Related papers: On equivariant and motivic slices

In this paper, we discuss the motivic stable homotopy type of abelian varieties. For an abelian variety over a perfect field $k$ with a rational point, it always splits off a top-dimensional cell in motivic stable homotopy category…

We construct a map from the suspension $G$-spectrum $\Sigma_G^\infty M$ of a smooth compact $G$-manifold to the equivariant $A$-theory spectrum $A_G(M)$, and we show that its fiber is, on fixed points, a wedge of stable $h$-cobordism…

A conjecture of Orlov predicts that derived equivalent smooth projective varieties over a field have isomorphic Chow motives. The conjecture is known for curves, and was recently observed for surfaces by Huybrechts. In this paper we focus…

The aim of this paper is to show that Besser syntomic cohomology is representable by a rational ring spectrum in the motivic homotopical sense. In fact, extending previous constructions, we exhibit a simple representability criterion and we…

Let $k$ be a field of characteristic zero and let $(F,R)$ be a Landweber exact formal group law. We consider a Landweber exact $T$-spectrum $\mathcal{E}:=R\otimes_{\mathbb{L}}\text{MGL}$ and its effective cover $f_0\mathcal{E}\to…

We analyze the spectrum of the tensor-triangulated category of Artin-Tate motives over the base field R of real numbers, with integral coefficients. Away from 2, we obtain the same spectrum as for complex Tate motives, previously studied by…

Let $X$ be a smooth irreducible projective variety over a field $\mathbf{k}$ of dimension $d.$ Let $\tau: \mathbb{Q}_l\to \mathbb{C}$ be any field embedding. Let $f: X\to X$ be a surjective endomorphism. We show that for every…

We found an explicit description of all $GL(n,\RR)$-Whittaker functions as oscillatory integrals and thus constructed equivariant mirrors of flag manifolds. As a consequence we proved the Virasoro conjecture for flag manifolds.

We discuss a conjecture saying that derived equivalence of simply connected smooth projective varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class.…

Without using the $L^2$ extension theorem, we provide a new proof of the equality part in Suita's conjecture, which states that for any open Riemann surface admitting a Green's function, the Bergman kernel and the logarithmic capacity…

The paper studies categories of definable subassignments with some category equivalences to semi-algebraic and constructible subsets of arc spaces of algebraic varieties. These materials allow us to compare the motivic measure of…

Over any field of characteristic not 2, we establish a 2-term resolution of the $\eta$-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is curiously similar to the resolution of the…

We study matrix semigroups in which ring commutators have real spectra. We prove that irreducible semigroups with this property are simultaneously similar to semigroups of real-entried matrices. We also obtain a structure theorem for…

For a Riemannian covering $\pi\colon M_1\to M_0$, the bottoms of the spectra of $M_0$ and $M_1$ coincide if the covering is amenable. The converse implication does not always hold. Assuming completeness and a lower bound on the Ricci…

Let $k$ be a field admitting resolution of singularities. We lift a number of motivic measures, such as the Gillet-Soul\'e measure and the compactly supported $\mathbb{A}^1$-Euler characteristic, to derived motivic measures in the sense of…

Over the past century, cohomology operations have played a crucial role in homotopy theory and its applications. A powerful framework for constructing such operations is the theory of commutative algebras in spectra. In this article, we…

We prove that the Almgren-Pitts and phase-transition half-volume spectra of a closed Riemannian manifold are equal. This confirms a conjecture of Liam Mazurowski and Xin Zhou.

If a tuple of matrices has a common invariant subspace, its projective joint spectrum has an algebraic component. In general, the converse is not true, and there might be algebraic components in the projective joint spectrum without…

We construct a theory of motivic cohomology for quasi-compact, quasi-separated schemes of equal characteristic, which is related to non-connective algebraic $K$-theory via an Atiyah--Hirzebruch spectral sequence, and to \'etale cohomology…

The author has already proven that the space $\Delta(\Pi_n)/G$ is homotopy equivalent to a wedge of spheres of dimension $n-3$ for all natural numbers $n\geq 3$ and all subgroups $G\subset S_1\times S_{n-1}$. We construct an $S_1\times…