2014 Fiscal Year Final Research Report
Localization of Ricci form and the existence of an anti-canonical divisor on asymptotically Chow stable Fano manifolds
| Project/Area Number |
23654025
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Nagoya University |
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Principal Investigator |
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| Project Period (FY) |
2011-04-28 – 2015-03-31
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| Keywords | 複素モンジュアンペール方程式 / ケーラー計量の空間 / モンジュアンペールエネルギー汎関数 / 修正リッチ流の離散化 / ケーラー計量の力学系 / 定スカラー曲率ケーラー計量 |
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| Outline of Final Research Achievements |
cscK metric is not directly expressed as a solution of a Monge-Ampere equation. However, cscK metric is a critical point of the Monge-Ampere energy functional on the space of Kaehler metrics. It is expected that the study of csck metric is deeply connected to that of Monge-Ampere equations. I discovered a dynamical system on the space of Kahler metrics which discretizes the modified KRF (the evolution equation obtained by replacing -Ric(ω)+ω in the RHS of KRF by -Ric(ω)+HRic(ω), where H means the harmonic part). This dynamical system consists of iteration of the following procedure. First we solve a Poisson equation for u so that the trace of Ric(ω)+ddc u becomes constant. Next we solve the Monge-Ampere equation Ric(ω(φ))=Ric(ω)+ddc u for φ to get a Kaehler metric whose Ricci form coincides with Ric(ω)+ddc u. This procedure constitutes the basic step of out dynamical system. The iteration of this procedure is our dynaical system. This dynakical system fixes cdcK metric.
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| Free Research Field |
数学（幾何学）
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