Fujita decomposition on families of abelian varieties

Let $$F:{\mathcal {V}}\rightarrow B$$ F : V → B be a smooth non-isotrivial $$1-$$ 1 - dimensional family of complex polarized abelian varieties and $$V_b=F^{-1}(b)$$ V b = F - 1 ( b ) be the general fiber. Let $${\mathcal {F}}^1\subset R^1F_*{\mathbb {C}}$$ F 1 ⊂ R 1 F ∗ C be the associated Hodge bundle filtration, $${\mathcal {F}}^1_b=H^{1.0}(V_b).$$ F b 1 = H 1.0 ( V b ) . Under the assumption that the Fujita decomposition for $${\mathcal {F}}^1$$ F 1 is non trivial, that is there is a non trivial flat sub-bundle $$0\ne {\mathbb {U}}\subset {\mathcal {F}}^1,$$ 0 ≠ U ⊂ F 1 , we show that $$V_b$$ V b has non-trivial endomorphism: $$End(V_b)\ne {\mathbb {Z}}.$$ E n d ( V b ) ≠ Z .