1
vote
0answers
102 views

Existence of open dense subset in a Lie group

Let $G=S\cdot R$ be a Levi-Malcev decomposition of a connected complex Lie group and $\Gamma$ a discrete subgroup of $G$ such that the subgroups $\Gamma_g := \Gamma\cap (gSg^{-1})$ are finite for all ...
1
vote
2answers
261 views

Abelian subgroups of ball quotient

Let $X$ be a compact complex surface of general type which a ball quotient. Is it true that $\pi_{1}(X)$ can not contain ${\mathbb{Z}}^{2}$ as a subgroup? What kind of infinite abelian groups can ...
0
votes
0answers
314 views

covariant derivative complex manifold

Assume we have $X$ a complex manifold and $Y = Y^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ and $Z = Z^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ two vector fields on $X$. Let $\nabla$ be the ...
2
votes
1answer
1k views

complex gradient of a function

Let $M$ be a complex $n-$dim manifold and $u : M \rightarrow \mathbb{R}$ be some smooth function. On $M$ assume that we have a Kaehler metric $h$. How is the complex gradient vectorfield defined with ...
1
vote
1answer
399 views

Lie Group Principal Embedding

I'm reading a paper on complex semi-simple algebraic group geometry at the moment, but finding the going a bit tough since I'm missing alot of the prerequisites. Firstly, the author refers to a ...