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243
Existence of minimal models for varieties of log general type
 J. AMER. MATH. SOC
, 2008
"... We prove that the canonical ring of a smooth projective variety is finitely generated. ..."
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Cited by 386 (34 self)
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We prove that the canonical ring of a smooth projective variety is finitely generated.
The Intrinsic Normal Cone
 Invent. Math
, 1997
"... We suggest a construction of virtual fundamental classes of certain types of moduli spaces. Contents 0 ..."
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Cited by 353 (9 self)
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We suggest a construction of virtual fundamental classes of certain types of moduli spaces. Contents 0
Deformations Of Calibrated Submanifolds
 Commun. Analy. Geom
, 1996
"... . Assuming the ambient manifold is Kahler, the theory of complex submanifolds can be placed in the more general context of calibrated submanifolds, see [HL]. It is therefore natural to try to extend some of the many results in complex geometry to the other calibrated geometries of [HL]. In particula ..."
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Cited by 223 (0 self)
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. Assuming the ambient manifold is Kahler, the theory of complex submanifolds can be placed in the more general context of calibrated submanifolds, see [HL]. It is therefore natural to try to extend some of the many results in complex geometry to the other calibrated geometries of [HL]. In particular, the question of deformability of calibrated submanifolds is addressed here (analogous to Kodaira's work on deformations of complex submanifolds [K]). Also, a formula for the second variation of volume of an arbitrary calibrated submanifolds which generalizes a result of Simons' in the complex category [S] is given. 1. Introduction and summary General Remarks. In this paper, we discuss the deformation theory of calibrated submanifolds of Riemannian manifolds with restricted holonomy. Most of the definitions of the terms used in this introduction can be found in the seminal paper of Harvey and Lawson ([HL]). Throughout, we denote the ambient manifold by M and the submanifold by X. Recall ...
GromovWitten invariants in algebraic geometry
, 1996
"... GromovWitten invariants for arbitrary projective varieties and arbitrary genus are constructed using the techniques from [K. Behrend, B. Fantechi. The Intrinsic Normal Cone.] ..."
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Cited by 200 (2 self)
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GromovWitten invariants for arbitrary projective varieties and arbitrary genus are constructed using the techniques from [K. Behrend, B. Fantechi. The Intrinsic Normal Cone.]
Hilbert schemes, polygraphs, and the Macdonald positivity conjecture
 J. Amer. Math. Soc
"... The Hilbert scheme of points in the plane Hn = Hilb n (C2) is an algebraic variety which parametrizes finite subschemes S of length n in C2. To each such subscheme S corresponds an nelement multiset, or unordered ntuple with possible repetitions, σ(S) =[P1,...,Pn] of points in C2,wherethePiare the ..."
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Cited by 158 (4 self)
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The Hilbert scheme of points in the plane Hn = Hilb n (C2) is an algebraic variety which parametrizes finite subschemes S of length n in C2. To each such subscheme S corresponds an nelement multiset, or unordered ntuple with possible repetitions, σ(S) =[P1,...,Pn] of points in C2,wherethePiare the points of S, repeated with
Instantons and affine algebras I: The Hilbert scheme and vertex operators
 Math. Res. Letters
, 1996
"... Abstract. This is the first in a series of papers which describe the action of an affine Lie algebra with central charge n on the moduli space of U(n)instantons on a four manifold X. This generalises work of Nakajima, who considered the case when X is an ALE space. In particular, this should describ ..."
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Cited by 123 (1 self)
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Abstract. This is the first in a series of papers which describe the action of an affine Lie algebra with central charge n on the moduli space of U(n)instantons on a four manifold X. This generalises work of Nakajima, who considered the case when X is an ALE space. In particular, this should describe the combinatorial complexity of the moduli space as beingprecisely that of representation theory, and thus will lead to a description of the Betti numbers of moduli space as dimensions of weight spaces. This Lie algebra acts on the space of conformal blocks (i.e., the cohomology of a determinant line bundle on the moduli space [LMNS]) generalising the “insertion ” and “deletion ” operations of conformal field theory, and indeed on any cohomology theory. In the particular case of U(1)instantons, which is essentially the subject of this present paper, the construction produces the basic representation after FrenkelKac. Then the well known quadratic nature of ch2, ch2 = 1 2 c1 · c1 − c2 becomes precisely the formula for the eigenvalue of the degree operator, i.e. the well known quadratic behaviour of affine Lie algebras.
Equivalences of triangulated categories and FourierMukai transforms
 Bull. London Math. Soc
, 1999
"... Abstract. We give a condition for an exact functor between triangulated categories to be an equivalence. Applications to FourierMukai transforms are discussed. In particular we obtain a large number of such transforms for K3 surfaces. 1. ..."
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Cited by 123 (7 self)
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Abstract. We give a condition for an exact functor between triangulated categories to be an equivalence. Applications to FourierMukai transforms are discussed. In particular we obtain a large number of such transforms for K3 surfaces. 1.
Groupoids: unifying internal and external symmetry. A tour through some examples
 Notices Amer. Math. Soc
, 1996
"... Mathematicians tend to think of the notion of symmetry as being virtually synonymous with the theory of groups and their actions, perhaps largely because of the wellknown Erlanger program ..."
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Cited by 104 (4 self)
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Mathematicians tend to think of the notion of symmetry as being virtually synonymous with the theory of groups and their actions, perhaps largely because of the wellknown Erlanger program
STABLE MORPHISMS TO SINGULAR SCHEMES AND RELATIVE STABLE MORPHISMS
"... Let W/C be a degeneration of smooth varieties so that the special fiber has normal crossing singularity. In this paper, we first construct the stack of expanded degenerations of W. We then construct the moduli space of stable morphisms to this stack, which provides a degeneration of the moduli spa ..."
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Cited by 101 (5 self)
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Let W/C be a degeneration of smooth varieties so that the special fiber has normal crossing singularity. In this paper, we first construct the stack of expanded degenerations of W. We then construct the moduli space of stable morphisms to this stack, which provides a degeneration of the moduli spaces of stable morphisms associated to W/C. Using a similar technique, for a pair (Z, D) of smooth variety and a smooth divisor, we construct the stack of expanded relative pairs and then the moduli spaces of relative stable morphisms to (Z, D). This is the algebrogeometric analogue of DonaldsonFloer theory in gauge theory. The construction of relative GromovWitten invariants and the degeneration formula of GromovWitten invariants will be treated in the subsequent paper.