1996 Fiscal Year Final Research Report Summary
Geometric Study of Algebraic Systems
Project/Area Number 
07454006

Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Osaka University 
Principal Investigator 
MIYANISHI Masayoshi Osaka University, Department of Mathematics, Professor > 大阪大学, 大学院・理学研究科, 教授 (80025311)

CoInvestigator(Kenkyūbuntansha) 
YAMANE Hiroyuki Osaka University, Mathematics, Associate Professor, 大学院・理学研究科, 講師 (10230517)
KONNO Kazuhiro Osaka University, Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (10186869)
MURAKAMI Jun Osaka University, Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (90157751)
USUI Sanpei Osaka University, Mathematics, Professor, 大学院・理学研究科, 教授 (90117002)
FUJIKI Akira Osaka University, Mathematics, Professor, 大学院・理学研究科, 教授 (80027383)

Project Period (FY) 
1995 – 1996

Keywords  Theorem of AbhyankarMoh / Open algebraic surfaces / Hyper Kahler manifold / Log geometry / Hodge structure / Quantum invariant / Surface of general type / BMW algebra 
Research Abstract 
1.The head investigator, Masayoshi Miyanishi, gave new proofs to the AbhyankarMoh Theorem and the LinZaidenberg Theorem which are based on the classification theory of open algebraic surfaces. He considered the embeddings of affine curves with oneplace points at infinity into the affine plane and classified such embeddings of possibly minimal degree when the genus is low. He, furthermore, simplified a proof in P.Roberts' counterexample to the fourteenth problem of Hilbert. 2.Akira Fujiki found a natural partial compactification of hyper kahler manifolds as quaternionic manifolds which are obtained as hyper Kahler quotinet spaces of certain kinds. He also investigated the variation via moment maps of Kahler quotient spaces by making use of equivariant cohomology groups. 3.Sanpei Usui proved that one can recover vanishing cycles by using log geometry. As applications of this result, he clarified the Zstructure of the variation of Hodge structures and the description of monodromies. 4.Jun Murakami constructed a new invariant for threedimensional manifolds which is based on the Kontsevich invariant for the knots and studied its properties. He constructed a topological quantum theory by making use of this invariant, and a family of mapping groups of surfaces as its application. 5.Kazuhiro Konno defined a new index for surfaces with nonhyperelliptic pencils and found some general method to investigate the lower bound of the slopes of surfaces. He also an index, called the Horikawa index, for degenerate fibers. 6.Hiroyuki Yamane obtained a new technique to calculate the index of the BMW algebras which are defined in a paper of Rham.

Research Products
(12 results)