Space of Extensions of Holomorphic Line Bundles

Question

On a compact complex manifold $X$, fix two holomorphic line bundles $L$ and $L'$. Consider a holomorphic vector bundle $V$ of rank 2 which fits in an exact sequence $$0\to L\to V\to L'\to0$$ I would like to understand why these $V$'s are parametrized by $H^1(X, (L')^\ast\otimes L)$.

Up to tensoring with $(L')^\ast$ the short exact sequence above, we can equivalently ask why, for fixed $L$, holomorphic vector bundles $V$ of rank 2 sitting in exact sequences of the form $$0\to L\to V\to \mathcal{O}_X\to0$$ are parametrized by $H^1(X,L)$.

I was thinking about reasoning with Cech cohomology. With respect to some covering $U_i$ of $X$, the transition functions of $V$ can be written in a upper-triangular matrix with the transition functions $\ell_{ij}$ of $L$ and $1$ on the diagonal and some $g_{ij}\in\mathcal{O}(U_i\cap U_j)$ at the top right entry. Thus, all the information should be encoded in the $g_{ij}$. The transition relations for the bundle then yield the following relations

1. $g_{ii}=0$ (on each $U_i$)
2. $g_{ij}=-\ell_{ij}g_{ji}$ (on each $U_i\cap U_j$)
3. $g_{ij}=\ell_{ik}g_{kj}+g_{ik}$ (on each $U_i\cap U_j\cap U_k$)

However, when interpreting $g=(g_{ij})$ as an element of the group $C^1(\mathcal{L})$ in the Cech complex, by using the above relations I get $$(dg)_{ijk}:=g_{jk}-g_{ik}+g_{ij}=g_{jk}(1-\ell_{ij})$$ and therefore it seems that my $g$ does not even define a class in $H^1$ in general (apart from the trivial case when $L$ is trivial). Any suggestion?

Answer 1

Given your exact sequence, the boundary map on cohomologies gives a map $H^0(\mathcal{O}_X)\to H^1(L)$ and since the first is just $k$, the element 1 gives an element in the $H^1$. Conversely, embed $L$ in a large bundle, for example, $\oplus_{i=1}^n M_i$, where $\deg M_i$s are very large as a subbundle and let $P$ the quotient vector bundle. Again taking the long exact sequence, we get a map $H^0(P)\to H^1(L)$ and this map is onto, since $\deg M_i>>0$. Thus, given $s\in H^1(L)$, we get an element $t\in H^0(P)$ and thus a map $\mathcal{O}_X\to P$. Take the pull back to get your extension.

Answer 2

(This should be a comment rather than an answer, but I do not have the right to make comments, since my rep is not yet established in this town.)

I personally found Atiyah's 1957 paper 'Complex analytic connections in fibre bundles' (see the proof of Proposition 2) very enlightening with respect to the problem you posed. You may want to take a look, it's always a good idea to learn from the masters.