Triviality of line bundle over complex manifold Is it possible to know whether the given line bundle over compact complex manifold or projective variety is trivial or not from its sheaf cohomology data?
I found this question when I trying to solve the exercise problem in voisin's book Hodge theory and complex algebraic geometry 1
(I haven't solve that problem yet)
 A: No. 
Let $X$ be the blow up of a smooth variety at a closed point and $E\simeq \mathbb P^r$ the exceptional divisor. Let $\mathscr L=\mathscr O_X(E)$. Then the section $E$ corresponds to a morphism $\mathscr O_X\to\mathscr L$ that sits in a short exact sequence:
$$
0\to \mathscr O_X \to \mathscr L\to \mathscr O_{\mathbb P^r}(-1)\to 0.
$$
All the cohomology groups of $\mathscr O_{\mathbb P^r}(-1)$ are zero, so all the cohomology groups of $\mathscr O_X$ and $\mathscr L$ are isomorphic, but $\mathscr L$ is not trivial.
A: There is one useful fact that might by relevant to you. A line bundle $L$ and a complex projective variety $X$ is trivial if and only if $H^0(X,L) \neq 0$ and $H^0(X,L^{-1})\neq0$. 
Indeed one implication is clear, and for the other choose two non-zero elements $s \in H^0(X,L)$ and $s' \in H^0(X,L^{-1})$. Then $ss' \in H^0(X,\mathcal{O})$ is non-zero. Since $H^0(X,\mathcal{O}) = \mathbb{C}$, we see that in fact $ss'$ is nowhere vanishing. Thus $L$ admits a nowhere vanishing section, and so is trivial.
Naturally a similar result holds for more general fields.
A: Given any holomorphic line bundle $L$ on a complex manifold $X$ , then we can get a element in $H^1(X,\mathcal{O}^\ast)$ . Then if it is zero in $H^1(X,\mathcal{O}^\ast)$ , it is trivial .
