2013 Fiscal Year Final Research Report
A study of compact complex manifolds admitting non-isomorphic surjective endomorphisms
| Project/Area Number |
23540055
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Nara Medical University |
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Principal Investigator |
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| Project Period (FY) |
2011 – 2013
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| Keywords | 自己正則写像 / 端射線 / コニック束 / 半安定ベクトル束 / 楕円繊維曲面 / 楕円ファイバー空間 / 極小モデル計画 |
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| Research Abstract |
We have studied structures of smooth projective 3-folds X with negative Kodaira dimension admitting non-isomorphic \'{e}tale endomorphisms. We can show that a suitable finite \'{e}tale covering X' of X is classified into 6 types. There exists some X where X' cannot be decomposed as a product of a uniruled surface and an elliptic curve. The contraction morphism associated to R is the inverse of a blow-up of another smooth 3-fold along an elliptic curve. The trouble is that R cannot necessarily be stabilized by a suitable power of f. Hence the minimal model program(MMP) cannot work so as to be compatible with endomorphisms. As a substitute, we consider the tower of non-isomorphic finite \'{e}tale covering of 3-folds (called ESP), on which MMP works well. Such a 3-fold X can be obtained as a succession of blow-ups along an elliptic curve from a characteristic ESP (called DESP). We can also give a simple criterion for certan conic bundles to have non-isomorphic \'{e}tale endomorphisms.
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