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Kollár's package for polystable locally abelian parabolic Higgs bundles

Junchao Shentu, Chen Zhao

TL;DR

The work addresses extending Kollár's package to polystable locally abelian parabolic Higgs bundles twisted by multiplier ideals, unifying several geometric settings. Using nonabelian Hodge theory (Simpson–Mochizuki) and $L^2$ methods, it proves that the sheaf $ ext{ω}_Xigl(P_{D-N,(2)}(H)igcap j_*(K)igr) ensor F ensor L$ satisfies torsion freeness, injectivity, vanishing, and a decomposition-type result with respect to any proper surjective morphism $f:X o Y$, under suitable semi-positivity and tameness hypotheses. The framework yields broad applications, including weak positivity of higher direct images in Viehweg’s sense and a generic vanishing theorem via Hacon–Pareschi–Popa criteria, for both parabolic Higgs and Hodge-theoretic settings. Concrete instances include twisted Kollár–Saito $S$-sheaves and multiplier Grauert–Riemenschneider sheaves, providing a unified approach to several classical results and enabling new vanishing/positivity statements in complex geometry.

Abstract

We generalize Kollár's package (including torsion freeness, injectivity theorem, vanishing theorem and decomposition theorem) to polystable locally abelian parabolic Higgs bundles twisted by a multiplier ideal sheaf associated with an $\mathbb{R}$-divisor. This gives a uniform treatment for various kinds of Kollár's package in different topics in complex geometry. As applications, the weakly positivity (in the sense of Viehweg) and the generic vanishing property for higher direct image sheaves are deduced.

Kollár's package for polystable locally abelian parabolic Higgs bundles

TL;DR

The work addresses extending Kollár's package to polystable locally abelian parabolic Higgs bundles twisted by multiplier ideals, unifying several geometric settings. Using nonabelian Hodge theory (Simpson–Mochizuki) and methods, it proves that the sheaf satisfies torsion freeness, injectivity, vanishing, and a decomposition-type result with respect to any proper surjective morphism , under suitable semi-positivity and tameness hypotheses. The framework yields broad applications, including weak positivity of higher direct images in Viehweg’s sense and a generic vanishing theorem via Hacon–Pareschi–Popa criteria, for both parabolic Higgs and Hodge-theoretic settings. Concrete instances include twisted Kollár–Saito -sheaves and multiplier Grauert–Riemenschneider sheaves, providing a unified approach to several classical results and enabling new vanishing/positivity statements in complex geometry.

Abstract

We generalize Kollár's package (including torsion freeness, injectivity theorem, vanishing theorem and decomposition theorem) to polystable locally abelian parabolic Higgs bundles twisted by a multiplier ideal sheaf associated with an -divisor. This gives a uniform treatment for various kinds of Kollár's package in different topics in complex geometry. As applications, the weakly positivity (in the sense of Viehweg) and the generic vanishing property for higher direct image sheaves are deduced.
Paper Structure (29 sections, 36 theorems, 100 equations)

This paper contains 29 sections, 36 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Example: parabolic bundle
  4. Example: twisted Kollár-Saito's $S$-sheaf
  5. Example: multiplier Grauert-Riemenschneider sheaf
  6. Applications
  7. Weakly positivity of higher direct images
  8. Generic vanishing theorem
  9. Organization of the article
  10. Acknowledgement
  11. Nonabelian Hodge theory on a smooth quasi-projective variety
  12. Parabolic Higgs bundle
  13. Tame harmonic bundle
  14. Analytic prolongations of tame harmonic bundles
  15. Norm estimate
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $K$ be a locally free subsheaf of $H|_{X\backslash D}$ satisfying the following conditions: Let $L$ be a line bundle on $X$ such that $L\simeq_{\mathbb{R}}B+N$, where $B$ is a semi-positive $\mathbb{R}$-divisor (see Definition defn_semipositive_divisor) and $N$ is an $\mathbb{R}$-divisor on $X$ supported on $D$. Let $F$ be a Nakano semi-positive vector bundle on $X$. Then, the sheaf $\omeg

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 60 more