Matroids and the space of torus-invariant subvarieties of the Grassmannian with given homology class

TL;DR

The paper develops a matroid-theoretic framework to classify $T$-invariant subvarieties of Grassmannians with prescribed homology, focusing on the space of orbit closures and their cohomology classes. It proves that subvarieties homologous to $T$-orbit closures are themselves orbit closures and constructs a complete set of orbit representatives for $\mathbb{G}(2,n)$, organizing the geometry via thin Schubert cells and torus bundles. By expressing cohomology classes of orbit closures in terms of partitions and Schubert classes, it provides an explicit algorithm to compute orbit classes and, consequently, to approach Euler--Chow series computations. The results yield a concrete calculation of the Euler--Chow series for $\mathbb{G}(2,4)$ and outline a path for higher cases, linking matroid subdivisions, tropical geometry, and Schubert calculus in a unified approach. This work advances the intersection of combinatorics and algebraic geometry, enabling more systematic Euler characteristic computations of Chow varieties in Grassmannians.

Abstract

Let $\mathbb{G}(d,n)$ be the complex Grassmannian of affine $d$-planes in $n$-space. We study the problem of characterizing the set of algebraic subvarieties of $\mathbb{G}(d,n)$ invariant under the action of the maximal torus $T$ and having given homology class $λ$. We give a complete answer for the case where $λ$ is the class of a $T$-orbit, and partial results for other cases, using techniques inspired by matroid theory. This problem has applications to the computation of the Euler-Chow series for Grassmannians of projective lines: we calculate the series for 3-cycles in $\mathbb{G}(2,4)$ and carry out partial calculations for $\mathbb{G}(2,5)$.

Theorems & Definitions (71)

• Definition 2.1
• Theorem 2.2: GGMS
• Remark 2.3
• Proposition 2.4
• Remark 2.5
• Definition 2.6
• Definition 2.7
• Proposition 2.8
• Proof 1
• Definition 2.9