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51
Orbifold cohomology for global quotients
 DUKE MATH. J
, 2003
"... Let X be an orbifold that is a global quotient of a manifold Y by a finite group G. We construct a noncommutative ring H ∗ (Y, G) with a Gaction such that H ∗ (Y, G) G is the orbifold cohomology ring of X defined by W. Chen and Y. Ruan [CR]. When Y = S n, with S a surface with trivial canonical cla ..."
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Cited by 68 (1 self)
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Let X be an orbifold that is a global quotient of a manifold Y by a finite group G. We construct a noncommutative ring H ∗ (Y, G) with a Gaction such that H ∗ (Y, G) G is the orbifold cohomology ring of X defined by W. Chen and Y. Ruan [CR]. When Y = S n, with S a surface with trivial canonical class and G = Sn, we prove that (a small modification of) the orbifold cohomology of X is naturally isomorphic to the cohomology ring of the Hilbert scheme S [n] , computed by M. Lehn and C. Sorger [LS2].
Cohomology ring of crepant resolutions of orbifolds
, 2001
"... Suppose that X is an orbifold. In general, KX is an orbifold vector bundle or a Qdivisor only. When the X is so called Gorenstein, KX is a bundle or a divisor. For Gorenstein orbifold, a resolution π: Y → X is called a crepant resolution if π ∗ KX = KY. Here, ”crepant ” can be viewed as a minimalit ..."
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Cited by 63 (4 self)
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Suppose that X is an orbifold. In general, KX is an orbifold vector bundle or a Qdivisor only. When the X is so called Gorenstein, KX is a bundle or a divisor. For Gorenstein orbifold, a resolution π: Y → X is called a crepant resolution if π ∗ KX = KY. Here, ”crepant ” can be viewed as a minimality condition with respect to canonical bundle. Crepant resolution always exists when
The cup product of the Hilbert scheme for K3 surfaces
"... Abstract. To any graded Frobenius algebra A we associate a sequence of graded Frobenius algebras A [n] so that there is canonical isomorphism of rings (H ∗ (X; Q)[2]) [n] ∼ = H (X ..."
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Cited by 47 (0 self)
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Abstract. To any graded Frobenius algebra A we associate a sequence of graded Frobenius algebras A [n] so that there is canonical isomorphism of rings (H ∗ (X; Q)[2]) [n] ∼ = H (X
Resolution of stringy singularities by noncommutative algebras
 JHEP 0106
"... Preprint typeset in JHEP style. PAPER VERSION ..."
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Twisted jet, motivic measure and orbifold cohomology
, 2001
"... We introduce the notion of twisted jet. For a DeligneMumford stack X of finite type over an algebraically closed field k, a twisted ∞jet on X is a representable morphism D → X such that D is a smooth DeligneMumford stack with the coarse moduli space Spec k[[t]]. We study the motivic measure on t ..."
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Cited by 34 (2 self)
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We introduce the notion of twisted jet. For a DeligneMumford stack X of finite type over an algebraically closed field k, a twisted ∞jet on X is a representable morphism D → X such that D is a smooth DeligneMumford stack with the coarse moduli space Spec k[[t]]. We study the motivic measure on the space of the twisted ∞jets on a smooth DeligneMumford stack. As an application, we prove that two birational minimal models with Gorenstein quotient singularities have the same orbifold cohomology with Hodge structure. Our main results are Theorem 0.4 and 0.5.
Orbifold cohomology of the symmetric product
 Comm. Anal. Geom
"... Abstract. Chen and Ruan’s orbifold cohomology of the symmetric product of a complex manifold is calculated. An isomorphism of rings (up to a change of signs) H ∗ orb (Xn /Sn; C) ∼ = H ∗ (X [n] ; C) between the orbifold cohomology of the symmetric product of a smooth projective surface with trivial ..."
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Cited by 31 (4 self)
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Abstract. Chen and Ruan’s orbifold cohomology of the symmetric product of a complex manifold is calculated. An isomorphism of rings (up to a change of signs) H ∗ orb (Xn /Sn; C) ∼ = H ∗ (X [n] ; C) between the orbifold cohomology of the symmetric product of a smooth projective surface with trivial canonical class X and the cohomology of its Hilbert scheme X [n] is obtained, yielding a positive answer to a conjecture of Ruan. 1.
On the motive of the Hilbert scheme of points on a surface
"... The Hilbert scheme S [n] of points on an algebraic surface S is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power S (n). For many phenomena expected for moduli spaces and nice resolutions of singular varieties it is a model case. Hilbert ..."
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Cited by 24 (0 self)
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The Hilbert scheme S [n] of points on an algebraic surface S is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power S (n). For many phenomena expected for moduli spaces and nice resolutions of singular varieties it is a model case. Hilbert schemes of points have connections to several fields of mathematics, including moduli spaces of sheaves, Donaldson invariants, enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras and also to theoretical physics. This talk will try to give an overview over these connections.
Motivic integration over DeligneMumford stacks
, 2004
"... The aim of this article is to develop the theory of motivic integration over DeligneMumford stacks and to apply it to the birational geometry of DeligneMumford stacks. ..."
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Cited by 18 (4 self)
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The aim of this article is to develop the theory of motivic integration over DeligneMumford stacks and to apply it to the birational geometry of DeligneMumford stacks.