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

Aleksey Zinger

Geometry & Topology 13 (2009) 2427–2522

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 (n,ω,J0) is the standard complex projective space, M¯1,k0(n,A;J0) is an irreducible component of M¯1,k(n,A;J0). We also show that if an almost complex structure J on n is sufficiently close to J0, the structure of the space M¯1,k0(n,A;J) is similar to that of M¯1,k0(n,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).

genus one Gromov–Witten invariant, pseudo-holomorphic map, Gromov compactness theorem, genus one
Mathematical Subject Classification 2000
Primary: 14D20
Secondary: 53D99
Received: 7 August 2007
Revised: 7 December 2008
Accepted: 8 May 2009
Published: 13 June 2009
Proposed: Gang Tian
Seconded: Jim Bryan, Yasha Eliashberg
Aleksey Zinger
Department of Mathematics
SUNY Stony Brook
Stony Brook, NY 11794-3651