The investigator plans to adapt his simplified algorithm of Hironaka's resolution of singularities to different types of resolution problems. By a suitable modification of the basic notions (like order, marked ideals and etc) introduced in the algorithm in characteristic zero the investigator plans to approach the problem of resolution of singularities in positive characteristic. In particular, the investigator plans to apply the resolution algorithm to some special classes like locally binomial varieties. Already in this case the existence of canonical embedded resolution is highly nontrivial and it is a good testing ground for the general approach. On the other hand by using the theory of Deligne Weight filtration on the cohomology groups the investigator plans to study new invariants of singularities describing the cohomological properties of the fibers of resolutions of singularities and their links. The investigator plans also to study generalizations of the Weak factorization theorem from the point of view of its further potential applications. The theorem allows to factor any birational maps between nonsingular varieties into a sequence of blow-ups and blow-downs with smooth centers. It is a very useful tool, especially for studying, so called, birational invariants. The problem is that the resolution of singularities is not unique. The Weak factorization theorem can be thus used to compare the invariants resulting from the desingularization process. The technique of a generalized Weak factorization theorem was used also in the recent study of the invariants of the fibers of resolutions.
Studying the singularities is of critical importance in many areas of Mathematics. The Hironaka theorem on resolution of singularities plays a central role in Algebraic Geometry. It constitutes a basis of proofs of many theorems not only in Algebraic and Analytic Geometry. The theorem on desingularization is proven in characteristic zero and is known only in special cases in positive characteristic. This is a major drawback for many theorems which rely on desingularization and which can be proven in characteristic zero only. The project deals with studying new ideas and developing new relatively simple approach to the resolution problem in positive characteristic. It also opens some new possibilities in general topic of singularities in positive characteristic. On the other hand the investigator plans to study new interesting topological invariants describing the resolutions of singularities in characteristic zero. He also plans to further develop and generalize birational techniques (like the Weak factorization theorems) relevant for studying topological properties of the resolutions.
