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

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 P^n in terms of topological Euler numbers over R and C.

Corrected statement and proof of Lemma 5.6 and Proposition 5.4

References in corpus (3)

• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• A comparison theorem for MW-motivic cohomology
• Topological models for stable motivic invariants of regular number rings

Cited by in corpus (6)

• Quadratic types and the dynamic Euler number of lines on a quintic threefold
• Applications to A1-enumerative geometry of the A1-degree
• Computing A1-Euler numbers with Macaulay2
• Conics meeting eight lines over perfect fields
• Compactly supported ${\mathbb{A}}^1$-Euler characteristic and the Hochschild complex
• An Introduction to ${\mathbf{A}}^1$-Enumerative Geometry