Related papers: Matrix factorizations and motivic measures

We study categories of matrix factorizations. These categories are defined for any regular function on a suitable regular scheme. Our paper has two parts. In the first part we develop the foundations; for example we discuss derived direct…

We propose a natural definition of a category of matrix factorizations for nonaffine Landau-Ginzburg models. For any LG-model we construct a fully faithful functor from the category of matrix factorizations defined in this way to the…

In this paper we establish a precise comparison between vanishing cycles and the singularity category of Landau-Ginzburg models over a complete discrete valuation ring. By using noncommutative motives, we first construct a motivic…

Given an element $f$ in a regular local ring, we study matrix factorizations of $f$ with $d \ge 2$ factors, that is, we study tuples of square matrices $(\varphi_1,\varphi_2,\dots,\varphi_d)$ such that their product is $f$ times an identity…

We study matrix factorization and curved module categories for Landau-Ginzburg models (X,W) with X a smooth variety, extending parts of the work of Dyckerhoff. Following Positselski, we equip these categories with model category structures.…

Branes and defects in topological Landau-Ginzburg models are described by matrix factorisations. We revisit the problem of deforming them and discuss various deformation methods as well as their relations. We have implemented these…

The bicategory $\mathcal{LG}$ of Landau-Ginzburg models has polynomials as objects and matrix factorisations as $1$-morphisms. The composition of these $1$-morphisms produces infinite rank matrix factorisations, which is a nuisance. In this…

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…

We consider matrix factorizations and homological mirror symmetry on the torus T^2 using a Landau-Ginzburg description. We identify the basic matrix factorizations of the Landau-Ginzburg superpotential and compute the full spectrum, taking…

We define a motivic analogue of the Haar measure for groups of the form G(k((t))), where k is an algebraically closed field of characteristic zero, and G is a reductive algebraic group defined over k. A classical Haar measure on such groups…

We investigate tensor products of matrix factorisations. This is most naturally done by formulating matrix factorisations in terms of bimodules instead of modules. If the underlying ring is C[x_1,...,x_N] we show that bimodule matrix…

This article generalizes the correspondence between matrix factorizations and maximal Cohen-Macaulay modules over hypersurface rings due to Eisenbud and Yoshino. We consider factorizations with several factors in a purely categorical…

Motivated by periodicity theorems for Real $K$-theory and Grothendieck--Witt theory and, separately, work of Hori-Walcher on the physics of Landau-Ginzburg orientifolds, we introduce and study categories of Real matrix factorizations. Our…

For a commutative ring $S$ and self-orthogonal subcategory $\mathsf{C}$ of $\mathsf{Mod}(S)$, we consider matrix factorizations whose modules belong to $\mathsf{C}$. Let $f\in S$ be a regular element. If $f$ is $M$-regular for every $M\in…

Ever since the introduction of motivic homotopy theory, as a well-proposed approximation of Grothendieck's dream, algebraic geometers then have the chance to study schemes via a homotopy theory. However topologists also found that lifting…

Right adjoints for the forgetful functors on $\lambda$-rings and bi-rings are applied to motivic measures and their zeta functions on the Grothendieck ring of $\mathbb{F}_1$-varieties in the sense of Lorscheid and Lopez-Pena (torified…

Given an isolated, quasi-homogeneous singularity $X$ we prove that there is a group isomorphism between the group of rank one reflexive sheaves on $X$ and the free abelian group generated by $\mathbb{C}^*$-divisors, modulo linear…

We consider the abelian group $PT$ generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semi-orthogonal decompositions of corresponding triangulated categories. We introduce an operation of…

We introduce a quotient of the Grothendieck ring of varieties by identifying classes of universally homeomorphic varieties. We show that the standard realization morphisms factor through this quotient, and we argue that it is the correct…

We show that the motivic vanishing cycles introduced by J. Denef and F. Loeser give rise to a motivic measure on the Grothendieck ring of varieties over the affine line. We discuss the relation of this motivic measure to the motivic measure…