[Paper Review] A remark on semipositivity theorems

Taro Fujisawa|arXiv (Cornell University)|Oct 3, 2017

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper introduces a new category of filtered vector bundles, FPVHS(X,D)_{\mathbb{R}}, to overcome flaws in a previously claimed semipositivity theorem. By redefining the category to ensure functoriality of the restriction map and preserving Higgs field conditions under restriction, the author proves a corrected semipositivity theorem (Theorem 4.2), which recovers Theorem 1.8 from [1] and resolves a counterexample that invalidated the original claim.

ABSTRACT

We propose a new class of filtered vector bundles, which is related to variation of (mixed) Hodge structures and give a slight generalization of the Fujita--Zucker--Kawamata semipositivity theorem.

Motivation & Objective

• To address a critical flaw in Theorem 4.5 of [1], which was shown to be false by Example 4.6.
• To resolve the issue that the kernel condition on the Higgs field is not preserved under restriction in the original framework.
• To construct a new category, FPVHS(X,D)_{\mathbb{R}}, as a full subcategory of filtered vector bundles to restore functoriality of the restriction functor.
• To prove a corrected semipositivity theorem (Theorem 4.2) that recovers Theorem 1.8 of [1] as a corollary.
• To ensure that the Higgs field condition on quotient bundles is preserved under restriction, enabling inductive proofs.

Proposed method

• Proposes a new category, FPVHS(X,D)_{\mathbb{R}}, defined as a full subcategory of filtered vector bundles on a log pair (X,D), with objects equipped with a filtered vector bundle (V,F) admitting additional data (W,...).
• Defines the category such that morphisms have no constraints on the weight filtration W, ensuring the restriction functor to strata of D is well-defined and functorial.
• Uses the refinement of the weight filtration in a controlled way, avoiding the use of the large index set Z^∞ that caused issues in [1].
• Constructs the restriction functor to strata of the boundary divisor D, ensuring compatibility with the Hodge and weight filtrations.
• Applies an inductive argument based on the dimension of strata, relying on the preservation of Higgs field conditions under restriction for quotient bundles.
• Employs canonical extensions and properties of polarized variations of Hodge structures to analyze the behavior of filtrations and Higgs fields on degenerations.

Experimental results

Research questions

• RQ1 Why does the original proof of Theorem 4.5 in [1] fail under restriction to subvarieties, particularly regarding the kernel condition of the Higgs field?
• RQ2 How can a new category of filtered vector bundles be defined to restore functoriality of the restriction map while preserving key geometric conditions?
• RQ3 Can a corrected semipositivity theorem be proven by ensuring that the Higgs field condition on quotient bundles is preserved under restriction?
• RQ4 Is Theorem 1.8 of [1] recoverable as a corollary of a corrected semipositivity theorem in the new framework?
• RQ5 What is the role of the weight filtration refinement in maintaining compatibility with the Hodge structure and the Higgs field?

Key findings

• The category FPVHS(X,D)_{\mathbb{R}} is defined as a full subcategory of filtered vector bundles, ensuring that restriction functors are well-defined and functorial.
• The restriction of a quotient bundle A of Gr_F V to a stratum preserves the condition that Gr^W A is contained in the kernel of the Higgs field, unlike in the original framework.
• The counterexample in Example 4.6 invalidates Theorem 4.5 of [1], as the dual of the quotient bundle A ≃ O_X(-C_0) is not semipositive due to C_0^2 = -n < 0.
• Theorem 4.2 establishes a corrected semipositivity theorem for objects in FPVHS(X,D)_{\mathbb{R}}, with the key condition preserved under restriction.
• Theorem 1.8 of [1] is recovered as a corollary of Theorem 4.2, validating its correctness in the new framework.
• The construction of the restriction functor, inspired by Brunebarbe, is made functorial by redefining the category to avoid constraints on the weight filtration W.