The $g$-fold symmetric product of a Riemann surface of genus $g$ naturally has both a symplectic structure as well as a complex structure. Is it in fact Calabi-Yau? If so, is anything known about a mirror for it (in the sense of mirror symmetry)?
My motivation for this comes from Mirror Symmetry and Heegaard-Floer Homology (neither of which I've done any serious work in so the following might be complete garbage). Heegaard-Floer homology is essentially the study of the Floer homology of various special Lagrangian tori in $Sym^g S$ for $S$ a surface of genus $g$. If $Sym^g S$ admits any sort of sensible mirror, under HMS one should be able to extract a lot of information about the Heegaard-Floer homology of 3-manifolds admitting a genus $g$ Heegaard splitting by looking at morphisms of sheaves on the mirror side.
So modulo the CYness of $Sym^g S$ a related question is what work has been done in this direction?
Best Answer
This is not a Calabi-Yau if $g\ne 1$ (for any definition of Calabi-Yau). Indeed, there is a degree one map from the symmetric power of the curve to a torus of dimension $g$. Pull-back of the volume form on the torus to $Sym^gS$ will have zeros at the set where the differential of the map is degenerate, this set is non-empty if $g\ne 1$.