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

For each compact almost Kahler manifold (X,ω, J) and an element A of H2(X ;Z), we describe a natural closed subspace M 0 1,k(X,A; J) of the moduli space M1,k(X,A; J) of stable J-holomorphic genus-one maps such that M 0 1,k(X,A; J) contains all stable maps with smooth domains. If (P, ω, J0) is the standard complex projective space, M 0 1,k(P , A; J0) is an irreducible component of M1,k(P, A; J0). We also show that if an almost complex structure J on P is sufficiently close to J0, the structure of the space M 0 1,k(P , A; J) is similar to that of M 0 1,k(P , 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 0 1,k(X,A; J) is useful for computing the genus-one Gromov-Witten invariants, which arise from the larger moduli space M1,k(X,A; J).