Topology of spaces of regular sections and applications to automorphism groups

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 $U\subset \mathrm{\Gamma }(X,E)$ 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 $O\text{\hspace{0.17em}⁣}:G\to U$ 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 $U\mathrm{/}G$ exists; (2) there is an isomorphism $H^(U,$\mathrm{Q}$)\cong H^(G,$\mathrm{Q}$)\otimes H^*(U/G,$\mathrm{Q}$)$ofcohomologyrings;(3)theorderofthestabiliser$G_s,s\in U$ 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.