Ng\^o support theorem and polarizability of quasi-projective commutative group schemes

Abstract

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 Ng\^o. This extends the applicability of Ng\^o's support theorem to new cases, for example to Lagrangian fibrations with integral fibers and has consequences to the construction of algebraic classes.