Geometry & Topology

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

Aleksey Zinger

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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).

Article information

Geom. Topol., Volume 13, Number 5 (2009), 2427-2522.

Received: 7 August 2007
Revised: 7 December 2008
Accepted: 8 May 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}
Secondary: 53D99: None of the above, but in this section

genus one Gromov–Witten invariant pseudo-holomorphic map Gromov compactness theorem genus one


Zinger, Aleksey. A sharp compactness theorem for genus-one pseudo-holomorphic maps. Geom. Topol. 13 (2009), no. 5, 2427--2522. doi:10.2140/gt.2009.13.2427.

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