Related papers: Some remarks on relative framed motives

Let $X$ be a variety. In this survey, we study (decompositions of) the motivic class, in the Grothendieck ring of stacks, of the stack $\mathscr{C}oh^n(X)$ of $0$-dimensional coherent sheaves of length $n$ on $X$. To do so, we review the…

In this article we further the study of non-commutative motives. Our main result is the construction of a simple model, given in terms of infinite matrices, for the suspension in the triangulated category of non-commutative motives. As a…

(This is an updated version; following an idea of Voevodsky, we have strengthened our results so all of them apply to one form of motivic homotopy theory). We give two general constructions for the passage from unstable to stable homotopy…

In this paper we prove over fields of characteristic zero that the zero slice of the motivic sphere spectrum is the motivic Eilenberg-Maclane spectrum. As a corollary one concludes that the slices of any spectrum are modules over the…

The category of framed correspondences $Fr_*(k)$ and framed sheaves were invented by Voevodsky in his unpublished notes [V2]. Based on the theory, framed motives are introduced and studied in [GP1]. These are Nisnivich sheaves of…

This paper has two aims. The former is to give an introduction to our earlier work on the Hodge theory of algebraic maps and more generally to some of the main themes of the theory of perverse sheaves and to some of its geometric…

Inspired by p-adic (and real) principal value integrals, we introduce motivic principal value integrals associated to multi-valued rational differential forms on smooth algebraic varieties. We investigate the natural question whether (for…

Let X_1, X_2 be symplectic 4-manifolds containing symplectic surfaces F_1,F_2 of identical positive genus and opposite squares. Let Z denote the symplectic sum of X_1 and X_2 along the F_k. Using relative Gromov--Witten theory, we determine…

The deepest arithmetic invariants attached to an algebraic variety defined over a number field $F$ are conjecturally captured by the integral part of its motivic cohomology. There are essentially two ways of defining it when $X$ is a smooth…

In this paper, we introduce the notions of motivic representation stability that is an algebraic counterpart of the notion of representation stability. In the process, we also introduce the notion of motivic decomposition for varieties…

Let $k$ be a field with resolution of singularities, and $X$ a separated $k$-scheme of finite type with structure map $g$. We show that the slice filtration in the motivic stable homotopy category commutes with pullback along $g$.…

Combining the idea of motivic zeta function, due to Kapranov, and Pellikaan's definition of a two- variable zeta function for curves over finite fields in the present note we introduce a motivic two- variable zeta function for curves over…

In this paper, we study twisted algebraic $K$-theory from a motivic viewpoint. For a smooth variety $X$ over a field of characteristic zero and an Azumaya algebra $\mathcal{A}$ over $X$, we construct the $\mathcal{A}$-twisted motivic…

The purpose of this note is to give a simple proof of the following theorem: Let $X$ be a normal projective variety over an algebraically closed field $k$, $\op{char} k = 0$ and let $D \subset X$ be a proper closed subvariety of $X$. Then…

This paper was motivated by the following question: Recall that for a smooth projective variety X whose polarized Hodge structure on H^n(X,Q)_{prim} leads to a period point ...

We prove the analog of the Morel-Voevodsky localization theorem for framed motivic spaces. We deduce that framed motivic spectra are equivalent to motivic spectra over arbitrary schemes, and we give a new construction of the motivic…

We compute the spectrum and Smith normal form of the incidence matrix of disjoint transversals, a combinatorial object closely related to the n-dimensional case of Rota's basis conjecture.

We consider the "limiting behavior" of *discriminants*, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on…

Given a smooth projective variety $X$ with a simple normal crossing divisor $D:=D_1+D_2+...+D_n$, where $D_i\subset X$ are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks $X_{D,\vec r}$ by constructing an…

We prove a deletion-contraction formula for motivic Feynman rules given by the classes of the affine graph hypersurface complement in the Grothendieck ring of varieties. We derive explicit recursions and generating series for these motivic…