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22
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.
Virtual pullbacks
, 805
"... We propose a generalization of Gysin maps for DMtype morphisms of stacks F → G that admit a perfect relative obstruction theory E • F/G. We prove functoriality properties of the generalized Gysin maps. As applications, we analyze GromovWitten invariants of blowups and ..."
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Cited by 14 (2 self)
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We propose a generalization of Gysin maps for DMtype morphisms of stacks F → G that admit a perfect relative obstruction theory E • F/G. We prove functoriality properties of the generalized Gysin maps. As applications, we analyze GromovWitten invariants of blowups and
Pointed admissible Gcovers and Gequivariant cohomological Field Theories
 Compositio Math
"... Abstract. For any finite group G we define the moduli space of pointed admissible Gcovers and the concept of a Gequivariant cohomological field theory (GCohFT), which, when G is the trivial group, reduce to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. ..."
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Cited by 14 (6 self)
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Abstract. For any finite group G we define the moduli space of pointed admissible Gcovers and the concept of a Gequivariant cohomological field theory (GCohFT), which, when G is the trivial group, reduce to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that taking the “quotient ” by Greduces a GCohFT to a CohFT. We also prove that a GCohFT contains a GFrobenius algebra, a Gequivariant generalization of a Frobenius algebra, and that the “quotient ” by G agrees with the obvious Frobenius algebra structure on the space of Ginvariants, after rescaling the metric. We then introduce the moduli space of Gstable maps into a smooth, projective variety X with G action. GromovWittenlike invariants of these spaces provide the primary source of examples of GCohFTs. Finally, we explain how these constructions generalize (and unify) the ChenRuan orbifold GromovWitten invariants of [X/G] as well as the ring H • (X,G) of Fantechi and Göttsche. 1.
Generating functions for HurwitzHodge integrals
, 2008
"... In this paper we describe explicit generating functions for a large class of HurwitzHodge integrals. These are integrals of tautological classes on moduli spaces of admissible covers, a (stackily) smooth compactification of the Hurwitz schemes. Admissible covers and their tautological classes are i ..."
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Cited by 10 (5 self)
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In this paper we describe explicit generating functions for a large class of HurwitzHodge integrals. These are integrals of tautological classes on moduli spaces of admissible covers, a (stackily) smooth compactification of the Hurwitz schemes. Admissible covers and their tautological classes are interesting mathematical objects on their own, but recently they have proved to be a useful tool for the study of the tautological ring of the moduli space of curves, and the orbifold GromovWitten theory of DM stacks. Our main tool is AtiyahBott localization: its underlying philosophy is to translate an interesting geometric problem into a purely combinatorial one.
GromovWitten theory of étale gerbes I: root gerbes
, 2009
"... Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r> 1 we study the GromovWitten theory of the stack rp L/X of rth root of L. We prove an exact formula expressing genus 0 GromovWitten invariants of rp L/X in terms of those of X. Assuming that ..."
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Cited by 9 (3 self)
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Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r> 1 we study the GromovWitten theory of the stack rp L/X of rth root of L. We prove an exact formula expressing genus 0 GromovWitten invariants of rp L/X in terms of those of X. Assuming that either rp L/X or X has semisimple quantum cohomology, we prove an exact formula between higher genus invariants. We also present constructions of moduli stacks of twisted stable maps to rp L/X starting from moduli stack of stable maps to X.
Comparison theorems for GromovWitten invariants of smooth pairs and of degenerations
, 2013
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