2007 Fiscal Year Final Research Report Summary

Splittings of degenerafions of Riemann surfaces

Research Project

Project/Area Number	16540062
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Single-year Grants
Section	一般
Research Field	Geometry
Research Institution	Kyoto University
Principal Investigator	TAKAMURA Shigeru Kyoto University, Graduate School of Science, Associate Professor (20362436)
Project Period (FY)	2004 – 2007
Keywords	Riemann surface / splitting / simgularity / monodromy / complex stracture / complex surface / moduli / theta function
Research Abstract	Consider a family fo Riemann surfaces with a slaguar fiber (a degeneration), Its topological type is determined by its topological monodromy (Matsumoto-Montesinos Theorem). If the topological monodromy is periodic, then the sluguar fiber is star-shaped ite core with branches, For a particular cure where the core is a Riemann sphere and the number of branches is exactly three, my joint work with Kazushi Ahara (Meiji Univ). Showed that such a degeneration necessarily hao a splitting deformation. The proof is based on my splittability criterion is terms of the existence of a shbdivisor (called a crust) of the singular fiber, The latter work is published an Springer Lecture Notes in Math 1886. We however remark that if the number of braches if at least four, then these is some cate where my criterion does not work, but it' may possibly have a splitting family. this is also a challenging problem to consider.

(6 results)