On the curvature of symmetric products of a compact Riemann surface

摘要

Let X be a compact connected Riemann surface of genus at least two. The main theorem of B?kstedt and Rom?o [3] says that for any positive integer n ≤ 2(genus(X) - 1), the symmetric product S~n(X) does not admit any K?hler metric satisfying the condition that all the holomorphic bisectional curvatures are nonnegative. Our aim here is to give a very simple and direct proof of this result of B?kstedt and Rom?o.