Related papers: Motivic integration over Deligne-Mumford stacks
We establish a conjecture of Mumford characterizing rationally connected complex projective manifolds in several cases.
Let R be a commutative ring with identity and M be an R-module. The purpose of this paper is to introduce and investigate the dual notion of morphic modules over a commutative ring.
We prove that the moduli spaces of twisted $\mathrm{SL}_n$ and $\mathrm{PGL}_n$-Higgs bundles on a smooth projective curve have the same (stringy) class in the Grothendieck ring of rational Chow motives. On the level of Hodge numbers this…
This work establishes the geometric component of Deligne's longstanding program on refined Grothendieck-Riemann-Roch formulas expressed through determinants of cohomology. The approach relies on a newly developed universal category of Chern…
This is the first in a series of papers devoted to foundations of topological stacks. We begin developing a homotopy theory for topological stacks along the lines of classical homotopy theory of topological spaces. In this paper we go as…
The main result of this paper amounts to a complete evaluation of the integral cohomological structure of the stable mapping class group. In particular it verifies the conjecture of D.Mumford about the rational cohomology of the stable…
In this paper, we introduce a birationally admissible stratification on the Deligne-Mumford stack of stable minimal models (e.g., the KSBA moduli stack), such that the universal family over each stratum admits a simple normal crossing log…
We define Reichstein transforms to be certain birational transformations of Artin stacks with good moduli spaces. Our main technical result is that the Reichstein transform of an Artin toric stack is again an Artin toric stack. This leads…
In this paper we study the integral Chow ring of toric Deligne-Mumford stacks. We prove that the integral Chow ring of a semi-projective toric Deligne-Mumford stack is isomorphic to the Stanley-Reisner ring of the associated stacky fan. The…
A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded…
Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi--particle dynamical system by finding polynomial solutions of a partial differential equations is…
The goal of this Section is to formulate some of the basic results on the theory of integral equations and mention some of its applications. The literature of this subject is very large. Proofs are not given due to the space restriction.…
Motivic homotopy theory is meant to play the role of algebraic topology, in particular homotopy theory, in the context of algebraic geometry. As proved by Oliver Rondigs and Paul Arne Ostvaer, this theory is closely connected to Voevodsky's…
In this paper we develop a generalization of foliated manifolds in the context of metric spaces. In particular we study dendritations of surfaces that are defined as maximal atlases of compatible upper semicontinuous local decompositions…
We prove the existence of the dualizing functor for a separated morphism of algebraic stacks with affine diagonal; then we explicitly develop duality for compact Deligne-Mumford stacks focusing in particular on the morphism from a stack to…
We calculate the motivic integral dual Steenrod algebra over base schemes for which the mod p motivic dual Steenrod algebra conforms with Voevodsky's formula.
This work brings Mellin transforms into the realm of motivic integration. The new, larger class of motivic functions is stable under motivic Mellin and Fourier transforms, with general Fubini results and change of variables formulas. It…
We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relation with Gromov--Witten theory and integrable systems.
The aim of this work is to show how we can decompose a module (if decomposable) into an indecomposable module with the help of the minimization process.
We illustrate the Arinkin-Deligne-Katz algorithm for rigid irreducible meromorphic bundles with connection on the projective line by giving motivicity consequences similar to those given by Katz for rigid local systems.