All Questions
Tagged with flag-varieties kahler-manifolds
10 questions
9
votes
2
answers
963
views
Hard Lefschetz Theorem for the Flag Manifolds
In the case of a generalized flag manifold $G/P$, we have an explicit description of their cohomology groups due to Borel.(See herehere for a description.) I would like to know what the hard Lefschetz ...
- 163
6
votes
1
answer
275
views
Equivariant Almost Complex Structures on the Full Flag Manifolds
On complex projective space ${\bf CP}^m$, there exists a unique $SU(m+1)$-equivariant almost-complex structure. What happens for the case of the full flag manifold of $SU(m+1)$, which is to say the ...
6
votes
1
answer
516
views
Lagrangian Grassmannian as a Spin Manifold
I am trying to better understand this nice answer to a question of mine, which states
Spin structures on a compact complex manifold $(M^{2n},J)$ are in bijective correspondence with isomorphism ...
- 1,453
4
votes
2
answers
252
views
Are Wolf spaces flag manifolds?
It's all in the title: Are Wolf spaces flag manifolds? Both are group quotients of semi-simple Lie groups. In the Grassmannian case this is so, and I always tacitly assumed it extended to the general ...
- 1,453
4
votes
1
answer
252
views
Flag manifolds as homogeneous Kahler manifolds
In this question it is asked if every flag manifold can be given the structure of a Kähler manifold. In the first answer it is written
Flag manifolds exhaust all compact homogeneous Kähler ...
4
votes
1
answer
185
views
Explicit formula for complex structure on flag manifold/isospectral matrices?
Consider the flag manifold $M = U(n, \mathbb{C})/T^n$, where $T^n$ is the maximal torus of unitary diagonal matrices. Fixing a diagonal matrix $D$ with distinct reals on its diagonal, we can identify ...
- 637
4
votes
0
answers
178
views
How the existence of holomorphic sections depends on the choice of complex structure
In this Mathoverflow question it is asked how many invariant complex structures exist on the full flag manifold of $SU(m)$. In this question it is asked when a line bundle over a flag manifold has ...
- 443
3
votes
1
answer
138
views
Euler characteristic of a holomorphic homogeneous vector bundle
Let $G/B$ be a compact homogeneous complex manifold, and let $E = G \times_{\rho} V$ be a hololmorphic homogeneous vector bundle over $G/B$. Does there exist a presentation of the Euler characteristic ...
- 643
3
votes
2
answers
360
views
Cohomology of Homogeneous Complex Manifolds
Let $M$ be compact $G$-homogeneous manifold, equipped with the equivariant complex structure, when $G$ is a semi-simple algebraic group. The obvious example is every flag manifold. In that case, all ...
- 163
0
votes
2
answers
446
views
Classifying compact homogeneous Kähler manifolds
In this comprehensive answer to an old question, it is stated that
Flag manifolds exhaust all compact homogeneous Kähler manifolds corresponding to a compact connected semi-simple Lie group.
...
- 163