Preliminary for Relative Monodromy Filtrations

Abstract

Let us recall the degeneration phenomena of polarizable variations of pure Hodge structure on a punctured disk Δ ∗, as a motivation to consider (relative) monodromy filtrations. Let $$(\mathcal{V}_{\mathbb{Q}},F)$$ be a polarizable variation of Hodge structure of weight w on Δ ∗, where $$\mathcal{V}_{\mathbb{Q}}$$ is a $$\mathbb{Q}$$ -local system on Δ ∗, and F is a Hodge filtration of the vector bundle $$V = \mathcal{O}_{\varDelta ^{{\ast}}}\otimes _{\mathbb{Q}}\mathcal{V}_{\mathbb{Q}}$$ . We have the induced connection ∇ on V. For simplicity, we assume that the local monodromy automorphism is unipotent. Let $$\tilde{V }$$ be the Deligne extension of V on the disk Δ. Then, F is extended to a filtration of $$\tilde{V }$$ by subbundles. Moreover, we have an isomorphism of the fiber $$\tilde{V }_{\vert 0}$$ and the space of multi-valued flat sections of V, which induces a $$\mathbb{Q}$$ -structure on $$\tilde{V }_{\vert 0}$$ , i.e., we obtain a $$\mathbb{Q}$$ -vector space $$H_{\mathbb{Q}}$$ with an isomorphism $$H_{\mathbb{Q}} \otimes _{\mathbb{Q}}\mathbb{C} \simeq \tilde{ V }_{\vert 0}$$ . In general, $${\bigl (H_{\mathbb{Q}},\tilde{F}_{\vert 0}\bigr )}$$ is not a pure Hodge structure of weight w. Instead, we should consider the weight filtration induced by the residue $$N =\mathop{ \mathrm{Res}}\nolimits (\nabla )$$ on $$\tilde{V }_{\vert 0}$$ . Namely, let W(N) denote the monodromy weight filtration of N on $$\tilde{V }_{\vert 0}$$ i.e., W(N) be the increasing filtration indexed by integers such that (1) W j (N) = 0 ( j < ​ < 0), $$W_{j}(N) = H_{\mathbb{Q}}$$ ( j > ​ > 0), (2) $$N \cdot W_{j}(N) \subset W_{j-2}(N)$$ $$(\,j \in \mathbb{Z})$$ (3) the induced morphisms $$N^{j}:\mathop{ \mathrm{Gr}}\nolimits _{j}^{W(N)}(\tilde{V }_{\vert 0})\longrightarrow \mathop{\mathrm{Gr}}\nolimits _{-j}^{W(N)}(\tilde{V }_{\vert 0})$$ are isomorphisms for any j ≥ 0. It turns out that W(N) is defined over $$H_{\mathbb{Q}}$$ . Let W be the filtration on $$H_{\mathbb{Q}}$$ determined by $$W_{j} = W_{j-w}(N)$$ . Then, $$(H_{\mathbb{Q}},W,F)$$ is a polarizable mixed Hodge structure.