Multiple covers and the integrality conjecture for rational curves in CalabiYau threefolds
Abstract
We study the contribution of multiple covers of an irreducible rational curve C in a CalabiYau threefold Y to the genus 0 GromovWitten 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/n^3 where n divides d. (2) For a smoothly embedded contractable curve C in Y we define schemes C_i for i=1,...,l where C_i is supported on C and has multiplicity i, and the integer l (0<l<7) is Kollar's invariant ``length''. We prove that the contribution of multiple covers of C of degree d is given by the sum of k_{d/n}/n^3 where n divides d and where k_i is the multiplicity of C_i in its Hilbert scheme (and k_i=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 socalled instanton numbers that are defined recursively in terms of the GromovWitten invariants and are conjectured to be integers.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 1999
 arXiv:
 arXiv:math/9911056
 Bibcode:
 1999math.....11056B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Differential Geometry;
 Mathematics  Symplectic Geometry;
 14N35;
 53D45
 EPrint:
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