Search

Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds, Klemm and Pandharipande for Calabi-Yau 4-folds, Pandharipande and Zinger for Calabi-Yau 5-folds. We conjecture that our invariants are integers and give a sheaf-theoretic interpretation in terms of reduced $4$-dimensional Donaldson-Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two $K3$ surfaces and for the cotangent bundle of $\mathbb{P}^2$. Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a $K3$ surface. This yields a conjectural formula for the number of isolated genus $2$ curves of minimal degree on a very general hyperk\"ahler $4$-fold of $K3^{[2]}$-type. The formula may be viewed as a $4$-dimensional analogue of the classical Yau-Zaslow formula concerning counts of rational curves on $K3$ surfaces. In the course of our computations, we also derive a new closed formula for the Fujiki constants of the Chern classes of tangent bundles of both Hilbert schemes of points on $K3$ surfaces and generalized Kummer varieties., Comment: 53 pages

As an analogy to Gopakumar-Vafa conjecture on Calabi-Yau 3-folds, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold $X$ using Gromov-Witten theory. When $X$ is holomorphic symplectic, Gromov-Witten invariants vanish and one can consider the corresponding reduced theory. In a companion work, we propose a definition of Gopakumar-Vafa type invariants for such a reduced theory. In this paper, we give them a sheaf theoretic interpretation via moduli spaces of stable pairs., 25 pages. Published version. arXiv admin note: text overlap with arXiv:2201.10878

We show the existence of semiorthogonal decompositions of Donaldson-Thomas categories for $(-1)$-shifted cotangent derived stacks associated with $��$-stratifications on them. Our main result gives an analogue of window theorem for categorical DT theory, which has applications to d-critical analogue of D/K equivalence conjecture, i.e. existence of fully-faithful functors (equivalences) under d-critical flips (flops). As an example of applications, we show the existence of fully-faithful functors of DT categories for Pandharipande-Thomas stable pair moduli spaces under wall-crossing at super-rigid rational curves for any relative reduced curve classes., 58 pages

We prove wall-crossing formula for categorical Donaldson-Thomas invariants on the resolved conifold, which categorifies Nagao-Nakajima wall-crossing formula for numerical DT invariants on it. The categorified Hall products are used to describe the wall-crossing formula as semiorthogonal decompositions. A successive application of categorical wall-crossing formula yields semiorthogonal decompositions of categorical Pandharipande-Thomas stable pair invariants on the resolved conifold, which categorify the product expansion formula of the generating series of numerical PT invariants on it., Comment: 52 pages

We prove two kinds of $\mathbb{Z}/2$-periodic Koszul duality equivalences for triangulated categories of matrix factorizations associated with $(-1)$-shifted cotangents over quasi-smooth affine derived schemes. We use this result to define $\mathbb{Z}/2$-periodic version of Donaldson-Thomas categories for local surfaces, whose $\mathbb{C}^{\ast}$-equivariant version was introduced and developed in the author's previous paper. We compare $\mathbb{Z}/2$-periodic DT category with the $\mathbb{C}^{\ast}$-equivariant one, and deduce wall-crossing equivalences of $\mathbb{Z}/2$-periodic DT categories from those of $\mathbb{C}^{\ast}$-equivariant DT categories., Comment: 26 pages. arXiv admin note: text overlap with arXiv:1907.09076

We show that the categorified cohomological Hall algebra structures on surfaces constructed by Porta-Sala descend to those on Donaldson-Thomas categories on local surfaces introduced in the author's previous paper. A similar argument also shows that Pandharipande-Thomas categories on local surfaces admit actions of categorified COHA for zero dimensional sheaves on surfaces. We also construct annihilator actions of its simple operators, and show that their commutator in the K-theory satisfies the relation similar to the one of Weyl algebras. This result may be regarded as a categorification of Weyl algebra action on homologies of Hilbert schemes of points on locally planar curves due to Rennemo, which is relevant for Gopakumar-Vafa formula of generating series of PT invariants., 48 pages, to appear in Selecta Mathematica

Let $X$ be a Calabi-Yau 4-fold and $D$ a smooth divisor on it. We consider tautological complex associated with $L=\mathcal{O}_X(D)$ on the moduli space of Le Potier stable pairs and define its counting invariant by integrating the Euler class against the virtual class. We conjecture a formula for their generating series expressed using genus zero Gopakumar-Vafa invariants of $D$ and genus one Gopakumar-Vafa type invariants of $X$, which we verify in several examples. When $X$ is the local resolved conifold, our conjecture reproduces a conjectural formula of Cao-Kool-Monavari in the PT chamber. In the JS chamber, we completely determine the invariants and confirm one of our previous conjectures., Comment: 26 pages. Published version

In this paper, we prove a window theorem for categorical Donaldson-Thomas theories on local surfaces as an analogue of window theorem for GIT quotient stacks. We give two applications of our main result. The first one is a proof of wall-crossing equivalences of DT categories for one dimensional stable sheave on local surfaces, under some technical condition on strictly semistable sheaves. The second one is to show the existence of fully-faithful functors from categorical PT theories to categorical MNOP theories, when the curve class is reduced. These results indicate categorifications of wall-crossing formulas of numerical DT invariants, and also regarded as d-critical analogue of D/K conjecture in birational geometry., Comment: This paper is withdrawn, as it is now incorporated into arXiv:1907.09076v3

We develope $\mathbb{C}^{\ast}$-equivariant categorical Donaldson-Thomas theory for local surfaces, i.e. the total spaces of canonical line bundles on smooth projective surfaces. We introduce $\mathbb{C}^{\ast}$-equivariant DT categories for local surfaces as Verdier quotients of derived categories of coherent sheaves on derived moduli stacks of coherent sheaves on surfaces, by subcategories of objects whose singular supports are contained in unstable loci. Via Koszul duality, our construction may be regarded as certain gluing of $\mathbb{C}^{\ast}$-equivariant derived categories of factorizations. We also develope $\mathbb{C}^{\ast}$-equivariant DT theory for stable D0-D2-D6 bound states on local surfaces, including categorical Pandharipande-Thomas theory. The key result toward the construction is the description of the stack of D0-D2-D6 bound states on the local surface as the dual obstruction cone over the moduli stack of pairs on the surface. We propose several conjectures on wall-crossing of PT categories, motivated by categorifications of wall-crossing formula of PT invariants and d-critical analogue of D/K equivalence conjecture in birational geometry. We establish three ways toward the categorical wall-crossing conjecture: semiorthogonal decomposition via linear Koszul duality, window theorem for DT categories, and categorified Hall products. These techniques indicate several implications, e.g. rationality of generating series of PT categories, wall-crossing equivalence of DT categories for one dimensional stable sheaves, and categorical MNOP/PT correspondence for reduced curve classes., Comment: 175 pages

We study the reduced Donaldson-Thomas theory of abelian threefolds using Bridgeland stability conditions. The main result is the invariance of the reduced Donaldson-Thomas invariants under all derived autoequivalences, up to explicitly given wall-crossing terms. We also present a numerical criterion for the absence of walls in terms of a discriminant function. For principally polarized abelian threefolds of Picard rank one, the wall-crossing contributions are discussed in detail. The discussion yield evidence for a conjectural formula for curve counting invariants by Bryan, Pandharipande, Yin, and the first author. For the proof we strengthen several known results on Bridgeland stability conditions of abelian threefolds. We show that certain previously constructed stability conditions satisfy the full support property. In particular, the stability manifold is non-empty. We also prove the existence of a Gieseker chamber and determine all wall-crossing contributions. A definition of reduced generalized Donaldson-Thomas invariants for arbitrary Calabi-Yau threefolds with abelian actions is given., Comment: 59 pages, 1 figure; changes to Section 2.3

In this paper, we generalize a mathematical definition of Gopakumar-Vafa (GV) invariants on Calabi-Yau 3-folds introduced by Maulik and the author, using an analogue of BPS sheaves introduced by Davison-Meinhardt on the coarse moduli spaces of one dimensional twisted semistable sheaves with arbitrary holomorphic Euler characteristics. We show that our generalized GV invariants are independent of twisted stability conditions, and conjecture that they are also independent of holomorphic Euler characteristics, so that they define the same GV invariants. As an application, we will show the flop transformation formula of GV invariants., Comment: 53 pages, Appendix A is removed and the proof of Prop 5.3 is modified

In this paper, we propose an ansatz for defining Gopakumar-Vafa invariants of Calabi-Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal is a modification of a recent approach of Kiem-Li, which is itself based on earlier ideas of Hosono-Saito-Takahashi. We conjecture that these invariants are equivalent to other curve-counting theories such as Gromov-Witten theory and Pandharipande-Thomas theory. Our main theorem is that, for local surfaces, our invariants agree with PT invariants for irreducible one-cycles. We also give a counter-example to the Kiem-Li conjectures, where our invariants match the predicted answer. Finally, we give examples where our invariant matches the expected answer in cases where the cycle is non-reduced, non-planar, or non-primitive., Comment: 63 pages, many improvements of the exposition following referee comments, final version to appear in Inventiones