Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds

Jim Bryan, Sheldon Katz, Naichung Conan Leung

Research output: Contribution to journalArticlepeer-review

Abstract

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 Σn|d1/n3. (2) For a smoothly embedded contractable curve C ⊂ Y we define schemes Ci for 1 ≤ i ≤ l where Ci is supported on C and has multiplicity i, the number l ∈ {1,...,6} being Kollár's invariant "length". We prove that the contribution of multiple covers of C of degree d is given by Σn|dkd/n/n3, where ki is the multiplicity of Ci in its Hilbert scheme (and ki = 0 if i > l). 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, which are defined recursively in terms of the Gromov-Witten invariants and are conjectured to be integers.

Original languageEnglish (US)
Pages (from-to)549-568
Number of pages20
JournalJournal of Algebraic Geometry
Volume10
Issue number3
StatePublished - Jul 1 2001
Externally publishedYes

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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