Singular Kahler-Einstein Metrics: Analytic and Algebraic Aspects

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

Complex spaces or complex manifolds are spaces that carry a certain additional structure that arises naturally in physical theories. Complex geometry aims to study the shapes of complex manifolds through distances and angles; mathematically these measurements are called Kähler metrics. In the physics context, manifolds that carry a metric that is both a Kähler metric and an Einstein metric are of interest; these are called Kähler-Einstein metrics. This project considers several central problems in complex geometry, both from analytic and algebraic points of view, and builds around the notion of a so-called singular Kähler-Einstein metric. This relatively recent notion has been introduced in an attempt to extendi Yau's theorem on the resolution of Calabi's conjecture to singular settings directly coming from algebraic geometry, and more precisely birational (complex) geometry. A major current challenge consists of constructing these objects in various contexts and trying to understand their behavior; a good understanding of the behavior of singular Kähler-Einstein metrics would have important consequences both in complex geometry as well as in areas of theoretical physics, such as string theory. Moreover, studying families of singular Kähler-Einstein metrics is very closely linked with the moduli theory, one of the main components of algebraic geometry. It is expected that results of the project concerning applications of singular Kähler-Einstein metrics will advance knowledge in the theory of singular spaces, in analytic and algebraic geometry, and in other areas of mathematics and physics. The project focuses on three central topics: behavior of the Kähler-Einstein metrics near the singularities of the variety, families and degenerations of singular Kähler-Einstein metrics, and applications of the Kähler-Einstein theory to algebraic geometry. The first circle of ideas is about finding models for Kähler-Einstein metrics near their singularities. The singularities can arise from those of the variety or from the boundary divisor of the pair to consider. The project examines the general case of (semi) log canonical pairs, through past results of the PI and others, as well as numerous questions that arise in the these works. The topics include improving the pluripotential methods to get sharper estimates, understanding the (Riemannian) geometry of the singular Kähler-Einstein metrics by relating singularities and completeness, and deriving higher regularity at the singular points. The second circle of ideas centers on families of singular Kähler-Einstein metrics, and in particular studying the moduli space of canonically polarized varieties from a differential geometric point of view through Weil-Petersson geometry. Other related questions are about Gromov-Hausdorff (or more refined) convergence of conic Kähler-Einstein metrics. The hope is that the recent developments in pluripotential theory should give a new angle to tackle these problems. The third circle of ideas focuses on various applications of singular Kähler-Einstein metrics, particularly in algebraic geometry. Typical questions that arise in the project concern vanishing/parallelism theorems or semi-stability properties of the tangent sheaf of singular varieties. Here again, a consequent part of the questions is motivated by recent progress in the theory of Monge-Ampère equations.