  Given a polarized abelian scheme with action by a ring, and a projective
finitely presented module over that ring, Serre's tensor construction produces
a new abelian scheme. We show that to equip these abelian schemes with
polarizations it's enough to equip the projective modules with non-degenerate
positive-definite hermitian forms. As an application, we relate certain moduli
spaces of principally polarized abelian schemes with action by the ring of
integers of a CM field. More specifically, we consider integral models of
zero-dimensional Shimura varieties associated to compact unitary groups. We
show that all abelian schemes in such moduli spaces are, \'etale locally over
their base schemes, Serre constructions of CM abelian schemes with
positive-definite hermitian modules. We also describe the morphisms between
such objects in terms of morphisms between the constituent data, and formulate
these relations as an isomorphism of algebraic stacks.
