A^1-Euler classes: six functors formalisms, dualities, integrality and
linear subspaces of complete intersections
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.
Bachmann, T; Wickelgren, K