Quaternionic Grassmannians and Borel classes in algebraic geometry

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.

💡 Research Summary

The paper develops a systematic framework for studying cohomology theories that are oriented with respect to symplectic (or “quaternionic”) structures, and applies this framework to compute the cohomology of quaternionic Grassmannians. The authors begin by defining the quaternionic Grassmannian HGr(r,n) as the open subscheme of the ordinary Grassmannian Gr(2r,2n) consisting of 2r‑dimensional subspaces on which a fixed non‑degenerate symplectic form remains non‑degenerate. In the special case r=1, HGr(1,n+1) coincides with the quaternionic projective space HPⁿ, providing a natural analogue of the real, complex and quaternionic projective tower.

A central notion introduced is that of a symplectically oriented cohomology theory A. This class includes all usual oriented theories (Chow groups, motivic cohomology, algebraic cobordism) as well as more exotic examples such as hermitian K‑theory, Witt groups, symplectic algebraic cobordism, and special‑linear cobordism. The defining feature of such a theory is the existence of Thom classes for rank‑2 symplectic bundles, which then generate Thom classes for arbitrary symplectic bundles.

Using this orientation, the authors first compute A(HPⁿ). They show that the cohomology ring is a truncated polynomial algebra \