Semistable Objects Research Articles

Spectral Networks and Stability Conditions for Fukaya Categories with Coefficients

Given a holomorphic family of Bridgeland stability conditions over a surface, we define a notion of spectral network which is an object in a Fukaya category of the surface with coefficients in a triangulated DG-category. These spectral networks are analogs of special Lagrangian submanifolds, combining a graph with additional algebraic data, and conjecturally correspond to semistable objects of a suitable stability condition on the Fukaya category with coefficients. They are closely related to the spectral networks of Gaiotto–Moore–Neitzke. One novelty of our approach is that we establish a general uniqueness results for spectral network representatives. We also verify the conjecture in the case when the surface is disk with six marked points on the boundary and the coefficients category is the derived category of representations of an A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_2$$\\end{document} quiver. This example is related, via homological mirror symmetry, to the stacky quotient of an elliptic curve by the cyclic group of order six.

Bridgeland stability of minimal instanton bundles on Fano threefolds

We prove that minimal instanton bundles on a Fano threefold $X$ of Picard rank one and index two are semistable objects in the Kuznetsov component $\mathsf{Ku}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macrì and Stellari. When the degree of $X$ is at least 3, we show torsion free generalizations of minimal instantons are also semistable objects. As a result, we describe the moduli space of semistable objects with same numerical classes as minimal instantons in $\mathsf{Ku}(X)$. We also investigate the stability of acyclic extensions of non-minimal instantons.

Geometric stability conditions under autoequivalences and applications: Elliptic surfaces

On a Weierstraß elliptic surface, we describe the action of the relative Fourier-Mukai transform on the geometric chamber of Stab(X), and in the K3 case we also study the action on one of its boundary components. Using new estimates for the Gieseker chamber we prove that Gieseker stability for polarizations on certain Friedman chamber is preserved by the derived dual of the relative Fourier-Mukai transform. As an application of our description of the action, we also prove projectivity for some moduli spaces of Bridgeland semistable objects.

On the period of Li, Pertusi, and Zhao’s symplectic variety

AbstractWe extend classical results of Perego and Rapagnetta on moduli spaces of sheaves of type OG10 to moduli spaces of Bridgeland semistable objects on the Kuznetsov component of a cubic fourfold. In particular, we determine the period of this class of varieties and use it to understand when they become birational to moduli spaces of sheaves on a K3 surface.

INTRINSIC STABILIZER REDUCTION AND GENERALIZED DONALDSON–THOMAS INVARIANTS

AbstractLet $\sigma $ be a stability condition on the bounded derived category $D^b({\mathop{\mathrm {Coh}}\nolimits } W)$ of a Calabi–Yau threefold W and $\mathcal {M}$ a moduli stack parametrizing $\sigma $ -semistable objects of fixed topological type. We define generalized Donaldson–Thomas invariants which act as virtual counts of objects in $\mathcal {M}$ , fully generalizing the approach introduced by Kiem, Li and the author in the case of semistable sheaves. We construct an associated proper Deligne–Mumford stack $\widetilde {\mathcal {M}}^{\mathbb {C}^{\ast }}$ , called the $\mathbb {C}^{\ast }$ -rigidified intrinsic stabilizer reduction of $\mathcal {M}$ , with an induced semiperfect obstruction theory of virtual dimension zero, and define the generalized Donaldson–Thomas invariant via Kirwan blowups to be the degree of the associated virtual cycle $[\widetilde {\mathcal {M}}]^{\mathrm {vir}} \in A_0(\widetilde {\mathcal {M}})$ . This stays invariant under deformations of the complex structure of W. Applications include Bridgeland stability, polynomial stability, Gieseker and slope stability.

Deformations of rational curves on primitive symplectic varieties and applications

We study the deformation theory of rational curves on primitive symplectic varieties and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus in the universal locally trivial deformation. As applications, we extend Markman's deformation invariance of prime exceptional divisors along their Hodge locus to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties which are locally trivial deformations of any moduli space of semistable objects on a projective $K3$ or fibers of the Albanese map of those on an abelian surface. We also present an application to the existence of prime exceptional divisors.

Stability conditions and moduli spaces for Kuznetsov components of Gushel–Mukai varieties

We prove the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel-Mukai varieties, and describe the structure of moduli spaces of Bridgeland semistable objects in these categories in the even-dimensional case. As applications, we construct a new infinite series of unirational locally complete families of polarized hyperk\"{a}hler varieties of K3 type, and characterize Hodge-theoretically when the Kuznetsov component of an even-dimensional Gushel-Mukai variety is equivalent to the derived category of a K3 surface.

Vertical Asymptotics for Bridgeland Stability Conditions on 3-Folds

Abstract Let $X$ be a smooth projective threefold of Picard number one for which the generalized Bogomolov–Gieseker inequality holds. We characterize the limit Bridgeland semistable objects at large volume in the vertical region of the geometric stability conditions associated to $X$ in complete generality and provide examples of asymptotically semistable objects. In the case of the projective space and $\operatorname {ch}^\beta (E)=(-R,0,D,0)$, we prove that there are only a finite number of nested walls in the $(\alpha ,s)$-plane. Moreover, when $R=0$ the only semistable objects in the outermost chamber are the 1-dimensional Gieseker semistable sheaves, and when $\beta =0$ there are no semistable objects in the innermost chamber. In both cases, the only limit semistable objects of the form $E$ or $E[1]$ (where $E$ is a sheaf) that do not get destabilized until the innermost wall are precisely the (shifts of) instanton sheaves.

Elliptic quintics on cubic fourfolds, O'Grady 10, and Lagrangian fibrations

For a smooth cubic fourfold Y, we study the moduli space M of semistable objects of Mukai vector 2λ1+2λ2 in the Kuznetsov component of Y. We show that with a certain choice of stability conditions, M admits a symplectic resolution M˜, which is a smooth projective hyperkähler manifold, deformation equivalent to the 10-dimensional examples constructed by O'Grady. As applications, we show that a birational model of M˜ provides a hyperkähler compactification of the twisted family of intermediate Jacobians associated to Y. This generalizes the previous result of Voisin [58] in the very general case. We also prove that M˜ is the MRC quotient of the main component of the Hilbert scheme of quintic elliptic curves in Y, confirming a conjecture of Castravet.

A note on Bridgeland moduli spaces and moduli spaces of sheaves on X_{14} and Y_3

We study Bridgeland moduli spaces of semistable objects of (-1)-classes and (-4)-classes in the Kuznetsov components on index one prime Fano threefold X_{4d+2} of degree 4d+2 and index two prime Fano threefold Y_d of degree d for d=3,4,5. For every Serre-invariant stability condition on the Kuznetsov components, we show that the moduli spaces of stable objects of (-1)-classes on X_{4d+2} and Y_d are isomorphic. We show that moduli spaces of stable objects of (-1)-classes on X_{14} are realized by Fano surface mathcal {C}(X) of conics, moduli spaces of semistable sheaves M_X(2,1,6) and M_X(2,-1,6) and the correspondent moduli spaces on cubic threefold Y_3 are realized by moduli spaces of stable vector bundles M^b_Y(2,1,2) and M^b_Y(2,-1,2). We show that moduli spaces of semistable objects of (-4)-classes on Y_{d} are isomorphic to the moduli spaces of instanton sheaves M^{inst}_Y when dne 1,2, and show that there are open immersions of M^{inst}_Y into moduli spaces of semistable objects of (-4)-classes when d=1,2. Finally, when d=3,4,5 we show that these moduli spaces are all isomorphic to M^{ss}_X(2,0,4).

Scattering diagrams, stability conditions, and coherent sheaves on ℙ²

We show that a purely algebraic structure, a two-dimensional scattering diagram, describes a large part of the wall-crossing behavior of moduli spaces of Bridgeland semistable objects in the derived category of coherent sheaves on P 2 \mathbb {P}^2 . This gives a new algorithm computing the Hodge numbers of the intersection cohomology of the classical moduli spaces of Gieseker semistable sheaves on P 2 \mathbb {P}^2 , or equivalently the refined Donaldson-Thomas invariants for compactly supported sheaves on local P 2 \mathbb {P}^2 . As applications, we prove that the intersection cohomology of moduli spaces of Gieseker semistable sheaves on P 2 \mathbb {P}^2 is Hodge-Tate, and we give the first non-trivial numerical checks of the general χ \chi -independence conjecture for refined Donaldson-Thomas invariants of one-dimensional sheaves on local P 2 \mathbb {P}^2 .

The stable cohomology of moduli spaces of sheaves on surfaces

Let $X$ be a smooth, irreducible, complex projective surface, $H$ a polarization on $X$. Let $\gamma = (r, c, \Delta)$ be a Chern character. In this paper, we study the cohomology of moduli spaces of Gieseker semistable sheaves $M_{X,H} (\gamma)$. When the rank $r = 1$, the Betti numbers were computed by Göttsche. We conjecture that if we fix the rank $r \geq 1$ and the first Chern class $c$, then the Betti numbers (and more generally the Hodge numbers) of $M_{X,H} (r, c, \Delta)$ stabilize as the discriminant $\Delta$ tends to infinity and that the stable Betti numbers are independent of $r$ and $c$. In particular, the conjectural stable Betti numbers are determined by Göttsche’s calculation. We present evidence for the conjecture. We analyze the validity of the conjecture under blowup and wall-crossing. We prove that when $X$ is a rational surface and $K_X \cdot H \lt 0$, then the classes $[M_{X,H} (\gamma)]$ stabilize in an appropriate completion of the Grothendieck ring of varieties as $\Delta$ tends to $\infty$. Consequently, the virtual Poincaré and Hodge polynomials stabilize to the conjectural value. In particular, the conjecture holds when $X$ is a rational surface, $H \cdot K_X \lt 0$ and there are no strictly semistable objects in $M_{X,H} (\gamma)$.

Uhlenbeck Compactification as a Bridgeland Moduli Space

Abstract Let $(X,H)$ be a smooth, projective, polarized surface over ${\mathbb {C}}$, and let $v \in K_{\textrm {num}}(X)$ be a class of positive rank. We prove that for certain Bridgeland stability conditions $\sigma = ({\mathcal {A}}, Z)$ “on the vertical wall” for $v$, the good moduli space $M^\sigma (v)$ parameterizing S-equivalence classes of $\sigma $-semistable objects of class $v$ in ${\mathcal {A}}$ is projective. Moreover, we construct a bijective morphism $M^{\textrm {Uhl}}(v) \to M^\sigma (v)$ from the Uhlenbeck compactification of $\mu $-stable vector bundles.

The generalized Franchetta conjecture for some hyper-Kähler varieties, II

We prove the generalized Franchetta conjecture for the locally complete family of hyper-K\"ahler eightfolds constructed by Lehn-Lehn-Sorger-van Straten (LLSS). As a corollary, we establish the Beauville-Voisin conjecture for very general LLSS eightfolds. The strategy consists in reducing to the Franchetta property for relative fourth powers of cubic fourfolds, by using the recent description of LLSS eightfolds as moduli spaces of semistable objects in the Kuznetsov component of the derived category of cubic fourfolds, together with its generalization to the relative setting due to Bayer-Lahoz-Macr\`i-Nuer-Perry-Stellari. As a by-product, we compute the Chow motive of the Fano variety of lines on a smooth cubic hypersurface in terms of the Chow motive of the cubic hypersurface.

Stability conditions in families

We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

A Harder–Narasimhan theory for Kisin modules

We develop a Harder-Narasimhan theory for Kisin modules generalizing a similar theory for finite flat group schemes due to Fargues. We prove the tensor product theorem, i.e., that the tensor product of semi-stable objects is again semi-stable. We then apply the tensor product theorem to the study of Kisin varieties for arbitrary connected reductive groups.

A construction of Frobenius manifolds from stability conditions

A finite quiver $Q$ without loops or 2-cycles defines a 3CY triangulated category $D(Q)$ and a finite heart $A(Q)$. We show that if $Q$ satisfies some (strong) conditions then the space of stability conditions $Stab(A(Q))$ supported on this heart admits a natural family of semisimple Frobenius manifold structures, constructed using the invariants counting semistable objects in $D(Q)$. In the case of $A_n$ evaluating the family at a special point we recover a branch of the Saito Frobenius structure of the $A_n$ singularity $y^2 = x^{n+1}$. We give examples where applying the construction to each mutation of $Q$ and evaluating the families at a special point yields a different branch of the maximal analytic continuation of the same semisimple Frobenius manifold. In particular we check that this holds in the case of $A_n$, $n \leq 5$.

Stratifications for moduli of sheaves and moduli of quiver representations

We study the relationship between two stratifications on parameter spaces for coherent sheaves and for quiver representations: a stratification by Harder–Narasimhan types and a stratification arising from the geometric invariant theory construction of the associated moduli spaces of semistable objects. For quiver representations, both stratifications coincide, but this is not quite true for sheaves. We explain why the stratifications on various Quot schemes do not coincide and that the correct parameter space to compare such stratifications is the stack of coherent sheaves, where we construct an asymptotic GIT stratification and prove that this coincides with the Harder–Narasimhan stratification. Then we relate these stratifications for sheaves and quiver representations using a generalisation of a construction of Alvarez-Consul and King.

Duality, Bridgeland wall-crossing and flips of secant varieties

Let [Formula: see text] denote the [Formula: see text]-uple Veronese surface. After studying some general aspects of the wall-crossing phenomena for stability conditions on surfaces, we are able to describe a sequence of flips of the secant varieties of [Formula: see text] by embedding the blow-up [Formula: see text] into a suitable moduli space of Bridgeland semistable objects on [Formula: see text].

Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface

We construct moduli spaces of semistable objects on an Enriques surface for generic Bridgeland stability condition and prove their projectivity. We further generalize classical results about moduli spaces of semistable sheaves on an Enriques surface to their Bridgeland counterparts. Using Bayer and Macr\`{i}'s construction of a natural nef divisor varying with the stability condition, we begin a systematic exploration of the relation between wall-crossing on the Bridgeland stability manifold and the minimal model program for these moduli spaces. We give three applications of our machinery to obtain new information about the classical moduli spaces of Gieseker-stable sheaves: 1) We obtain a region in the ample cone of the moduli space of Gieseker-stable sheaves which works for all unnodal Enriques surfaces. 2) We determine the nef cone of the Hilbert scheme of $n$ points on an unnodal Enriques surface in terms of the classical geometry of its half-pencils and the Cossec-Dolgachev $\phi$-function. 3) We recover some classical results on linear systems on Enriques surfaces and obtain some new ones about $n$-very ample line bundles.