# Ampleness of Hodge bundles over complex curves

Let $C$ be a smooth, proper and connected curve over the complex numbers $\bf C$. Let ${\cal G}\to C$ be a smooth group scheme over $C$ and let $\epsilon_{\cal G}:C\to{\cal G}$ be its zero-section. Suppose that the fibre of $\cal G$ over the generic point of $C$ is an abelian variety.

By a theorem of Griffiths we then have $\mu_{\rm min}(\epsilon_{\cal G}^*\Omega^1_{{\cal G}/C})\geq 0$. Here $\mu_{\rm min}(\epsilon_{\cal G}^*\Omega^1_{{\cal G}/C})$ is the minimal slope of $\epsilon_{\cal G}^*\Omega^1_{{\cal G}/C}$ (for the Harder-Narasimhan filtration).

My question is: are there simple known geometric criteria that ensure that $\mu_{\rm min}(\epsilon_{\cal G}^*\Omega^1_{{\cal G}/C})>0$, or in other words, that ensure that $\epsilon_{\cal G}^*\Omega^1_{{\cal G}/C}$ is an ample vector bundle ?

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I would expect that the quotient of $\epsilon_{\mathcal{G}}^*\Omega^1_{\mathcal{G}/C}$ by the maximal ample subsheaf is integrable, i.e., it is $\epsilon_{\mathcal{H}}^*\Omega_{\mathcal{H}/C}$ for an isotrivial subgroup scheme $\mathcal{H}$ of $\mathcal{G}$. Thus, if you insist that $\mathcal{G}$ has no isotrivial subgroup scheme, presumably that implies ampleness. –  Jason Starr May 14 '14 at 21:14
@Jason Starr. Thank you for your remark. I would expect something like that but I cannot find any coherent bibliographical reference for this kind of thing. –  Damian Rössler May 14 '14 at 21:24