Quaternionic Grassmannians and Borel classes in algebraic geometry

Ivan Panin; Charles Walter

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arXiv · 1011.0649

Quaternionic Grassmannians and Borel classes in algebraic geometry

Ivan Panin·Charles Walter

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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.

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Ivan Panin, Charles Walter. 2010-11-02. Quaternionic Grassmannians and Borel classes in algebraic geometry. https://arxiv.org/abs/1011.0649

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