Northwestern, Notre Dame, UIC Complex Geometry Seminar

Abstract: In a lot of geometric situations we need to work with families of varieties. In this talk we focus on families of singular Kähler-Einstein metric. In particular we study the case of a family of Kähler varieties and we develop the first steps of pluripotential theory in family, which will allow us to have a control on the C^0 estimate when the complex structure varies. This type of result will be applied in different geometric contexts. This is a joint work with V. Guedj and H. Guenancia.

The laws of nature at its most fundamental level have long been a source of inspiration for geometry and partial differential equations. With unified string theories and particularly supersymmetry, a particularly important new requirement has emerged, which is that of special holonomy. The earliest manifestation was identified by Candelas, Horowitz, Strominger, and Witten in 1985 as the Calabi-Yau condition, but more general spaces have emerged since, which can be interpreted as generalizations of Calabi-Yau spaces to both non-Kaehler complex geometry and symplectic geometry. The corresponding equations are interesting in their own right from the point of view of the theory of non-linear partial differential equations. We shall survey some of these developments, with emphasis on the analytic open problems.