Higher discriminants and the topology of algebraic maps

Abstract

We show that the way in which Betti cohomology varies in a proper family of complex algebraic varieties is controlled by certain in the base. These discriminants are defined in terms of transversality conditions, which in the case of a morphism between smooth varieties can be checked by a tangent space calculation. They control the variation of cohomology in the following two senses: (1) the support of any summand of the pushforward of the IC sheaf along a projective map is a component of a higher discriminant, and (2) any component of the characteristic cycle of the proper pushforward of the constant function is a conormal variety to a component of a higher discriminant. The same would hold for the Whitney stratification of the family, but there are vastly fewer higher discriminants than Whitney strata. For example, in the case of the Hitchin fibration, the stratification by higher discriminants gives exactly the {\delta} stratification introduced by Ngo.