Cohomology of line bundles on Schubert varieties: The rank two case
Let \(G\) be a connected, simply-connected, semisimple algebraic group defined over an algebraically closed field \(k\) of characteristic 0, \(B\subset G\) a fixed Borel subgroup, \(T\subset B\) a maximal torus and \(W = N_G(T)/T\) the Weyl group of \(G\). Consider the complete flag variety \(G/B\). The fixed points of the action from the left of \(T\) on \(G/B\) are \(e_w := wB\), \(w\in W\). The Schubert variety \(X_w\) is defined as the Zariski closure of \(Be_w\) in \(G/B\). If \(B\) acts from the right (resp., left) on a variety \(X\) (resp., \(Y\)) then \(B\) acts from the right on the product \(X\times Y\) by : \((x,y)b := (xb,b^{-1}y)\) and one defines \(X{\times}^BY\) as \(X\times Y/B\). If \(V\) is a rational representation of \(B\) one gets a vector bundle \({\mathcal L}(V):=G{\times}^BV\) over \(G/B\). In particular, if \(\lambda \in X^{\ast}(B)\simeq X^{\ast}(T)\) is a character and if \(k_{\lambda}\) is the representation associated to \(\lambda \) then \({\mathcal L}(\lambda ):={\mathcal L}(k_{\lambda})\) is a line bundle on \(G/B\). In this paper, the author gives explicitly the vanishing and non-vanishing of the cohomology modules \(\text{H}^i(X_w,{\mathcal L}(\lambda ))\) for \(w\in W\) and ``most'' \(\lambda \in X^{\ast}(T)\) when \(G\) has rank 2 (i.e., it is of type \(A_2\), \(B_2\), or \(G_2\)). For his computations, the author uses the Bott-Samelson-Demazure-Hansen desingularizations of \(X_w\), the fact that the cohomology of \({\mathcal L}(\lambda )\) on \(X_w\) is the same as the cohomology of its pull-back on the BSDH desingularisation, the realization of the BSDH desingularisations as towers of \({\mathbb P}^1\)-bundles starting with \({\mathbb P}^1\) and the Leray spectral sequence. The author's explicit tables reveal nice patterns. For example, the author conjectures that, for \(G\) of arbitrary rank, if \(\text{H}^m(X_w,{\mathcal L}(\lambda ))\neq 0\) and \(\text{H}^n(X_w,{\mathcal L}(\lambda ))\neq 0\) for some \(m\leq n\) then \(\text{H}^i(X_w,{\mathcal L}(\lambda ))\neq 0\) for all \(i\) with \(m\leq i\leq n\).
scientific article; zbMATH DE number 2139582
scientific article