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Results tagged with complex-geometry
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user 391
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
6
votes
Accepted
Deformation of $\mathbb{P}^1 \times \mathbb{P}^1$
Rigidity means, if you have a family $F$ of surfaces such that one $f\in F$ of them is $\mathbb P^1 \times \mathbb P^1$, then there is an open set $U\subseteq F, U \ni f$ each of whom is $\mathbb P^1 …
- 27.9k
3
votes
Analogy of a Fano manifold with anticanonical divisor
I'd say it's closer to an oriented manifold with corners (corners happening where the divisor is singular), or even that times a coefficient. In these papers Khesin, Rosly, and later Thomas build a ho …
- 27.9k
2
votes
Accepted
Searching for resolutions of generalized determinental varieties
These are "type A quiver cycles" (a name chosen so as not to collide with type A quiver varieties, which involve taking quotients; here the quotient would be a point). Your guess for the closure is co …
- 27.9k
4
votes
Compact Kaehler manifolds that are isomorphic as symplectic manifolds but not as complex man...
If $M \to X$ is smooth and proper, and $M$ is K\"ahler, then the fibers are all symplectomorphic. (Proof: the Levi-Civita connection generates symplectomorphisms.) The family of elliptic curves was al …
- 27.9k
5
votes
Hermitian symmetric spaces vs Hermitian homogeneous spaces
If you throw in "compact", so that the space is $G/P$ for some parabolic $P$, here's an alternate characterization: the unipotent radical of $P$ should be abelian. Equivalently, $P$ should be maximal, …
- 27.9k
9
votes
Structure of Kähler cone
Flag manifolds $G/B$ are nice: the Kähler cone is the positive Weyl chamber, with edges coming from the Poincaré duals of the Schubert divisors.
- 27.9k
4
votes
Accepted
Unique Equivariant Symplectic Structure for the Full Flag Manifold of $SU(3)$?
If $G$ acts on $M$ (both compact and finite-dimensional) preserving the symplectic form, and $M$ is simply-connected, the action is Hamiltonian. Then $M$ maps symplectomorphically to a coadjoint orbit …
- 27.9k
6
votes
Accepted
What is the Explicit Relationship between Coadjoint Orbits and Flag Manifolds?
Let $K$ be a maximal compact subgroup of $G$. Then $K$ acts transitively on each $G/P$, and up to $K$-isomorphism, the $K$-spaces obtained exactly match those occurring as coadjoint orbits of $K$ (act …
- 27.9k
5
votes
Accepted
Weyl group action on complexified Iwasawa decomposition
You need more data before you can say "the" complexified Iwasawa decomposition, namely the choice of real group of which $G$ is the complexification (as in your reference). For example, if your $GL_2( …
- 27.9k
2
votes
Degeneration of coadjoint orbits
By "coadjoint orbits" I assume you mean "of a compact Lie group, but then endowed with invariant complex structures". In which case the answer is no. There is a flat family whose general fiber is $\ma …
- 27.9k
12
votes
What are parabolic bundles good for?
The paper by Agnihotri and Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, uses a Narasimhan-Seshadri correspondence between parabolic bundles and unitary connecti …
- 27.9k
24
votes
Rep Theory Consequences of Bott--Weil--Borel
What I'm writing here seems more like a contribution to a
big-list than an "answer", but since you've already chosen one anyway...
Say you're interested in which irreps $V_\nu$ occur in
$V_\lambda \ …
- 27.9k