Topology of spaces of regular sections and applications to automorphism groups

TL;DR

This paper investigates the topology of spaces of regular sections of equivariant vector bundles over complex varieties, providing conditions for the existence of geometric quotients and cohomological decompositions, with applications to various algebraic varieties.

Contribution

It establishes topological criteria ensuring the existence of quotients and cohomological isomorphisms for spaces of regular sections, extending understanding of automorphism groups in algebraic geometry.

Findings

Surjectivity of orbit maps in rational cohomology under certain conditions

Existence of geometric quotients for regular sections

Cohomology ring isomorphisms between spaces and 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.