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38
Polynomial Bridgeland stability conditions and the large volume limit
 Geom. Topol
"... ABSTRACT. We introduce the notion of a polynomial stability condition, generalizing Bridgeland stability conditions on triangulated categories. We construct and study a family of polynomial stability conditions for any normal projective variety. This family includes both Simpsonstability, and large ..."
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Cited by 38 (4 self)
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ABSTRACT. We introduce the notion of a polynomial stability condition, generalizing Bridgeland stability conditions on triangulated categories. We construct and study a family of polynomial stability conditions for any normal projective variety. This family includes both Simpsonstability, and large volume limits of Bridgeland stability conditions. We show that the PT/DTcorrespondence relating stable pairs to DonaldsonThomas invariants (conjectured by Pandharipande and Thomas) can be understood as a wallcrossing in our family of polynomial stability conditions. Similarly, we show that the relation between stable pairs and invariants of onedimensional torsion sheaves (proven recently by the same authors) is a wallcrossing formula. CONTENTS
Moduli stacks and invariants of semistable objects on K3
, 2007
"... 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 Arti ..."
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Cited by 32 (9 self)
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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.
Limit stable objects on CalabiYau 3folds
"... In this paper, we introduce new enumerative invariants of curves on CalabiYau 3folds via certain stable objects in the derived category of coherent sheaves. We introduce the ..."
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Cited by 28 (8 self)
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In this paper, we introduce new enumerative invariants of curves on CalabiYau 3folds via certain stable objects in the derived category of coherent sheaves. We introduce the
Spaces of stability conditions
"... Abstract. Stability conditions are a mathematical way to understand Πstability for Dbranes in string theory. Spaces of stability conditions seem to be related to moduli spaces of conformal field theories. This is a survey article describing what is currently known about spaces of stability conditi ..."
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Cited by 27 (3 self)
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Abstract. Stability conditions are a mathematical way to understand Πstability for Dbranes in string theory. Spaces of stability conditions seem to be related to moduli spaces of conformal field theories. This is a survey article describing what is currently known about spaces of stability conditions, and giving some pointers for future research. 1.
The space of stability conditions on the local projective plane
 Duke Math. J
"... ABSTRACT. We study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions, and show that its translates via autoequivalences cover a whole connected component. We prove that this conne ..."
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Cited by 27 (3 self)
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ABSTRACT. We study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions, and show that its translates via autoequivalences cover a whole connected component. We prove that this connected component is simplyconnected. We determine the group of autoequivalences preserving this connected component. Finally, we show that there is a submanifold isomorphic to the universal covering of a moduli space of elliptic curves with level structure, with the morphism given by solutions of PicardFuchs equations. This result is motivated by the notion of Πstability and by mirror symmetry. 1.
Stability conditions and crepant small resolutions
, 2007
"... In this paper, we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions. The resulting spaces have chamber structures such that each chamber corresponds to a birational model together with a special FourierMukai transfor ..."
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Cited by 22 (9 self)
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In this paper, we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions. The resulting spaces have chamber structures such that each chamber corresponds to a birational model together with a special FourierMukai transform. We observe that these spaces are covering spaces over certain open subsets of finite dimensional vector spaces, and determine their deck transformations.
STABILITY CONDITIONS FOR GENERIC K3 CATEGORIES
, 2007
"... K3 categories occur as bounded derived categories of (twisted) coherent sheaves on K3 or abelian surfaces. A K3 category is generic if there are no spherical objects (or just one up to shift). We study stability conditions on K3 categories as introduced by Bridgeland and prove his conjecture about t ..."
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Cited by 19 (6 self)
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K3 categories occur as bounded derived categories of (twisted) coherent sheaves on K3 or abelian surfaces. A K3 category is generic if there are no spherical objects (or just one up to shift). We study stability conditions on K3 categories as introduced by Bridgeland and prove his conjecture about the topology of the stability manifold and the autoequivalences group for generic twisted projective K3, abelian surfaces, and K3 surfaces with trivial Picard group.
Reconstruction algebras of type A
, 2007
"... Abstract. This is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin diagrams. This paper deals with dihedral groups G = ..."
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Cited by 18 (7 self)
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Abstract. This is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin diagrams. This paper deals with dihedral groups G =�n,q for which all special CM modules have rank one, and we show that all but four of the relations on such a reconstruction algebra are given simply as the relations arising from a reconstruction algebra of type A. As a corollary, the reconstruction algebra reduces the problem of explicitly understanding the minimal resolution (=GHilb) to the same level of difficulty as the toric case. Contents
On the C n /Zm fractional branes
, 2008
"... We construct several geometric representatives for the C n /Zm fractional branes on either a partially or the completely resolved orbifold. In the process we use large radius and conifoldtype monodromies, and provide a strong consistency check. In particular, for C 3 /Z5 we give three different set ..."
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Cited by 14 (4 self)
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We construct several geometric representatives for the C n /Zm fractional branes on either a partially or the completely resolved orbifold. In the process we use large radius and conifoldtype monodromies, and provide a strong consistency check. In particular, for C 3 /Z5 we give three different sets of geometric representatives. We also find the explicit Seibergduality which connects our fractional branes to the ones given by the McKay correspondence.
INDUCING STABILITY CONDITIONS
, 705
"... Abstract. We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a smooth projective variety we prove that the subset of invariant stability conditions embeds as a closed submanifold into the stability manifold of the equivariant derived cat ..."
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Cited by 12 (2 self)
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Abstract. We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a smooth projective variety we prove that the subset of invariant stability conditions embeds as a closed submanifold into the stability manifold of the equivariant derived category. As an application we examine stability conditions on Kummer and Enriques surfaces and we give a derived version of the Torelli Theorem for the latter surfaces. We also study the relationship between stability conditions on projective spaces and those on their canonical bundles. 1.