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172
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 78 (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.
Relative virtual localization and vanishing of tautological classes on moduli spaces of curves
 Duke Math. J
"... ABSTRACT. We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i − g + 1 genus 0 components. As ..."
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Cited by 68 (6 self)
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ABSTRACT. We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i − g + 1 genus 0 components. As consequences, we prove and generalize various conjectures and theorems about various moduli spaces of curves (due to Getzler, Ionel, Faber, Looijenga, Pandharipande, Diaz, and others). This theorem appears to be the geometric content behind these results; the rest is straightforward graph combinatorics. The theorem also suggests the importance of the stratification of the moduli space by number of rational components. CONTENTS
Toric degenerations of toric varieties and tropical curves
 Duke Math. J
"... Abstract. We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraicgeometric and relies on degeneration techniques and log deformation theory. Conte ..."
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Cited by 65 (5 self)
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Abstract. We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraicgeometric and relies on degeneration techniques and log deformation theory. Contents
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 64 (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
Relative maps and tautological classes
"... 0.1. Tautological classes. Let Mg,n be the moduli space of stable curves of genus g with n marked points defined over C. Let A ∗ (Mg,n) denote the Chow ring (always taken here with Qcoefficients). The system of tautological rings ..."
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Cited by 62 (9 self)
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0.1. Tautological classes. Let Mg,n be the moduli space of stable curves of genus g with n marked points defined over C. Let A ∗ (Mg,n) denote the Chow ring (always taken here with Qcoefficients). The system of tautological rings
Algebraic cobordism revisited
"... Abstract. We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provid ..."
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Cited by 52 (7 self)
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Abstract. We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative DonaldsonThomas theory. We use double point cobordism to prove all the degree 0 conjectures in DonaldsonThomas theory: absolute, relative, and equivariant. 0.1. Overview. A first idea for defining cobordism in algebraic geometry is to impose the relation
On a proof of a conjecture of MarinoVafa on Hodge integrals
"... Abstract. We prove a remarkable formula for Hodge integrals conjectured by Mariño and Vafa [23] based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable morphisms. 1. ..."
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Cited by 46 (15 self)
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Abstract. We prove a remarkable formula for Hodge integrals conjectured by Mariño and Vafa [23] based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable morphisms. 1.
Towards the geometry of double Hurwitz numbers
 Advances Math
"... ABSTRACT. Double Hurwitz numbers count branched covers of CP 1 with fixed branch points, with simple branching required over all but two points 0 and ∞, and the branching over 0 and ∞ points specified by partitions of the degree (with m and n parts respectively). Single Hurwitz numbers (or more usua ..."
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Cited by 44 (6 self)
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ABSTRACT. Double Hurwitz numbers count branched covers of CP 1 with fixed branch points, with simple branching required over all but two points 0 and ∞, and the branching over 0 and ∞ points specified by partitions of the degree (with m and n parts respectively). Single Hurwitz numbers (or more usually, Hurwitz numbers) have a rich structure, explored by many authors in fields as diverse as algebraic geometry, symplectic geometry, combinatorics, representation theory, and mathematical physics. A remarkable formula of Ekedahl, Lando, M. Shapiro, and Vainshtein (the ELSV formula) relates single Hurwitz numbers to intersection theory on the moduli space of curves. This connection has led to many consequences, including Okounkov and Pandharipande’s proof of Witten’s conjecture (Kontsevich’s theorem) connecting intersection theory on the moduli space of curves to integrable systems. In this paper, we determine the structure of double Hurwitz numbers using techniques from geometry, algebra, and representation theory. Our motivation is geometric: we give strong evidence that double Hurwitz numbers are top intersections on a moduli space of curves with a line bundle (a universal Picard variety). In particular, we prove a piecewisepolynomiality
Holomorphic disks, link invariants, and the multivariable Alexander polynomial
, 2007
"... We define a Floerhomology invariant for links in S 3, and study its properties. ..."
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Cited by 44 (9 self)
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We define a Floerhomology invariant for links in S 3, and study its properties.
Absolute and relative GromovWitten invariants of very ample hypersurfaces
 Duke Math. J
"... For any smooth complex projective variety X and any smooth very ample hypersurface Y ⊂ X, we develop the technique of genus zero relative GromovWitten invariants of Y in X in algebrogeometric terms. We prove an equality of cycles in the Chow groups of the moduli spaces of relative stable maps whic ..."
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Cited by 43 (3 self)
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For any smooth complex projective variety X and any smooth very ample hypersurface Y ⊂ X, we develop the technique of genus zero relative GromovWitten invariants of Y in X in algebrogeometric terms. We prove an equality of cycles in the Chow groups of the moduli spaces of relative stable maps which relates these relative invariants to the GromovWitten invariants of X and Y. Given the GromovWitten invariants of X, we show that these relations are sufficient to compute all relative invariants, as well as all genus zero GromovWitten invariants of Y whose homology and cohomology classes are induced by X. Much work has been done recently on GromovWitten invariants related to hypersurfaces. There are essentially two different problems that have been studied. The first one is the question, how can one compute the GromovWitten invariants of a hypersurface from those of the ambient variety (see [Be], [G], [K], [LLY])? The second