# 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 ?

• 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
• @DamianRössler. Do you know a reference for the positivity theorem due to Griffiths that you cited? – François Sep 25 '18 at 21:17
• @François. See Th. 5.2 in P. Griffiths, Periods of integrals on algebraic manifolds, III (some global differ- ential properties of the period mapping). Inst. Hautes Études Sci. Publ. Math. 38 (1970), 125–180 and also Cor. 2.7 in "Germs of analytic varieties in algebraic varieties" by JB Bost in the Dwork Festschrift. – Damian Rössler Sep 28 '18 at 10:37