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 Bökstedt and Romão [3] says that for any positive integer n ≤ 2(genus(X) − 1), the symmetric product Sn(X) does not admit any Kähler 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 Bökstedt and Romão.