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A birational description of the minimal exponent

Qianyu Chen, Mircea Mustaţă

TL;DR

This work provides a birational description of the minimal exponent $\tilde{\alpha}(Z)$ for a hypersurface $Z$ in a smooth variety $X$ by relating it to higher direct images of twists of logarithmic differential forms on a log resolution. It distinguishes the integer and non-integer cases, giving vanishing and isomorphism criteria for $R^q\pi_*\Omega_Y^i(\log E)$ and related twists, and expresses the minimal exponent in terms of the $V$-filtration and nearby cycles, including a sharp dual formulation connected to Du Bois and rational singularities. A central technical achievement is a filtered resolution of nearby cycles in the SNC setting, enabling explicit identifications of graded de Rham pieces with push-forwards of logarithmic forms. The results yield a precise criterion for when $\tilde{\alpha}(Z)\ge p$ or $>p+\alpha$, with a dual interpretation in terms of morphisms among twisted logarithmic forms, and they establish the constancy of the minimal exponent in families with simultaneous log resolutions. Together, these contributions advance the understanding of how birational and Hodge-theoretic structures govern refined singularity invariants beyond the log canonical threshold.

Abstract

We give a description of the minimal exponent of a hypersurface using higher direct images of suitably twisted sheaves of log forms on a log resolution.

A birational description of the minimal exponent

TL;DR

This work provides a birational description of the minimal exponent for a hypersurface in a smooth variety by relating it to higher direct images of twists of logarithmic differential forms on a log resolution. It distinguishes the integer and non-integer cases, giving vanishing and isomorphism criteria for and related twists, and expresses the minimal exponent in terms of the -filtration and nearby cycles, including a sharp dual formulation connected to Du Bois and rational singularities. A central technical achievement is a filtered resolution of nearby cycles in the SNC setting, enabling explicit identifications of graded de Rham pieces with push-forwards of logarithmic forms. The results yield a precise criterion for when or , with a dual interpretation in terms of morphisms among twisted logarithmic forms, and they establish the constancy of the minimal exponent in families with simultaneous log resolutions. Together, these contributions advance the understanding of how birational and Hodge-theoretic structures govern refined singularity invariants beyond the log canonical threshold.

Abstract

We give a description of the minimal exponent of a hypersurface using higher direct images of suitably twisted sheaves of log forms on a log resolution.

Paper Structure

This paper contains 7 sections, 13 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Acknowledgment
  3. Background concerning Hodge modules, $V$-filtrations, and the minimal exponent
  4. The condition $\widetilde{\alpha}(Z)>p$ for $p\in {\mathbf Z}_{>0}$
  5. A filtered resolution of $\psi_{g,\alpha}(\mathcal{O}_Y)$, when ${\rm div}(g)$ is an SNC divisor
  6. The description of the minimal exponent
  7. An application to families with simultaneous log resolutions

Key Result

Theorem 1.1

If $Z$ is a reduced hypersurface in $X$ and $p\in {\mathbf Z}_{>0}$, then $\widetilde{\alpha}(Z)>p$ if and only if the following two conditions hold: We have We have ${\rm codim}_Z(Z_{\rm sing})\geq 2p$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 29 more