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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?

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Is ${\rm Sym}^g(S)$ even simply connected? It's birational to the Jacobian of $S$, which has nontrivial $H^1$. –  Noam D. Elkies Jun 22 '11 at 21:07
    
If $j\colon S\to J$ is the Abel-Jacobi map, the induced map $j+\cdots+j\colon S^g\to J$ is obviously surjective on $\pi_1$, and factors through $\text{Sym}^g(S)$. –  Tom Church Jun 22 '11 at 21:25
    
True, it's not simply connected but that's not so important for what I want (elliptic curves aren't simply connected but are still CY enough for MS) –  Steve Jun 22 '11 at 22:19
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The idea seems right but the question doesn't: mirror symmetry is not restricted to CY varieties (or put another way, the Fukaya category is always CY). Conjecturally any compact symplectic manifold has a mirror, which is a Landau-Ginzburg model (category of matrix factorizations). See papers of Katzarkov, Orlov, Seidel and Efimov (et al) for curves of higher genus and more general varieties of general type. –  David Ben-Zvi Jun 23 '11 at 2:07
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If you could figure out what the vanishing locus of that form looks like, you could try to remove it, and then compute the mirror of that CY complement. You could hope that this would then be the base space of your mirror to Sym^g(S) and that there would then be just one extra piece of data to compute, the superpotential w. Usually people do this for e.g. toric varieties or more generally in the presence of a special Lagrangian fibration, but maybe it's enough that there are enough Lagrangians in the complement to generate Fuk(Sym^g(S))... www-math.mit.edu/~auroux/papers/slagmirror.pdf –  Daniel Pomerleano Jun 23 '11 at 10:54
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up vote 13 down vote accepted

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 $g\ne 1$.

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Ahh.. the final nail in the coffin. Thanks –  Steve Jun 22 '11 at 22:18
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A simple counterexample is $Sym^2(S_2)$, which is a once blown up $T^4$, hence $c_1\ne 0$. –  Paul Jun 23 '11 at 15:19
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