A sharp compactness theorem for genus-one pseudo-holomorphic maps

Zinger, Aleksey

doi:10.2140/gt.2009.13.2427

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A sharp compactness theorem for genus-one pseudo-holomorphic maps

Aleksey Zinger

Geometry & Topology 13 (2009) 2427–2522

DOI: 10.2140/gt.2009.13.2427

Abstract

For every compact almost Kahler manifold $(X, ω, J)$ and an integral second homology class $A$ , we describe a natural closed subspace ${\bar{M}}_{1, k}^{0} (X, A; J)$ of the moduli space ${\bar{M}}_{1, k} (X, A; J)$ of stable $J$ –holomorphic genus-one maps such that ${\bar{M}}_{1, k}^{0} (X, A; J)$ contains all stable maps with smooth domains. If $(ℙ^{n}, ω, J_{0})$ is the standard complex projective space, ${\bar{M}}_{1, k}^{0} (ℙ^{n}, A; J_{0})$ is an irreducible component of ${\bar{M}}_{1, k} (ℙ^{n}, A; J_{0})$ . We also show that if an almost complex structure $J$ on $ℙ^{n}$ is sufficiently close to $J_{0}$ , the structure of the space ${\bar{M}}_{1, k}^{0} (ℙ^{n}, A; J)$ is similar to that of ${\bar{M}}_{1, k}^{0} (ℙ^{n}, A; J_{0})$ . This paper’s compactness and structure theorems lead to new invariants for some symplectic manifolds, which are generalized to arbitrary symplectic manifolds in a separate paper. Relatedly, the smaller moduli space ${\bar{M}}_{1, k}^{0} (X, A; J)$ is useful for computing the genus-one Gromov–Witten invariants, which arise from the larger moduli space ${\bar{M}}_{1, k} (X, A; J)$ .

Keywords

genus one Gromov–Witten invariant, pseudo-holomorphic map, Gromov compactness theorem, genus one

Mathematical Subject Classification 2000

Primary: 14D20

Secondary: 53D99

References

Publication

Received: 7 August 2007

Revised: 7 December 2008

Accepted: 8 May 2009

Published: 13 June 2009

Proposed: Gang Tian

Seconded: Jim Bryan, Yasha Eliashberg

Authors

Aleksey Zinger
Department of Mathematics SUNY Stony Brook Stony Brook, NY 11794-3651 USA
http://www.math.sunysb.edu/~azinger

Volume 13, issue 5 (2009)