[Paper Review] Moduli of log mixed Hodge structures

Kazuya Katô, Chikara Nakayama+1 more|ArXiv.org|Oct 23, 2009

Algebraic Geometry and Number Theory7 references4 citations

TL;DR

This paper constructs toroidal partial compactifications of moduli spaces for mixed Hodge structures with polarized graded quotients by introducing log mixed Hodge structures, using nilpotent orbits and fans in the Lie algebra of the monodromy group. The key result establishes that these compactifications represent the moduli functor of log mixed Hodge structures with neat monodromy, enabling applications to the analyticity of zero loci of period maps.

ABSTRACT

We announce the construction of toroidal partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. They are moduli spaces of log mixed Hodge structures with polarized graded quotients. We include an application to the analyticity of zero loci of some functions.

Motivation & Objective

• To construct moduli spaces for mixed Hodge structures with polarized graded quotients by extending the pure case construction of KU09.
• To introduce log mixed Hodge structures as a framework for compactifying moduli spaces of mixed Hodge structures.
• To prove that the constructed spaces represent the moduli functor of log mixed Hodge structures with neat monodromy group.
• To apply the compactified moduli spaces to prove the analyticity of zero loci of period maps.

Proposed method

• Uses a fan Σ in the rational Lie algebra 𝔤ℚ of the monodromy group to define nilpotent orbits in the compact dual Ď.
• Defines σ-nilpotent orbits Z = exp(σℂ)F for F ∈ Ď and F in the closure of the classifying space D.
• Constructs DΣ as the union of all σ-nilpotent orbits for σ ∈ Σ, equipped with a topology making it a complex analytic space.
• Imposes admissibility conditions on fans to ensure existence of relative monodromy filtrations and independence of the orbit from the choice of nilpotent generators.
• Uses the mixed Hodge theoretic version of the SL(2)-orbit theorem (KNU08) as a key technical tool, replacing the pure case's CKS theorem.
• Applies the construction to period maps of admissible variations of mixed Hodge structure, showing that their extensions to DΣ yield closed analytic subvarieties.

Experimental results

Research questions

• RQ1 Can the moduli space of mixed Hodge structures with polarized graded quotients be compactified in a way that parametrizes log mixed Hodge structures?
• RQ2 How can the construction of KU09 for pure log Hodge structures be extended to the mixed case?
• RQ3 What conditions on the monodromy fan ensure that the compactified space represents a moduli functor?
• RQ4 Is the zero locus of a period map between such moduli spaces analytic?
• RQ5 Can the analyticity of zero loci be proven using the compactified moduli space as a compactification tool?

Key findings

• The moduli functor LMHΦ of log mixed Hodge structures with polarized graded quotients is represented by the quotient Γ\DΣ when Γ is neat.
• The space DΣ is a toroidal partial compactification of the classifying space D, constructed via nilpotent orbits and admissible fans in the Lie algebra.
• The period map of any admissible variation of mixed Hodge structure with polarized graded quotients extends to a morphism S → Γ\DΣ in the category B(log).
• The zero locus V of n period maps f1,…,fn on a quasi-projective base S* extends to a closed analytic subset V̄ in the compactification S, proving its analyticity.
• The construction provides a new approach to proving that zero loci of admissible normal functions are analytic, recovering results of Saito and Brosnan-Pearlstein in a broader framework.