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

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 Σ_n|d1/n³. (2) For a smoothly embedded contractable curve C ⊂ Y we define schemes C_i for 1 ≤ i ≤ l where C_i 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 Σ_n|dkd/n/n³, 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 so-called instanton numbers, which are defined recursively in terms of the Gromov-Witten invariants and are conjectured to be integers.