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

TL;DR

This paper characterizes torus-invariant algebraic subvarieties of Grassmannians with specific homology classes, employing matroid theory, and computes related Euler-Chow series for certain cases.

Contribution

It provides a complete classification for subvarieties with homology class of a torus orbit and partial results for other classes, advancing understanding of Grassmannian invariants.

Findings

Complete classification for T-orbit homology classes

Partial results for other homology classes

Calculated Euler-Chow series for 3-cycles in G(2,4)

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 $\lambda$. We give a complete answer for the case where $\lambda$ 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 $\mathbb{G}(2,5)$.