Let $M$ be a compact K\"ahler manifold, which is assumed to be projective, i.e. there exists an ample line bundle over $M$ giving an embedding into $\mathbb{C}P^n$.
Let $\mathcal{L}$ be a smooth line bundle over $M$. Can $\mathcal{L}$ always be endowed with a holomorphic structure? Is such a structure necessarily unique, or can there exist more than one?
The group of isomorphism classes of complex line bundles on $M$ is isomorphic to $H^2(M,\mathbb{Z})$, via the map $L\mapsto c_1(L)$. The analogous group $\operatorname{Pic}(M) $ for holomorphic line bundles fits into an exact sequence $$0\rightarrow T\rightarrow \operatorname{Pic}(M)\xrightarrow{\ c_1\ } H^2(M,\mathbb{Z})\cap H^{1,1} \rightarrow 0$$ where $T$ is a complex torus (usually denoted $\operatorname{Pic}^{\mathrm{o}}(M)$), of dimension $h^{1,0}$ (both statements follow from the cohomology exact sequence associated to the exact sequence of sheaves $0\rightarrow \mathbb{Z}\rightarrow \mathcal{O}_M\xrightarrow{\ \mathbf{e}\ }\mathcal{O}_M^*\rightarrow 1$, where $\mathcal{O}_M$ is the sheaf of $C^{\infty}$ or holomorphic functions, and $\mathbf{e}(f)=\exp(2\pi if)$). Thus, if $h^{1,0}$ and $h^{2,0}$ are $\neq 0$, there are complex line bundles which do not admit holomorphic structures, and those which do admit a whole family, parametrized by $T$.
