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Relative Gromov-Witten invariants and the mirror formula
Authors:Andreas Gathmann
Affiliation:1.Fachbereich Mathematik, Universit?t Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany (e-mail: andreas@rhrk.uni-kl.de),Germany
Abstract:Let X be a smooth complex projective variety, and let be a smooth very ample hypersurface such that is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called “mirror transformation”). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry. Received: 11 July 2001 / Published online: 4 February 2003 Funded by the DFG scholarships Ga 636/1–1 and Ga 636/1–2.
Keywords:Mathematics Subject Classification (1991): 14N35  14N10  14J70
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