The Hard Lefschetz Theorem

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The pioneering study of the topology of algebraic varieties was carried out by Lefschetz in his monograph [79] published in 1924. He suggested working inductively, by comparing the homology of a smooth projective variety with its intersection with a single hyperplane and eventually a suitable family of them. There were two basic results, nowadays called the weak and hard Lefschetz theorems. In this chapter, we discuss both of these but focus on the latter. In one of its incarnations, it gives the structure of cohomology under cup product with a hyperplane class. The first correct proof of this was due to Hodge using harmonic forms, and we present a version of this. The hard Lefschetz theorem has a number of important consequences for the topology of projective, and more generally Kähler, manifolds, and we will discuss a number of these. We also want to interpret this using Lefschetz’s original more geometric point of view, which, remarkably, played a role in Deligne’s proof of the Weil conjectures [25] and in his subsequent arithmetic proof of hard Lefschetz [27].