Moduli Spaces Of Stable Objects Research Articles

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).

Equivariant categories of symplectic surfaces and fixed loci of Bridgeland moduli spaces

Given an action of a finite group $G$ on the derived category of a smooth projective variety $X$ we relate the fixed loci of the induced $G$-action on moduli spaces of stable objects in $D^b(\mathrm{Coh}(X))$ with moduli spaces of stable objects in the equivariant category $D^b(\mathrm{Coh}(X))_G$. As an application we obtain a criterion for the equivariant category of a symplectic action on the derived category of a symplectic surface to be equivalent to the derived category of a surface. This generalizes the derived McKay correspondence, and yields a general framework for describing fixed loci of symplectic group actions on moduli spaces of stable objects on symplectic surfaces.

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].

On the birational motive of hyper-Kähler varieties

We introduce a new ascending filtration, that we call the co-radical filtration in analogy with the basic theory of co-algebras, on the Chow groups of pointed smooth projective varieties. In the case of zero-cycles on projective hyper-Kähler manifolds, we conjecture it agrees with a filtration introduced by Voisin. This is established for moduli spaces of stable objects on K3 surfaces, for generalized Kummer varieties and for the Fano variety of lines on a smooth cubic fourfold. Our overall strategy is to view the birational motive of a smooth projective variety as a co-algebra object with respect to the diagonal embedding and to show in the aforementioned cases the existence of a so-called strict grading whose associated filtration agrees with the filtration of Voisin. As results of independent interest, we upgrade to rational equivalence Voisin's notion of “surface decomposition” and show that the birational motive of some projective hyper-Kähler manifolds is determined, as a co-algebra object, by the birational motive of a surface. We also relate our co-radical filtration on the Chow groups of abelian varieties to Beauville's eigenspace decomposition.

Some Remarks on Fano Three-Folds of Index Two and Stability Conditions

AbstractWe prove that ideal sheaves of lines in a Fano three-fold $X$ of Picard rank one and index two are stable objects in the Kuznetsov component ${\operatorname{\mathsf{Ku}}}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macrì, and Stellari, giving a modular description to the Hilbert scheme of lines in $X$. When $X$ is a cubic three-fold, we show that the Serre functor of ${\operatorname{\mathsf{Ku}}}(X)$ preserves these stability conditions. As an application, we obtain the smoothness of nonempty moduli spaces of stable objects in ${\operatorname{\mathsf{Ku}}}(X)$. When $X$ is a quartic double solid, we describe a connected component of the stability manifold parametrizing stability conditions on ${\operatorname{\mathsf{Ku}}}(X)$.

Birational geometry of moduli spaces of stable objects on Enriques surfaces

Using wall-crossing for K3 surfaces, we establish birational equivalence of moduli spaces of stable objects on generic Enriques surfaces for different stability conditions. As an application, we prove in the case of a Mukai vector of odd rank that they are birational to Hilbert schemes. The argument makes use of a new Chow-theoretic result, showing that moduli spaces on an Enriques surface give rise to constant cycle subvarieties of the moduli spaces of the covering K3.

Derived categories of surfaces, O’Grady’s filtration, and zero-cycles on holomorphic symplectic varieties

Moduli spaces of stable objects in the derived category of a $K3$ surface provide a large class of holomorphic symplectic varieties. In this paper, we study the interplay between Chern classes of stable objects and zero-cycles on holomorphic symplectic varieties which arise as moduli spaces. First, we show that the second Chern class of any object in the derived category lies in a suitable piece of O’Grady’s filtration on the $\text{CH}_{0}$-group of the $K3$ surface. This solves a conjecture of O’Grady and improves on previous results of Huybrechts, O’Grady, and Voisin. Second, we propose a candidate for the Beauville–Voisin filtration on the $\text{CH}_{0}$-group of the moduli space of stable objects. We discuss its connection with Voisin’s recent proposal via constant cycle subvarieties, and prove a conjecture of hers on the existence of special algebraically coisotropic subvarieties for the moduli space.

CATEGORIES, ONE-CYCLES ON CUBIC FOURFOLDS, AND THE BEAUVILLE–VOISIN FILTRATION

We explore the connection between$K3$categories and 0-cycles on holomorphic symplectic varieties. In this paper, we focus on Kuznetsov’s noncommutative$K3$category associated to a nonsingular cubic 4-fold.By introducing a filtration on the$\text{CH}_{1}$-group of a cubic 4-fold$Y$, we conjecture a sheaf/cycle correspondence for the associated$K3$category ${\mathcal{A}}_{Y}$. This is a noncommutative analog of O’Grady’s conjecture concerning derived categories of$K3$surfaces. We study instances of our conjecture involving rational curves in cubic 4-folds, and verify the conjecture for sheaves supported on low degree rational curves.Our method provides systematic constructions of (a) the Beauville–Voisin filtration on the$\text{CH}_{0}$-group and (b) algebraically coisotropic subvarieties of a holomorphic symplectic variety which is a moduli space of stable objects in${\mathcal{A}}_{Y}$.

Remarks on groups of bundle automorphisms over the Riemann sphere

A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff-Grothendieck bundle splitting $\bigoplus_{i=1}^{r} \mathcal(m_{i})$ over $\mathbb{C}\mathbb{P}^{1}$ is provided, in terms of its action on a suitable space of generalized flags in the fibers over a finite subset $S\subset\mathbb{C}\mathbb{P}^{1}$. The relevance of such characterization derives from the possibility of constructing geometric models for diverse moduli spaces of stable objects in genus 0, such as parabolic bundles, parabolic Higgs bundles, and logarithmic connections, as collections of orbit spaces of parabolic structures and compatible geometric data satisfying a given stability criterion, under the actions of the different splitting types' automorphism groups, that are glued in a concrete fashion. We illustrate an instance of such idea, on the existence of several natural representatives for the induced actions on the corresponding vector spaces of (orbits of) logarithmic connections with residues adapted to a parabolic structure.

Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects

We revisit the work of Lehn–Lehn–Sorger–van Straten on twisted cubic curves in a cubic fourfold not containing a plane in terms of moduli spaces. We show that the blow-up Z′ along the cubic of the irreducible holomorphic symplectic eightfold Z, described by the four authors, is isomorphic to an irreducible component of a moduli space of Gieseker stable torsion sheaves or rank three torsion free sheaves.For a very general such cubic fourfold, we show that Z is isomorphic to a connected component of a moduli space of tilt-stable objects in the derived category and to a moduli space of Bridgeland stable objects in the Kuznetsov component. Moreover, the contraction between Z′ and Z is realized as a wall-crossing in tilt-stability.Finally, Z is birational to an irreducible component of a moduli space of Gieseker stable aCM bundles of rank six.

Induced automorphisms on irreducible symplectic manifolds:

We introduce the notion of induced automorphisms in order to state a criterion to determine whether a given automorphism on a manifold of K 3 [ n ] -type is, in fact, induced by an automorphism of a K 3 surface, and the manifold is a moduli space of stable objects on the K 3 . This criterion is applied to the classification of non-symplectic prime order automorphisms on manifolds of K 3 [ 2 ] -type, and we prove that almost all cases are covered. Variations of this notion and the above criterion are introduced and discussed for the other known deformation types of irreducible symplectic manifolds. Furthermore, we provide a description of the picard lattice of several irreducible symplectic manifolds having a lagrangian fibration.

Interactions between autoequivalences, stability conditions, and moduli problems

We formulate a strong compatibility between autoequivalences and Bridgeland stability conditions and derive a sufficiency criterion. We apply this criterion to an extension of classical slope on the derived category associated to any Galois cover of the nodal cubic. In particular, we give an explicit description of the moduli space of stable objects, its compactification (under S-equivalence), and the group of all autoequivalences compatible with the choice of stability condition.

Extensions of Higgs bundles

We prove a Hitchin-Kobayashi correspondence for extensions of Higgs bundles. The results generalize known results for extensions of holomorphic bundles. Using Simpson’s methods, we construct moduli spaces of stable objects. In an appendix we construct Bott-Chern forms for Higgs bundles ∗The first author thanks the Tata Institute for Fundamental Research for its hospitality. The second author was supported by a postdoctoral fellowship of Ministerio de Educacion y Cultura (Spain).