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Pseudoholomorphic tori in the Kodaira–Thurston manifold

Part of: Projective and enumerative geometry Symplectic geometry, contact geometry Complex manifolds

Published online by Cambridge University Press:  16 July 2015

Jonathan David Evans  and
Jarek Kędra
Jonathan David Evans
Affiliation:
Department of Mathematics, University College London, Gower Street, LondonWC1E 6BT, UK email j.d.evans@ucl.ac.uk
Jarek Kędra
Affiliation:
University of Aberdeen, UK University of Szczecin, Poland email kedra@abdn.ac.uk
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Abstract

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The Kodaira–Thurston manifold is a quotient of a nilpotent Lie group by a cocompact lattice. We compute the family Gromov–Witten invariants which count pseudoholomorphic tori in the Kodaira–Thurston manifold. For a fixed symplectic form the Gromov–Witten invariant is trivial so we consider the twistor family of left-invariant symplectic forms which are orthogonal for some fixed metric on the Lie algebra. This family defines a loop in the space of symplectic forms. This is the first example of a genus one family Gromov–Witten computation for a non-Kähler manifold.

Keywords

family Gromov–Witten invariantpseudoholomorphic curvenon-KählerKodaira–Thurstonnilpotent Lie group

MSC classification

Primary: 32Q65: Pseudoholomorphic curves 53D35: Global theory of symplectic and contact manifolds 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 12 , December 2015 , pp. 2212 - 2250
Copyright
© The Authors 2015 

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