#### Volume 13, issue 5 (2009)

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

### Aleksey Zinger

Geometry & Topology 13 (2009) 2427–2522
##### Abstract

For every compact almost Kahler manifold $\left(X,\omega ,J\right)$ and an integral second homology class $A$, we describe a natural closed subspace ${\overline{\mathfrak{M}}}_{1,k}^{0}\left(X,A;J\right)$ of the moduli space ${\overline{\mathfrak{M}}}_{1,k}\left(X,A;J\right)$ of stable $J$–holomorphic genus-one maps such that ${\overline{\mathfrak{M}}}_{1,k}^{0}\left(X,A;J\right)$ contains all stable maps with smooth domains. If $\left({ℙ}^{n},\omega ,{J}_{0}\right)$ is the standard complex projective space, ${\overline{\mathfrak{M}}}_{1,k}^{0}\left({ℙ}^{n},A;{J}_{0}\right)$ is an irreducible component of ${\overline{\mathfrak{M}}}_{1,k}\left({ℙ}^{n},A;{J}_{0}\right)$. We also show that if an almost complex structure $J$ on ${ℙ}^{n}$ is sufficiently close to ${J}_{0}$, the structure of the space ${\overline{\mathfrak{M}}}_{1,k}^{0}\left({ℙ}^{n},A;J\right)$ is similar to that of ${\overline{\mathfrak{M}}}_{1,k}^{0}\left({ℙ}^{n},A;{J}_{0}\right)$. 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 ${\overline{\mathfrak{M}}}_{1,k}^{0}\left(X,A;J\right)$ is useful for computing the genus-one Gromov–Witten invariants, which arise from the larger moduli space ${\overline{\mathfrak{M}}}_{1,k}\left(X,A;J\right)$.

##### Keywords
genus one Gromov–Witten invariant, pseudo-holomorphic map, Gromov compactness theorem, genus one
Primary: 14D20
Secondary: 53D99