A remark on K\"ahler forms on symmetric products of Riemann surfaces

Abstract

Users of Heegaard Floer homology may be reassured to know that it can be made to conform exactly to the standard analytic pattern of Lagrangian Floer homology. This follows from the following remark, which we prove using an argument of J. Varouchas: the natural singular K\"ahler form $\sym^n(\omega)$ on the $n$th symmetric product of a K\"ahler curve $(\Sigma,\omega)$ admits a cohomologous smoothing to a K\"ahler form which equals $\sym^n(\omega)$ away from a chosen neighbourhood of the diagonal.