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72
Lagrangian Floer theory on compact toric manifolds: Survey
, 2010
"... This is a survey of a series of papers [FOOO3, FOOO4, FOOO5]. We discuss the calculation of the Floer cohomology of Lagrangian submanifold which is a T n orbit in a compact toric manifold. Applications to symplectic topology and to mirror symmetry are also discussed. ..."
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Cited by 79 (8 self)
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This is a survey of a series of papers [FOOO3, FOOO4, FOOO5]. We discuss the calculation of the Floer cohomology of Lagrangian submanifold which is a T n orbit in a compact toric manifold. Applications to symplectic topology and to mirror symmetry are also discussed.
The crepant resolution conjecture
, 2006
"... Abstract. For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the GromovWitten theories of the orbifold and the resolution. We prove the conjecture for the equivariant GromovWitten theories of Sym n C 2 and Hilb n C ..."
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Cited by 42 (8 self)
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Abstract. For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the GromovWitten theories of the orbifold and the resolution. We prove the conjecture for the equivariant GromovWitten theories of Sym n C 2 and Hilb n C 2. 1.
WALLCROSSINGS IN TORIC GROMOV–WITTEN THEORY I: CREPANT EXAMPLES
, 2006
"... Graber asserts that certain generating functions for genuszero Gromov–Witten invariants of an orbifold X can be obtained from their counterparts for a crepant resolution of X by analytic continuation followed by specialization of parameters. In this paper we use mirror symmetry to determine the rel ..."
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Cited by 18 (4 self)
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Graber asserts that certain generating functions for genuszero Gromov–Witten invariants of an orbifold X can be obtained from their counterparts for a crepant resolution of X by analytic continuation followed by specialization of parameters. In this paper we use mirror symmetry to determine the relationship between the genuszero Gromov–Witten invariants of the weighted projective spaces P(1, 1, 2), P(1, 1, 1, 3) and those of their crepant resolutions. Our methods are applicable to other toric birational transformations. Our results verify the Crepant Resolution Conjecture when X = P(1, 1, 2) and suggest that it needs modification when
On the crepant resolution conjecture in the local case
"... Abstract. In this paper we analyze four examples of birational transformations between local Calabi– Yau 3folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genuszero Gromov–Witten invariants, proving the Coates–Corti–Iritani– ..."
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Cited by 15 (2 self)
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Abstract. In this paper we analyze four examples of birational transformations between local Calabi– Yau 3folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genuszero Gromov–Witten invariants, proving the Coates–Corti–Iritani– Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan’s original Crepant Resolution Conjecture should be modified, by including appropriate “quantum corrections”, and that there is no straightforward generalization of either Ruan’s original Conjecture or the Cohomological Crepant Resolution Conjecture to the case of crepant partial resolutions. Our methods are based on mirror symmetry for toric orbifolds. 1.
Gross fibrations, SYZ mirror symmetry, and open GromovWitten invariants for toric CalabiYau orbifolds
, 2013
"... Given a toric CalabiYau orbifold X whose underlying toric variety is semiprojective, we construct and study a nontoric Lagrangian torus fibration on X, which we call the Gross fibration. We apply the StromingerYauZaslow recipe to the Gross fibration of (a toric modification of) X to construct ..."
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Cited by 10 (6 self)
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Given a toric CalabiYau orbifold X whose underlying toric variety is semiprojective, we construct and study a nontoric Lagrangian torus fibration on X, which we call the Gross fibration. We apply the StromingerYauZaslow recipe to the Gross fibration of (a toric modification of) X to construct its instantoncorrected mirror, where the instanton corrections come from genus 0 open orbifold GromovWitten invariants, which are virtual counts of holomorphic orbidisks in X bounded by fibers of the Gross fibration. We explicitly evaluate all these invariants by first proving an open/closed equality and then employing the toric mirror theorem for suitable toric (parital) compactifications of X. Our calculations are then applied to (1) prove a conjecture of GrossSiebert on a relation between genus 0 open orbifold GromovWitten invariants and mirror maps of X – this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and (2) demonstrate how open (orbifold) GromovWitten invariants for toric CalabiYau orbifolds change under toric crepant resolutions – this is an open analogue of Ruan’s crepant resolution conjecture.
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.