2007 Fiscal Year Final Research Report Summary
Existence Problem of Fxtremal Metran and Degeneration of Balanced Metrics
| Project/Area Number |
16204006
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Osaka University |
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Principal Investigator |
MABUCHI Toshiki Osaka University, Graduate School of Science, Professor (80116102)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAJIMA Hiraku Kyoto University, Graduate Schcol of Science, Professor (00201666)
ONO Kaoru Hokkaido University, Graduate School of Science, Professor (20204232)
ITOH Mitsuhiro University of Thukuba, Graduate Schcol of Pure and Applied Sciences, Professor (40015912)
UMEHARA Masaaki Osaka University, Graduate Schcol of Science, Professor (90193945)
GOTO Ryushi Osaka University, Graduate School of Science, Associate Professor (30252571)
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| Project Period (FY) |
2004 – 2007
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| Keywords | geometry / extremal metric / stability / balanced metric / degeneration / constant scalar curvature / test configuration / obstruction |
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| Research Abstract |
(1) Related to the existence problem of extremal metrics, we studied various kinds of stabilities for manifolds For instance, we succeeded in showing that Chow-Mumford stability and Hilbert-Mumford stability are asymptotically equivalent (Chow-Mumford stability implies HiItert-Mumford stability by the work of Fogarty, while nothing was known even asymptotically about its converse).
(2) Associated to the Monge-Ampere equation for the Kahler-Einstein metric on an Einstein toric surface, we have a hyperbolic affine sphere equation with Dirichlet condition defined on a bounded convex C^<2> domain in R^<2>, and for the solution of the equation, we obtain a very explicit asymptotic expansion along the boundary.
(3) It is known that the anticanonical bundle of a toric or Kahler Einstein Fano manifold admits a Ricci-flat Kahler metric inducing a Sasaki-Einstein metric on a suitable quotient. We studied analogous examples for anticanonical bundles of Fano manifolds with Kahler-Ricci solitons.
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