# Holomorphic structures for line bundles over projective manifolds

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 $$T$$.
• So if $h^{(1,0)} = h^{(2,0)} = 0$, then every line bundle admits one, and only one, holomorphic structure? Oct 11 '19 at 5:54