1999/11/09 by Jim Bryan, Sheldon Katz, Bryan, Jim +3 · 2 citations
Mathematics · #14N35 #53D45 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Symplectic Geometry (math.SG) #math.AG #math.DG #math.SG #msc:14N35 #msc:53D45
paper · pdf · doi:10.48550/arxiv.math/9911056
uses diagrams.sty
arxiv created 1999/11/09 · openalex publication_date 1999/11/09 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the contribution of multiple covers of an irreducible rational curve C in a Calabi-Yau threefold Y to the genus 0 Gromov-Witten invariants in the following cases. (1) If the curve C has one node and satisfies a certain genericity condition, we prove that the contribution of multiple covers of degree d is given by the sum of all 1/n3 where n divides d. (2) For a smoothly embedded contractable curve C in Y we define schemes Ci for i=1,...,l where Ci is supported on C and has multiplicity i, and the integer l (0l). In the latter case we also get a formula for arbitrary genus. These results show that the curve C contributes an integer amount to the so-called instanton numbers that are defined recursively in terms of the Gromov-Witten invariants and are conjectured to be integers.