Geometry & Topology
- Geom. Topol.
- Volume 13, Number 5 (2009), 2427-2522.
A sharp compactness theorem for genus-one pseudo-holomorphic maps
For every compact almost Kahler manifold and an integral second homology class , we describe a natural closed subspace of the moduli space of stable –holomorphic genus-one maps such that contains all stable maps with smooth domains. If is the standard complex projective space, is an irreducible component of . We also show that if an almost complex structure on is sufficiently close to , the structure of the space is similar to that of . 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 is useful for computing the genus-one Gromov–Witten invariants, which arise from the larger moduli space .
Geom. Topol., Volume 13, Number 5 (2009), 2427-2522.
Received: 7 August 2007
Revised: 7 December 2008
Accepted: 8 May 2009
First available in Project Euclid: 20 December 2017
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Zinger, Aleksey. A sharp compactness theorem for genus-one pseudo-holomorphic maps. Geom. Topol. 13 (2009), no. 5, 2427--2522. doi:10.2140/gt.2009.13.2427. https://projecteuclid.org/euclid.gt/1513800316