Analytic Methods in Complex Algebraic Geometry

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Award: DMS 1707661, Principal Investigator: Mihai Paun Algebraic varieties are objects defined by polynomial equations. Their classification is the far-reaching problem in this field, and this project aims to answer a few central questions in this framework. After the ground-breaking work of B. Riemann in dimension one, the theory of higher dimensional algebraic and analytic varieties developed impressively fast under the impetus of differential geometry and global analysis methods. The concept of positivity, with its twofold incarnations - in algebraic terms (positivity of divisors and algebraic cycles), or in analytic terms (plurisubharmonicity, positive curvature) - played a key role in such development. These investigations will pursue further this circle of ideas by treating problems arising from algebraic geometry via analytic techniques. The main motivation is to show that the methods developed in analysis can offer a very powerful complement to the purely algebraic methods. A successful implementation of this project would lead to important progress towards classification problems in both algebraic and analytic geometry. These projects are structured in three main parts. The first concerns generalizations of the Ohsawa-Takegoshi theorem on analytic extensions, with connections to questions arising in the minimal model program of algebraic geometry. The second line of investigation concerns regularity properties of degenerate Monge-Ampere equations and planned applications to certain relative canonical bundles. Given an algebraic fiber space, one of the main objects of study is its corresponding relative canonical bundle. In many instances, this bundle carries some positivity, and the project is to use the Monge-Ampere theory in order to quantify the amount of positivity this bundle has. The third research direction is an analysis of the positivity properties of the canonical bundle of rank one holomorphic foliations defined on the jet spaces corresponding to a projective variety. This is in connection to the notion of Kobayashi hyperbolicity, where such foliations appear naturally.