The moduli space of such objects does not naturally determine an enumerative theory. Instead, we build a zero-dimensional virtual fundamental class by truncating a deformation-obstruction theory coming from the moduli of objects in the derived categ…
For a quasi-projective scheme M which carries a perfect obstruction theory, we construct the virtual cobordism class of M. If M is projective, we prove that the corresponding Chern numbers of the virtual cobordism class are given by integrals of the Chern classes of the virtual tangent bundle. Further, we study cobordism invariants of the moduli space of stable pairs introduced by Pandharipande-Thomas. Rationality of the partition function is conjectured together with a functional equation, which can be regarded as a generalization of the rationality and 1/q <-> q symmetry of the Calabi-Yau ca…
A refinement of the stable pair invariants of Pandharipande and Thomas for non-compact Calabi-Yau spaces is introduced based on a virtual Bialynicki-Birula decomposition with respect to a C* action on the stable pair moduli space, or alternatively the equivariant index of Nekrasov and Okounkov. This effectively calculates the refined index for M-theory reduced on these Calabi-Yau geometries. Based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local P^1. We explicitly compute …
We show how a quasi-smooth derived enhancement of a Deligne-Mumford stack X naturally endows X with a functorial perfect obstruction theory in the sense of Behrend-Fantechi. This result is then applied to moduli of maps and perfect complexes on a smooth complex projective variety. For moduli of maps, we consider X=S an algebraic K3-surface, $g\geq 0$, and $β$ a curve class, and we construct a derived stack whose truncation is the usual stack of pointed stable maps from curves of genus g to S hitting the class $β$, and such that the inclusion of the trunaction induces on a perfect obstruction t…
