Quaternionic Grassmannians and Borel classes in algebraic geometry

TL;DR

This paper studies quaternionic Grassmannians and introduces Borel classes in algebraic geometry, providing calculations of their cohomology in symplectically oriented theories and establishing foundational properties of these classes.

Contribution

It defines Borel classes for symplectic bundles, proves their key properties, and computes the cohomology of quaternionic Grassmannians in various cohomology theories.

Findings

Borel classes satisfy splitting principle and Cartan sum formula.

Cohomology of quaternionic Grassmannians computed explicitly.

Thom classes determine Borel classes for symplectic bundles.

Abstract

The quaternionic Grassmannian HGr(r,n) is the affine open subscheme of the ordinary Grassmannian parametrizing those 2r-dimensional subspaces of a 2n-dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular there is HP^{n} = HGr(1,n+1). For a symplectically oriented cohomology theory A, including oriented theories but also hermitian K-theory, Witt groups and symplectic and special linear algebraic cobordism, we have A(HP^{n}) = A(pt)[p]/(p^{n+1}). We define Borel classes for symplectic bundles. They satisfy a splitting principle and the Cartan sum formula, and we use them to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 symplectic bundles determine Thom and Borel classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Borel classes.