A vanishing theorem ``à la Kawamata-Viehweg''
In this article, the author considers nef vector bundles (i.e., the ones whose tautological bundle is nef) of arbitrary rank over smooth projective varieties. Let \(E\) be an algebraic vector bundle of rank \(r\) on a smooth projective variety \(X\) of dimension \(n\). Let \(s = \{i | 0 \leq i \leq r\}\), Fl\(_ s(E)\) = flag variety on \(E\) of form \(s\), \(N_ s\) = relative dimension on \(X\), and \(\rho =\) a partition of length \(l(\rho)\) \((\leq r)\). Now Fl\(_ s(E)\) admits a line bundle \({\mathfrak L}^ \rho\) whose direct image on \(X\) is the fibre \(S^ \rho (E)\) associated to \(E\) for the irreducible representation of \(GL_ r (\mathbb{C})\) defined by \(\rho\). Using the following result: \[ {c_ 1 ({\mathfrak L}^ \rho)^{n + N_ s} \over (n + N_ s)!} = C_{\rho,r} \sum_{| \pi | = n,l (\pi)\leq l(\rho)} \int_ X {s_ \pi (\rho) s_ \pi (E) \over \prod^{l(\rho)}_{k = 1} (r + \pi_ k - k)!} \] where \(C_{\rho,r} \in \mathbb{Z}^ +\) and \(s_ \pi\) is the Schur polynomial associated to the partition \(\pi\) and \(s_ \pi (E) = \text{det} (c_{\pi^*_ i - i + j} (E))\), \(\pi^*\) being the conjugate partition of \(\pi\) in the Young diagram, the author notes that the coefficient of \(s_ \pi (E)\) is positive, and that \({\mathfrak L}^ \rho\) is nef and big if and only if the Schur polynomial admits strictly positive values on \(X\), whence the Kawamata-Viehweg vanishing theorem gives the following main result of the paper: If \(E\) is an algebraic vector bundle of rank \(r\), and \(E\) is nef on a smooth projective variety \(X\) of dimension \(n\), and if \({\mathfrak L}\) is a nef line bundle on \(X\) with a partition \(\mu\) such that \(\int s_ \mu (E) > 0\) (which is weaker than \({\mathfrak L}\) big on \(X)\), then for all partitions \(\rho\) of length \(l(\rho) \geq l(\mu)\), \[ H^ q (X,K_ X \otimes S^ \rho E \otimes (\text{det} E)^ l \otimes {\mathfrak L}) = 0,\quad\text{if}\quad q > 0\quad\text{and}\quad l \geq l (\rho). \] Along the way, the author discusses classes of symmetric powers and gives multidimensional versions of Worpitzky's identity, and in the third section he discusses the self-intersections of line bundles on flag varieties and some expressions for their Euler-Poincaré characteristics. He concludes the section by exploring connections via the Gysin map, on flag varieties, and establishes one between the Chow groups of \(X\) and that of the flag variety Fl\(_ s (E)\). The conclusion of the paper is the proof of a vanishing theorem as noted in the title.
self- intersections of line bundles on flag varieties
scientific article; zbMATH DE number 624944
scientific article