Flag Bott manifolds and the toric closure of a generic orbit associated to a generalized Bott manifold

Borel-Hirzebruch type formula for graph equivariant cohomology of projective bundle over GKM-graph

“…The equivariant cohomology ring H * T 2 (P(T CP 2 )) of the T 2 -action on the projectivization P(T CP 2 ) of the tangent bundle of CP 2 is isomorphic to the following ring: Z[τ 1 , τ 2 , τ 3 , κ]/(τ 1 τ 2 τ 3 , κ 2 − c T 1 (ξ) • κ + c T 2 (ξ)). One can also prove that P(T CP 2 ) is equivariantly diffeomorphic to the flag manifold F l(C 3 ) Therefore, this gives the computation of the equivariant cohomology of effective torus actions on flag manifolds by using the Leray-Hirsh theorem and the Borel-Hirzebruch formula (also see [KLSS20]). This establish the statement.…”

$K$-theory of Flag Bott manifolds

“…In particular, F lag(E) = x∈X F lag(E x ). We recall the definition of flag Bott manifolds from [17,18]. We broadly follow their notations and conventions.…”

“…The construction of a Bott tower has recently been generalized in another natural direction of a flag Bott manifold by Kaji, Kuroki, Lee, Song and Suh (see [15], [18], [17]) by replacing P 1 C (which can be identified with the variety of full flags in C 2 ) at every stage with the variety F lag(C n ) of full flags in C n for any n ≥ 2. More precisely, at the jth stage we let B j to be the flagification of direct sum of n j + 1 complex line bundles over B j−1 .…”

“…There is a natural effective action of a complex torus D and the corresponding compact torus T ⊆ D on the flag Bott manifold (see [15], [18,Section 3.1]).…”

“…Organisation of the sections. In section 2 we recall from [18] and [17] the definition and construction of flag Bott manifolds. In particular, in section 2.1 we recall the construction of tautological line bundles on these manifolds which generate its Picard group.…”