Related papers: Singular varieties and infinitesimal non-commutati…

We present a framework that connects three interesting classes of groups: the twisted groups (also known as Suzuki-Ree groups), the mixed groups and the exotic pseudo-reductive groups. For a given characteristic p, we construct categories…

In this note several computations of equivariant cohomology groups are performed. For the compactly supported equivariant cohomology, the notion of infinitesimal index developed in arXiv:1003.3525, allows to describe these groups in terms…

The purpose of this largely expository note is to give some examples of smooth projective varieties with zero first Betti number but nontrivial families of local systems of higher ranks. The most interesting examples are due to Simpson.

Let $X$ be an algebraic variety over the field of real numbers $\mathbb{R}$. We use the signature of a quadratic form to produce "higher" global signatures relating the derived Witt groups of $X$ to the singular cohomology of the real…

Given an algebraic surface $X$, the Hilbert scheme $X^{[n]}$ of $n$-points on $X$ admits a contraction morphism to the $n$-fold symmetric product $X^{(n)}$ with the extremal ray generated by a class $\beta_n$ of a rational curve. We…

The vertex PI index $PI(G) = \sum_{xy \in E(G)} [n_{xy}(x) + n_{xy}(y)]$ is a distance-based molecular structure descriptor, where $n_{xy}(x)$ denotes the number of vertices which are closer to the vertex $x$ than to the vertex $y$ and…

Let $X$ and $Y$ be irreducible normal projective varieties, of same dimension, defined over an algebraically closed field, and let $f : Y \rightarrow X$ be a finite generically smooth morphism such that the corresponding homomorphism…

Let R be a semi-local regular domain containing an infinite perfect field k, and let K be the field of fractions of R. Let G be a reductive semi-simple simply connected R-group scheme such that each of its R-indecomposable factors is…

A theorem of G\"ottsche establishes a connection between cohomological invariants of a complex projective surface $S$ and corresponding invariants of the Hilbert scheme of $n$ points on $S.$ This relationship is encoded in certain infinite…

This article studies an extended Nori and local fundamental group schemes of Abelian varieties. We also discuss the birational invariance of these group schemes and study their behaviour under the Albanese and \'{e}tale morphisms.

We state results from noncommutative deformation theory of modules over an associative $k$-algebra $A,$ $k$ a field, necessary for this work. We define a set of $A$-modules $\operatorname{aSpec}A$ containing the simple modules, whose…

We establish a variant of the splitting principle of Garibaldi-Merkurjev-Serre for invariants taking values in a strictly homotopy invariant sheaf. As an application, we prove the folklore result of Morel that pi_0 of the motivic…

Let $X/\mathbb{F}_{q}$ be a smooth geometrically connected variety. Inspired by work of Corlette-Simpson over $\mathbb{C}$, we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on $X$ come from…

We study an analytically irreducible algebroid germ (X, 0) of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the divisorial valuations associated to the irreducible…

The Matsushita fundamental groups of a graph $X$, denoted $\pi_1^r(X)$, are certain discrete versions of the fundamental group for topological spaces. For $r=2$, these groups have a nice combinatorial description, due to Sankar. In this…

All spaces below are Tychonov. We define the projective pi-character p(X) of a space X as the supremum of the values $\pi\chi(Y)$ where Y ranges over all continuous images of X. Our main result says that every space X has a pi-base whose…

Let $X$ be a smooth projective real algebraic variety. We give new positive and negative results on the problem of approximating a submanifold of the real locus of $X$ by real loci of subvarieties of $X$, as well as on the problem of…

We show that Witt groups of spinor varieties (aka.\ maximal isotropic Grassmannians) can be presented by combinatorial objects called even shifted young diagram. Our method relies on the Blow-up setup of Balmer-Calm\`es, and we investigate…

In a previous paper we have classified the smooth projective symmetric G-varieties with Picard number one (and G semisimple). In this work we give a geometrical description of such varieties. In particular, we determine their group of…

We consider the ring of invariants of n points on the projective line. The space (P^1)^n // PGL_2 is perhaps the first nontrivial example of a Geometry Invariant Theory quotient. The construction depends on the weighting of the n points.…