A¹-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections

2020/02/05 by Tom Bachmann, Bachmann, Tom, Kirsten Wickelgren +1 · 3 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)

paper · pdf · doi:10.48550/arxiv.2002.01848

openalex publication_date 2020/02/05 · openalex created_date 2020/02/14 · openalex updated_date 2026/07/28

Abstract

We equate various Euler classes of algebraic vector bundles, including those of [BM, KW, DJK], and one suggested by M.J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class, and give formulas for local indices at isolated zeros, both in terms of 6-functor formalism of coherent sheaves and as an explicit recipe in commutative algebra of Scheja and Storch. As an application, we compute the Euler classes associated to arithmetic counts of d-planes on complete intersections in Pⁿ in terms of topological Euler numbers over R and C.

Citations

• Quaternionic Grassmannians and Borel classes in algebraic geometry
• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• The six operations in equivariant motivic homotopy theory
• η-periodic motivic stable homotopy theory over Dedekind domains

Cited by

• Motivic nearby cycles and their monodromy at a singular point
• The Evolution of Enumerative Geometry: A Narrative from Classical Problems to Enriched Invariants
• η-periodic motivic stable homotopy theory over fields

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