Automorphism of integral Hodge structures

Question

Let $(V,V^{p,q},Q)$ be a polarized integral Hodge strucutre of weight $n$. I would like to understand the automorphism of this datum better. Specifically, I'm wondering if there are good conditions where we can show that the automorphism group of polarized integral Hodge structures is finite.

Since an automorphism of $(V,V^{p,q},Q)$ is given by automorphism of $V$ respecting the additional datum, if each $V^{p,q}$ is one-dimensional, then the automorphism group would be finite. Of course this argument does not work anymore when we assume that some of the $V^{p,q}$'s are of higher dimension. However, we still have the polarization at our disposal.

My motivation for this is that I know that for hyperbolic surfaces, we have an injective map $$Aut(X)\rightarrow Aut_{\mathbb{Z}-HS}(H^1(X,\mathbb{Z})$$ where we consider $H^1(X,\mathbb{Z})$ as polarized integral Hodge structure. Hence if we want to prove the finiteness of $Aut(X)$, it would suffice to show that polarized integral Hodge structures coming from hyperbolic surfaces have a finite automorphism group. However I don't really understand the polarization, or rather specific properties thereof.

Answer 1

The polarization $Q$ gives rise to a positive definite form $Q'$. Automorphisms preserve $Q'$, so they lie in a compact group. On the other hand, automorphisms preserve the lattice, so they lie in a discrete subgroup of a compact group. This gives finiteness. In the case of a Riemann surface, the polarization comes from the cup product, so it's canonical. Putting together with with your remarks gives finiteness for the automorphism group for a compact hyperbolic curve. As far as I can tell, this argument doesn't the Hurwitz bound of $84(g-1)$, but it works in other situations.