Birational Geometry and Hodge Theory

点击翻译按钮获取中文摘要

The project will be broken into various subprojects in algebraicgeometry. In the first subproject, Arapura intends to study vanishingtheorems, which are very roughly a collection of techniques forcontrolling homological invariants. In the second subproject, Arapuraintends to study some problems related to the Hodge conjecture, whichpredicts that certain homological entities called Hodge classes can berealized in a geometric way. In the fifth subproject, Matsuki plans tostudy the homological aspects of the minimal model program, and alsoto study the automorphisms of the three dimensional space. Severalsubproject involve finding good models, or resolutions, for varietiesand maps between them. For maps this can be formulated more preciselyas the toroidalization problem, and this will be studied by Matsuki inthe fourth subproject. Finding good models for a variety amounts toresolution of singularities. Various aspects of the problem will bestudied by Matsuki and Wlodarczyk in the seventh subproject. Inparticular, Wlodarczyk has found a simplified algorithm forresolutions of singularities in characteristic zero, which he plans torefine. In the eighth project, Wlodarczyk will further develop histheory of stratified toroidal varieties, which extends the theory oftoroidal embeddings.Algebraic varieties are basic objects in mathematics; they are sets ofsolutions of systems of algebraic equations. They have foundapplications in areas as diverse as mathematical physics andcryptography. The goal of this project is to further the understandingof these objects. A standard technique involves expressing the objectsby their homological invariants, which are usually more accessible andoften computable. Some of the subproject involve this approach. Theremaining subproject involves finding good models for algebraicvarieties.