Topology and geometry of flag bott-samelson varieties and bott towers = 플래그 보트-사멜슨 다양체와 보트 다양체의 위상과 기하에 관한 연구

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One of the most natural way to construct a new manifold from a given one is considering fiber bundles. In this paper we take the projective space and the full flag manifold as fibers. Iterating these procedures the former gives Bott-Samelson manifolds, Bott manifolds, and generalized Bott manifolds, and the later produces flag Bott-Samelson manifolds, and flag Bott manifolds. Grossberg and Karshon define a family of complex structures on a Bott-Samelson variety degenerates to a Bott tower, which is a toric variety. This connection defines a virtual polytope, called a twisted cube, which encodes the character of a representation. We study a sufficient and necessary condition for untwistedness of certain twisted cubes. We define flag Bott-Samelson manifolds and flag Bott manifolds, and we show that flag Bott-Samelson manifolds degenerate flag Bott towers, and also this relation gives a combinatorial object which encodes the character of a representation. Moreover we show that Newton-Okounkov bodies of flag Bott-Samelson varieties are generalized string polytopes. On the other hand there is a different extended notion of Bott tower, called a generalized Bott tower introduced by Masuda and Suh. We show that for a given generalized Bott tower we can find the associated flag Bott tower so that the closure of a generic torus orbit in the latter is a blow-ups of the former along certain invariant submanifolds. We use the GKM structure of a flag Bott tower together with some toric topological arguments to prove it.
Suh, Dong Youpresearcher서동엽researcher
한국과학기술원 :수리과학과,
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학위논문(박사) - 한국과학기술원 : 수리과학과, 2018.2,[v, 104 p. :]


flag Bott-Samelson variety▼aflag Bott tower▼ageneralized Bott tower▼aGrossberg-Karshon twisted cube▼aNewton-Okounkov body▼aGKM theory; 플래그 보트-사멜슨 다양체▼a플래그 보트 다양체▼a일반화된 보트 다양체▼aGrossberg-Karshon 꼬인 입방체▼a뉴튼-오쿤코프 체▼aGKM 이론

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