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 M0 1,k(X,A; J) of the moduli space M⁰_1,k(X,A; J) of stable J-holomorphic genus-one maps such that M⁰ _1,k(X,A; J) contains all stable maps with smooth domains. If (Pⁿ, ω, J₀) is the standard complex projective space, M⁰ _1,k(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 M⁰ _1,k(Pⁿ,A; J) is similar to that of M⁰ _1,k(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 M⁰ _1,k(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).