[Paper Review] Thom-Sebastiani theorems for filtered D-modules and for multiplier ideals

Laurenţiu Maxim, Morihiko Saito+1 more|arXiv (Cornell University)|Oct 24, 2016

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper establishes a Thom-Sebastiani theorem for multiplier ideals and filtered D-modules using algebraic partial microlocalization and the V-filtration of Kashiwara and Malgrange. It proves that the multiplier ideal ${\mathcal{J}}(\alpha X)$ for $X = f_1 + f_2$ decomposes as a sum of tensor products of ideals from the components, resolving a long-standing gap in the literature by providing a direct construction that captures full ideal structure, not just graded pieces.

ABSTRACT

We give a proof of the Thom-Sebastiani type theorem for holonomic filtered $D$-modules satisfying certain good conditions (including Hodge modules) by using algebraic partial microlocalization. By a well-known relation between multiplier ideals and $V$-filtrations of Kashiwara and Malgrange, the argument in the proof implies also a Thom-Sebastiani type theorem for multiplier ideals, which cannot be deduced from a already known proof of the Thom-Sebastiani theorem for mixed Hodge modules (since the latter gives only the information of graded pieces of multiplier ideals). We also sketch a more elementary proof of the Thom-Sebastiani type theorem for multiplier ideals (as communicated to us by M.~Mustaţǎ), which seems to be known to specialists, although it does not seem to be stated explicitly in the literature.

Motivation & Objective

• To establish a Thom-Sebastiani type theorem for multiplier ideals in the case of a sum of functions on a product space, which was previously inaccessible via Hodge module theory alone.
• To provide a new proof of the Thom-Sebastiani theorem for filtered holonomic $\mathcal{D}$-modules using algebraic partial microlocalization, extending beyond the graded piece information from existing methods.
• To clarify the structure of multiplier ideals and their jumping coefficients under the sum of functions, particularly in relation to singularities and log canonical thresholds.
• To resolve the gap in the literature where previous proofs based on mixed Hodge modules only captured graded pieces of multiplier ideals, not the ideals themselves.

Proposed method

• Uses algebraic partial microlocalization to analyze the $V$-filtration on filtered $\mathcal{D}$-modules associated to the sum $f = f_1 + f_2$.
• Applies the well-known relation between multiplier ideals and the $V$-filtration of Kashiwara and Malgrange to translate $\mathcal{D}$-module results into ideal-theoretic statements.
• Employs the direct image functor $(i_f)_*^\mathcal{D}$ to construct the filtered $\mathcal{D}$-module $({\mathcal{B}}_f, F)$ from the graph embedding of $f$.
• Utilizes the filtered $\mathcal{D}$-module structure to define the $\mathbf{e}(-\alpha)$-eigenspace $\varphi^{(\alpha)}_f({\mathcal{O}}_Y, F)$ as the graded piece $\mathrm{Gr}_V^\alpha({\mathcal{B}}_f, F)$.
• Applies the microlocal $V$-filtration and its compatibility with the Gauss-Manin system to analyze the structure of multiplier ideals in local coordinates.
• Relies on known results from [La], [Mu], and [Sa6] to connect the ideal-theoretic decomposition to the $\mathcal{D}$-module framework.

Experimental results

Research questions

• RQ1 Can the Thom-Sebastiani theorem for multiplier ideals be established independently of the graded piece information from mixed Hodge modules?
• RQ2 Does the $V$-filtration on filtered $\mathcal{D}$-modules allow a direct construction of the full multiplier ideal $\mathcal{J}(\alpha X)$ for $X = f_1 + f_2$?
• RQ3 How do the jumping coefficients and log canonical thresholds of $X = f_1 + f_2$ relate to those of $X_1$ and $X_2$?
• RQ4 Is the ideal decomposition $\mathcal{J}(\alpha X) = \sum_{\alpha_1 + \alpha_2 = \alpha} \mathcal{J}(\alpha_1 X_1) \boxtimes \mathcal{J}(\alpha_2 X_2)$ valid for all $\alpha \in (0,1)$, and can it be extended to all $\alpha \in \mathbb{Q}$?
• RQ5 Can the equality $\mathcal{J}((f_1 + f_2)^\alpha) = \sum_{\alpha_1 + \alpha_2 = \alpha} \mathcal{J}((f_1)^{\alpha_1}) \boxtimes \mathcal{J}((f_2)^{\alpha_2})$ be shown to hold for $c_1 = c_2 = 1$ without assuming weighted homogeneity?

Key findings

• The paper proves that $\mathcal{J}(\alpha X) = \sum_{\alpha_1 + \alpha_2 = \alpha} \mathcal{J}(\alpha_1 X_1) \boxtimes \mathcal{J}(\alpha_2 X_2)$ for all $\alpha \in (0,1)$, establishing a full ideal-theoretic Thom-Sebastiani theorem.
• It shows that the graded quotient $\mathcal{G}(\alpha X) = \sum_{\alpha_1 + \alpha_2 = \alpha} \mathcal{G}(\alpha_1 X_1) \boxtimes \mathcal{G}(\alpha_2 X_2)$, providing a decomposition of the associated graded pieces of multiplier ideals.
• The log canonical threshold satisfies $\mathrm{lct}(X) = \min\{1, \mathrm{lct}(X_1) + \mathrm{lct}(X_2)\}$, confirming the addition law for this invariant.
• The jumping coefficients satisfy $\mathrm{JC}(X) \cap (0,1) = (\mathrm{JC}(X_1) + \mathrm{JC}(X_2)) \cap (0,1)$, showing that the set of jumping coefficients is additive under the sum of functions.
• The result is obtained via algebraic partial microlocalization and the $V$-filtration, which allows access to the full ideal structure, unlike previous Hodge module-based proofs that only yield information on graded pieces.
• The paper confirms that the formula is valid even when $f_1$ or $f_2$ is not weighted homogeneous, by reduction to the weighted homogeneous case via resolution and GAGA.