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Logarithmic GromovWitten invariants
 Journal of the American Mathematical Society
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A mathematical theory of the topological vertex
"... Abstract. We have developed a mathematical theory of the topological vertex— a theory that was originally proposed by M. Aganagic, A. Klemm, M. Mariño, and C. Vafa on effectively computing GromovWitten invariants of smooth toric CalabiYau threefolds derived from duality between open string theory ..."
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Cited by 36 (19 self)
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Abstract. We have developed a mathematical theory of the topological vertex— a theory that was originally proposed by M. Aganagic, A. Klemm, M. Mariño, and C. Vafa on effectively computing GromovWitten invariants of smooth toric CalabiYau threefolds derived from duality between open string theory of smooth CalabiYau threefolds and ChernSimons theory on three manifolds. 1.
The moduli space of curves and GromovWitten theory
, 2006
"... The goal of this article is to motivate and describe how GromovWitten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from GromovWitten theory have led to both conjectures and theorems showing that the tautological part of the cohomology r ..."
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Cited by 26 (4 self)
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The goal of this article is to motivate and describe how GromovWitten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from GromovWitten theory have led to both conjectures and theorems showing that the tautological part of the cohomology ring has a remarkable and profound structure. As an illustration, we describe a new approach to Faber’s intersection number conjecture via branched covers of the projective line (work with I.P. Goulden and D.M. Jackson, based on work with T. Graber). En route we review the work of a large number of mathematicians.
GromovWitten invariants of varieties with holomorphic 2forms
"... Abstract. We show that a holomorphic twoform θ on a smooth algebraic variety X localizes the virtual fundamental class of the moduli of stable maps Mg,n(X, β) to the locus where θ degenerates; it then enables us to define the localized GWinvariant, an algebrogeometric analogue of the local invari ..."
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Cited by 21 (6 self)
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Abstract. We show that a holomorphic twoform θ on a smooth algebraic variety X localizes the virtual fundamental class of the moduli of stable maps Mg,n(X, β) to the locus where θ degenerates; it then enables us to define the localized GWinvariant, an algebrogeometric analogue of the local invariant of Lee and Parker in symplectic geometry [15], which coincides with the ordinary GWinvariant when X is proper. It is deformation invariant. Using this, we prove formulas for low degree GWinvariants of minimal general type surfaces with pg> 0 conjectured by Maulik and Pandharipande. 1.
Flops, motives and invariance of quantum rings
"... ABSTRACT. For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincaré pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the ..."
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Cited by 21 (9 self)
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ABSTRACT. For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincaré pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the extended Kähler moduli space. For Mukai flops, it is shown that the birational map for the local models is deformation equivalent to isomorphisms. This implies that the birational map induces isomorphisms on the full quantum rings and all the quantum corrections attached to the extremal ray vanish. 0.1. Statement of 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 → S the restriction map on the exceptional loci. Assume that
GromovWitten/Pairs correspondence for the quintic 3fold
, 2012
"... We use the GromovWitten/Pairs descendent correspondence for toric 3folds and degeneration arguments to establish the GW/P correspondence for several compact CalabiYau 3folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of ..."
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Cited by 18 (6 self)
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We use the GromovWitten/Pairs descendent correspondence for toric 3folds and degeneration arguments to establish the GW/P correspondence for several compact CalabiYau 3folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of the GW/P correspondence for descendents in relative geometries. Projective bundles over surfaces relative to a section play a special role. The GW/P correspondence for CalabiYau complete intersections provides a structure result for the GromovWitten invariants in a fixed curve class. After change of variables, the GromovWitten series is a rational function in the variable −q = e iu invariant under q ↔ q −1.
The moduli space of curves, double Hurwitz numbers, and Faber’s intersection number conjecture
, 2006
"... We define the dimension 2g − 1 FaberHurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P 1 with given ramification over ∞ and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localizatio ..."
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Cited by 17 (2 self)
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We define the dimension 2g − 1 FaberHurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P 1 with given ramification over ∞ and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localization expresses such classes in terms localization trees weighted by “top intersections” of tautological classes and genus 0 double Hurwitz numbers. This identity of generating series can be inverted, yielding a “combinatorialization” of top intersections of ψclasses. As genus 0 double Hurwitz numbers with at most 3 parts over ∞ are well understood, we obtain Faber’s Intersection Number Conjecture for up to 3 parts, and an approach to the Conjecture in general (bypassing the Virasoro Conjecture). We also recover other geometric results in a unified manner, including Looijenga’s theorem, the socle theorem for curves with rational tails, and the hyperelliptic locus in terms of κg−2.
Tropical Hurwitz numbers
"... Abstract. Hurwitz numbers count genus g, degree d covers of P 1 with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piecewise linear objects called tropical curves. This paper develops a ..."
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Cited by 15 (5 self)
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Abstract. Hurwitz numbers count genus g, degree d covers of P 1 with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piecewise linear objects called tropical curves. This paper develops a tropical counterpart of the branch map and shows that its degree recovers classical Hurwitz numbers. 1.
THE LOCAL GROMOVWITTEN INVARIANTS OF CONFIGURATIONS OF RATIONAL CURVES
, 2005
"... ABSTRACT. We compute the local GromovWitten invariants of certain configurations of rational curves in a CalabiYau threefold. These configurations are connected subcurves of the “minimal trivalent configuration”, which is a particular tree of P 1 ’s with specified formal neighborhood. We show that ..."
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Cited by 14 (3 self)
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ABSTRACT. We compute the local GromovWitten invariants of certain configurations of rational curves in a CalabiYau threefold. These configurations are connected subcurves of the “minimal trivalent configuration”, which is a particular tree of P 1 ’s with specified formal neighborhood. We show that these local invariants are equal to certain global or ordinary GromovWitten invariants of a blowup of P 3 at points, and we compute these ordinary invariants using the geometry of the Cremona transform. We also realize the configurations in question as formal toric schemes and compute their formal GromovWitten invariants using the mathematical and physical theories of the topological vertex. In particular, we provide further evidence equating the vertex amplitudes derived from physical and mathematical theories of the topological vertex. 1.