Birational Geometry and Hodge Theory

Purdue University, West Lafayette IN

Investigators

Abstract

The project will be broken into various subprojects in algebraic geometry. In the first subproject, Arapura intends to study vanishing theorems, which are very roughly a collection of techniques for controlling homological invariants. In the second subproject, Arapura intends to study some problems related to the Hodge conjecture, which predicts that certain homological entities called Hodge classes can be realized in a geometric way. In the fifth subproject, Matsuki plans to study the homological aspects of the minimal model program, and also to study the automorphisms of the three dimensional space. Several subproject involve finding good models, or resolutions, for varieties and maps between them. For maps this can be formulated more precisely as the toroidalization problem, and this will be studied by Matsuki in the fourth subproject. Finding good models for a variety amounts to resolution of singularities. Various aspects of the problem will be studied by Matsuki and Wlodarczyk in the seventh subproject. In particular, Wlodarczyk has found a simplified algorithm for resolutions of singularities in characteristic zero, which he plans to refine. In the eighth project, Wlodarczyk will further develop his theory of stratified toroidal varieties, which extends the theory of toroidal embeddings. Algebraic varieties are basic objects in mathematics; they are sets of solutions of systems of algebraic equations. They have found applications in areas as diverse as mathematical physics and cryptography. The goal of this project is to further the understanding of these objects. A standard technique involves expressing the objects by their homological invariants, which are usually more accessible and often computable. Some of the subproject involve this approach. The remaining subproject involves finding good models for algebraic varieties.