  We prove a cohomological splitting result for Hamiltonian fibrations over
enumeratively rationally connected symplectic manifolds As a key application,
we prove that the cohomology of a smooth, projective family over a smooth
(stably) rational projective variety splits additively over any field. The main
ingredients in our arguments include the theory of Fukaya-Ono-Parker (FOP)
perturbations developed by the first and third author, which allows one to
define integer-valued Gromov-Witten type invariants, and variants of
Abouzaid-McLean-Smith's global Kuranishi charts tailored to concrete geometric
problems.
