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Modular interpretation of the Weil-Petersson metric asymptotics for abelian varieties

Andres Gomez

Abstract

As a first step towards a refined description of the asymptotic of the Weil-Petersson metric on the moduli space of polarized Calabi-Yau manifolds we investigate the concrete case of abelian varieties by linking such asymptotic with the multi-scale collapsing limits of the parametrized flat tori, as explicitly classified by Odaka.

Modular interpretation of the Weil-Petersson metric asymptotics for abelian varieties

Abstract

As a first step towards a refined description of the asymptotic of the Weil-Petersson metric on the moduli space of polarized Calabi-Yau manifolds we investigate the concrete case of abelian varieties by linking such asymptotic with the multi-scale collapsing limits of the parametrized flat tori, as explicitly classified by Odaka.
Paper Structure (9 sections, 12 theorems, 140 equations)

This paper contains 9 sections, 12 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Moduli space of principally polarized abelian varieties.
  4. Boundary components of .
  5. Geometric compactifications of .
  6. Asymptotic behavior of the Weil--Petersson metric
  7. The Weil--Petersson metric on .
  8. Pointed Gromov-Hasudorff limits of .
  9. Final remarks

Key Result

Theorem 1.1

Let $(A_g,\mathcal{g}_{\scaleto{W\!P}{3.5pt}})$ be the moduli space of complex $g$-dimensional principally polarized abelian varieties endowed with its canonical Kähler structure given by the Weil--Petersson metric. Then for every non-degenerate direction at infinity in $A_g$, there exists an intege where

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3: Odaka2019 Proposition $2.13$
  • Theorem 2.4: Odaka2019 Theorem $2.3$
  • Theorem 2.5: Odaka2019 Theorem $3.1$
  • ...and 23 more