## Geometry & Topology

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

Aleksey Zinger

#### Abstract

For every compact almost Kahler manifold $(X,ω,J)$ and an integral second homology class $A$, we describe a natural closed subspace $M¯1,k0(X,A;J)$ of the moduli space $M¯1,k(X,A;J)$ of stable $J$–holomorphic genus-one maps such that $M¯1,k0(X,A;J)$ contains all stable maps with smooth domains. If $(ℙn,ω,J0)$ is the standard complex projective space, $M¯1,k0(ℙn,A;J0)$ is an irreducible component of $M¯1,k(ℙn,A;J0)$. We also show that if an almost complex structure $J$ on $ℙn$ is sufficiently close to $J0$, the structure of the space $M¯1,k0(ℙn,A;J)$ is similar to that of $M¯1,k0(ℙn,A;J0)$. 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 $M¯1,k0(X,A;J)$ is useful for computing the genus-one Gromov–Witten invariants, which arise from the larger moduli space $M¯1,k(X,A;J)$.

#### Article information

Source
Geom. Topol., Volume 13, Number 5 (2009), 2427-2522.

Dates
Revised: 7 December 2008
Accepted: 8 May 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800316

Digital Object Identifier
doi:10.2140/gt.2009.13.2427

Mathematical Reviews number (MathSciNet)
MR2529940

Zentralblatt MATH identifier
1174.14012

#### Citation

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

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