Chamber Decompositions of Moment Polytopes for Torus Actions of Positive Complexity

Abstract: The present work develops the results of the series of papers by Buchstaber and Terzić on the standard actions of the compact torus $Tⁿ = (S<sup>1)<sup>n$ on the complex Grassmann manifolds $G_{n,2}$. In those works, a hyperplane arrangement in $\mathbb{R}<sup>n$ was introduced that determines the chamber decomposition of the hypersimplex $Δ<em>{n,2}$ for the $T<sup>n$-action on $G</em>{n,2}$. We introduce a notion of admissible graph for the standard action of the torus $T<sup>n$ on the complex Grassmannian $G_{n,2}$. In terms of admissible graphs, we give a complete inductive description (with respect to $n \ge 4$) of the admissible polytopes in $Δ<em>{n,2}$, as well as of the toric varieties arising as closures of $(\mathbb{C}<sup>*)<sup>n$-orbits on $G</em>{n,2}$ under the standard $(\mathbb{C}<sup>*)<sup>n$-action. We consider the $T<sup>n$-equivariant Plücker embedding $G_{n,2} \hookrightarrow \mathbb{C}P<sup>{N_2}$, where $N_2 = \binom{n}{2}-1$. Using admissible graphs, for the considered $T<sup>n$-actions, we describe hyperplane arrangements in $\mathbb{R}<sup>n$ that determine the chambers in $Δ<em>{n,2}$ for the $T<sup>n$-actions on $G</em>{n,2}$ and $\mathbb{C}P<sup>{N_2}$. Gel'fand, Kapranov, and Zelevinsky introduced the notions of secondary polytopes and secondary fans in connection with the problem of describing triangulations of a given convex polytope, which is closely related to the Newton polytopes of discriminants and resultants. For the $T<sup>n$-action on $\mathbb{C}P<sup>{N_2}$, we show that the cones in $\mathbb{R}<sup>n$ with vertex at the origin spanned by the chambers form the secondary fan of the cone spanned by the vertices of $Δ_{n,2}$.

• The paper introduces a combinatorial framework using admissible graphs to classify chamber decompositions of moment polytopes in torus actions.
• It establishes explicit dimension formulas and stratification criteria for polytopes arising from bipartite and multipartite graph structures.
• The study extends toric geometry methods by linking GKZ fan analysis and hyperplane arrangements to the orbit structure of positive complexity actions.

Chamber Decompositions of Moment Polytopes for Torus Actions of Positive Complexity

Introduction and Context

This work addresses the fine structure of moment polytopes for torus actions of positive complexity, focusing on the standard $T^n$-actions on the complex Grassmannians $G_{n,2}$ and their Plücker embeddings in projective space. The main objective is to provide a comprehensive, inductive description of the combinatorics and geometry underlying chamber decompositions of the hypersimplex $\Delta_{n,2}$—the moment polytope for these actions—using admissible graphs and, for the projective case, secondary/GKZ fans. The study extends classical toric geometry, where moment polytopes capture the orbit structure, to the setting where the action's complexity (manifold dimension minus twice the torus dimension) is strictly positive.

The theoretical framework synthesizes tools from equivariant algebraic topology, the theory of toric and orbit closures, convex polytopes, and combinatorics (notably Johnson and multipartite graphs), as well as the categorical and moduli interpretations central to previous work by Buchstaber, Terzić, and others.

Admissible Graphs and Polytope Decomposition

The core conceptual advance is the introduction of admissible graphs as a means to encode the stratification of $G_{n,2}$ under torus action. For each admissible collection $\sigma \subseteq \binom{[n]}{2}$, the associated graph $G_\sigma$ on $n$ vertices with edges in $\sigma$ encodes which Plücker coordinates are non-vanishing on the stratum $W_\sigma$. The full classification demonstrates that admissible sets correspond precisely to graphs which are disjoint unions of complete multipartite components (possibly with isolated vertices). This combinatorial model enables an explicit, inductive (in $n$) description of all admissible polytopes and strata.

A critical result is the dimension formula for admissible polytopes:

• For bipartite graphs ($G_{n,2}$0 parts): $G_{n,2}$1
• For $G_{n,2}$2 multipartite graphs: $G_{n,2}$3

Here, $G_{n,2}$4 is the set of isolated vertices. Maximal admissible polytopes (of dimension $G_{n,2}$5) correspond to complete $G_{n,2}$6-partite graphs with $G_{n,2}$7.

Geometric dualities are emphasized: the vertices of $G_{n,2}$8 are indexed by $G_{n,2}$9 pairs ($\Delta_{n,2}$0), i.e., by edges of the complete graph $\Delta_{n,2}$1. Thus, admissible polytopes correspond to subgraphs of $\Delta_{n,2}$2, and faces are characterized by subgraph containment, aligning the stratification structure with graphical combinatorics.

Hyperplane Arrangements and Chamber Structure

The chambers—the connected components of the complement of relevant hyperplane arrangements within $\Delta_{n,2}$3—are indexed by collections $\Delta_{n,2}$4 of admissible sets subject to certain intersection and maximality constraints. The explicit description of facet-supporting hyperplanes, and the combinatorial mechanism (vertex isolation, merging of parts) to describe face lattices of admissible polytopes, is detailed.

Hyperplane arrangements $\Delta_{n,2}$5 correspond structurally to coordinate and bipartition-type equations:

• $\Delta_{n,2}$6, yielding the faces isomorphic to $\Delta_{n,2}$7
• $\Delta_{n,2}$8 ($\Delta_{n,2}$9), yielding faces isomorphic to $G_{n,2}$0
• $G_{n,2}$1-splits: $G_{n,2}$2, $G_{n,2}$3

These equations encode the locations of codimension-one admissible polytopes, and more generally the full intersection lattice $G_{n,2}$4 underpins the chamber structure. The analysis aligns with observed dualities in the literature between the stratification of moment polytopes under torus action and moduli-theoretic decompositions of configuration spaces.

GKZ (Secondary) Fan and Chambers for the Plücker Embedding

For the $G_{n,2}$5-action on $G_{n,2}$6 via the exterior square representation, the corresponding moment map image is also $G_{n,2}$7, but the chamber structure is governed not by admissibility but by intersections of the GKZ (secondary) fan with $G_{n,2}$8. The fans arise from the cones generated by the vector configuration $G_{n,2}$9, and their combinatorics reflect generalized subdivisions, triangulations, and regular values for the moment map.

The work provides an explicit combinatorial and geometric criterion for the maximal cells (chambers), linking them to certain graphs whose components are a $\sigma \subseteq \binom{[n]}{2}$0 and a $\sigma \subseteq \binom{[n]}{2}$1, covering both admissible and non-admissible polytopes. The set of regular values for the moment map are the points in $\sigma \subseteq \binom{[n]}{2}$2 which avoid all hyperplanes in the refined arrangement $\sigma \subseteq \binom{[n]}{2}$3, which strictly contains $\sigma \subseteq \binom{[n]}{2}$4 for $\sigma \subseteq \binom{[n]}{2}$5 (notably incorporating the type-$\sigma \subseteq \binom{[n]}{2}$6 Coxeter/braid arrangement for $\sigma \subseteq \binom{[n]}{2}$7).

Inductive and Structural Results

The admissible polytopes of full dimension and their facets are classified in terms of partitions of $\sigma \subseteq \binom{[n]}{2}$8, and explicit normal form models for the corresponding orbit closures are given (products and joins of simple polytopes). In particular:

• Admissible polytopes cut out by $\sigma \subseteq \binom{[n]}{2}$9 are combinatorially $G_\sigma$0, their closures diffeomorphic to $G_\sigma$1
• Lower-dimensional polytopes correspond to joins with lower-dimensional hypersimplices, reflecting the underlying multipartite structure.

Furthermore, the arrangement and face-lattice perspective allows for a functorial, combinatorially transparent description of the boundary operator for the complex of admissible polytopes, encoding the poset structure of the stratification.

Implications and Future Directions

This research clarifies the combinatorics and topology governing orbit stratifications and orbit closure geometry for torus actions of positive complexity, systems which do not reduce to toric varieties. The results synthesize stratification theory, polytopal and graphical combinatorics, GKZ theory, and moduli techniques (notably embedding moduli spaces of weighted stable curves in the orbit spaces), suggesting several directions:

• Extension to $G_\sigma$2 for $G_\sigma$3 and identification of the underlying combinatorial types of strata and polytopes.
• Exploitation of the categorical correspondences (e.g., Hassett categories) to relate chamber decompositions to moduli of curves and their birational models.
• Applications in symplectic and equivariant topology, particularly the study of invariants of orbit spaces of non-toric complexity.
• Deeper connection to geometric invariant theory via the identification of secondary polytopes/fans, with potential for interpreting wall-crossing and birational phenomena in these terms.

Conclusion

The paper presents a detailed, combinatorially rich description of chamber decompositions for $G_\sigma$4 under standard and induced torus actions, harnessing admissible graph theory and GKZ fans. These advancements illuminate the structure and geometry of orbit spaces for torus actions of positive complexity, bridging algebraic, topological, and combinatorial approaches, and provide a foundation for generalizations to more intricate actions and spaces.