The Lefschetz (1,1) Theorem as the First Case of the Hodge Conjecture

The Hodge conjecture predicts that for a smooth projective complex variety, every rational Hodge class of type $(p,p)$ is algebraic. The only fully proven case remains $p=1$, known as the Lefschetz $(1,1)$ theorem. This article presents a complete, self-contained proof of that theorem for compact K\"ahler manifolds. The proof combines the exponential sheaf sequence with the Hodge decomposition: the connecting map of the exponential sequence sends holomorphic line bundles to their first Chern class, and the Hodge decomposition shows that every integral $(1,1)$-class lies in the image of this map. We conclude with a discussion of why this argument cannot be generalized to higher p, highlighting the fundamental obstructions discovered by Griffiths.