Kähler-Ricci solitons and generalized Tian-Zhu’s invariant. (English) Zbl 1297.53052
Summary: In this paper, we introduce the notion of modified \(K\)-stability (\(K\)-semistability) associated with the soliton field on a Fano manifold \(M\). Then we prove that if the modified \(K\)-energy associated with the soliton vector field is bounded from below, \(M\) is modified \(K\)-semistable associated with the soliton field. This result generalizes the corresponding result in a paper of Ding and Tian.
MSC:
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
Keywords:
Kähler-Ricci solitons; Tian-Zhu’s invariant; modified \(K\)-energy; modified \(K\)-stability; Fano manifoldReferences:
| [1] | DOI: 10.1007/s00039-005-0522-y · Zbl 1084.53035 · doi:10.1007/s00039-005-0522-y |
| [2] | DOI: 10.1007/BF01231335 · Zbl 0779.53044 · doi:10.1007/BF01231335 |
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| [4] | DOI: 10.1002/9781118032527 · doi:10.1002/9781118032527 |
| [5] | DOI: 10.1112/blms/bdl015 · Zbl 1111.53057 · doi:10.1112/blms/bdl015 |
| [6] | DOI: 10.1007/s002220050176 · Zbl 0892.53027 · doi:10.1007/s002220050176 |
| [7] | DOI: 10.1007/BF02392630 · Zbl 1036.53052 · doi:10.1007/BF02392630 |
| [8] | DOI: 10.1007/s00014-002-8341-3 · Zbl 1036.53053 · doi:10.1007/s00014-002-8341-3 |
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| [10] | DOI: 10.1007/BF02921996 · Zbl 1036.53054 · doi:10.1007/BF02921996 |
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