Stable Sheaves Research Articles

Stable sheaves on K3 surfaces via wall-crossing

Stable sheaves on K3 surfaces via wall-crossing

Descent of tautological sheaves from Hilbert schemes to Enriques manifolds

Let X be a K3 surface which doubly covers an Enriques surface S. If n∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document} is an odd number, then the Hilbert scheme of n-points X[n]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X^{[n]}$$\\end{document} admits a natural quotient S[n]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{[n]}$$\\end{document}. This quotient is an Enriques manifold in the sense of Oguiso and Schröer. In this paper we construct slope stable sheaves on S[n]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{[n]}$$\\end{document} and study some of their properties.

Gopakumar–Vafa Type Invariants of Holomorphic Symplectic 4-Folds

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, and 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 P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^2$$\\end{document}. 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 hyperkähler 4-fold of K3[2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K3^{[2]}$$\\end{document}-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.

Sixfolds of generalized Kummer type and K3 surfaces

We prove that any hyper-Kähler sixfold $K$ of generalized Kummer type has a naturally associated manifold $Y_K$ of $\mathrm {K}3^{[3]}$ type. It is obtained as crepant resolution of the quotient of $K$ by a group of symplectic involutions acting trivially on its second cohomology. When $K$ is projective, the variety $Y_K$ is birational to a moduli space of stable sheaves on a uniquely determined projective $\mathrm {K}3$ surface $S_K$. As an application of this construction we show that the Kuga–Satake correspondence is algebraic for the K3 surfaces $S_K$, producing infinitely many new families of $\mathrm {K}3$ surfaces of general Picard rank $16$ satisfying the Kuga–Satake Hodge conjecture.

Wall crossing for moduli of stable sheaves on an elliptic surface

Wall crossing for moduli of stable sheaves on an elliptic surface

Perverse filtrations, Chern filtrations, and refined BPS invariants for local [formula omitted

We explore connections between three structures associated with the cohomology of the moduli of 1-dimensional stable sheaves on P2: perverse filtrations, tautological classes, and refined BPS invariants for local P2. We formulate the P=C conjecture identifying the perverse filtration with the Chern filtration for the free part of the cohomology. This can be viewed as an analog of de Cataldo–Hausel–Migliorini's P=W conjecture for Hitchin systems. Our conjecture is compatible with the enumerative invariants of local P2 calculated by refined Pandharipande–Thomas theory or Nekrasov partition functions. It provides a cohomological lift of a conjectural product formula of the asymptotic refined BPS invariants. We prove the P=C conjecture for degrees ≤4.

Counting sheaves on Calabi–Yau 4-folds, I

Borisov and Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi–Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localization of Edidin and Graham’s square root Euler class for SO(2n,C)-bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localization formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a K-theoretic refinement by defining K-theoretic square root Euler classes and their localized versions. In a sequel, we prove that our invariants reproduce those of Borisov and Joyce.

Localizing virtual cycles for Donaldson-Thomas invariants of Calabi-Yau 4-folds

In 2020, Oh and Thomas constructed a virtual cycle [ X ] v i r ∈ A ∗ ( X ) [X]^{\mathrm {vir}} \in A_*(X) for a quasi-projective moduli space X X of stable sheaves or complexes over a Calabi-Yau 4-fold against which DT4 invariants may be defined as integrals of cohomology classes. In this paper, we prove that the virtual cycle localizes to the zero locus X ( σ ) X(\sigma ) of an isotropic cosection σ \sigma of the obstruction sheaf O b X Ob_X of X X and construct a localized virtual cycle [ X ] l o c v i r ∈ A ∗ ( X ( σ ) ) [X]^{\mathrm {vir}} _\mathrm {loc}\in A_*(X(\sigma )) . This is achieved by further localizing the Oh-Thomas class which localizes Edidin-Graham’s square root Euler class of a special orthogonal bundle. When the cosection σ \sigma is surjective so that the virtual cycle vanishes, we construct a reduced virtual cycle [ X ] r e d v i r [X]^{\mathrm {vir}} _{\mathrm {red}} . As an application, we prove DT4 vanishing results for hyperkähler 4-folds. All these results hold for virtual structure sheaves and K-theoretic DT4 invariants.

A note on the Kuznetsov component of the Veronese double cone

This note describes moduli spaces of complexes in the derived category of a Veronese double cone $Y$. Focusing on objects with the same class $\kappa_1$ as ideal sheaves of lines, we describe the moduli space of Gieseker stable sheaves and show that it has two components. Then, we study the moduli space of stable complexes in the Kuznetsov component of $Y$ of the same class, which also has two components. One parametrizes ideal sheaves of lines and it appears in both moduli spaces. The other components are not directly related by a wall-crossing: we show this by describing an intermediate moduli space of complexes as a space of stable pairs in the sense of Pandharipande and Thomas.

Virasoro constraints for moduli spaces of sheaves on surfaces

AbstractWe introduce a conjecture on Virasoro constraints for the moduli space of stable sheaves on a smooth projective surface. These generalise the Virasoro constraints on the Hilbert scheme of a surface found by Moreira and Moreira, Oblomkov, Okounkov and Pandharipande. We verify the conjecture in many nontrivial cases by using a combinatorial description of equivariant sheaves found by Klyachko.

Numerical characterization of complex torus quotients

This article gives a characterization of quotients of complex tori by finite groups acting freely in codimension two in terms of a numerical vanishing condition on the first and second Chern class. This generalizes results previously obtained by Greb–Kebekus–Peternell in the projective setting, and by Kirschner and the second author in dimension three. As a key ingredient to the proof, we obtain a version of the Bogomolov–Gieseker inequality for stable sheaves on singular spaces, including a discussion of the case of equality.

Moduli of Stable Sheaves on a K3 Surface of Picard Number 1

Moduli of Stable Sheaves on a K3 Surface of Picard Number 1

Universality of Descendent Integrals over Moduli Spaces of Stable Sheaves on K3 Surfaces

We interprete results of Markman on monodromy operators as a universality statement for descendent integrals over moduli spaces of stable sheaves on K3 surfaces. This yields effective methods to reduce these descendent integrals to integrals over the punctual Hilbert scheme of the K3 surface. As an application we establish the higher rank Segre-Verlinde correspondence for K3 surfaces as conjectured by Göttsche and Kool.

The Moduli Space of Stable Coherent Sheaves via Non-archimedean Geometry

We provide a construction of the moduli space of stable coherent sheaves in the world of non-archimedean geometry, where we use the notion of Berkovich non-archimedean analytic spaces. The motivation for our construction is Tony Yue Yu’s non-archimedean enumerative geometry in Gromov—Witten theory. The construction of the moduli space of stable sheaves using Berkovich analytic spaces will give rise to the non-archimedean version of Donaldson—Thomas invariants. In this paper we give the moduli construction over a non-archimedean field $${\mathbb{K}}$$ . We use the machinery of formal schemes, that is, we define and construct the formal moduli stack of (semi)-stable coherent sheaves over a discrete valuation ring R, and taking generic fiber we get the non-archimedean analytic moduli space of semistable coherent sheaves over the fractional non-archimedean field $${\mathbb{K}}$$ . We generalize Joyce’s d-critical scheme structure in [37] or Kiem—Li’s virtual critical manifolds in [38] to the world of formal schemes, and Berkovich non-archimedean analytic spaces. As an application, we provide a proof for the motivic localization formula for a d-critical non-archimedean $${\mathbb{K}}$$ -analytic space using global motive of vanishing cycles and motivic integration on oriented formal d-critical schemes. This generalizes Maulik’s motivic localization formula for the motivic Donaldson—Thomas invariants.

Cohomology bounds and Chern class inequalities for stable sheaves on a smooth projective variety

Cohomology bounds and Chern class inequalities for stable sheaves on a smooth projective variety

Virtual Segre and Verlinde numbers of projective surfaces

Recently, Marian–Oprea–Pandharipande established (a generalization of) Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending the work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of any rank. Using Mochizuki's formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg–Witten invariants and intersection numbers on products of Hilbert schemes of points. We prove that certain canonical virtual Segre and Verlinde numbers of general type surfaces are topological invariants and we verify our conjectures in examples. The power series in our conjectures are algebraic functions, for which we find expressions in several cases and which are permuted under certain Galois actions. Our conjectures imply an algebraic analog of the Mariño–Moore conjecture for higher rank Donaldson invariants. For ranks 3 and 4, we obtain explicit expressions for Donaldson invariants in terms of Seiberg–Witten invariants.

Beauville–Voisin Filtrations on Zero-Cycles of Moduli Space of Stable Sheaves on K3 Surfaces

Abstract The Beauville–Voisin conjecture predicts the existence of a filtration on a projective hyper-Kähler manifold opposite to the conjectural Bloch–Beilinson filtration, called the Beauville–Voisin filtration. In [13], Voisin has introduced a filtration on zero-cycles of an arbitrary projective hyper-Kähler manifold. On the moduli space of stable objects on a projective K3 surface, there are other candidates constructed by Shen–Yin–Zhao and Barros–Flapan–Marian–Silversmith in [1, 10] and more recently by Vial in [11] from a different point of view. According to the work in [11], all of them are proved to be equivalent except Voisin’s filtration. In this paper, we show that Voisin’s filtration is the same as the other filtrations. As an application, we prove a conjecture in [1].

Hecke correspondences for smooth moduli spaces of sheaves

We define functors on the derived category of the moduli space ℳ of stable sheaves on a smooth projective surface (under Assumptions A and S below), and prove that these functors satisfy certain commutation relations. These relations allow us to prove that the given functors induce an action of the elliptic Hall algebra on the K-theory of the moduli space ℳ, thus generalizing the action studied by Nakajima, Grojnowski and Baranovsky in cohomology.

Geometry of Genus One Fine Compactified Universal Jacobians

Abstract We introduce a general abstract notion of fine compactified Jacobian for nodal curves of arbitrary genus. We focus on genus $1$ and prove combinatorial classification results for fine compactified Jacobians in the case of a single nodal curve and in the case of the universal family $\overline {{\mathcal {C}}}_{1,n} / \overline {{\mathcal {M}}}_{1,n}$ over the moduli space of stable pointed curves. We show that if the fine compactified Jacobian of a nodal curve of genus $1$ can be extended to a smoothing of the curve, then it can be described as the moduli space of stable sheaves with respect to some polarisation. In the universal case we construct new examples of genus $1$ fine compactified universal Jacobians. Then we give a formula for the Hodge and Betti numbers of each genus $1$ fine compactified universal Jacobian $\overline {{\mathcal {J}}}^{d}_{1,n}$ and prove that their even cohomology is algebraic.

W$W$‐algebras associated to surfaces

We define an integral form of the deformed W $W$ -algebra of type gl r ${\mathfrak {gl}}_r$ , and construct its action on the K $K$ -theory groups of moduli spaces of rank r $r$ stable sheaves on a smooth projective surface S $S$ , under certain assumptions. Our construction generalizes the action studied by Nakajima, Grojnowski and Baranovsky in cohomology, although the appearance of deformed W $W$ -algebras by generators and relations is a new feature. Physically, this action encodes the Alday–Gaiotto–Tachikawa correspondence for 5-dimensional supersymmetric gauge theory on S × $S \times$ circle.