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Logarithmic Gromov-Witten invariants

Authors: Mark Gross and Bernd Siebert
Journal: J. Amer. Math. Soc. 26 (2013), 451-510
MSC (2010): Primary 14D20, 14N35
Published electronically: November 20, 2012
MathSciNet review: 3011419
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Abstract: The goal of this paper is to give a general theory of logarithmic Gromov-Witten invariants. This gives a vast generalization of the theory of relative Gromov-Witten invariants introduced by Li-Ruan, Ionel-Parker, and Jun Li and completes a program first proposed by the second named author in 2002. One considers target spaces $X$ carrying a log structure. Domains of stable log curves are log smooth curves. Algebraicity of the stack of such stable log maps is proven, subject only to the hypothesis that the log structure on $X$ is fine, saturated, and Zariski. A notion of basic stable log map is introduced; all stable log maps are pull-backs of basic stable log maps via base-change. With certain additional hypotheses, the stack of basic stable log maps is proven to be proper. Under these hypotheses and the additional hypothesis that $X$ is log smooth, one obtains a theory of log Gromov-Witten invariants.

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Additional Information

Mark Gross
Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112
MR Author ID: 308804

Bernd Siebert
Affiliation: FB Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany

Received by editor(s): March 16, 2011
Received by editor(s) in revised form: August 26, 2011, and July 30, 2012
Published electronically: November 20, 2012
Additional Notes: This work was partially supported by NSF grants 0505325 and 0805328.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.