[Paper Review] Logarithmic Comparison with smooth boundary divisor in Mixed Hodge Modules

Chuanhao Wei|arXiv (Cornell University)|Oct 20, 2017

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper establishes a logarithmic comparison theorem for mixed Hodge modules with respect to a smooth boundary divisor, proving canonical quasi-isomorphisms between the spectral and de Rham complexes of localized modules. It generalizes the classical logarithmic comparison theorem to the setting of mixed Hodge modules and shows that the localization functors $\mathcal{M}[\ast\log D]$ and $\mathcal{M}[!\log D]$ behave well under direct image and duality, extending results of Popa–Schnell and Viehweg.

ABSTRACT

We use filtered log-$\mathscr{D}$-modules to represent the (dual) localization of Saito's Mixed Hodge Modules along a smooth hypersurface, and show that they also behave well under the direct image functor and the dual functor in the derived category of filtered log-$\mathscr{D}$-modules. The results of this paper can be used to generalize the result of M. Popa and C. Schnell about Kodaira dimension and zeros of holomorphic one-forms into the log setting.

Motivation & Objective

• To extend the logarithmic comparison theorem to the setting of mixed Hodge modules with respect to a smooth boundary divisor.
• To establish the behavior of $\mathcal{M}[\ast\log D]$ and $\mathcal{M}[!\log D]$ under the direct image and duality functors in the derived category of filtered $\widetilde{\mathscr{D}}$-modules.
• To generalize results of Popa–Schnell and Viehweg to the log-smooth setting using the framework of mixed Hodge modules.
• To show that the localized modules $\mathcal{M}[\ast\log D]$ and $\mathcal{M}[!\log D]$ admit V-compatible multi-indexed Kashiwara–Malgrange filtrations.
• To provide a simplified and generalized proof of Viehweg’s hyperbolicity for families of log-smooth varieties of general type.

Proposed method

• The paper uses the Rees algebra construction to define filtered $\widetilde{\mathscr{D}}$-modules underlying mixed Hodge modules.
• It introduces the functors $\mathcal{M}[\ast\log D] := \mathbf{V}^{D}_{\mathbf{0}}(\mathcal{M}[\ast D])$ and $\mathcal{M}[!\log D] := \mathbf{V}^{D}_{<\mathbf{0}}(\mathcal{M}[!D])$ via the multi-indexed Kashiwara–Malgrange filtration.
• The key technical tool is the derived tensor product over $\widetilde{\mathscr{D}}_{(X,D_I)}$, showing that $\mathcal{M}[\ast D][\ast\log D_I] \otimes^{\mathbf{L}} \widetilde{\mathscr{D}}_{(X,D_S)} \simeq \mathcal{M}[\ast D][\ast\log D_S]$.
• The proof relies on the strictness of the Kashiwara–Malgrange filtration and the compatibility of the $\mathbf{V}^{D}$-filtration with duality and direct image functors.
• The spectral functor $\text{Sp}_{(X,D)}$ is used to relate the log-de Rham complex to the classical de Rham complex.
• Graded pieces of the filtered modules are analyzed using $\text{Gr}^F$ and $\mathcal{G}$ functors, leading to duality isomorphisms on the cotangent bundle.

Experimental results

Research questions

• RQ1 Does the logarithmic comparison theorem extend to mixed Hodge modules with respect to a smooth boundary divisor?
• RQ2 How do the localized modules $\mathcal{M}[\ast\log D]$ and $\mathcal{M}[!\log D]$ behave under the direct image functor in the derived category of filtered $\widetilde{\mathscr{D}}$-modules?
• RQ3 Can the duality functor preserve the $\mathbf{V}^{D}$-filtration structure in the mixed Hodge module setting?
• RQ4 Is the Kashiwara–Malgrange filtration compatible with the dualizing complex in the log-smooth setting?
• RQ5 Can the results be applied to generalize Viehweg’s hyperbolicity theorem to log-smooth families of general type varieties?

Key findings

• The paper proves that $\text{Sp}_{(X,D)}(\mathcal{M}[\ast\log D]) \simeq \text{Sp}_X(\mathcal{M}[\ast D])$ and $\text{Sp}_{(X,D)}(\mathcal{M}[!\log D]) \simeq \text{Sp}_X(\mathcal{M}[!D])$ in $DG(\widetilde{\mathbb{C}}_X)$, establishing the logarithmic comparison theorem in the mixed Hodge module setting.
• The derived tensor product formula $\mathcal{M}[\ast D][\ast\log D_I] \otimes^{\mathbf{L}}_{\widetilde{\mathscr{D}}_{(X,D_I)}} \widetilde{\mathscr{D}}_{(X,D_S)} \simeq \mathcal{M}[\ast D][\ast\log D_S]$ holds, showing compatibility of localization with restriction to sub-divisors.
• The dual module $\mathcal{M}'$ of a mixed Hodge module $\mathcal{M}$ admits a $\mathbf{V}^{D}$-compatible multi-indexed Kashiwara–Malgrange filtration, extending Saito’s results to the mixed case.
• The associated graded modules satisfy $\text{Gr}^F \mathcal{M}'[\ast\log D] \simeq \mathbf{R}\mathcal{H}om_{\mathcal{A}_{(X,D)}}(\text{Gr}^F \mathcal{M}[!\log D], \omega_X[d_X] \otimes_{\mathscr{O}_X} \mathcal{A}_{(X,D)})$, providing a duality isomorphism.
• The isomorphism $\mathcal{G}(\mathcal{M}'[\ast\log D]) \simeq (-1)^*_{T^*_{(X,D)}} \mathbf{R}\mathcal{H}om_{\mathscr{O}_{T^*_{(X,D)}}}(\mathcal{G}(\mathcal{M}[!\log D]), p_X^*\omega_X[d_X] \otimes \mathscr{O}_{T^*_{(X,D)}})$ holds on the cotangent bundle, generalizing known duality results.
• The results provide a simplified and generalized proof of Viehweg’s hyperbolicity for log-smooth families of general type, extending to the log-setting.