Degree Formulae for Grassmann Bundles, II

Let $X$ be a non-singular quasi-projective variety over a field, and let $\mathcal E$ be a vector bundle over $X$. Let $\mathbb G_X({d}, \mathcal E)$ be the Grassmann bundle of $\mathcal E$ over $X$ parametrizing corank $d$ subbundles of $\mathcal E$ with projection $\pi : {\mathbb G_X({d}, \mathcal E)} \to X$, and let $\mathcal Q \gets \pi^*\mathcal E$ be the universal quotient bundle of rank $d$. In this article, a closed formula for $\pi_{*}\operatorname{ch} (\det \mathcal Q)$, the push-forward of the Chern character of the Pl\"ucker line bundle $\det \mathcal Q$ by $\pi$ is given in terms of the Segre classes of $\mathcal E$. Our formula yields a degree formula for $\mathbb G_X({d}, \mathcal E)$ with respect to $\det \mathcal Q$ when $X$ is projective and $\wedge ^d \mathcal E$ is very ample. To prove the formula above, a push-forward formula in the Chow rings from a partial flag bundle of $\mathcal E$ to $X$ is given.