Birational Geometry and Hodge Theory

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This project, which is divided into several parts,is concerned with several interrelated areas ofalgebraic 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 tothe study of algebraic cycles. In the second partthe investigators, in collaboration with D. Abramovichand K. Karu, intend to extend their previous work to theprove the strong factorization conjecture for birational maps.This conjecture says that any birational map between smoothcomplete varieties has a particularly simple structure: itis a sequence blow ups followed by a sequence of blow downswith smooth centers. In the third part, the investigatorswill 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 vanishingtheorems and apply them to the study of birationalinvariants. 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 thesixth part, one of the investigators intends to study a class of surface singularities, which are important for birationalgeometry, over fields of positive characteristic. In thesixth and final part, one of the investigators intends to extend the theory of toroidal embeddings by taking into accountcertain stratifications. Algebraic varieties are geometric objects which provide arich set of models for a number of phenomena within mathematicsas well as in neighboring fields of science such as physics andcomputer science. They have the advantage of being describable in finite terms, as solutions to a finite system of algebraicequations. 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 ofbirational 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 certainnatural birational invariants, that is, measuresof the geometric complexity of algebraic varieties. Some ofthese invariants count the number of harmonic (energyminimizing) objects associated to the algebraic variety.These two goals are related since the investigators expect that thefine structure of the birational maps will yield insights intothe properties of these invariants.