Abstract
AbstractWe 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 ofd-planes on complete intersections in$\mathbb P^n$in terms of topological Euler numbers over$\mathbb {R}$and$\mathbb {C}$.
Highlights
For algebraic vector bundles with an appropriate orientation, there are Euler classes and numbers enriched in bilinear forms
When x is a simple zero of σ with k(x) = k, the index is given by a well-defined Jacobian Jacσ of σ, indPxHσ = Jacσ(x), illustrating the relation with the Poincare–Hopf formula for topological vector bundles. (For the definition of the Jacobian, see the beginning of §6.2.) In [49, Section 4, Corollary 36], it was shown that nPH(V,σ) = nPH(V,σ′) when σ and σ′ are in a family over A1L of sections with only isolated zeros, where L is a field extension with [L : k] odd
We offer three ways around this: (1) If 1/2 ∈ S, we could look at the image of φ in the Balmer–Witt group of S. (2) If φ happens to be concentrated in degree 0, it corresponds to a symmetric bilinear form on a vector bundle on S, which is a sensible invariant
Summary
For algebraic vector bundles with an appropriate orientation, there are Euler classes and numbers enriched in bilinear forms. Note that βi,j ⊕ βn−i,n−j in GW(k) is an integer multiple of h, where h denotes the hyperbolic form h = 1 + −1 , with Gram matrix h= This notion of the Euler number was suggested by M. For a relatively oriented vector bundle V equipped with a section σ with only isolated zeros, an Euler number nPH(V ,σ) was defined in [49, Section 4] as a sum of local indices: nPH(V ,σ) =. (For the definition of the Jacobian, see the beginning of §6.2.) In [49, Section 4, Corollary 36], it was shown that nPH(V ,σ) = nPH(V ,σ′) when σ and σ′ are in a family over A1L of sections with only isolated zeros, where L is a field extension with [L : k] odd We strengthen this result by equating nPH(V ,σ) and nGS(V ); this is the main result of §2. Zeros are of codimension 2, as in [49, Lemmas 54, 56, and 57] and in [72, Lemma 1], because nPH(V ,σ) is independent of σ
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