Polarizable variations of (mixed) Hodge structures

Question

I am trying to come to grips with Saito's theory of mixed Hodge modules (slightly) beyond just the basic axiomatic formalism. I will take my Hodge structures and sheaves to be rational, but I would be grateful if the experts would point out any subtleties with my questions if rational is replaced by real. Let me at the moment focus on questions related to polarizability:

1. Let $\mathcal{L}$ be a local system underlying a polarizable variation of Hodge structure on a smooth variety. Does the polarizability imply $\mathcal{L}$ is self-dual? If yes, then does every direct summand of this local system also have to be self-dual (I am guessing no to the latter)?

2. More generally, let $M$ be a polarizable mixed Hodge module on some variety (not necessarily smooth). Is polarizability equivalent to Verdier self-duality (up to Tate twist) in the derived category of mixed Hodge modules? If not equivalent, does it at least imply Verdier self-duality?

Answer 1

You have my sympathies for trying come to grips with this stuff. Fortunately the answer, which is both yes and no, can be explained in the simplest case of a point. A polarization on a pure Hodge structure $H$ of weight $w$ is a pairing $$\langle, \rangle: H\otimes H\to \mathbb{Q}(-w)$$ such that $(2\pi i)^n\langle -, C -\rangle$ is positive definite and symmetric, where $C$ is the Weil operator which acts by $i^{p-q}$ on $H^{pq}$. These conditions known as the Hodge-Riemann bilinear relations imply that $H\cong H^*(-w)$ as Hodge structures, so in this sense $H$ self dual. However, just having such an isomorphism would not give everything else.

If $H$ is replaced by a variation of Hodge structures, then a polarization is flat pairing as above satisfying the Hodge-Riemann conditions on the fibres. For Hodge modules the story is similar but more delicate. Since the whole theory is constructed by induction on dimension of support, a polarization is also defined in this manner. Given Hodge module $H$ of weight $w$ on $X$, a polarization is a pairing $H\otimes H\to \mathbb{Q}(\dim X-w)[2\dim X]$ satisfying the inductive conditions (0.8)-(0.10) of Saito's "Modules Hodge Polarizables". Such a pairing should induce an isomorphism $H\cong DH(?)$ up to twist, but it's not equivalent. (Added I don't have a precise reference or proof for this, just a feeling that one could prove it as follows: Buried in Saito's second paper, "Mixed Hodge modules", is a proof that any simple polarizable Hodge module is $j_{!*}V$, where $V$ is a VHS. This effectively reduces the duality statement to the previous case.)