Families of abelian varieties over curves in char p with small l-adic local systems have non-nef Hodge bundles and are non-liftable to W_2(k).
Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part
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Non-liftability of Families of Abelian Varieties with Small $l$-adic Local System
Families of abelian varieties over curves in char p with small l-adic local systems have non-nef Hodge bundles and are non-liftable to W_2(k).