Incompressibility of orthogonal Grassmannians of rank 2

TL;DR

This paper establishes a sufficient condition for the 2-incompressibility of the variety of 2-dimensional totally isotropic subspaces in a quadratic form, extending known results for other cases using advanced motivic and Chow ring techniques.

Contribution

It generalizes the criteria for 2-incompressibility of orthogonal Grassmannians of rank 2, employing motivic decomposition and Chow ring analysis.

Findings

Provides a new sufficient condition for 2-incompressibility of X_2

Extends known results for X_1 and X_n to X_2

Utilizes motivic decomposition and Chow ring characterization

Abstract

For a nondegenerate quadratic form phi on a vector space V of dimension 2n + 1, let X_d be the variety of d-dimensional totally isotropic subspaces of V. We give a sufficient condition for X_2 to be 2-incompressible, generalizing in a natural way the known sufficient conditions for X_1 and X_n. Key ingredients in the proof include the Chernousov-Merkurjev method of motivic decomposition as well as Pragacz and Ratajski's characterization of the Chow ring of (X_2)_E, where E is a field extension splitting phi.