Quaternionic Grassmannians and Borel classes in algebraic geometry

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 algebraic symplectic cobordism, we have A(HP n ) = A(pt)[p]/(p n+1 ). We define Borel classes for symplectic bundles. They satisfy the 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.The cell structure of the HGr(r, n) exists in the cohomology, but it is difficult to see more than part of it geometrically. The exception is HP n where the cell of codimension 2i is a quasi-affine quotient of A 4n−2i+1 by a nonlinear action of Ga.

A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections

“…We will use the notation Õ(−) (a) to mean either Õ(a) and Õ− (a) when a claim holds for both possibilities. We likewise use the EM(W)-Pontryagin (or Borel) classes of a vector bundle with trivialized determinant of Panin and Walter[PW10c]. See [Ana15, Introduction, Section 3] or [Lev19, Section 3] [Wen18, Section 2] for background on these classes.Let e i ∈ EM(W) * (N r+1 ) denote the pullback of e(S * 2 , EM(W)) under the ith projection N r+1 → N composed with the canonical map N → BSL 2 .…”

Cellularity of hermitian K-theory and Witt theory

“…The normal bundle N of the embedding HGr(E) ⊂ HGr(F) is the tensor product U ∨ E ⊗O ⊕2 for the dual of the tautological symplectic subbundle of rank 2r on HGr(E). Theorem 4.1 in [10] shows that N is naturally isomorphic to an open subscheme of Gr(2r, F) and there is a decomposition N = N + ⊕ N − ; here, N + = HGr(F) ∩ Gr(2r, O ⊕ E) and N − = HGr(F) ∩ Gr(2r, E ⊕ O) have intersection HGr(E). Thus there are natural vector bundle isomorphisms N + ∼ = N − ∼ = U r,n−1 and the normal bundle N of N + in HGr(F) is isomorphic to π * + U r,n−1 for the bundle projection π + : N + → HGr(E).…”

“…Thus Lemma 2.3 and (1) reduce the proof to showing that Σ ∞ Th(U ⊕m r,n |Y ) is a finite cellular spectrum. To this end we recall parts of Theorem 5.1 in [10]: There exists maps…”

The special linear version of the projective bundle theorem

“…for the natural map z : X → T h(E). Then one may define the lower Borel classes using the symplectic version of the projective bundle theorem, see [PW1] for the details. Note that since these Borel classes are similar to the symplectic Borel classes in topology and not to the Pontryagin classes, we omit the sign in the above formula for the top Borel class.…”

“…There are analogous computations for symplectically oriented cohomology theories [PW1] with appropriately chosen varieties: quaternionic projective spaces HP n instead of the ordinary ones and quaternionic Grassmannian and flag varieties. The answers are essentially the same, algebras of truncated polynomials in characteristic classes.…”