Rational curves on hypersurfaces [after A. Givental]

Details Rational curves on hypersurfaces [after A. Givental] Rational curves on hypersurfaces [after A. Givental]

1998-06-23

arxiv.org/abs/math/9806133v1

Mathematics

Algebraic Geometry

33 pages, Latex2e

Scientific paper

This article accompanies my June 1998 seminaire Bourbaki talk on Givental's work. After a quick review of descendent integrals in Gromov-Witten theory, I discuss Givental's formalism relating hypergeometric series to solutions of quantum differential equations arising from hypersurfaces in projective space. A particular case of this relationship is a proof of the Mirror prediction for the numbers of rational curves on the Calabi-Yau quintic 3-fold. The approach taken here is entirely algebro-geometric and relies upon a localization formula on the moduli space of stable genus 0 maps to projective space. A different proof of the quintic Mirror prediction may be found in the work of Lian, Liu, and Yau.

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