Limits of Calabi-Yau metrics when the Kahler class degenerates

Abstract

We study the behaviour of families of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold when the Kahler classes degenerate to the boundary of the ample cone. We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ricci-flat metric. The limit can also be understood from algebraic geometry.

22 pages; a new L^infty estimate in section 4; to appear in J.Eur.Math.Soc. 2009

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