Marc Levine: Refined enumerative geometry (Lecture 1)

The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields.

Marc Levine: Refined enumerative geometry

Abstract:
Lecture 1: Milnor-Witt sheaves, motivic homotopy theory and Chow-Witt groups
We review the Hoplins-Morel construction of the Milnor-Witt ring of a field, its extension to the unramified Milnor-Witt sheaves and its relation with the classical Grothendieck-Witt ring of quadratic forms. We discuss Morel's theorem identifying the 0th stable homotopy sheaf of the motivic sphere spectrum with the Milnor-Witt sheaves. We introduce the twisted Chow-Witt ring of a smooth scheme as the cohomology of the twisted Milnor-Witt sheaves and describe its functorial properties.

Lecture 2: Euler classes, Euler characteristics and Riemann-Hurwicz formulas
The Euler class of a vector bundle is defined in the twisted Chow-Witt ring and gives rise to an Euler characteristic for a smooth projective variety over a field k with values in the Grothendieck-Witt ring GW(k). Morel's theorem on the endomorphisms of the motivic sphere spectrum gives a categorically defined Euler characteristic in GW(k) for a larger class of smooth schemes over k. We show how these two classes agree when both are defined and derive a number of consequences, including a quadritc forms version of the classical Riemann-Hurwicz formula.

Lecture 3: Virtual fundamental classes in motivic homotopy theory
Using the formalism of algebraic stacks, Behrend-Fantechi define the intrinsic normal cone, its fundamental class in the Chow group and a virtual fundamental class [Z;φ]vir ∈ CHr(Z) associated to a perfect obstruction theory φ of virtual rank r. Using the six-functor formalism for the motivic stable homotopy category, as developed by Ayoub and Cisinski-Déglise, we define motivic analogs of these constructions (for Z a quasi-projective scheme or G-scheme), which recover the fundamental class and virtual fundamental class of Behrend-Fantechi as a special case. This makes available the "degree" of the motivic virtual fundamental class as an element of GW(k) for a perfect obstruction theory of virtual rank 0 and virtual determinant a square.

Lecture 4: Characteristic classes in Witt-cohomology
Classical enumerative geometry relies heavily on the theory of Chern classes of vector bundles and the splitting principle, which makes possible the computation of the Chern classes of associated bundles (symmetric powers, exterior powers, tensor products, etc.) in terms of the Chern classes of the "input" bundles. This does not seem to be possible in general for the Euler classes described in Lecture 2: a principle obstruction is the lack of classes which correspond to the Chern classes in degree less than the rank of the bundle. However, symplectic bundles and SLn-bundles do admit a good theory of characteristic classes (Borel classes and Pontryagin classes, respectively) when one passes from the Milnor-Witt sheaves to the classical Witt sheaves. We will discuss the construction of the Borel and Pontryagin classes, and the results of A. Ananyevskiy computing the Witt cohomology of B SLn, as well as his SL2-splitting principle. Finally, we discuss a further reduction of an SL2-bundle to an NT-bundle, where NT is the normalizer of the standard torus in SL2, which reduces the computation to the situation reminiscent of the case of real SO(2)-bundles.

References:
[1] A. Ananyevskiy, The special linear version of the projective bundle theorem. Compos. Math. 151 (2015), no. 3, 461-501.
[2] J. Ayoub, Les six opérations de Grothendieck et le formalisme des cycles evanescents dans le monde motivique. I+II. Astérisque Nos. 314, 315 (2007).
[3] J. Barge and F. Morel, Groupe de Chow des cycles orientés et classe d'Euler des fibrés vectoriels. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 4, 287-290.
[4] D.C. Cisinski and F. Déglise, Triangulated categories of mixed motives. Preprint 2012, arXiv:0912.2110.
[5] J. Fasel, Groupes de Chow-Witt. Mém. Soc. Math. Fr. (N.S.) No. 113 (2008).
[6] F. Morel, An introduction to A1-homotopy theory. Contemporary developments in algebraic K-theory, 357-441, ICTP Lect. Notes, XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004.
[7] F. Morel, On the motivic π0 of the sphere spectrum. Axiomatic, enriched and motivic homotopy theory, 219-260, NATO Sci. Ser. II Math. Phys. Chem., 131, Kluwer Acad. Publ., Dordrecht, 2004.
[8] F. Morel and V. Voevodsky, A1-homotopy theory of schemes. Inst. Hautes Études Sci. Publ. Math. No. 90 (1999), 45-143 (2001).