The Hodge number $h^{2,0}$ of (finite) quotient variety of a K3 surface

Question

Let $X$ be an (algebraic) K3 surface, then we have $H^{2,0}(X)=\langle \omega_X\rangle$, where $\omega_X$ is the period. Suppose $G=\langle g\rangle$ is a finite group acting on $X$ and $g$ as an automorphism of $X$ doesn't fix $\omega_X$(e.g. $g$ is purely non-symplectic), then why is $h^{2,0}(X/G)=0$?

What's in my head is $$ H^k(X,\mathbb{Q})^G\cong H^k(X/G,\mathbb{Q}). $$ by the Leray-Cartan-Serre spectral sequence. But it requires free action and it's of coefficient $\mathbb Q$. Do we have $$ H^{i,j}(X)^G\cong H^{i,j}(X/G) $$ in general?

Also, I am not sure if we always have Hodge structure on normal varieties in general. Here $X/G$ is a normal variety, I guess we can define $H^{i,j}(X/G):=H^j(X/G,\omega_{X/G}^i)$? I know the canonical divisor is well-defined for normal varieties, so for my case that $X$ is K3, $H^{2,0}(X/G)$ is well defined but is $H^j(X/G,\omega_{X/G}^i)$ well-defined in general?

Answer 1

Your variety $X/G$ is an orbifold; on a singular variety, the Hodge decomposition does not work, but on an orbifold, it works just as well. Then $G$ acts on $H^*(X,{\Bbb Q})$, and $H^*(X/G,{\Bbb Q})$ is the space of $G$-invariants (this is more or less a definition of $H^*(X/G)$ for an orbifold). Similarly, $H^{p,q}(X/G)$ is the space of $G$-invariants in $H^{p,q}(X)$. Then $H^{2,0}(X/G)= H^{2,0}(X)^G=0$ because your $G$-action is not holomorphically symplectic.

To use this argument, you would probably need to compare the orbifold cohomology with the usual cohomology of $X/G$. This is not very hard to do, see https://www.pnas.org/content/42/6/359 I. Satake, "ON A GENERALIZATION OF THE NOTION OF MANIFOLD", PNAS June 1, 1956 42 (6) 359-363.

Answer 2

Going off Misha's comment above, here are three useful notions of differentials for a normal variety $X$:

1. The exterior powers $\Omega_X^p$ of the sheaf of K"ahler differentials.

2. The reflexive differentials $\Omega_X^{[p]}: = (\Omega_X^p)^{**} \cong j_*\Omega_{X_{\mathrm{reg}}}^p$, where $j:X_{\mathrm{reg}} \hookrightarrow X$ is the inclusion of the regular locus. The isomorphism requires normality.

3. $\pi_*\Omega_{\tilde X}^p$, where $\pi:\tilde X \to X$ is a resolution of singularities.

The sheaves $\Omega_X^p$ often have torsion, so we look at the other options (especially when studying Hodge theory). For each $p$, there is an inclusion $\pi_*\Omega_{\tilde X}^p \hookrightarrow j_*\Omega_{X_{\mathrm{reg}}}^p$. To see that these are in general different, the classic example is to let $X$ be the affine cone of an elliptic curve. Then the singular point is Cohen-Macaulay but not rational, and so by Kempf's criterion the inclusion $\pi_*\omega_{\tilde X} \hookrightarrow j_*\omega_{X_{\mathrm{reg}}}$ is strict. What's less classical, and maybe unintuitive, is that $\pi_*\Omega_{\tilde X}^1 \cong j_*\Omega_{X_{\mathrm{reg}}}^1$ in this case.

In your case of quotient singularities, these notions agree for all $p$.