Hodge Theory of Maps

The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar constraints on algebraic maps? Let f : X → Y a proper morphism. Our goal is going to be to linearize the problem. For this lecture, we will assume f is projective and smooth, to simplify the problem. In fact, we will assume X and Y are nonsingular. So our map factors X → P × Y → Y and df is surjective. Equivalently, there exists a line bundle L on X which restricts on every fiber to an ample line bundle. If X ⊆ P is a projective, nonsingular variety, tehn there’s a universal hyperplane section {(x,H)|x ∈ H} ⊂ X × (Pn)∗ and call it X , we have a map X → (Pn)∗. This is not smooth, and we call this map h. Define X∨ to be the set of hyperplanes tangent to X. Then h−1((Pn)∗ \X∨)→ (Pn)∗ \X∨, we have a smooth projective morphism.