Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFlag Bott manifolds of general Lie type and their equivariant cohomology rings
182
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this article we introduce flag Bott manifolds of general Lie type as the total spaces of iterated flag bundles. They generalize the notion of flag Bott manifolds and generalized Bott manifolds, and admit nice torus actions. We calculate the torus equivariant cohomology rings of flag Bott manifolds of general Lie type.
rate research
Read More
In this article the generic torus orbit closure in a flag Bott manifold is shown to be a non-singular toric variety, and its fan structure is explicitly calculated.
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.
We demonstrate how by using the intersection theory to calculate the cohomology of $G_2$-manifolds constructed by using the generalized Kummer construction. For one example we find the generators of the rational cohomology ring and describe the product structure.
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.
In the present paper, we characterize Fano Bott manifolds up to diffeomorphism in terms of three operations on matrix. More precisely, we prove that given two Fano Bott manifolds $X$ and $X$, the following conditions are equivalent: (1) the upper triangular matrix associated to $X$ can be transformed into that of $X$ by those three operations; (2) $X$ and $X$ are diffeomorphic; (3) the integral cohomology rings of $X$ and $X$ are isomorphic as graded rings. As a consequence, we affirmatively answer the cohomological rigidity problem for Fano Bott manifolds.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
