Canonical metric on moduli spaces of log Calabi-Yau varieties
Résumé
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. 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. 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.
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