I am learning the variations of Hodge structure, yet getting stuck at the very first beginning.
Let $S$ be a projective nonsingular variety over $\mathbb{C}$. I have seen that a variation of Hodge structure of weight $k$ on $S$ is the data
- A $\mathbb{Q}$-local system $V$,
- A decreasing separated exhaustive filtration $\{F^p\mathcal{V}\}_{p \in \mathbb{Z}}$ of holomorphic subbundles of $\mathcal{V} := V \otimes_{\underline{\mathbb{Q}}} \mathcal{O}_S$ such that for any $s \in S$, the fiber $(V_s, F^pV_s)$ is a Hodge structure of weight $k$ and the canonical flat connection $\nabla$ satisfies the Griffith transversality.
Question 1: I am quite confused on the meaning of the "fiber" $(V_s, F^pV_s)$. Could someone tell me the definition or provide a reference on what it means?
As I understand it, a $\mathbb{Q}$-local system $V$ is a locally constant sheaf of finite dimensional $\mathbb{Q}$-vector space. (I guess the dimensions locally are constant since $S$ is irreducible? Am I right?)
I have been searching, and guessing that it might be $V_s := \mathcal{V}_s \otimes_{\mathcal{O}_{S,s}} \kappa(s)$, where $\kappa(s)$ is the residue field of $s \in S$, and similar for $F^p \mathcal{V}$. Yet trying to be a Hodge structure, it has to be a $\mathbb{Q}$-vector space, but I cannot see why it is a $\mathbb{Q}$-vector space. Moreover,
Question 2: what does the subbundle in the definition mean? And what does people mean by saying a "holomorphic bundle"?
As far as I can see, $\mathcal{V}$ is a locally free $\mathcal{O}_S$-module of finite rank since locally it looks like $\mathbb{Q}^{\oplus N} \otimes_{\mathbb{Q}} \mathcal{O}_S \cong \mathcal{O}_S^{\oplus n}$ if we ignore the sheafification. (I am a little uncertain on this, since I have heard people saying that it is coherent, yet seldomly they say it is locally free.) So does the "subbundle" mean a sub-$\mathcal{O}_S$-module that is still locally free?
Thank you all for answering and commenting! :) Sorry for being so stupid.