2006 Fiscal Year Final Research Report Summary
Study of moduli spaces of projective varieties of general type
| Project/Area Number |
15340018
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Sophia University |
|---|
Principal Investigator |
TSUJI Hajime Sophia University, Department of Mathematics, Professor, 理工学部, 教授 (30172000)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
FUTAKI Akito Tokyo Institute of Technology, Department of Mathematics, Professor, 理工学部, 教授 (90143247)
ISHII Shihoko Tokyo Institute of Technology, Department of Mathemtaics, Professor, 理工学部, 教授 (60202933)
|
|---|
| Project Period (FY) |
2003 – 2006
|
| Keywords | pluricanonical systems / varieties of general type / Kaehler-Einstein metrics / closed positive current / singular hermitian metrics / analytic Zariski decomposition / Bergman kernel / plurisubharmonicity |
|---|
| Research Abstract |
I obtain that there exists a positive number m(n) dependeing only on n such that for every smooth projective n-fold of general type and for every positive integer m> m(n), mK_{X} gives a birtaional embedding of X into a projective space.
This is a furthere generalization of the result of E. Bombieri for surfaces of general type.
Next I proved that the for a smooth projective family f X longrightarrow S and a semipositive singular hermitian line bundle L,
The adjoint bundle K {X/S} + L has a Bergman kernel which is logarithmically plurisubharmonic on $X$.
Moreover the curvature current extends to the whole completion as a closed positive current.
3rd I have proven a new construction of Kaehler-Einstein metric with negative Ricci curvature. This enables us to study the variation of Kaehler-Einstein volume form on a smooth projective family of canonically polarized varieties.
As a consequence I have obtained the plurisubharmonic variation of the Kaehler-Einstein volume form on a smooth projective Family.
4^<th>. I have constructed a canonical singular hermitian metric on the canonical bundle of any nonuniruled projective varieties with semipositive curvature current. And minimal singularities (AZD).
This enables us to obtain a very short proof of the invariance of plurigenera.
|
