Proposition

Lemma 2 ([Weight-one operations]). For complex abelian varieties P,Q , pullback identifies Hom(Q,P)⊗ℚ ≃ Hom_ HS (H^1(P,ℚ),H^1(Q,ℚ)). Thus arbitrary rational weight-one Hodge maps can act algebraically and independently on the two factors …

Lemma 2 ([Weight-one operations]). For complex abelian varieties P,Q , pullback identifies Hom(Q,P)⊗ℚ ≃ HomHS (H1(P,ℚ),H1(Q,ℚ)). Thus arbitrary rational weight-one Hodge maps can act algebraically and independently on the two factors of an ordered tensor product of first cohomology groups.

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Corollary 1. If the hypothesis of Theorem 1.2 ([Specialization in a polarized…Theorem 1.2 ([Specialization in a polarized component]). Fix a component of the…Proposition
Dualize a rational Hodge map to a map on first homology. After multiplication by a positive integer it preserves the integral lattices. The Hodge condition says that its real extension commutes with the complex structures on the universal covers. It therefore induces a holomorphic group homomorphism Q→P , which is algebraic by projectivity. Conversely pullback by every such homomorphism is a Hodge map. This is the usual equivalence between abelian varieties up to isogeny and polarizable rational Hodge structures of weight one, written contravariantly . Transpose graphs realize the pullbacks. Products of two such correspondences, and their rational linear combinations, act independently on the two tensor factors. Preservation of the chosen polarizations is not required.
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