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.

• 14C20 - Divisors, linear systems, invertible sheaves
• 14C25 - Algebraic cycles
• 14K05 - Algebraic theory of abelian varieties
• 14K20 - Analytic theory of abelian varieties; abelian integrals and differentials
• 14L15 - Group schemes

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