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)
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.
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.
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 .
