Birational Geometry and Hodge Theory

Purdue University, West Lafayette IN

Investigators

Abstract

This project, which is divided into several parts, is concerned with several interrelated areas of algebraic geometry centered around the birational geometry and the Hodge theory of algebraic varieties. In the first part, the investigators intend to construct differentials on certain universal spaces arising in algebraic geometry, and apply these to the study of algebraic cycles. In the second part the investigators, in collaboration with D. Abramovich and K. Karu, intend to extend their previous work to the prove the strong factorization conjecture for birational maps. This conjecture says that any birational map between smooth complete varieties has a particularly simple structure: it is a sequence blow ups followed by a sequence of blow downs with smooth centers. In the third part, the investigators will apply the previously established weak factorization conjecture to compare the Hodge structure of two birationally equivalent minimal models. In the fourth part, one of the investigators intends to extend their previous vanishing theorems and apply them to the study of birational invariants. In the fifth part, one of the investigators will attempt to relate the Hodge theory of higher homotopy groups to the intersection theory of algebraic cycles. In the sixth part, one of the investigators intends to study a class of surface singularities, which are important for birational geometry, over fields of positive characteristic. In the sixth and final part, one of the investigators intends to extend the theory of toroidal embeddings by taking into account certain stratifications. Algebraic varieties are geometric objects which provide a rich set of models for a number of phenomena within mathematics as well as in neighboring fields of science such as physics and computer science. They have the advantage of being describable in finite terms, as solutions to a finite system of algebraic equations. However, these descriptions are often complicated and not unique; deciding when two such descriptions lead to equivalent, or even approximately equivalent, varieties is very difficult. Approximate equivalence is made precise by the notion of birational equivalence. One of the goals of this project is to study the finer structure of the birational equivalence relation. Another goal of this project is to introduce and study certain natural birational invariants, that is, measures of the geometric complexity of algebraic varieties. Some of these invariants count the number of harmonic (energy minimizing) objects associated to the algebraic variety. These two goals are related since the investigators expect that the fine structure of the birational maps will yield insights into the properties of these invariants.