Euler Classes: Six-Functors Formalism, Dualities, Integrality and Linear Subspaces of Complete Intersections

Abstract

We equate various Euler classes of algebraic vector bundles, including those of [12] 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 the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

Compactly supported $\mathbb{A}^{1}$-Euler characteristic and the Hochschild complex

“…In the following theorem, we prove that [B X ] = χ A 1 (X) by identifying the former with the pairing on (Hdg(X/k), Tr). It is proved analogously to the proof of [BW,Proposition 2.4]. However, we also need a key result of Neeman, which gives us a better understand of the constituents used to define B X .…”

“…) is equipped with the natural duality given by composing multiplication of forms with projection off of the top wedge power of T * X . After pushforward to Spec k, this complex with duality represents χ A 1 for smooth projective varieties, by [BW,Proposition 2.4].…”

Applications to A1-enumerative geometry of the A1-degree

“…This Euler number is an element of GW(k) and equals the sum of local A 1 -degrees at the isolated zeros of a general section [15]. It can be shown [4] to equal a pushfoward in oriented Chow of the Euler class of Barge and Morel [5] [11]. This was also studied by M. Levine in [25] where particular attention is given to the GW(k)-valued Euler characteristic which is the Euler number of the tangent bundle.…”

“…As an application Kass and the second named author get an enriched count of lines on a cubic surface as the Euler number of the vector bundle Sym 3 S * → Gr (2,4) Let X ⊂ P 3 k be a smooth cubic surface. It is a classical result that Xk contains 27 lines.…”

“…The Euler numbers corresponding to counts of lines on generic hypersurfaces of degree 2n− 1 in P n+1 was computed in [27]. The Euler numbers corresponding to counts of d-planes on generic complete intersections was computed in [4].…”