2015/08/07 by Hajime Kaji, Kaji, Hajime, Tomohide Terasoma +1
Mathematics · #14M15 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14M15
paper · pdf · doi:10.48550/arxiv.1508.01663
arxiv created 2015/08/07 · arxiv updated 2015/08/10
Let X be a non-singular quasi-projective variety over a field, and let \mathcal E be a vector bundle over X. Let \mathbb GX(d, \mathcal E) be the Grassmann bundle of \mathcal E over X parametrizing corank d subbundles of \mathcal E with projection π: \mathbb GX(d, \mathcal E) → X, and let \mathcal Q \gets π^*\mathcal E be the universal quotient bundle of rank d. In this article, a closed formula for π*ch (det \mathcal Q), the push-forward of the Chern character of the Plücker line bundle det \mathcal Q by π is given in terms of the Segre classes of \mathcal E. Our formula yields a degree formula for \mathbb GX(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.