Families of Calabi-Yau manifolds and canonical singularities

Abstract

Given a polarized family of varieties over the unit disc, smooth except over the origin and with smooth fibers Calabi-Yau, we show that the origin lies at finite Weil-Petersson distance if and only if after a finite base change the family is birational to one with central fiber a Calabi-Yau variety with at worst canonical singularities, answering a question of C.-L. Wang. This condition also implies that the Ricci-flat Kahler metrics in the polarization class on the smooth fibers have uniformly bounded diameter, or are uniformly volume non-collapsed.

8 pages; final version to appear in IMRN

References in corpus (2)

• Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, II
• Continuity of Extremal Transitions and Flops for Calabi-Yau Manifolds

Cited by in corpus (12)

• Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, II
• Geometry of twisted Kähler-Einstein metrics and collapsing
• Hybrid convergence of Kähler-Einstein measures
• Canonical metric on moduli spaces of log Calabi-Yau varieties
• Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures
• Degeneration of Ricci-flat Calabi-Yau manifolds and its applications
• On log minimality of weak K-moduli compactifications of Calabi-Yau varieties
• Diameter bounds for degenerating Calabi-Yau metrics
• Generic regularity of intermediate complex structure limits
• Relative Kähler-Einstein metric on Kähler varieties of positive Kodaira dimension
• Completion of the moduli space for polarized Calabi-Yau manifolds
• Deformations of Calabi-Yau varieties with isolated log canonical singularities