Recent Trends in Algebraic Cycles, Algebraic K-Theory and Motives

Let $G$ be a split semisimple algebraic group over a field $k$ and let $X$ be the flag variety (i.e., the variety of Borel subgroups) of $G$ twisted by a generic $G$-torsor. We study the conjecture that the canonical epimorphism of the Chow ring of $X$ onto the associated graded ring of the topological filtration on the Grothendieck ring of $X$ is an isomorphism. Since the topological filtration in this case is known to coincide with the computable gamma filtration, this conjecture indicates a way to compute the Chow ring.

Let $B$ be a reasonable base-scheme and $Z$ a quasi-projective $B$-scheme. Relying on the Grothendieck 6-functor formalism for the motivic stable homotopy category, we define an object $C^{st}_{Z/B}$ in the motivic stable homotopy category $\text{SH}(B)$, which we call the intrinsic stable normal cone of $Z$ over $B$. For a motivic ring spectrum $\mathcal{E}$, we construct a fundamental class $[C^{st}_{Z/B}]_\mathcal{E}$ in $\mathcal{E}^{0,0}(C^{st}_{Z/B})$ and use this to construct for each perfect obstruction theory $\phi:E\to L_{Z/B}$ a virtual fundamental class $[Z,\phi]^{vir}_\mathcal{E}\in \mathcal{E}^{0,0}(\pi_{Z!}\Sigma^{E^\vee}1_Z)$. Here $\pi_Z:Z\to B$ is the structure morphism and we assume that $B$ is affine. There are also $G$-equivariant versions of these constructions for $G$ a ``tame'' algebraic group over $B$. Taking $B=\text{Spec} k$ and $\mathcal{E}=H\mathbb{Z}$, the spectrum representing motivic cohomology, we recover the definition of the fundamental class $[C_{Z/B}]\in \text{CH}_0(C_{Z/B})$ of the intrinsic normal cone $C_{Z/B}$ of $Z$ and the virtual fundamental class $[Z,\phi]^{vir}\in \text{CH}_r(Z)$, $r=\text{rank}E$, as defined by Behrend-Fantechi. Taking $\mathcal{E}=EM(K^{MW}_*)$, we get a virtual fundamental class $[Z,\phi]^{vir}_{K^{MW}_*}\in \tilde{\text{CH}}_r(Z,\text{det}^{-1}E)$, with $\tilde{\text{CH}}$ the Chow-Witt theory of Barge-Morel and Fasel. In case $r=0$, $\text{det}E=\mathcal{O}_Z$, and $Z$ projective over $k$, we can push this class forward to get a Grothendieck-Witt degree $\tilde{deg}[Z,\phi]^{vir}_{K^{MW}_*}\in \text{GW}(k)$.

Recent work of Röndigs-Spitzweck-Østvær sharpens the connection between the slice and Novikov spectral sequences. Using classical vanishing lines for the $E_2$-page of the Adams-Novikov spectral sequence and the work of Andrews-Miller on the $\alpha_1$-periodic ANSS, I will deduce some new vanishing theorems in the bigraded homotopy groups of the $\eta$-complete motivic sphere spectrum. In particular, I will show that the $m$-th $\eta$-complete Milnor-Witt stem is bounded above (by an explicit piecewise linear function) when $m \equiv 1$ or $2 \pmod{4}$, and then lift this result to integral Milnor-Witt stems (where an additional constraint on $m$ appears). This is joint work with Oliver Röndigs and Paul Arne Østvær.

We will present an approach to the Bloch-Beilinson filtration in the context of Voevodsky's triangulated category of motives.

A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, the number of lines depends on the surface, but work of Finashin-Kharlamov, Okonek-Teleman, and Segre shows that a certain signed count is always 3. We extend this count to an arbitrary field using A1-homotopy theory: we define an Euler number in the Grothendieck-Witt group for a relatively oriented algebraic vector bundle as a sum of local degrees, and then generalize the count of lines to a cubic surface over an arbitrary field. This is joint work with Jesse Leo Kass.

We will describe the Nil groups of the ring $\mathbb{Z}Q_8$, the integral group ring of the quaternion group, we will give applications for the calculation of $K$ theory groups of some infinite groups.

Barwick proved that the $K$-theory of a stable infinity-category with a bounded $t$-structure agrees with the $K$-theory of its heart in non-negative degrees. Joint work with David Gepner and Jeremiah Heller extends this to an equivalence of nonconnective K-theory spectra when the heart satisfies certain finiteness conditions such as noetherianity. Applications to negative $K$-theory and homotopy $K$-theory of ring spectra are provided, which were the original motivation for our work.

Motivic measures can be thought of as homomorphisms out of the Grothendieck ring of varieties. Two well-known such measures are the Larsen--Lunts measure (over $\mathbf{C}$) and the Hasse--Weil zeta function (over a finite field). In this talk we will show how to lift the Hasse--Weil zeta function to a map of $K$-theory spectra which restricts to the usual zeta function on $K_0$. As an application we will show that the Grothendieck spectrum contains nontrivial elements in the higher homotopy groups.