The number of rational curves on K3 surfaces

Abstract

Let X be a K3 surface with a primitive ample divisor H, and let $β=2[H]\in H_2(X, \mathbf Z)$. We calculate the Gromov-Witten type invariants $n_β$ by virtue of Euler numbers of some moduli spaces of stable sheaves. Eventually, it verifies Yau-Zaslow formula in the non primitive class $β$.