Upper bounds for the dimension of tori acting on GKM manifolds
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- Kuroki Shintarรด
- Department of Applied Mathematics Okayama University of Science
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Abstract
<p>The aim of this paper is to give an upper bound for the dimension of a torus ๐ which acts on a GKM manifold ๐ effectively. In order to do that, we introduce a free abelian group of finite rank, denoted by ๐(Γ, ๐ผ, ∇), from an (abstract) (๐, ๐)-type GKM graph (Γ, ๐ผ, ∇). Here, an (๐, ๐)-type GKM graph is the GKM graph induced from a 2๐-dimensional GKM manifold ๐2๐ with an effective ๐-dimensional torus ๐๐-action which preserves the almost complex structure, say (๐2๐, ๐๐). Then it is shown that ๐(Γ, ๐ผ, ∇) has rank โ(> ๐) if and only if there exists an (๐, โ)-type GKM graph (Γ, \widetilde{๐ผ}, ∇) which is an extension of (Γ, ๐ผ, ∇). Using this combinatorial necessary and sufficient condition, we prove that the rank of ๐(Γ๐, ๐ผ๐, ∇๐) for the GKM graph (Γ๐, ๐ผ๐, ∇๐) induced from (๐2๐, ๐๐) gives an upper bound for the dimension of a torus which can act on ๐2๐ effectively. As one of the applications of this result, we compute the rank associated to ๐(Γ, ๐ผ, ∇) of the complex Grassmannian of 2-planes ๐บ2(โ๐+2) with the natural effective ๐๐+1-action, and prove that this action on ๐บ2(โ๐+2) is the maximal effective torus action which preserves the standard complex structure.</p>
Journal
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- Journal of the Mathematical Society of Japan
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Journal of the Mathematical Society of Japan 71 (2), 483-513, 2019
The Mathematical Society of Japan
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Details ่ฉณ็ดฐๆ ๅ ฑใซใคใใฆ
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- CRID
- 1390845713064055936
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- NII Article ID
- 130007636387
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- NII Book ID
- AA0070177X
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- ISSN
- 18811167
- 18812333
- 00255645
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- NDL BIB ID
- 029647272
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- Text Lang
- en
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- Data Source
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- JaLC
- NDL
- Crossref
- CiNii Articles
- KAKEN
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- Abstract License Flag
- Disallowed