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19
RUAN’S CONJECTURE AND INTEGRAL STRUCTURES IN QUANTUM COHOMOLOGY
, 2008
"... This is an expository article on the recent studies [23, 24, 44, 19] of Ruan’s crepant resolution/flop conjecture [59, 60] and its possible relations to the Ktheory integral structure [44, 50] in quantum cohomology. ..."
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Cited by 9 (4 self)
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This is an expository article on the recent studies [23, 24, 44, 19] of Ruan’s crepant resolution/flop conjecture [59, 60] and its possible relations to the Ktheory integral structure [44, 50] in quantum cohomology.
Invariance of Gromov–Witten theory under simple flops
 J. Reine Angew. Math
"... ABSTRACT. We show that the generating functions of Gromov–Witten invariants with ancestors are invariant under a simple flop, for all genera, after an analytic continuation in the extended Kähler moduli space. This is a sequel to [14]. 0.1. Statement of the main results. Let X be a smooth complex p ..."
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Cited by 7 (5 self)
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ABSTRACT. We show that the generating functions of Gromov–Witten invariants with ancestors are invariant under a simple flop, for all genera, after an analytic continuation in the extended Kähler moduli space. This is a sequel to [14]. 0.1. Statement of the main results. Let X be a smooth complex projective manifold and ψ: X → X ̄ a flopping contraction in the sense of minimal model theory, with ψ ̄ : Z ∼ = Pr → pt the restriction map to the extremal contraction. Assume that NZ/X ∼ = OPr(−1)⊕(r+1). It was shown
Notes on axiomatic Gromov–Witten theory and applications
"... The purpose of these notes is to give their readers some idea of Givental’s axiomatic Gromov–Witten theory, and a few applications. Due to the scope of these notes, some statements are not precisely formulated and almost all proofs are omitted. However, we try to ..."
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Cited by 7 (4 self)
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The purpose of these notes is to give their readers some idea of Givental’s axiomatic Gromov–Witten theory, and a few applications. Due to the scope of these notes, some statements are not precisely formulated and almost all proofs are omitted. However, we try to
INVARIANCE OF QUANTUM RINGS UNDER ORDINARY FLOPS
"... ABSTRACT. For ordinary flops over a smooth base, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. If the flop is of splitting type, the big quantum cohomology ring is also shown to b ..."
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Cited by 4 (3 self)
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ABSTRACT. For ordinary flops over a smooth base, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. If the flop is of splitting type, the big quantum cohomology ring is also shown to be invariant after an analytic continuation in the Kähler moduli space. Viewed from the context of the Kequivalence (crepant transformation) conjecture, there are two new features of our results. First, there is no semipositivity assumption on the varieties. Second, the local structure of the exceptional loci can not be deformed to any explicit (e.g. toric) geometry and the analytic continuation is nontrivial. This excludes the possibility of an ad hoc comparison by explicit computation of both sides. To achieve that, we have to clear a few technical hurdles. One technical breakthrough is a quantum Leray–Hirsch theorem for the local models (a certain toric bundle) which extends the quantum D modules of Dubrovin connection on the base by a Picard–Fuchs system of the toric fibers. Nonsplit flops as well as further applications of the quantum Leray– Hirsch theorem will be discussed in subsequent papers.
On stratified Mukai flops
, 2006
"... In recent studies (see [Na2], [Fu]) of birational geometry of symplectic resolutions of nilpotent orbit closures, three types of flops (which will be called stratified Mukai flops of type A,D,E6) are shown to be fundamental, in the sense that others can be decomposed into a sequence of these flops. ..."
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Cited by 3 (1 self)
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In recent studies (see [Na2], [Fu]) of birational geometry of symplectic resolutions of nilpotent orbit closures, three types of flops (which will be called stratified Mukai flops of type A,D,E6) are shown to be fundamental, in the sense that others can be decomposed into a sequence of these flops. Stratified
The Crepant Transformation Conjecture for toric complete intersections
, 2014
"... Let X and Y be Kequivalent toric Deligne–Mumford stacks related by a single toric wallcrossing. We prove the Crepant Transformation Conjecture in this case, fullyequivariantly and in genus zero. That is, we show that the equivariant quantum connections for X and Y become gaugeequivalent after a ..."
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Let X and Y be Kequivalent toric Deligne–Mumford stacks related by a single toric wallcrossing. We prove the Crepant Transformation Conjecture in this case, fullyequivariantly and in genus zero. That is, we show that the equivariant quantum connections for X and Y become gaugeequivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Givental spaces for X and Y, with a Fourier–Mukai transformation between the Kgroups of X and Y, via an equivariant version of the Gammaintegral structure on quantum cohomology. We prove similar results for toric complete intersections. We impose only very weak geometric hypotheses on X and Y: they can be noncompact, for example, and need not be weak Fano or have Gorenstein coarse moduli space. Our main tools are the Mirror Theorems for toric Deligne–Mumford stacks and toric complete intersections, and the Mellin–Barnes method for analytic continuation of hypergeometric functions.
MOTIVIC AND QUANTUM INVARIANCE UNDER STRATIFIED MUKAI FLOPS
, 801
"... ABSTRACT. For stratified Mukai flops of type A n,k, D 2k+1 and E6,I, it is shown the fiber product induces isomorphisms between Chow motives, cohomology rings and full GromovWitten theory. 1. ..."
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ABSTRACT. For stratified Mukai flops of type A n,k, D 2k+1 and E6,I, it is shown the fiber product induces isomorphisms between Chow motives, cohomology rings and full GromovWitten theory. 1.