Related papers: Uniform approximation of Betti numbers
In this paper, we prove that the $L^2$ Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, proving a conjecture that we made in an earlier paper. We also prove that an…
At the end of the twentieth century, J.-B. Bost developped a slope theory of hermitian vector bundles over number fields. A new method of diophantine approximation, the so-called slope method, has emerged from his research. Our article…
This book contains notes of a seminar on Ofer Gabber's work on the etale cohomology and uniformization of quasi-excellent schemes. His main results include (cf. introduction) constructibility theorems (for abelian or non-abelian…
We show that a natural isomorphism between the rational cohomology groups of the two zero-dimensional Hilbert schemes of $n$-points of two surfaces, the affine plane minus the axes and the cotangent bundle of an elliptic curve, exchanges…
We define twelve variants of a Reifenberg's affine approximation property, which are known to be connected with the singular sets of minimal surfaces. With this motivation we investigate the regularity of the sets possessing these. We…
Using Saito's theory of mixed Hodge modules, we study a generalization of Hellus-Schenzel's "cohomologically complete intersection" property. This property is equivalent to perversity of the shifted constant sheaf. We relate the generalized…
We describe a new relation between the topology of hypersurface complements, Milnor fibers and degree of gradient mappings. In particular we show that any projective hypersurface has affine parts which are bouquets of spheres. The main…
This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the…
We identify the moduli space of complex affine surfaces with the moduli space of regular meromorphic connections on Riemann surfaces and show that it satisfies a corresponding universal property. As a consequence, we identify the tangent…
Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close…
The Dolbeault resolution of the sheaf of holomorphic vector fields $Lie$ on a complex manifold $M$ relates $Lie$ to a sheaf of differential graded Lie algebras, known as the Fr\"olicher-Nijenhuis algebra $g$. We establish - following B. L.…
The purpose of this paper is to study motivic aspects of the Hitchin system for $\mathrm{GL}_n$. Our results include the following. (a) We prove the motivic decomposition conjecture of Corti-Hanamura for the Hitchin system; in particular,…
We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by…
By relying on a new approach to Lefschetz type questions based on Beilinson's singular support and Saito's characteristic cycle, we prove an instance of the wild Lefschetz theorem envisioned by Deligne. Our main tool are new finiteness…
On the complement $X= {\mathbb C}^2 - \bigcup_{i=1}^n L_i$ to a central plane line arrangement $\bigcup_{i=1}^n L_i \subset {\mathbb C}^2$, a locally constant sheaf of complex vector spaces $\mathcal L_a$ is associated to any multi-index $a…
Given a very ample line bundle on a smooth projective variety, the variation of Hodge structure associated to the universal family of hyperplane sections can be thought of as a $D$-module with action generated by the Gauss-Manin connection.…
We prove a criterion for determining whether the normalization of a complex analytic space on which the constant sheaf is perverse is a rational homology manifold, using a perverse sheaf known as the multiple-point complex. This perverse…
Scheme-theoretic methods are used to classify ternary quadratic forms with values in line bundles over arbitrary schemes and to canonically determine the isomorphisms between them. The association of a quadratic bundle to its even Clifford…
These notes are an account of a series of lectures I gave at the LMS-CMI Research School `Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects', in July 2018, at the Imperial College London. The goal of these notes is to…
We introduce a cohomology theory for a class of projective varieties over a finite field coming from the canonical trace on a C*-algebra attached to the variety. Using the cohomology, we prove the rationality, functional equation and the…