Journal of the European Mathematical Society

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Volume 11, Issue 4, 2009, pp. 755–776

DOI: 10.4171/JEMS/165

Limits of Calabi–Yau metrics when the Kähler class degenerates

Valentino Tosatti (1)

(1) Department of Mathematics, Harvard University, 1 Oxford Street, MA 02138-2901, CAMBRIDGE, UNITED STATES

We study the behavior of families of Ricci-ﬂat Kähler metrics on a projective Calabi– Yau manifold when the Kähler 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.

Keywords: Calabi–Yau manifolds, Ricci-ﬂat metrics, degenerate complex Monge–Ampère equations