Compact smooth projective pushforwards have semisimple perverse cohomology in characteristic zero

Given: The projective morphism $f:X\to Y$ of complex projective varieties with $X$ smooth of pure complex dimension $n$, the characteristic-zero coefficient field $k$ with its canonical inclusion of $\mathbb{Q}$, and the Axiom of Choice inherited from the suppliers.

The rational compact projective theorem Relative hard Lefschetz and perverse splitting for compact projective algebraic maps gives a finite perverse splitting of $R{f}_{\ast }{\mathbb{Q}}_X[n]$, the IC description of its perverse cohomology objects with simple rational local systems, and semisimplicity of the complexifications of those rational local systems.

A finite-rank rational local system whose complexification is semisimple remains semisimple after extension of scalars to every characteristic-zero field, by Characteristic-zero scalar extension of geometric semisimplicity.

Ordinary constructible six operations, including proper pushforward and Verdier duality, commute with field extension by Actual analytic constructible sheaves admit coherent six operations. Exact field extension preserves the perverse upper support bounds, and its compatibility with duality preserves the lower bounds, so it is perverse-exact. Uniqueness of truncation triangles then gives compatibility with perverse truncation. On the hearts it preserves images, hence also the image of ${}^p{H}^0{j}_{!}\to {}^p{H}^{0}R{j}_{\ast }$ defining intermediate extension. These deductions use the ordinary perverse structure of [F4].

Simple perverse sheaves are the intersection complexes of simple finite-rank local systems on smooth dense opens of irreducible supports, and the IC construction is additive in the local system, by Ordinary middle perverse t structure, recollement and intersection complexes.