The Hodge Theory of Maps

Abstract

This chapter summarizes the classical results of Hodge theory concerning algebraic maps. Hodge theory gives nontrivial restrictions on the topology of a nonsingular projective variety, or, more generally, of a compact Kähler manifold: the odd Betti numbers are even, the hard Lefschetz theorem, the formality theorem, stating that the real homotopy type of such a variety is, if simply connected, determined by the cohomology ring. Similarly, Hodge theory gives nontrivial topological constraints on algebraic maps. This chapter focuses on the latter, as it considers how the existence of an algebraic map f : X → Y of complex algebraic varieties is reflected in the topological invariants of X.