We study adiabatic limits of Ricci-flat K¨ahler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Amp`ere equation, we show that the Ricci-flat metrics collapse (away from the singular fibers) to a metric on the base of the fibration. This metric has Ricci curvature equal to a WeilPetersson metric that measures the variation of complex structure of the Calabi-Yau fibers. This generalizes results of Gross-Wilson for K3 surfaces to higher dimensions.
Duncan, JohnCase Western Reserve UniversityEtingof, PavelMITIvan Chi-Ho IpUniversity of TokyoKhovanov, MikhailColumbia UniversityLibine, MatveiIndiana UniversityLicata, AnthonyAustralian National UniversitySavage, AlistairUniversity of OttawaSchlosser, MichaelUniversity of Vienna
Perspectives in Representation Theory: A Conference in Honor of Igor Frenkel’s 60th Birthday on Perspectives in Representation Theory, Yale University, New Haven, 12-17 May 2012