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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 ?

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It is the combination of the following two theorems, which can all be found in Griffiths and Harris, Principles of Algebraic Geomtry:

  1. 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).

  2. 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).

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