Degree Formulae for Grassmann Bundles, II

Abstract

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 $\mathcal{Q}\leftarrow {\pi }^{\ast }\mathcal{E}$ be the universal quotient bundle of rank $d$. In this article, a closed formula for ${\pi }_{\ast }\mathrm{ch}(\det \mathcal{Q})$, the push-forward of the Chern character of the Plücker 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.

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