Semistable Objects Research Articles

Projectivity of Bridgeland Moduli Spaces on Del Pezzo Surfaces of Picard Rank 2

We prove that, for a natural class of Bridgeland stability conditions on $\mathbb{P}^1\times\mathbb{P}^1$ and the blow-up of $\mathbb{P}^2$ at a point, the moduli spaces of Bridgeland semistable objects are projective. Our technique is to find suitable regions of stability conditions with hearts that are (after rotation) equivalent to representations of a quiver. The helix and tilting theory is well-behaved on Del Pezzo surfaces and we conjecture that this program (begun in arXiv:1203.0316) runs successfully for all Del Pezzo surfaces, and the analogous Bridgeland moduli spaces are projective.

Moduli of Bridgeland semistable objects on 3-folds and Donaldson–Thomas invariants

Abstract In this paper we show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are quasi-proper algebraic stacks of finite type if they satisfy the Bogomolov–Gieseker (BG for short) inequality conjecture proposed by Bayer, Macrì and the second author. The key ingredients are the equivalent form of the BG inequality conjecture and its generalization to arbitrary very weak stability conditions. This result is applied to define Donaldson–Thomas invariants counting Bridgeland semistable objects on smooth projective Calabi–Yau 3-folds satisfying the BG inequality conjecture, for example on étale quotients of abelian 3-folds.

The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds

We describe a connected component of the space of stability conditions on abelian threefolds, and on Calabi-Yau threefolds obtained as (the crepant resolution of) a finite quotient of an abelian threefold. Our proof includes the following essential steps: Important in our approach is a more systematic understanding on the behaviour of quadratic inequalities for semistable objects under wall-crossing, closely related to the support property.

Birational geometry of singular moduli spaces of O'Grady type

Following Bayer and Macrì, we study the birational geometry of singular moduli spaces M of sheaves on a K3 surface X which admit symplectic resolutions. More precisely, we use the Bayer–Macrì map from the space of Bridgeland stability conditions Stab(X) to the cone of movable divisors on M to relate wall-crossing in Stab(X) to birational transformations of M. We give a complete classification of walls in Stab(X) and show that every minimal birational model of M in the sense of the log minimal model program appears as a moduli space of Bridgeland semistable objects on X. An essential ingredient of our proof is an isometry between the orthogonal complement of a Mukai vector inside the algebraic Mukai lattice of X and the Néron–Severi lattice of M which generalises results of Yoshioka, as well as Perego and Rapagnetta. Moreover, this allows us to conclude that the symplectic resolution of M is deformation equivalent to the 10-dimensional irreducible holomorphic symplectic manifold found by O'Grady.

Stability and Fourier–Mukai transforms on higher dimensional elliptic fibrations

We consider elliptic fibrations with arbitrary base dimensions, and generalise most of the results in [Lo1, Lo2, Lo5]. In particular, we check universal closedness for the moduli of semistable objects with respect to a polynomial stability that reduces to PT-stability on threefolds. We also show openness of this polynomial stability. On the other hand, we write down criteria under which certain 2-term polynomial semistable complexes are mapped to torsionfree semistable sheaves under a Fourier-Mukai transform. As an application, we construct an open immersion from a moduli of complexes to a moduli of Gieseker stable sheaves on higher dimensional elliptic fibrations.

Wall-crossing and invariants of higher rank Joyce–Song stable pairs

We introduce a higher rank analog of the Joyce–Song theory of stable pairs. Given a nonsingular projective Calabi–Yau threefold $X$, we define the higher rank Joyce–Song pairs given by ${O}^{\oplus r}_{X}(-n)\rightarrow F$ where $F$ is a pure coherent sheaf with one dimensional support, $r>1$ and $n\gg0$ is a fixed integer. We equip the higher rank pairs with a Joyce–Song stability condition and compute their associated invariants using the wallcrossing techniques in the category of “weakly” semistable objects.

A note on Bogomolov-Gieseker type inequality for Calabi-Yau 3-folds

The conjectural Bogomolov-Gieseker (BG) type inequality for tilt semistable objects on projective 3-folds was proposed by Bayer, Macri and the author. In this note, we prove our conjecture for slope stable sheaves with the smallest first Chern class on certain Calabi-Yau 3-folds, e.g. quintic 3-folds.

Mini-walls for Bridgeland stability conditions on the derived category of sheaves over surfaces

For the derived category of bounded complexes of sheaves on a smooth projective surface, Bridgeland and Arcara-Bertram constructed Bridgeland stability conditions $(Z_m, \mathcal P_m)$ parametrized by $m \in (0, +\infty)$. In this paper, we show that the set of mini-walls in $(0, +\infty)$ of a fixed numerical type is locally finite. In addition, we strengthen a result of Bayer by proving that the moduli of polynomial Bridgeland semistable objects of a fixed numerical type coincides with the moduli of $(Z_m, \mathcal P_m)$-semistable objects whenever $m$ is larger than a universal constant depending only on the numerical type. We further identify the moduli of polynomial Bridgeland semistable objects with the Gieseker/Simpson moduli spaces and the Uhlenbeck compactification spaces.

Some Moduli Spaces of Bridgeland’s Stability Conditions

We shall study some moduli spaces of Bridgeland's semi-stable objects on abelian surfaces and K3 surfaces with Picard number 1. Under some conditions, we show that the moduli spaces are isomorphic to the moduli spaces of Gieseker semi-stable sheaves. We also study the ample cone of the moduli spaces.

Moduli of PT-semistable objects II

We generalise the techniques of semistable reduction for flat families of sheaves to the setting of the derived category D b ( X ) D^b(X) of coherent sheaves on a smooth projective three-fold X X . Then we construct the moduli of PT-semistable objects in D b ( X ) D^b(X) as an Artin stack of finite type that is universally closed. In the absence of strictly semistable objects, we construct the moduli as a proper algebraic space of finite type.

The minimal model program for the Hilbert scheme of points on [formula omitted] and Bridgeland stability

In this paper, we study the birational geometry of the Hilbert scheme P2[n] of n-points on P2. We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n≤9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions.

Curve counting invariants around the conifold point

In this paper, we investigate the space of certain weak stability conditions on the triangulated category of D0-D2-D6 bound states on a smooth projective Calabi-Yau 3-fold. In the case of a quintic 3-fold, the resulting space is interpreted as a universal covering space of an infinitesimal neighborhood of the conifold point in the stringy Kahler moduli space. We then construct the DT type invariants counting semistable objects in our triangulated category, which are new curve counting invariants on a Calabi-Yau 3-fold. We also investigate the wall-crossing formula of our invariants and their interplay with the Seidel-Thomas twist.

Noncrossing partitions and representations of quivers

AbstractWe situate the noncrossing partitions associated with a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated with a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiverQwithout oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show that these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.

Moduli stacks and invariants of semistable objects on K3 surfaces

For a K3 surface X and its bounded derived category of coherent sheaves D ( X ) , we have the notion of stability conditions on D ( X ) in the sense of T. Bridgeland. In this paper, we show that the moduli stack of semistable objects in D ( X ) with a fixed numerical class and a phase is represented by an Artin stack of finite type over C . Then following D. Joyce's work, we introduce the invariants counting semistable objects in D ( X ) , and show that the invariants are independent of a choice of a stability condition.

A Universal Construction for Moduli Spaces of Decorated Vector Bundles over Curves

Let $X$ be a smooth projective curve over the field of complex numbers, and fix a homogeneous representation $\rho\colon \mathop{\rm GL}(r)\rightarrow \mathop{\rm GL}(V)$. Then one can associate to every vector bundle $E$ of rank $r$ over $X$ a vector bundle $E_\rho$ with fibre $V$. We would like to study triples $(E,L,\phi)$ where $E$ is a vector bundle of rank $r$ over $X$, $L$ is a line bundle over $X$, and $\phi\colon E_\rho\rightarrow L$ is a nontrivial homomorphism. This setup comprises well known objects such as framed vector bundles, Higgs bundles, and conic bundles. In this paper, we will formulate a general (parameter dependent) semistability concept for such triples, which generalizes the classical Hilbert--Mumford criterion, and we establish the existence of moduli spaces for the semistable objects. In the examples which have been studied so far, our semistability concept reproduces the known ones. Therefore, our results give in particular a unified construction for many moduli spaces considered in the literature.

Joyce invariants for K3 surfaces and mock theta functions

For stability conditions on K3 surfaces, we study moduli stacks of semistable objects with Donaldson–Thomas type invariants, introduced by Joyce, and mock theta functions, introduced by Ramanujan. In particular, we will show invariance of moduli stacks on faithful stability conditions and motivic invariants, and in terms of mock theta functions, study generating functions obtained by moduli-stack counting and differential equations.