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Sections of Hodge bundles I: Global theory and applications to period maps

Kefeng Liu, Yang Shen

Abstract

We study global sections of Hodge bundles arising from two complementary constructions: a deformation-theoretic construction, which yields global geometric consequences for period maps, and a construction from the matrix representation of the image of the period map, which provides an explicit Euclidean realization. Combining these perspectives, we prove that the image of the lifted period map on the universal cover is contained in a complex Euclidean subspace of the period domain, thereby giving a partial solution to a conjecture of Griffiths on the global behavior of period maps. As an application, we construct a global complex affine structure on the Teichmüller space of Calabi--Yau type manifolds.

Sections of Hodge bundles I: Global theory and applications to period maps

Abstract

We study global sections of Hodge bundles arising from two complementary constructions: a deformation-theoretic construction, which yields global geometric consequences for period maps, and a construction from the matrix representation of the image of the period map, which provides an explicit Euclidean realization. Combining these perspectives, we prove that the image of the lifted period map on the universal cover is contained in a complex Euclidean subspace of the period domain, thereby giving a partial solution to a conjecture of Griffiths on the global behavior of period maps. As an application, we construct a global complex affine structure on the Teichmüller space of Calabi--Yau type manifolds.
Paper Structure (12 sections, 27 theorems, 226 equations)

This paper contains 12 sections, 27 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Period domains, period maps and unipotent orbits in period domains
  3. Sections of Hodge bundles from Hodge theory
  4. Sections of Hodge bundles from deformation theory
  5. Beltrami differentials
  6. Harmonic theory on compact Kähler manifolds
  7. Extension equations and closed formulas
  8. Closed formula for global section of Hodge bundles
  9. A conjecture of Griffiths
  10. Affine structures on the moduli spaces and examples
  11. Moduli spaces and period maps
  12. Affine structure on the Teichmüller space of Calabi--Yau type manifolds

Key Result

Theorem 1

Let the notations be as above. Then the sections $\Omega_{(p)}(t)$ in intr lm section, defined over $t \in \widetilde{S}^{\vee}$ via the matrix representation of the image $\Phi(t)$ in $N_-$, can be holomorphically extended to the sections $\widetilde{\Omega}_{(p)}(t)$ in intr def section, defined o

Theorems & Definitions (60)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 1.7
  • ...and 50 more