We study the topology of toric maps. We show that if f : X → Y is a proper toric morphism, with X simplicial, then the cohomology of every fiber of f is pure and of Hodge-Tate type. When the map is a fibration, we give an explicit formula for the Betti numbers of the fibers in terms of a relative version of the f -vector, extending the usual formula for the Betti numbers of a simplicial complete toric variety. We then describe the Decomposition Theorem for a toric fibration, giving in particular a nonnegative combinatorial invariant attached to each cone in the fan of Y , which is positive precisely when the corresponding closed subset of Y appears as a support in the Decomposition Theorem. The description of this invariant involves the stalks of the intersection cohomology complexes on X and Y , but in the case when both X and Y are simplicial, there is a simple formula in terms of the relative f -vector.

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For G = GL 2 , PGL 2 , SL 2 we prove that the perverse filtration associated with the Hitchin map on the rational cohomology of the moduli space of twisted G-Higgs bundles on a compact Riemann surface C agrees with the weight filtration on the rational cohomology of the twisted G character variety of C, when the cohomologies are identified via non-Abelian Hodge theory. The proof is accomplished by means of a study of the topology of the Hitchin map over the locus of integral spectral curves.

Abstract. This article concerns properties of mixed ℓ-adic complexes on varieties over finite fields, related to the action of the Frobenius automorphism. We establish a fiberwise criterion for the semisimplicity and Frobenius semisimplicity of the direct image complex under a proper morphism of varieties over a finite field. We conjecture that the direct image of the intersection complex on the domain is always semisimple and Frobenius semisimple; this conjecture would imply that a strong form of the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber is valid over finite fields. We prove our conjecture for (generalized) convolution morphisms associated with partial affine flag varieties for split connected reductive groups over finite fields. As a crucial tool, we develop a new schematic theory of big cells for loop groups. With suitable reformulations, the main results are valid over any algebraically closed ground field.

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