Topology of Kempf-Ness sets for algebraic torus actions

TL;DR

This paper extends the concept of Kempf-Ness sets to algebraic torus actions on quasiaffine varieties, linking toric topology with geometric invariant theory to analyze their cohomology and geometric structure.

Contribution

It introduces a new notion of Kempf-Ness sets for torus actions on quasiaffine varieties, connecting toric topology with GIT and describing their cohomology and geometric properties.

Findings

Kempf-Ness sets exist for certain torus actions on quasiaffine varieties.

Cohomology of these sets can be studied using toric topology techniques.

Kempf-Ness sets for smooth projective toric varieties are complete intersections of real quadrics.

Abstract

In the theory of algebraic group actions on affine varieties, the concept of a Kempf-Ness set is used to replace the categorical quotient by the quotient with respect to a maximal compact subgroup. By making use of the recent achievements of "toric topology" we show that an appropriate notion of a Kempf-Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate subspace arrangement complements) arising in the Batyrev-Cox "geometric invariant theory" approach to toric varieties. We proceed by studying the cohomology of these "toric" Kempf-Ness sets. In the case of projective non-singular toric varieties the Kempf-Ness sets can be described as complete intersections of real quadrics in a complex space.