Higher discriminants and the topology of algebraic maps

TL;DR

This paper introduces higher discriminants as a tool to understand how Betti cohomology varies in families of complex algebraic varieties, linking geometric transversality conditions to cohomological behavior.

Contribution

It defines higher discriminants via transversality, demonstrating their role in controlling cohomology variation and characteristic cycles in algebraic families, and relates them to known stratifications.

Findings

Higher discriminants are characterized by transversality conditions.

They control the support of pushforward sheaves and characteristic cycles.

In the Hitchin fibration, higher discriminants match Ngo's delta stratification.

Abstract

We show that the way in which Betti cohomology varies in a proper family of complex algebraic varieties is controlled by certain "higher discriminants" 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.