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

TL;DR

The paper proves that the natural singular K"ahler form on symmetric products of Riemann surfaces can be smoothed to a genuine K"ahler form, aligning with standard Lagrangian Floer homology techniques.

Contribution

It demonstrates a method to smooth the singular K"ahler form on symmetric products, facilitating the application of Floer homology in this context.

Findings

Existence of a cohomologous smoothing of the K"ahler form.

The smoothing matches the original form away from the diagonal.

Application to Heegaard Floer homology frameworks.

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.