Quaternionic Grassmannians and Borel classes in algebraic geometry

Abstract

The quaternionic Grassmannian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H upper G r left-parenthesis r comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>H Gr</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {H Gr}(r,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the affine open subscheme of the usual Grassmannian parametrizing those <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 r"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>r</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional subspaces of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 n"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular, we have <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H upper P Superscript n Baseline equals upper H upper G r left-parenthesis 1 comma n plus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>HP</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>H Gr</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {HP}^n = \operatorname {H Gr}(1,n+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For a symplectically oriented cohomology theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , including oriented theories but also the Hermitian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper K"> <mml:semantics> <mml:mi mathvariant="normal">K</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -theory, Witt groups, and algebraic symplectic cobordism, we have <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis upper H upper P Superscript n Baseline right-parenthesis equals upper A left-parenthesis p t right-parenthesis left-bracket p right-bracket slash left-parenthesis p Superscript n plus 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>HP</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>pt</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A(\operatorname {HP}^n) = A(\operatorname {pt})[p]/(p^{n+1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Borel classes for symplectic bundles are introduced in the paper. They satisfy the splitting principle and the Cartan sum formula, and they are used to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H upper G r left-parenthesis r comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>H Gr</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {H Gr}(r,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> exists in cohomology, but it is difficult to see more than part of it geometrically. An exception is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H upper P Superscript n"> <mml:semantics> <mml:msup> <mml:mi>HP</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\operatorname {HP}^n</mml:annotation> </mml:semantics>

Topics & Concepts