The Chow Characteristic Image of \(\Spin(10)\) via the Affine Cone over the Spinor Variety

2026/07/26 by Sanghoon Baek
#math.AG

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Abstract

Let \(G=\Spin(10)\) be the split spin group over a field of characteristic different from \(2\), and let \(T⊂ G\) be a split maximal torus. We determine the image of the integral Chow restriction map \(\CH(BG)→ \CH(BT)^W\), equivalently \(\CH(BG)\) modulo torsion. The main new geometric ingredient in the proof is a construction of the class \(c₂c₃c₅\), where the \(c_i\) are the elementary Chern classes after restriction to \(T\). This class is obtained from the proper equivariant push-forward associated with the affine cone over the spinor variety in its half-spin embedding for the special Clifford group \(Γ⁺(10)\). Modulo two, the image is the subring generated, over the smallest Steenrod-stable subring of \(\F[c₂,c₃,c₄,c₅]\) containing \(c₂²,c₃²,c₄²,c₅\) and \(c₂c₃c₅\), by the torus restriction of the top Chern class of a half-spin representation. The integral characteristic image is the full inverse image of this mod-two subring under reduction modulo \(2\).

Citations

• Integral Weyl Invariants in Chow Characteristic Images of Spin and Special Clifford Groups
• The Chow rings of G₂ and Spin₇
• The Chow ring of a classifying space
• Equivariant intersection theory
• Localization in equivariant intersection theory and the Bott residue formula

Related

• Integral Weyl Invariants in Chow Characteristic Images of Spin and Special Clifford Groups
• Clifford Multiplication on Spinor Abelian Varieties
• The Chow rings of G₂ and Spin₇
• Group theoretic properties of Clifford multiplication on 2-torsion points on the Dirac Spinor Abelian Variety
• Holomorphic disc, spin structures and Floer cohomology of the Clifford torus