A Sharp Compactness Theorem for Genus-One Pseudo-Holomorphic Maps

Abstract

For each compact almost Kahler manifold $(X,\om,J)$ and an element A of $H_2(X;Z)$, we describe a closed subspace $\ov{\frak M}_{1,k}^0(X,A;J)$ of the moduli space $\ov{\frak M}_{1,k}(X,A;J)$ of stable J-holomorphic genus-one maps such that $\ov{\frak M}_{1,k}^0(X,A;J)$ contains all stable maps with smooth domains. If $(P^n,\om,J_0)$ is the standard complex projective space, $\ov{\frak M}_{1,k}^0(P^n,A;J_0)$ is an irreducible component of $\ov{\frak M}_{1,k}(P^n,A;J_0)$. We also show that if an almost complex structure $J$ on $P^n$ is sufficiently close to J_0, the structure of the space $\ov{\frak M}_{1,k}^0(P^n,A;J)$ is similar to that of $\ov{\frak M}_{1,k}^0(P^n,A;J_0)$. 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 $\ov\M_{1,k}^0(X,A;J)$ is useful for computing the genus-one Gromov-Witten invariants, which arise from the larger moduli space $\ov\M_{1,k}(X,A;J)$.