Let $V$ be a vector bundle given by a direct sum $V=\bigoplus_{i=1}^n L_i$ of $n$ line bundles. The transition functions are then diagonal matrices, which have $n$ free parameters.
Is it also the case that $\text{dim Hom}(V,V)=n$? What is a good way to think about this?
Edit:
In a comment below, peterag gave a counterexample for the statement that $\text{dim Hom}(V,V)=n$ for $V$ a sum of line bundles.
This being the case, I thought I'd add some additional queries:
- Are there situations when one would expect $\text{dim Hom}(V,V)=n$?
- Is this in some sense more likely for a line bundle sum than for a general vector bundle?
$\DeclareMathOperator{\Hom}{Hom}$If $V = \bigoplus_{i} L_{i}$ is an $n$-fold direct sum, you can identify $$ \Hom(V, V) = \bigoplus_{i, j} \Hom(L_{i}, L_{j}); $$ the $n^{2}$ summands in the direct sum correspond to the matrix entries in peter's comment.
While $\dim \Hom(L_{i}, L_{j})$ may be finite (e.g., if you're speaking of algebraic or holomorphic bundles over a projective manifold), there's no reason to expect $\dim \Hom(L_{i}, L_{j}) = \delta_{ij}$, so no reason to expect $\dim \Hom(V, V) = n$.
For example, if $H$ denotes the hyperplane bundle over complex projective $m$-space and $L_{i} = H^{i}$, then $$ \dim\Hom(L_{i}, L_{j}) \simeq \dim\Gamma(L_{-i} \otimes L_{j}) \simeq \dim\Gamma(L_{j - i}) = \binom{m + j - i + 1}{j - i}. $$
Generally, of course, $\dim\Hom(L_{i}, L_{j})$ can be infinite.