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Variation of mixed Hodge structure and its applications

Osamu Fujino, Taro Fujisawa

TL;DR

The paper develops a robust complex-analytic framework for variations of $\mathbb{R}$-mixed Hodge structures arising from projective morphisms of complex analytic spaces, proving an analytic analogue of Kollár's torsion-freeness, vanishing, and semipositivity results without invoking Saito's mixed Hodge modules. It constructs canonical extensions along simple normal crossing divisors, shows local freeness and dualities for $R^i f_*(\omega_{X/Y}(D))$ and related sheaves, and extends Kollár-type results to reducible spaces to support minimal model theory in the analytic setting. The approach relies on the theory of variations of mixed Hodge structure, spectral sequence degeneration, and a semisimplicial resolution framework, complemented by a supplementary Kostant-type construction for rational structures. The results provide foundational tools for semipositivity, vanishing, and torsion-freeness in the analytic MMP and have implications for moduli and quasi-log structures in complex analytic geometry.

Abstract

We discuss variations of mixed Hodge structure arising from projective morphisms of complex analytic spaces. Then we treat generalizations of Kollár's torsion-free theorem, vanishing theorem, and so on, for reducible complex analytic spaces as an application. The results will play a crucial role in the theory of minimal models for projective morphisms between complex analytic spaces.

Variation of mixed Hodge structure and its applications

TL;DR

The paper develops a robust complex-analytic framework for variations of -mixed Hodge structures arising from projective morphisms of complex analytic spaces, proving an analytic analogue of Kollár's torsion-freeness, vanishing, and semipositivity results without invoking Saito's mixed Hodge modules. It constructs canonical extensions along simple normal crossing divisors, shows local freeness and dualities for and related sheaves, and extends Kollár-type results to reducible spaces to support minimal model theory in the analytic setting. The approach relies on the theory of variations of mixed Hodge structure, spectral sequence degeneration, and a semisimplicial resolution framework, complemented by a supplementary Kostant-type construction for rational structures. The results provide foundational tools for semipositivity, vanishing, and torsion-freeness in the analytic MMP and have implications for moduli and quasi-log structures in complex analytic geometry.

Abstract

We discuss variations of mixed Hodge structure arising from projective morphisms of complex analytic spaces. Then we treat generalizations of Kollár's torsion-free theorem, vanishing theorem, and so on, for reducible complex analytic spaces as an application. The results will play a crucial role in the theory of minimal models for projective morphisms between complex analytic spaces.
Paper Structure (9 sections, 29 theorems, 148 equations)

This paper contains 9 sections, 29 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lemmas on analytic simple normal crossing pairs
  4. Complex analytic generalization of Kollár's package
  5. On variations of mixed Hodge structure
  6. Proofs of Theorems \ref{['z-thm1.1']}, \ref{['z-thm1.3']}, and \ref{['z-thm1.4']}
  7. Proof of Theorem \ref{['z-thm1.5']}
  8. Proof of Theorem \ref{['z-thm1.7']}
  9. Supplement to steenbrink2

Key Result

Theorem 1.1

Let $(X, D)$ be an analytic simple normal crossing pair such that $D$ is reduced and let $f\colon X\to Y$ be a proper surjective morphism onto a smooth complex variety $Y$. Assume that every stratum of $(X, D)$ is dominant onto $Y$. Let $\Sigma$ be a normal crossing divisor on $Y$ such that every st We put for every $k$. The Hodge filtration and the weight filtration on $\mathcal{V}^k_{Y^*}$ are

Theorems & Definitions (67)

  • Theorem 1.1: Canonical extensions of Hodge bundles, see fujino-fujisawa
  • Remark 1.2
  • Theorem 1.3: Semipositivity
  • Theorem 1.4: Weight spectral sequence
  • Theorem 1.5: Torsion-freeness and vanishing theorem
  • Remark 1.6
  • Theorem 1.7: see fujino-analytic-vanishing
  • Theorem 1.8: see fujino-analytic-vanishing
  • Theorem 1.9: Vanishing theorem of Reid--Fukuda type, see fujino-analytic-vanishing
  • Remark 1.10
  • ...and 57 more