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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.
5
votes
1
answer
377
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Holomorphic extension of an action by a compact Lie group on a complex homogeneous manifold
Let $G$ be a compact Lie group and let $M$ be a $G$-homogeneous manifold. Suppose that $M$ is endowed with a complex structure invariant by the action of $G$. Denote by $G_{\mathbb C}$ the complexific …
2
votes
1
answer
107
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Holomorphic local trivialization of a principal toric bundle
Let $G$ be an even-dimensional compact Lie group with Lie algebra $\mathfrak{g}$ and let $T \subset G$ be a maximal torus with Lie algebra $\mathfrak{t}$.
We can construct a left-invariant complex st …