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A reference for a proof that a bijective endomorphism of an algebraic variety over a field of characteristic zero is an automorphism (which is a corrected version of Mariano's statement in the comment …

The partial flag manifold $X$ which you mentioned is the set of flags $0\subset V_1\subset..\subset V_m=\Bbb C^n$, such that ${\rm dim}(V_j/V_{j-1})=k_j$. So we have vector bundles $V_j$ on $X$ and …

Check out the paper arXiv:0710.3392 - it talks about such generalization.

As was noted in the comment, $d$ must be $0$ (bundles with a flat connection can have only degree $0$), and different connections can lead to the same bundle. However, every line bundle of degree zero …

Suppose we are over an algebraically closed field, and $G$ is connected. Then, we have an exact sequence $$ 1\to U\to G\to G_r\to 1, $$ where $U$ is the unipotent radical of $G$, and $G_r$ is a red …

You probably mean that you are interested in principal bundles on a projective variety $X$ with fiber being a compact Lie group $G$. If you are interested in the topological classification of such bun …

The branched cover (defined by the Weierstrass function) has degree 2. To obtain $\Bbb C\Bbb P^1$, we need to quotient the elliptic curve $\Bbb C/\Lambda$ by the transformation $z\to -z$, which has or …

Let $K$ be a compact connected Lie group, and $L$ a Levi subgroup of $K$ (the centralizer of an element of the Lie algebra). Then $X=K/L$ is a complex manifold (a coadjoint orbit). So one can ask abo …

The statements about the group and Lie algebra in the question are special cases of a more general fact. Namely, if $A$ is an associative algebra and $V$ an $A$-module, then obstructions to deformat …

In question 1, it seems that the subspace $H^*(g(Q))$ in $H^*(g(C))$ indeed depends on the rational form of g. Consider the example of $sl(3)$ and two rational forms, $su(3,Q)$ and $sl(3,Q)$, and look …

Here is a definition that I heard: a family of additive categories over a scheme $X$ (say, defined over a field $k$) is an $k$-linear additive category ${\mathcal C}$ equipped with a structure of a mo …

Maybe it is helpful to look at these lectures by Yekutieli: arXiv:0801.3233. There he discusses twisted deformations of algebraic varieties $X$ suggested by Kontsevich, i.e. deformations of the catego …