Over the past few months, CMSA member Valentino Tosatti has authored the following papers. Preprints can be found in ArXiv.
Diameter Bounds for Degenerating Calabi-Yau Metrics
Yang Li and Valentino Tosatti
Abstract. We obtain sharp upper and lower bounds for the diameter of Ricciflat Kähler metrics on polarized Calabi-Yau degeneration families, as conjectured by Kontsevich-Soibelman.
Smooth Asymptotics for Collapsing Calabi-Yau Metrics
Hans-Joachin Hein and Valentino Tosatti
Abstract. We prove that Calabi-Yau metrics on compact Calabi-Yau manifolds whose Kähler classes shrink the fibers of a holomorphic fibration have a priori estimates of all orders away from the singular fibers. This follows from a stronger statement which gives an asymptotic expansion of these metrics as the fibers shrink, in terms of explicit functions on the total space and with k-th order remainders that satisfy uniform C k -estimates with respect to a collapsing family of background metrics.
Canonical Currents and Heights for K3 Surfaces
Simion Filip and Valentino Tosatti
Abstract. We construct canonical positive currents and heights on the boundary of the ample cone of a K3 surface. These are equivariant for the automorphism group and fit together into a continuous family, defined over an enlarged boundary of the ample cone. Along the way, we construct preferred representatives for certain height functions and currents on elliptically fibered surfaces.
