Cohomological splitting over rationally connected bases

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.