This is a question I asked at Math.SE but got no answers: http://math.stackexchange.com/q/396217/7110/ The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann m …

As we all know, for any homogeneous space $G/H$ we have that the equivariant vector bundles over $G/H$ are characterized by the representations of $H$. Thus, for the the complex pr …

Is there anything that can be said in general about the Cech cohomology of the sheaf $\mathcal{F}$ of sections of a holomorphic bundle over a contractible space $X$? I know that …

Let $X$ be a nonsingular complex algebraic variety whose tangent bundle is $T_X$. I use Lazarsfeld's book for the definition of a big vector bundle. A line bundle $L$ is big if it …

Question. Is it known/easy to see that every smooth projective variety $X$ (over an algebraically closed field), except for the point and $\mathbb{P}^1$, has a vector bundle whi …

Let $X$ and $Y$ be connected smooth manifolds. Let $L$ be a topological real line bundle over $X\times Y$. Then we know that the isomorphism class of such a line bundle is determin …

There are of course lots of definitions and references for this, but in the same way that, on a manifold $M$, a Riemannian metric is a section of positive definite symmetric bili …

Let $X$ a smooth projective curve over $\mathbb{C}$, $S$ a finite subscheme of $X$. $E$ a vector bundle over $X$ with a divisor $D$. We look at the sections $A:=H^{0}(X,E)$ with …

In the last weeks I have been think of the transition from topological vector bundles to smooth and holomorphic vector bundles. This has resulted in a few questions (with a common …

For a complex manifold $M$, I'm trying to understand (from a differential geometry point of view) what its category of coherent sheaves is. If I understand correctly, then the shea …

Recall the usual definition of a $k$-dimensional vector bundle (everything is assumed to be continuous/smooth/etc depending on the category): A $k$-dimensional vector bundle is …

I was recently looking back at an old question of mine, where I asked about the classification of the line bundles over a general complex flag manifold. Pavel Etingof gave the foll …

For a holomorphic vector bundle $E$ over a complex manifold $M$, we denote its space of smooth sections by $\Gamma^{\infty}(E)$, and its space of holomorphic sections by $\Gamma^{h …

Recall a definition. Let $V\subset \mathbb CP^n$ be a projective variety and $E$ be a holomorphic vector bundle on it. We call $E$ linearly trivial if the restriction of $E$ to any …

Let $X$ a smooth projective connected curve over $\mathbb{C}$. Let $E$ a vector bundle and $E'$ a subbundle of $E$. Let $(x_{1},\dots,x_{n})$ $n$ closed points on the curve $X$ wit …