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Hodge theory for twisted log-differential forms

Junyan Cao

TL;DR

This survey develops a comprehensive framework to extend Hodge theory to twisted log-differential forms and currents with singular metrics. By combining conic and Poincaré singularities, $L^2$ techniques, and current theory, it establishes Hodge decompositions and $\partial\bar{\partial}$-lemmas for $L$-valued forms, logarithmic flat bundles, and their hypercohomology, then applies these tools to extension of pluricanonical forms, jumping loci for local system cohomology, and deformation theory of log Calabi–Yau pairs. The approach yields new proofs and generalizations of classical results in a quasi-compact setting, and sets the stage for higher-rank extensions and broader singular space applications. Key innovations include the conic Hodge theory, the regularity lemma for logarithmic currents, and the hypercohomology resolutions that relate log de Rham complexes to rank-one local systems. These advances have significant implications for extension problems, cohomology structure, and unobstructed deformations in complex geometry.

Abstract

In this survey, we review recent developments in extending Hodge theory to differential forms with values in bundles equipped with singular metrics, based on joint work with Ya Deng, Christopher D. Hacon, and Mihai Păun.

Hodge theory for twisted log-differential forms

TL;DR

This survey develops a comprehensive framework to extend Hodge theory to twisted log-differential forms and currents with singular metrics. By combining conic and Poincaré singularities, techniques, and current theory, it establishes Hodge decompositions and -lemmas for -valued forms, logarithmic flat bundles, and their hypercohomology, then applies these tools to extension of pluricanonical forms, jumping loci for local system cohomology, and deformation theory of log Calabi–Yau pairs. The approach yields new proofs and generalizations of classical results in a quasi-compact setting, and sets the stage for higher-rank extensions and broader singular space applications. Key innovations include the conic Hodge theory, the regularity lemma for logarithmic currents, and the hypercohomology resolutions that relate log de Rham complexes to rank-one local systems. These advances have significant implications for extension problems, cohomology structure, and unobstructed deformations in complex geometry.

Abstract

In this survey, we review recent developments in extending Hodge theory to differential forms with values in bundles equipped with singular metrics, based on joint work with Ya Deng, Christopher D. Hacon, and Mihai Păun.
Paper Structure (12 sections, 41 theorems, 198 equations)

This paper contains 12 sections, 41 theorems, 198 equations.

Table of Contents

  1. Hodge theory for $L$-valued forms
  2. Hodge theory for logarithmic holomorphic flat line bundles
  3. Harmonic metrics and basic Hodge identities
  4. Hodge theory for logarithmic holomorphic flat line bundles
  5. Resolution of the Hypercohomology
  6. Hodge decomposition for currents and a regularity lemma
  7. De Rham--Kodaira decomposition and the $\partial\bar{\partial}$-lemma for currents
  8. Regularity lemma
  9. A log canonical vanishing theorem
  10. Applications, I: extension of the pluri-canonical forms
  11. Applications, II: cohomology jumping loci property.
  12. Applications, III: deformation of log Calabi--Yau manifolds

Key Result

Proposition 1.2

JCMP Let $\phi$ be an $L$-valued $(p,q)$-form which is smooth in the orbifold sense. Then its "natural" derivatives $D'_{h_L}$, $D_{h_L}^\star$, etc. are also smooth in the orbifold sense. Moreover:

Theorems & Definitions (78)

  • Definition 1.1
  • Proposition 1.2
  • Example 1.3
  • Remark 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 68 more