# perverse sheaves – Decomposition of direct image of a smooth morphism, Deligne’s theorem, motives

Let $$f : X to Y$$ be projective and smooth morphism of complex algebraic varieties. Here we care about the algebraic topology of $$X$$ and $$Y$$, so use classical topology for simplicity.

I can take the constant sheaf $$mathbb{Q}_X$$ and (derived) push it forward to get $$f_* mathbb{Q}_X in D^b_c(Y,mathbb{Q})$$. There is a celebrated theorem of Deligne that $$f_* mathbb{Q}_X$$ is semi-simple, i.e. isomorphic to a direct sum of its cohomology sheaves. The argument uses hard Lefschetz along the fibres. (It is also true that each summand is a semi-simple local system, as a polarizable VHS, however I want to ignore that extra piece of information below.)

Suppose I replace $$mathbb{Q}$$ with $$k := mathbb{F}_p$$.

Question: Is it true that $$f_* k_X in D^b_c(Y,k)$$ is always semi-simple? That is, does it always split as a direct sum of its (not-necessarily semi-simple) cohomology sheaves.

I had always assumed the answer was no, but woke up this morning feeling unusually optimistic. (I have tried several times to produce a counter-example.) I understand that this is deep water, and I am happy with a heuristic answer either way (potentially using motives).