[Paper Review] Hodge modules on complex tori and generic vanishing for compact Kähler manifolds

Giuseppe Pareschi, Mihnea Popa+1 more|arXiv (Cornell University)|May 4, 2015

Algebraic Geometry and Number Theory23 references8 citations

TL;DR

This paper extends generic vanishing theory to polarizable real Hodge modules on compact complex tori, proving that higher direct images of canonical bundles under holomorphic maps to tori decompose into sums of M-regular sheaves tensored with torsion line bundles. This leads to a bimeromorphic characterization of compact Kähler manifolds as tori, and establishes the coincidence of the Leray and cup-product filtrations on cohomology, resolving a key technical gap in classical Hodge theory.

ABSTRACT

We extend the results of generic vanishing theory to polarizable real Hodge modules on compact complex tori, and from there to arbitrary compact Kähler manifolds. As applications, we obtain a bimeromorphic characterization of compact complex tori among compact Kähler manifolds, semi-positivity results, and a description of the Leray filtration for maps to tori.

Motivation & Objective

• To extend generic vanishing theorems from projective varieties to arbitrary compact Kähler manifolds using Hodge module theory.
• To establish a decomposition theorem for higher direct images of canonical bundles under maps to complex tori.
• To provide a bimeromorphic characterization of compact Kähler manifolds as complex tori via cohomological invariants.
• To describe the Leray filtration on cohomology of canonical bundles in terms of cup-product actions, resolving a gap in classical Hodge theory.

Proposed method

• Use of polarizable real Hodge modules on compact complex tori as a generalization of classical Hodge theory.
• Application of Saito's theory of Hodge modules and the Fourier-Mukai transform to study cohomological properties of canonical bundles.
• Leveraging the GV-sheaf and M-regularity conditions to analyze the support and positivity of higher direct images.
• Proof of the coincidence of the Leray filtration and the cup-product filtration via 0-regularity of graded modules over exterior algebras.
• Use of Hodge conjugation and Serre duality to dualize the filtration description to holomorphic forms.
• Adaptation of results from [LPS] on 0-regularity to the Kähler setting, ensuring the filtrations coincide under the new framework.

Experimental results

Research questions

• RQ1 Can generic vanishing theorems for canonical bundles be extended from projective varieties to arbitrary compact Kähler manifolds?
• RQ2 What is the precise structure of higher direct images $ R^j f_* ω_X $ for maps from compact Kähler manifolds to complex tori?
• RQ3 Under what cohomological conditions is a compact Kähler manifold bimeromorphically equivalent to a complex torus?
• RQ4 How do the Leray filtration and the cup-product filtration on $ H^k(X, ω_X) $ relate in the Kähler setting?
• RQ5 Can the Leray filtration be described explicitly in terms of wedge products with holomorphic 1-forms?

Key findings

• The higher direct images $ R^j f_* ω_X $ decompose as finite direct sums of pullbacks of M-regular sheaves from quotients of the target torus, tensored with torsion line bundles in $ ×^0(T) $.
• The sheaf $ R^j f_* ω_X $ is a GV-sheaf, hence nef, and under submersion assumptions, it is semi-positive.
• A compact Kähler manifold $ X $ is bimeromorphically equivalent to a complex torus if and only if $ \dim H^1(X,\mathbb{C}) = 2\dim X $ and $ P_1(X) = P_2(X) = 1 $.
• The Leray filtration on $ H^k(X, ω_X) $ coincides with the filtration induced by the cup-product action of $ H^1(T, \mathcal{O}_T) $, resolving a long-standing technical gap.
• The dual filtration on $ H^0(X, \Omega_X^j) $ is characterized by vanishing of wedge products with $ \bigwedge^{n+1-i-j} H^0(X, \Omega_X^1) $, providing an explicit geometric description.
• The proof relies on Hodge module theory, as classical Hodge theory alone cannot establish the filtration coincidence due to lack of a priori control over the Leray filtration.