Non-liftability of Families of Abelian Varieties with Small $l$-adic Local System

We study families of abelian varieties over smooth proper curves with small $l$-adic local system over characteristic $p$. We show that such abelian schemes have a non-nef Hodge bundle and cannot be lifted to $W_2(k)$. We also establish an Arakelov-type inequality for families of abelian varieties over smooth proper curves in characteristic $p$, assuming $W_2(k)$-liftability.