Why is every holomorphic line bundle over $\mathbb{C}$ necessarily trivial? I am having a hard time finding a proof to this seemingly innocuous fact. I have tried showing that such a line bundle will have a nowhere-zero global section (this is one of the few criterion I currently have at my disposal for showing a […]

Could someone give me a definition of globally generated vector bundle? A rapid search gives me the definition of globally generated sheaves, but I am in the middle of a long work and don’t really have time to learn all basics of sheaves theory and the connection to vector bundles. I just need a definition […]

Let $X$ be a compact connected Riemann surface (not $\mathbb{P}^1$), $p\in X$ be a point on it. Let $L$ be the holomorphic line bundle associated to the divisor $D=p$. By construction $L$ comes with a holomorphic section $\sigma$ vanishing exactly at $p$; since $$ H^0(X,\mathcal{O}(L)) \cong \mathcal{L}(D)=\{ f \in \mathcal{M}(X) \ | \ (f)+p \geq […]

I have two highly-coupled questions concerning holomorphic line bundles, and so I will go ahead and ask them together. The first concerns line bundles on $\mathbb{CP}^1$ and the other concerns line bundles on a complex torus (elliptic curve). On a Riemann surface $X$, I can always define a so-called “point bundle”, to use the terminology […]

In this question, $X$ will be a smooth complex projective variety. This question will be about comparing two different ways of calculating the degree of a vector bundle on such an $X$. I understand that, unless $X$ is a curve or $\mbox{Pic}(X)=\mathbb Z$, there is no well-defined notion of “degree” for vector bundles. I would […]

Given $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ $c_2(V\otimes L)=c_2(V)+(r-1)c_1(V).c_1(L)+{r \choose 2}c_1(L)^2 $ Friedman book on Algebraic Surfaces and Holomorphic Vector Bundles says there are similar formulas for $\operatorname{Sym}^kV$, $\bigwedge^kV$, $\operatorname{Hom}(V,W)$, $V\otimes […]

Let $V$ be a $n$-dimensional complex vector space and consider the Grassmannian of complex $k$-planes $Gr(k,V)$. The Plücker embedding is an embedding $p:Gr(k,V) \to \mathbb P^M$ where $M = \left(\begin{array}{c} n \\ k \end{array}\right)$, given by $$ \text{span}\{ u_1,\ldots,u_k\} \mapsto [u_1\wedge u_2 \wedge \cdots \wedge u_k].$$ The Grassmannian comes equipped with a tautological vector bundle […]

What are holomorphic and anti-holomorphic components? Why don’t we call them complex components and their conjugates? What is holomorphic coordinate transformation?

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