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27
Log Crepant Birational Maps and Derived Categories
, 2003
"... The purpose of this paper is to extend the conjecture stated in the paper [5] to the logarithmic case and prove some supporting evidences. [5] Conjecture ..."
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Cited by 22 (3 self)
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The purpose of this paper is to extend the conjecture stated in the paper [5] to the logarithmic case and prove some supporting evidences. [5] Conjecture
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 15 (6 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.
Dualizing complexes and perverse sheaves on noncommutative ringed schemes
, 2002
"... Let (X, A) be a separated differential quasicoherent ringed scheme of finite type over a field k. We prove that there exists a rigid dualizing complex over A. The proof consists of two main parts. In the algebraic part we study differential filtrations on rings, and use the results obtained to sh ..."
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Cited by 13 (6 self)
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Let (X, A) be a separated differential quasicoherent ringed scheme of finite type over a field k. We prove that there exists a rigid dualizing complex over A. The proof consists of two main parts. In the algebraic part we study differential filtrations on rings, and use the results obtained to show that a rigid dualizing complex exists on every affine open set in X. In the geometric part of the proof we construct a perverse tstructure on the derived category of bimodules, and this allows us to glue the affine rigid dualizing complexes to
Cluster tilting for higher Auslander algebras
"... Abstract. The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the ca ..."
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Cited by 13 (4 self)
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Abstract. The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the category Mn of preinjectivelike modules obtained by applying τn to injective modules repeatedly. We call a finite dimensional algebra Λ ncomplete if Mn = add M for an ncluster tilting object M. Our main result asserts that the endomorphism algebra EndΛ(M) is (n + 1)complete. This gives an inductive construction of ncomplete algebras. For example, any representationfinite hereditary algebra Λ (1) is 1complete. Hence the Auslander algebra Λ (2) of Λ (1) is 2complete. Moreover, for any n ≥ 1, we have an ncomplete algebra Λ (n) which has an ncluster tilting object M (n) such that Λ (n+1) = End Λ (n)(M (n)). We give the presentation of Λ (n) by a quiver with relations. We apply our results to construct ncluster tilting subcategories of derived categories of ncomplete algebras. Contents 1. Our results 3 1.1. ncluster tilting in module categories 4
Derived categories of coherent sheaves
 Proceedings of the 2006 ICM
, 2006
"... Abstract. We discuss derived categories of coherent sheaves on algebraic varieties. We focus on the case of nonsingular CalabiYau varieties and consider two unsolved problems: proving that birational varieties have equivalent derived categories, and computing the group of derived autoequivalences. ..."
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Cited by 13 (0 self)
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Abstract. We discuss derived categories of coherent sheaves on algebraic varieties. We focus on the case of nonsingular CalabiYau varieties and consider two unsolved problems: proving that birational varieties have equivalent derived categories, and computing the group of derived autoequivalences. We also introduce the space of stability conditions on a triangulated category and explain its relevance to these two problems. 1.
Crystal Melting and Toric CalabiYau Manifolds
"... We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric CalabiYau threefold. The threedimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low ene ..."
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Cited by 10 (1 self)
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We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric CalabiYau threefold. The threedimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram, and the chemical bond is dictated by the arrows of the quiver diagram. BPS states are constructed by removing atoms from the crystal. This generalizes the earlier results on the BPS state counting to an arbitrary noncompact toric CalabiYau manifold. We point out that a proper understanding of the relation between the topological In type IIA superstring theory, supersymmetric bound states of D branes wrapping holomorphic cycles on a CalabiYau manifold give rise to BPS particles in four dimensions. In the past few years, remarkable connections have been found between the counting of such bound states and the topological string theory:
Birational CalabiYau 3folds and BPS state counting
, 2008
"... This paper contains some applications of BridgelandDouglas stability conditions on triangulated categories, and Joyce’s work on counting invariants of semistable objects, to the study of birational geometry. We introduce the notion of motivic GopakumarVafa invariants as counting invariants of D2b ..."
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Cited by 8 (3 self)
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This paper contains some applications of BridgelandDouglas stability conditions on triangulated categories, and Joyce’s work on counting invariants of semistable objects, to the study of birational geometry. We introduce the notion of motivic GopakumarVafa invariants as counting invariants of D2branes, and show that they are invariant under birational transformations between CalabiYau 3folds. The result is similar to the fact that birational CalabiYau 3folds have the same betti numbers or Hodge numbers.
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
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Cited by 8 (7 self)
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Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
LEFSCHETZ DECOMPOSITIONS AND CATEGORICAL RESOLUTIONS OF SINGULARITIES
, 2006
"... Abstract. Let Y be a singular algebraic variety and let ˜ Y be a resolution of singularities of Y. Assume that the exceptional locus of ˜Y over Y is an irreducible divisor ˜Z in ˜Y. For every Lefschetz decomposition of ˜ Z we construct a triangulated subcategory ˜ D ⊂ D b ( ˜ Y) which gives a desin ..."
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Cited by 6 (5 self)
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Abstract. Let Y be a singular algebraic variety and let ˜ Y be a resolution of singularities of Y. Assume that the exceptional locus of ˜Y over Y is an irreducible divisor ˜Z in ˜Y. For every Lefschetz decomposition of ˜ Z we construct a triangulated subcategory ˜ D ⊂ D b ( ˜ Y) which gives a desingularization of D b (Y). If the Lefschetz decomposition is generated by a vector bundle tilting over Y then ˜ D is a noncommutative resolution, and if the Lefschetz decomposition is rectangular, then ˜ D is a crepant resolution. 1.
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 4 (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.