Independence conjectures for GKM manifolds

• Applicant: Dr. Grigory Solomadin, Ph.D.
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 561158824

Toric geometry and its derivatives, such as toric topology, constitute a well-established area of research that has many profound mathematical results. In this context, the term “complexity” refers to the quantity n − k, where k is the real dimension of a compact torus acting effectively on a smooth manifold of dimension 2n, or the complex dimension for an algebraic complex torus acting effectively on an n-dimensional complex non-singular variety. Actions of complexity 0 and 1 (with isolated fixed points) are relatively well understood in both algebraic topology and geometry. There is a growing interest in the research field to scrutinize actions of higher complexity. A key principle guiding this area of study is that generic actions of high complexity exhibit properties analogous to those of a complexity zero action restricted to a generic subtorus. Here, “genericity” can be defined in terms of j-independence, which requires that the tangential weights of the action at any fixed point form a tuple of vectors such that any distinct j elements (or fewer) are linearly independent. A nice class of torus actions is formed by that of GKM-manifolds. This is because many homotopy invariants and topological properties of a space with a torus action can be studied in case of a GKM-manifold in terms of the associated GKM-graph (a finite graph with labellings on its edges by elements of Z^k satisfying a number of properties). This principle fails among the wider class of non-generic actions (by the results of Luna, Vust; Ayzenberg, Cherepanov). The objective of the current project is to elucidate a seemingly paradoxical dichotomy regarding non-extendible torus actions on smooth manifolds within the class of 4-independent GKM torus actions. The goals of this project are of geometrical and topological nature. The expected results would providewith classification of 4-independent GKM-manifolds and a more effective description for the respective equivariant cohomology (compared to GKM-theorem). In addition, this would prove or disprove the conjectures proposed by Masuda and Kuroki. As a secondary objective, we expect to find new striking examples for GKM-manifolds with perfect fundamental groups.

• DFG Programme: WBP Position