On the curvature of symmetric products of a compact Riemann surface

Let $X$ be a compact connected Riemann surface of genus at least two. The main theorem of arxiv:1010.1488 says that for any positive integer $n \leq 2({\rm genus}(X)-1)$, the symmetric product $S^n(X)$ does not admit any K\"ahler 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\"okstedt and Rom\~ao.