The role of curvature in algebraic geometry: some history and current issues 1/2

Plática dada por Phillip Griffiths (Institute for Advanced Study, Princeton) en el evento Algebraic Geometry in Mexico 2016 el martes 1 de noviembre del 2016.

Curvature has historically guided some of the most important results in algebraic geometry: vanishing theorems leading to existence results, holomorphic mappings to algebraic varieties (hyperbolicity), and global geometric applications of Hodge theory. Typically curvature methods prove a result when the variety is smooth and the curvature form is definite, and then considerable skill and technical effort extends these results to a general algebro-geometric setting when the variety is singular and the curvature is only definite on a Zariski open set. There has recently been work using curvature properties of the Hodge bundles in a family of algebraic varieties to establish results in the smooth and definite case such as the well-known conjectures “The Hodge bundle is ample on the Satake-Baily-Borel completion of a period mapping” and “The moduli space for varieties of general type is itself of log-general type”. The proofs of these results in general require non-trivial technical extensions in the directions suggested by curvature considerations, including extending the methods to include when the metric and its curvature become singular. In this talk I will briefly recall some of the classical results and then discuss some more recent results and issues just mentioned.

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