Motivic Euler Characteristics and Witt-Valued Characteristic Classes

Abstract

This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker-Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from GLn or SLn to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy's splitting principle reduces questions about characteristic classes of vector bundles in SL-oriented, η-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for SL 2 to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.2010 Mathematics Subject Classification. 14F42, 55N20, 55N35.

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

“…For example, combining with results of Finashin and Kharlamov over R [34] are arithmetic counts of the 3-planes in a 7-dimensional cubic hypersurface and in a 16dimensional degree 5 hypersurface, respectively (see Example 6.13). This builds on results of Finashin and Kharlamov [34], J. L. Kass and the second author of the present paper [49], M. Levine [54], S. McKean [58], Okonek and Teleman [62], S. Pauli [66], J. Solomon [70], P. Srinivasan and the second author [72], and M. Wendt [74].…”

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

“…This builds on results of Finahin-Kharlamov [FK13], J.L. Kass and the second-named author [KW17], M. Levine [Lev19] [Lev17], S. McKean [McK19], Okonek-Teleman [OT14], S. Pauli [Pau19], J. Solomon [Sol06], P.Srinivasan and the second-named author [SW18], and M. Wendt [Wen18].…”

“…By Lemma 6.6, it follows that disc n(V, Gr(d, n)) = 1 in k * /(k * ) 2 as desired. M. Levine [Lev19] uses the normalizer N of the standard torus of BSL 2…”

On quadratically enriched excess and residual intersections

“…By this we mean that we will give each of the points of Γ a weight in GW(k) associated to its field of definition and the quadrics Q i and compute the sum of these weights in GW(k). See also [KW21], [SW21], [Lev19], [McK21], [Pau20b], [Pau20c] and [BW20] [CDH20c] for other arithmetic or quadratically enriched counts. In this context, the following definition of a relative orientation of a vector bundle is useful.…”