Topology of spaces of regular sections and applications to automorphism groups

• Published in: arXiv.org 2021-05
• Main Authors: Gorinov, Alexey, Konovalov, Nikolay
• Format: Article
• Language: English
• Subjects: Bundling; Colon; Degeneration; Homology; Lie groups; Topology
• ISSN: 2331-8422

Let \(G\) be a complex connected reductive algebraic group that acts on a smooth complex algebraic variety \(X\), and let \(E\) be a \(G\)-equivariant algebraic vector bundle over \(X\). A section of \(E\) is regular if it is transversal to the zero section. Let \(U\subset\Gamma(X,E)\) be the subset of regular sections. We give a sufficient condition in terms of topological invariants of \(E\) and \(X\) that implies that every orbit map \(O\colon G\to U\) induces a surjection in rational cohomology. Under natural assumptions on \(X\) and \(E\) this condition is also necessary. If the condition is satisfied, then (1) the geometric quotient \(U/G\) exists; (2) there is an isomorphism \(H^*(U,\mathrm{Q})\cong H^*(G,\mathrm{Q})\otimes H^*(U/G,\mathrm{Q})\) of cohomology rings; (3) the order of the stabiliser \(G_s,s\in U\) divides a certain expression that can be explicitly calculated e.g. if \(X\) is a compact homogeneous space. In some cases (e.g. if \(E\) is a line bundle) we also prove similar statements for the space of the zero loci of \(s\in U\). We apply these results to several explicit examples which include hypersurfaces in projective spaces, non-degenerate quadrics and complete flag varieties of the simple Lie groups of rank 2, and also certain Fano varieties of dimension 3 and 4.