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On the continuity of Weil-Petersson volumes of the moduli space weighted points on the projective line

Salvatore Tambasco

Abstract

In this work we show that the Weil-Petersson volume (which coincides with the CM degree) in the case of weighted points in the projective line is continuous when approaching the Calabi-Yau geometry from the Fano geometry. More specifically, the CM volume computed via localization converges to the geometric volume, computed by McMullen with different techniques, when the sum of the weights approaches the Calabi-Yau geometry.

On the continuity of Weil-Petersson volumes of the moduli space weighted points on the projective line

Abstract

In this work we show that the Weil-Petersson volume (which coincides with the CM degree) in the case of weighted points in the projective line is continuous when approaching the Calabi-Yau geometry from the Fano geometry. More specifically, the CM volume computed via localization converges to the geometric volume, computed by McMullen with different techniques, when the sum of the weights approaches the Calabi-Yau geometry.

Paper Structure

This paper contains 7 sections, 11 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. The log CM line bundle
  3. The CM line bundle for pairs
  4. The intersection number of the log CM line bundle
  5. A first study: the case of four points on the complex projective line.
  6. The general case
  7. Final remarks

Key Result

Theorem 1.1

The Fano CM volumes of $M_d$ with $d_i \in (0,1) \cap \mathbb{Q}$, for all $i \in \{1,2,....,n\}$ converge to the volume computed by McMullen in MM00 (Theorem 8.1) when the sum of the weights approaches 2 from below.

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.1
  • Definition 2.1
  • Proposition 2.1
  • Remark 2.1
  • Corollary 2.1
  • Corollary 2.2
  • Proposition 2.2
  • Remark 2.2
  • Proposition 2.3
  • ...and 12 more