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125
Configurations in abelian categories. IV. Invariants and changing stability conditions
, 2007
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Bridgelandstable moduli spaces for Ktrivial surfaces (with an appendix by M. Lieblich), arXiv:0708.2247
"... ABSTRACT. We give a natural family of Bridgeland stability conditions on the derived category of a smooth projective complex surface S and describe “wallcrossing behavior ” for objects with the same invariants as OC(H) when H generates Pic(S) and C ∈ H. If, in addition, S is a K3 or Abelian surfac ..."
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Cited by 53 (5 self)
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ABSTRACT. We give a natural family of Bridgeland stability conditions on the derived category of a smooth projective complex surface S and describe “wallcrossing behavior ” for objects with the same invariants as OC(H) when H generates Pic(S) and C ∈ H. If, in addition, S is a K3 or Abelian surface, we use this description to construct a sequence of fine moduli spaces of Bridgelandstable objects via Mukai flops and generalized elementary modifications of the universal coherent sheaf. We also discover a natural generalization of Thaddeus’ stable pairs for curves embedded in the moduli spaces.
Mirror symmetry and integral variations of Hodge structure underlying oneparamameter families of Calabi–Yau threefolds
 IN “MIRROR SYMMETRY V”, THE BIRS PROC. ON CALABI–YAU VARIETIES AND MIRROR SYMMETRY, AMS/IP
, 2005
"... This proceedings note introduces aspects of the authors’ work relating mirror symmetry and integral variations of Hodge structure. The emphasis is on their classification of the integral variations of Hodge structure which can underly families of CalabiYau threefolds over P 1 \ {0, 1, ∞} with b 3 = ..."
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Cited by 45 (6 self)
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This proceedings note introduces aspects of the authors’ work relating mirror symmetry and integral variations of Hodge structure. The emphasis is on their classification of the integral variations of Hodge structure which can underly families of CalabiYau threefolds over P 1 \ {0, 1, ∞} with b 3 = 4, or equivalently h 2,1 = 1, and the related issues of geometric realization of these variations. The presentation parallels that of the first author’s talk at the BIRS workshop.
Stability conditions on a noncompact CalabiYau threefold
"... Abstract. We study the space of stability conditions on the noncompact CalabiYau threefold X which is the total space of the canonical bundle of P 2. We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the ..."
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Cited by 38 (1 self)
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Abstract. We study the space of stability conditions on the noncompact CalabiYau threefold X which is the total space of the canonical bundle of P 2. We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the quantum cohomology of P 2. We give some evidence from mirror symmetry for this conjecture. 1.
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
Projectivity and birational geometry of Bridgeland moduli spaces
, 2012
"... ABSTRACT. We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby gene ..."
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Cited by 35 (2 self)
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ABSTRACT. We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka. Our result also gives a systematic explanation of the relation between wallcrossing for Bridgelandstability and the minimal model program for the moduli space. We give three applications of our method for classical moduli spaces of sheaves on a K3 surface: 1. We obtain a region in the ample cone in the moduli space of Giesekerstable sheaves only depending on the lattice of the K3. 2. We determine the nef cone of the Hilbert scheme of n points on a K3 surface of Picard rank one when n is large compared to the genus. 3. We verify the “HassettTschinkel/Huybrechts/Sawon ” conjecture on the existence of a birational Lagrangian fibration for the Hilbert scheme in a new family of cases.
MMP FOR MODULI OF SHEAVES ON K3S VIA WALLCROSSING: NEF AND MOVABLE CONES, LAGRANGIAN FIBRATIONS
"... ABSTRACT. We use wallcrossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X: (a) We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X. (b) ..."
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Cited by 35 (2 self)
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ABSTRACT. We use wallcrossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X: (a) We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X. (b) We establish a longstanding conjecture that predicts the existence of a birational Lagrangian fibration on M whenever M admits an integral divisor class D of square zero (with respect to the BeauvilleBogomolov form). These results are proved using a natural map from the space of Bridgeland stability conditions Stab(X) to the cone Mov(X) of movable divisors on M; this map relates wallcrossing in Stab(X) to birational transformations of M. In particular, every minimal model of M appears as a moduli space of Bridgelandstable objects on X. CONTENTS
THE MINIMAL MODEL PROGRAM FOR THE HILBERT SCHEME OF POINTS ON P 2 AND BRIDGELAND STABILITY
"... Abstract. In this paper, we study the birational geometry of the Hilbert scheme P 2[n] of npoints on P 2. 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 Bridgela ..."
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Cited by 35 (11 self)
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Abstract. In this paper, we study the birational geometry of the Hilbert scheme P 2[n] of npoints on P 2. 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 semistable 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 wallcrossings in the Bridgeland stability manifold and wallcrossings between Mori cones. For n ≤ 9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions. Contents
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 (7 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