Let $X$ be a projective algebraic manifold. 92' Singular hermitian metrics on positive line bundles demailly wrote:
An integral cohomology class in $H^2(X,\mathbb{Z})$ is the first Chern class of a holomorphic (or algebraic) line bundle if and only if this class is of type (1, 1).
Qustion: How to prove the above claim ? Or where can I get the detailed proof ?
Best Answer
It is the combination of the following two theorems, which can all be found in Griffiths and Harris, Principles of Algebraic Geomtry:
Lefschetz $(1,1)$ theorem, i.e., classes in $H^2(X,\mathbb Z)\cap H^{1,1}(X,\mathbb C)$ can be represented by Poincare duality of divisors (page 163).
If the line bundle $\mathcal{L}$ is defined by a divisor $D$, then its first Chern class $c_1(\mathcal{L})$ is the Poincare duality of $D$ (page 141, Prop. 2).