[Paper Review] Topology of Kempf-Ness sets for algebraic torus actions

Taras Panov|ArXiv.org|Mar 23, 2006

Advanced Combinatorial Mathematics14 references3 citations

TL;DR

This paper introduces a new notion of Kempf–Ness sets for algebraic torus actions on quasiaffine varieties arising from the Batyrev–Cox geometric invariant theory construction of toric varieties. Using moment-angle complexes and toric topology, it establishes that these Kempf–Ness sets are complete intersections of real quadrics in complex space for projective non-singular toric varieties, and computes their cohomology rings explicitly, revealing non-trivial Massey products and non-formality in the topological structure.

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.

Motivation & Objective

• To extend the concept of Kempf–Ness sets—originally defined for reductive group actions on affine varieties—to algebraic torus actions on quasiaffine varieties arising from toric geometry.
• To establish that these new Kempf–Ness sets retain key properties: being a deformation retract and yielding a quotient homeomorphic to the categorical quotient.
• To provide a topological description of these sets in terms of moment maps and real quadric intersections for projective non-singular toric varieties.
• To compute the cohomology ring of the Kempf–Ness set using moment-angle complex techniques and identify non-trivial topological invariants such as Massey products.

Proposed method

• Utilizes the geometric invariant theory (GIT) framework of Batyrev–Cox to realize toric varieties as quotients of coordinate subspace arrangement complements under torus actions.
• Constructs the Kempf–Ness set as a $K$-invariant subset via a moment map construction, analogous to the affine case but adapted to quasiaffine settings.
• Identifies the Kempf–Ness set with a complete intersection of real quadratic hypersurfaces in complex space when the toric variety is projective and non-singular.
• Applies moment-angle complex theory to model the topology of the Kempf–Ness set, using a differential graded algebra structure on $\Lambda[u_1,\dots,u_m] \otimes \mathbb{Z}[\mathcal{K}_\Sigma]$.
• Employs Poincaré duality and cohomological computations via full subcomplexes of the simplicial complex $\mathcal{K}_\Sigma$ to determine Betti numbers and cohomology classes.
• Analyzes Massey products in the cohomology ring using triple products of 3-cocycles, demonstrating non-triviality and non-formality of the manifold.

Experimental results

Research questions

• RQ1 Can the Kempf–Ness set construction be generalized from reductive group actions on affine varieties to torus actions on quasiaffine varieties in the context of toric geometry?
• RQ2 What is the topological structure of the Kempf–Ness set for projective non-singular toric varieties, and how does it relate to moment maps and real quadrics?
• RQ3 How can the cohomology ring of the Kempf–Ness set be computed explicitly using moment-angle complex techniques?
• RQ4 Are there non-trivial Massey products in the cohomology of these Kempf–Ness sets, and what does this imply about their topological formality?

Key findings

• The Kempf–Ness set for a projective non-singular toric variety is a complete intersection of real quadric hypersurfaces in a complex space, corresponding to a level set of the moment map.
• The cohomology ring of the Kempf–Ness set $\mathcal{Z}_P$ for a 3-dimensional simple polytope $P$ has Betti vector $(1,0,0,10,16,5,5,16,10,0,0,1)$, indicating a non-trivial topological structure.
• The 3rd cohomology group $H^3(\mathcal{Z}_P) \cong \mathbb{Z}^{10}$ is generated by 10 specific 3-cocycles involving products of $u_i$ and $v_j$.
• The 4th cohomology group $H^4(\mathcal{Z}_P) \cong \mathbb{Z}^{16}$ arises from 16 distinct 4-cocycles, primarily from subcomplexes of type 'an edge and a point'.
• The 5th cohomology group $H^5(\mathcal{Z}_P) \cong \mathbb{Z}^5$ is generated by 5 cocycles, including a non-monomial one, indicating complex structure.
• The manifold $\mathcal{Z}_P$ is non-formal, as demonstrated by a non-trivial triple Massey product $\langle\alpha,\beta,\gamma\rangle$ containing a non-zero cohomology class.