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

TL;DR

This paper proves that Weil-Petersson volumes of weighted points on the projective line are continuous when transitioning from Fano to Calabi-Yau geometries, showing convergence of different volume computations.

Contribution

It establishes the continuity of Weil-Petersson volumes in the weighted points setting and connects two different volume computation methods.

Findings

Weil-Petersson volume converges to the Calabi-Yau volume as weights approach the Calabi-Yau case.

Localization and McMullen's techniques yield consistent volume limits.

The work bridges Fano and Calabi-Yau geometries through volume analysis.

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.