Project Details
Projekt
Geometry, topology, algebra and combinatorics of the GKM correspondence
Subject Area
Mathematics
Term
since 2020
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 452427095
In this project we will take the recent proof of the 6-dimensional smooth GKM correspondence as a starting point, and explore the various new lines of research it opens. It turned out that various purely graph-theoretical notions play a crucial role when it comes to the question of realizability of GKM graphs, such as freeness of equivariant graph cohomology, as well as several natural orientability notions for GKM graphs. We will improve our knowledge on the GKM correspondence by understanding possible interrelations of these algebro-combinatorial properties, both in dimension 6 and beyond. We will investigate variants of the GKM rigidity question, such as a version of the Petrie conjecture for GKM manifolds. GKM rigidity can also be phrased as the realization problem for graph automorphisms; we ask if GKM realization can be extended to twisted automorphisms. Also, we will extend the existing realization results to GKM fiber bundles in higher dimensions as well as to non-orientable GKM graphs and coverings. Throughout the project we have in mind that invariant geometric structures on GKM manifolds leave their mark on the GKM graph, and we consider these and other questions on the GKM correspondence in presence of invariant spin structures, almost complex, symplectic, Kähler and nearly Kähler structures.
DFG Programme
Research Grants
