Given a proper morphism $f:X\rightarrow Y$ where $X$ and $Y$ are smooth projective varieties. The proper pushforward $f_!$ is the homomorphism that sends the class of a coherent sheaf $M$ in $K_0(X)$ to $\sum_i(-1)^i[R^if_*M]$. Furthermore assume $X$ and $Y$ are abelian varieties so they have trivial tangent bundles. Let $M$ be the structure sheaf $\mathcal{O}_X$. By the application of GRR it implies that higher chern classes of $f_![\mathcal{O}_X]$ are zero. Is it also possible $f_![\mathcal{O}_X]$ to have the generic rank equal to zero? In other words is it possible for $f_![\mathcal{O}_X]$ to be zero in $K_0(Y)\otimes \mathbb{Q}$? (I am interested in the case that $f$ is the covering from a Jacobian of a curve to the arbitrary abelian variety.)

Given a proper surjective morphism $f:X\rightarrow Y$ where $X$ and $Y$ are smooth projective varieties. The proper pushforward $f_!$ is the homomorphism that sends the class of a coherent sheaf $M$ in $K_0(X)$ to $\sum_i(-1)^i[R^if_*M]$. Furthermore assume $X$ and $Y$ are abelian varieties so they have trivial tangent bundles. Let $M$ be the structure sheaf $\mathcal{O}_X$. By the application of GRR it implies that higher chern classes of $f_![\mathcal{O}_X]$ are zero. Is it also possible $f_![\mathcal{O}_X]$ to have the generic rank equal to zero? In other words is it possible for $f_![\mathcal{O}_X]$ to be zero in $K_0(Y)\otimes \mathbb{Q}$? (I am interested in the case that $f$ is the covering from a Jacobian of a curve to the arbitrary abelian variety.)