The number of rational curves on K3 surfaces

Abstract

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

15 pages, added references, to appear in Asian J. Math