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Let X be a smooth projective connected curve over $\mathbb{C}$ and let $n>1$ be an integer. Let $Y= Sym^n_X$ be the $n$-th symmetric product of $X$.

Is there, for every $i$, a nice formula for the Hodge decomposition of $H^i(Y,\mathbb{C})$?

If not, what part of the Hodge diamond can be described easily?

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  • $\begingroup$ Presumably you are aware of Macdonald's formula for the Poincaré polynomial of $Y$. There is an analogous formula for the Hodge polynomial. $\endgroup$
    – ssx
    Commented Sep 26, 2018 at 0:13

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Look at Example $1.1$ in this paper for a nice formula.

You can also compute them by hands (and, hopefully, prove the desired formula) by identifying $\mathrm{H}^{p,q}(\operatorname{Sym}^n X)$ with $S_n$-invariant part of $\mathrm{H}^{p,q}(X^n)$ (use the Kunneth's formula to compute the latter group).

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