  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.
