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 ?
