Ngô support theorem and polarizability of quasi-projective commutative group schemes

We prove that any commutative group scheme over an arbitrary base scheme of finite type over a field with connected fibers and admitting a relatively ample line bundle is polarizable in the sense of Ngô. This extends the applicability of Ngô’s support theorem to new cases, for example to Lagrangian fibrations with integral fibers and has consequences to the construction of algebraic classes.