Geometric study of linear differential equations in complex variables
Grant-in-Aid for General Scientific Research (B)
|Allocation Type||Single-year Grants |
|Research Institution||KYUSHU UNIVERSITY |
YOSHIDA Masaaki Kyushu Univ., Graduate School of Math., Professor, 大学院・数理学研究科, 教授 (30030787)
NAKAO Mitsuhiro Kyushu Univ., Graduate School of Math., Professor, 大学院・数理学研究科, 教授 (10136418)
KAJIWARA Jouji Kyushu Univ., Graduate School of Math., Professor, 大学院・数理学研究科, 教授 (90037169)
CHOU Kanchi Kyushu Univ., Graduate School of Math., Associ.Prof., 大学院・数理学研究科, 助教授 (10197634)
MIMACHI Katsuhisa Kyushu Univ., Graduate School of Math., Associ.Prof., 大学院・数理学研究科, 助教授 (40211594)
白谷 克巳 九州大学, 大学院数理学研究科, 教授 (80037168)
若山 正人 九州大学, 大学院数理学研究科, 助教授 (40201149)
|Project Period (FY)
1994 – 1995
Completed (Fiscal Year 1995)
|Budget Amount *help
¥7,200,000 (Direct Cost: ¥7,200,000)
Fiscal Year 1995: ¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 1994: ¥3,900,000 (Direct Cost: ¥3,900,000)
|Keywords||hypergeometric / intersection theory / twisted (co) homology / configuration space / modular interpretation / twisted(co)homology / twisted cycles / twisted cohomology / 交叉理論 / Selberg型積分 / Capelli恒等式 / 量子群 / pfaffian / Hasse不変量|
Though the Gauus's hypergeometric differential equation has a fruitful relationship with the theory of automorphic forms and differential geometry, the theory of ordinary differential equations in complex variables has been restricted to analytic and local studies. The present research aims generalizations and specializations of the Gauss's theory regarding the hypergeometric differential equation as the uniformizing equation of the obifolds with three singular points on the projective line. In 1994 and 1995, we got the following results.
(1) Equivalence problem for projective submanifolds. Let D be a symmetric space equivariantly embedded in the minimal dimensional projective space P^N. We consider the problem : For a submanifold V * P^N, state a differential geometrical condition for V so that it is locally projectively equivalent to D.This problem is crucial for studing uniformizing equations. When D is of type VI (i.e.a hyperquadric), the problem is solved.
(2) Geometry of configurat
ion spaces. Let X (k, n) be the configuration space of n points in general postion in P^<k-1>. This space is one of the main objects in algebraic geometry in the beginning of this century. Since this space turned out to be the natural domain of definition of the hypergeometric differential equation of type (k, n), this became a hot topic again. We clarified the combinatorial topological properties of the configuration spaces of type (2, n) and (3,6).
(3) Modular interpretation of configuration spaces. Problem : Is a configuration space isomorphic to the quotient space D/GAMMA for a symmetric domain D and a discontinuous group GAMMA? There are many studies about the space X (2, n). We succeeded to solve positively the problem for X (3,6),
(4) Intersection theory for twisted (co) homolgies. We established the intersection thery for twisted homolgies. We are about to establish it for twisted cohomolgies. Comparison of (co) homolgies of a variety and those of a branched covering is also studied.
(5) Studies of Selberg-type integrals. A kind of Selberg-type integrals is studied. The theory stated in (4) is useful.
(6) The image of the hypergeometric maps. We proved that if (k, n) * (2, n), (3,6), then the image of the hypergeometric map is not locally projective equivalent to the Grassmannian embedded in the projective space by the Plucker map. This negative result destroyed the dream of many mathematicians. Less
Report (3 results)
Research Products (29 results)