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Register a new user$mathbb{Q}$-trivial generalized Bott manifolds
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When the cohomology ring of a generalized Bott manifold with $mathbb{Q}$-coefficient is isomorphic to that of a product of complex projective spaces $mathbb{C}P^{n_i}$, the generalized Bott manifold is said to be $mathbb{Q}$-trivial. We find a necessary and sufficient condition for a generalized Bott manifold to be $mathbb{Q}$-trivial. In particular, every $mathbb{Q}$-trivial generalized Bott manifold is diffeomorphic to a $prod_{n_i>1}mathbb{C}P^{n_i}$-bundle over a $mathbb{Q}$-trivial Bott manifold.
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To a direct sum of holomorphic line bundles, we can associate two fibrations, whose fibers are, respectively, the corresponding full flag manifold and the corresponding projective space. Iterating these procedures gives, respectively, a flag Bott tower and a generalized Bott tower. It is known that a generalized Bott tower is a toric manifold. However a flag Bott tower is not toric in general but we show that it is a GKM manifold, and we also 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-up of the former along certain invariant submanifolds. We use GKM theory together with toric geometric arguments.
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