Apr
15
Everyone is welcome. Coffee will be served in Math Lounge before the exam. Departmental PhD Thesis Exam ---------------------------- Wednesday, April 21, 2010, 1:10 p.m., in BA 6183, 40 St. George Street PhD Candidate: Mikhail Mazin PhD Advisor: Askold Khovanskii Thesis Title: Geometric Theory of Parshin Residues. (http://www.math.utoronto.ca/mmazin/thesis.pdf) Thesis Abstract: In the early 70's Parshin introduced his notion of the multidimensional residues of meromorphic top-forms on algebraic varieties. Parshin's theory is a generalization of the classical one-dimensional residue theory. The main difference between the Parshin's definition and the one-dimensional case is that in higher dimensions one computes the residue not at a point but at a complete flag of irreducible subvarieties X=X_n\supset ... \supset X_0, dim(X_k)=k. Parshin, Beilinson, and Lomadze also proved the Reciprocity Law for residues: if one fixes all elements of the flag, except for X_k, where 0<k<n, and consider all possible choices of X_k, then only finitely many of these choices give non-zero residues, and the sum of these residues is zero. Parshin's constructions are completely algebraic. In fact, they work in very general settings, not only over complex numbers. However, in the complex case one would expect a more geometric variant of the theory. In my thesis I study Parshin residues from the geometric point of view. In particular, the residue is expressed in terms of the integral over a smooth cycle. Parshin-Lomadze Reciprocity Law for residues in the complex case is proved via a homological relation on these cycles. The thesis consists of two parts. In the first part the theory of Leray coboundary operators for stratified spaces is developed. These operators are used to construct the cycle and prove the homological relation. In the second part resolution of singularities techniques are applied to study the local geometry near a complete flag of subvarieties.
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