The Converse of Abel's Theorem

• Main Author: Kissounko, Veniamine
• Other Authors: Khovanskii, Askold
• Language: en_ca
• Published: 2009
• Subjects: 0405
• Online Access: http://hdl.handle.net/1807/17785

In my thesis I investigate an algebraization problem. The simplest, but already nontrivial, problem in this direction is to find necessary and sufficient conditions for three graphs of smooth functions on a given interval to belong to an algebraic curve of degree three. The analogous problems were raised by Lie and Darboux in connection with the classification of surfaces of double translation; by Poincare and Mumford in connection with the Schottky problem; by Griffiths and Henkin in connection with a converse of Abel’s theorem; by Bol and Akivis in the connection with the algebraization problem in the theory of webs. Interestingly, the complex-analytic technique developed by Griftiths and Henkin for the holomorphic case failed to work in the real smooth setting. In the thesis I develop a technique of, what I call, complex moments. Together with a simple differentiation rule it provides a unified approach to all the algebraization problems considered so far (both complex-analytic and real smooth). As a result I prove two variants (’polynomial’ and ’rational’) of a converse of Abel’s theorem which significantly generalize results of Griffiths and Henkin. Already the ’polynomial’ case is nontrivial leading to a new relation between the algebraization problem in the theory of webs and the converse of Abel’s theorem. But, perhaps, the most interesting is the rational case as a new phenomenon occurs: there are forms with logarithmic singularities on special algebraic varieties that satisfy the converse of Abel’s theorem. In the thesis I give a complete description of such varieties and forms.