Canonical metric on moduli spaces of log Calabi-Yau varieties

Hassan Jolany (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

In this paper, we give a short proof of closed formula [9],[18] of logarithmic Weil-Petersson metric on moduli space of log Calabi-Yau varieties (if exists!) of conic and Poincare singularities and its connection with Bismut-Vergne localization formula. Moreover we give a relation between logarithmic Weil-Petersson metric and the logarithmic version of semi Ricci flat metric on the family of log Calabi-Yau pairs with conical singularities. We highlight Bismut-Gillet-Soule fiberwise integral formula for logarithmic Weil-Petersson metric on moduli space of polarized log Calabi-Yau spaces by using fiberwise Ricci flat metric. In final we consider the semi-positivity of singular logarithmic Weil-Petersson metric on the moduli space of log-Calabi-Yau varieties. Moreover, we show that Song-Tian-Tsuji measure is bounded along Iitaka fibration if and only if central fiber has log terminal singularities and we consider the goodness of fiberwise Calabi-Yau metric in the sense of Mumford and goodness of singular Hermitian metric corresponding to Song-Tian-Tsuji measure. We also mention that the Song-Tian-Tsuji measure is bounded near origin if and only if after a finite base change the Calabi-Yau family is birational to one with central fiber a Calabi-Yau variety with at worst canonical singularities.

Mots clés: Moduli Spaces; Kahler-Ricci flow; Singularites; Kahler-Einstein metrics; Calabi-Yau manifolds; Weil-Petersson metric

Domaines: Mathématiques [math]

Hassan Jolany. Canonical metric on moduli spaces of log Calabi-Yau varieties . 2017. ⟨hal-01413746v4⟩