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.
Cite this article
Valentino Tosatti, Limits of Calabi–Yau metrics when the Kähler class degenerates. J. Eur. Math. Soc. 11 (2009), no. 4, pp. 755–776DOI 10.4171/JEMS/165