[Paper Review] Hodge ideals and minimal exponents of ideals

Mircea Mustaţă, Mihnea Popa|arXiv (Cornell University)|Dec 17, 2019

Algebraic Geometry and Number Theory18 references4 citations

TL;DR

This paper introduces Hodge ideals for coherent ideal sheaves on smooth complex varieties, generalizing the theory of Hodge ideals for Q-divisors. It defines the generic minimal exponent of an ideal, proves it is a root of the Bernstein-Sato polynomial, and establishes a direct link between Hodge ideals and this invariant via a condition on ideal membership related to the exponent and Hodge filtration level.

ABSTRACT

We define and study Hodge ideals associated to a coherent ideal sheaf J on a smooth complex variety, via algebraic constructions based on the already existing concept of Hodge ideals associated to Q-divisors. We also define the generic minimal exponent of J, extending the standard invariant for hypersurfaces. We relate it to Hodge ideals, and show that it is a root of the Bernstein-Sato polynomial of J.

Motivation & Objective

• To extend the theory of Hodge ideals from Q-divisors to arbitrary coherent ideal sheaves on smooth complex varieties.
• To define and study the generic minimal exponent of an ideal, generalizing the minimal exponent of a hypersurface.
• To establish a precise relationship between Hodge ideals and the generic minimal exponent, mirroring the known link for divisors.
• To prove that the generic minimal exponent is a root of the Bernstein-Sato polynomial of the ideal, extending a classical result for hypersurfaces.

Proposed method

• Define Hodge ideals $I_p(a^\lambda)$ for rational $\lambda \leq 1$ using the Hodge ideals of general linear combinations of generators of $\fa$, ensuring invariance under choice of generators.
• Use the fiberwise restriction of the Hodge filtration on $\cO_{X \times \bbA^r}(\ast G)$, where $G$ is the divisor defined by $g = \sum y_i f_i$, to characterize $I_p(\fa^\lambda)$ as the coefficient ideal of $I_p(\lambda G)$.
• Establish the equivalence of three definitions: via general divisors, via the coefficient ideal of a universal family, and via the Hodge filtration on a total space.
• Define the generic minimal exponent $\bar\alpha_x(\fa)$ as the minimal exponent of a general hypersurface containing the subscheme $V(\fa)$ near $x$.
• Prove that $I_p(\fa^\lambda)_x = \sO_{X,x}$ if and only if $p + \lambda \leq \bar\alpha_x(\fa)$, generalizing the divisorial case.
• Use results from Saito and Mustata on the Bernstein-Sato polynomial and $V$-filtration to show $-\bar\alpha_x(\fa)$ is a root of $b_{\fa,x}(s)$.

Experimental results

Research questions

• RQ1 How can the theory of Hodge ideals, previously defined for Q-divisors, be extended to arbitrary coherent ideal sheaves on smooth varieties?
• RQ2 What is the correct generalization of the minimal exponent invariant for ideals, and how does it relate to Hodge theory?
• RQ3 Is the generic minimal exponent of an ideal a root of its Bernstein-Sato polynomial, as in the hypersurface case?
• RQ4 Can the Hodge ideal $I_p(\fa^\lambda)$ be characterized uniformly across different geometric constructions, such as coefficient ideals or restriction to general fibers?

Key findings

• The Hodge ideal $I_p(\fa^\lambda)$ is well-defined and independent of the choice of generators of $\fa$, provided they define reduced divisors in codimension one.
• The ideal $I_p(\fa^\lambda)$ is equal to the coefficient ideal of $I_p(\lambda G)$, where $G$ is the divisor defined by $g = \sum y_i f_i$ on $X \times \bbA^r$.
• The generic minimal exponent $\bar\alpha_x(\fa)$ is equal to the minimal exponent of a general linear combination of the generators of $\fa$ near $x$.
• The condition $I_p(\fa^\lambda)_x = \sO_{X,x}$ holds if and only if $p + \lambda \leq \bar\alpha_x(\fa)$, generalizing the divisorial case.
• The negative of the generic minimal exponent $-\bar\alpha_x(\fa)$ is a root of the Bernstein-Sato polynomial $b_{\fa,x}(s)$, extending a classical result from hypersurfaces to arbitrary ideals.
• The proof relies on the fact that $b_{\fa}(s) = b_g(s)/(s+1)$ and that $b_{g,(x,\lambda)}(s)$ divides $b_g(s)$, allowing the transfer of root information from $g$ to $\fa$.