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Hodge theory for local systems and cohomological support loci

Junyan Cao, Ya Deng, Christopher D. Hacon, Mihai Paun

TL;DR

The paper develops an $L^2$-Hodge-theoretic framework for rank-one local systems on quasi-compact Kähler manifolds and unifies Green–Lazarsfeld generic vanishing with Budur–Wang cohomology-jumping loci through suitable $\partial\bar{\partial}$-lemma techniques. It constructs a universal line bundle over $X\times\mathrm{Jac}(X)$ and extends key extension-of-sections results from order-one infinitesimal deformations to arbitrary order, with higher-rank analogues via Higgs-type deformations. The work provides two proofs that the pluricanonical jumping loci $Z_{k,m}$ are finite unions of translates of subtori, and develops a quasi-compact extension theory, including logarithmic connections, harmonic metrics, and detailed elliptic estimates for the generalized Laplacian $\Delta_K$. Collectively, these tools advance non-abelian Hodge theory on quasi-projective spaces and open avenues for representation-variety and mixed Hodge-structure applications.

Abstract

In this article, we pursue two main objectives. The first is to show that the fundamental results of Green-Lazarsfeld (1987, 1991) on generic vanishing theorems, and works of Budur-Wang (2015, 2020) on cohomology jumping loci, can be established within a unified framework based on suitable versions of the $\partial\bar{\partial}$-lemma. Our second-and primary-goal is to develop the technical tools required for this approach, namely an $L^2$-Hodge theory for the cohomology of rank-one local systems on quasi-compact Kähler manifolds. Further developments concerning higher-rank local systems, as well as several geometric applications, will be presented in a companion paper and are briefly outlined in the introduction.

Hodge theory for local systems and cohomological support loci

TL;DR

The paper develops an -Hodge-theoretic framework for rank-one local systems on quasi-compact Kähler manifolds and unifies Green–Lazarsfeld generic vanishing with Budur–Wang cohomology-jumping loci through suitable -lemma techniques. It constructs a universal line bundle over and extends key extension-of-sections results from order-one infinitesimal deformations to arbitrary order, with higher-rank analogues via Higgs-type deformations. The work provides two proofs that the pluricanonical jumping loci are finite unions of translates of subtori, and develops a quasi-compact extension theory, including logarithmic connections, harmonic metrics, and detailed elliptic estimates for the generalized Laplacian . Collectively, these tools advance non-abelian Hodge theory on quasi-projective spaces and open avenues for representation-variety and mixed Hodge-structure applications.

Abstract

In this article, we pursue two main objectives. The first is to show that the fundamental results of Green-Lazarsfeld (1987, 1991) on generic vanishing theorems, and works of Budur-Wang (2015, 2020) on cohomology jumping loci, can be established within a unified framework based on suitable versions of the -lemma. Our second-and primary-goal is to develop the technical tools required for this approach, namely an -Hodge theory for the cohomology of rank-one local systems on quasi-compact Kähler manifolds. Further developments concerning higher-rank local systems, as well as several geometric applications, will be presented in a companion paper and are briefly outlined in the introduction.

Paper Structure

This paper contains 40 sections, 91 theorems, 770 equations.

Table of Contents

  1. Families of flat line bundles
  2. Preliminaries
  3. An extension result
  4. Extension to the infinitesimal neighborhood of $1^{\rm st}$ order
  5. Extension to the infinitesimal neighborhood of $2^{\rm nd}$ order
  6. Higher order extensions
  7. Higher rank version
  8. Adjoint version
  9. Pluricanonical version
  10. Further results involving infinitesimal extension of sections
  11. Quasi-compact version
  12. Differential operators and harmonic metrics in log setting
  13. Technicalities I: elliptic estimates for $\Delta_K$
  14. Laplace operators on smooth/log forms
  15. Metric structure on the (orbifold) tangent bundle of $X_0$
  16. ...and 25 more sections

Key Result

Theorem 1

Let $u$ be a holomorphic section of $K_X$, which extends to order one in the direction $\alpha$. Then there exists a smooth family $(u_t)_{t\in (\mathbb C, 0)}$ of $(n, 0)$-forms, such that the following hold so that $u$ can be extended to a holomorphic section for the operator $\bar{\partial}_t$.

Theorems & Definitions (211)

  • Theorem 1
  • Theorem 2
  • Theorem 3: HPS18
  • Conjecture 4
  • Theorem 5: BW20
  • Theorem 1.1
  • proof
  • Remark 1.2
  • Theorem 2.1
  • proof
  • ...and 201 more