EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS
Tom Bachmann, Kirsten Wickelgren
Abstract
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 $\mathbb {C}$ .
Topics & Concepts
MathematicsLinear subspaceEuler's formulaFunctorAlgebraic numberFormalism (music)Pure mathematicsVector bundleEuler characteristicAlgebra over a fieldMathematical analysisMusicalArtVisual artsAlgebraic Geometry and Number TheoryAdvanced Algebra and GeometryHomotopy and Cohomology in Algebraic Topology