Topology of spaces of regular sections and applications to automorphism groups

Abstract

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.