Related papers: Fields of cohomological dimension one versus C_1-f…

We prove that the category of Laumon 1-motives up isogenies over a field of characteristic zero is of cohomological dimension $\le 1$. As a consequence this implies the same result for the category of formal Hodge structures of level $\le…

Based on the celebrated result on zeros of holomorphic 1-forms on complex varieties of general type by Popa and Schnell, we study holomorphic 1-forms on $n$-dimensional varieties of Kodaira dimension $n-1$. We show that a complex minimal…

This article contains a new argument which proves vanishing of the first cohomology for negative vector bundles over a complex projective variety if the rank of the bundle is smaller than the dimension of the base. Similar argument is…

We investigate the first two Galois cohomology groups of $p$-extensions over a base field which does not necessarily contain a primitive $p$th root of unity. We use twisted coefficients in a systematic way. We describe field extensions…

Over a field of characteristic $p>2,$ the first cohomology of the 3-dimensional simple Lie algebra $\frak{sl}(2)$ with coefficients in all simple modules is determined, which implies Whitehead's first lemma is not true in prime…

Differential $p$-forms and $q$-vector fields with constant coefficients are studied. Differential $p$-forms of degrees $p=1,2,n-1,n$ with constant coefficients on a smooth $n$-dimensional manifold $M$ are characterized. In the contravariant…

Let $q$ be a non-negative integer. We prove that a perfect field $K$ has cohomological dimension at most $q+1$ if, and only if, for any finite extension $L$ of $K$ and for any homogeneous space $Z$ under a smooth linear connected algebraic…

We give three examples of fields concerning the Northcott property on elements of small height: The first one has the Northcott property but its Galois closure does not satisfy the Bogomolov property. The second one has the Northcott…

We study various aspects of codimension one defects in free scalar field theory, with particular emphasis on line defects in two-dimensions. These defects are generically non-conformal, but include conformal and topological defects as…

We show that the Heisenberg Lie algebras over a field $\mathbb{F}$ of characteristic $p>0$ admit a family of restricted Lie algebras, and we classify all such non-isomorphic restricted Lie algebra structures. We use the ordinary 1- and…

A way to characterize the space of leaves of a foliation in terms of connections is proposed. A particular example of vertex algebra cohomology of codimension one foliations on complex curves is considered.

We give some general properties of good and bad reduction, and some recent examples (worked out with Dipendra Prasad) of varieties having bad reduction not accounted for by their cohomology. We include some consequences of our remarks for…

For a field $\mathbb{F}$, let $L_k(\mathbb{F})$ be the Lie algebra of derivations $f(t)\frac{d}{dt}$ of the polynomial ring $\mathbb{F}[t]$, where $f(t)$ is a polynomial of degree $\geqslant k$. For any $k\geqslant -1$, we present a basis…

We prove a version of Ihara's Lemma for degree q=1,2 cuspidal cohomology of the symmetric space attached to automorphic forms of arbitrary weight (k\geq 2) over an imaginary quadratic field with torsion (prime power) coefficients. This…

The reduction of singularities of codimension one foliations is known in the case of ambient dimension 2 (Seidenberg, A. (1968). Reduction of singularities of the differential equation Ady= Bdx. American Journal of Mathematics, 90(1),…

Critical collapse of a self-gravitating scalar field in a (2+1)-dimensional spacetime with negative cosmological constant seems to be dominated by a continuously self-similar solution of the field equations without cosmological constant.…

For slowly varying fields the vacuum functional of a quantum field theory may be expanded in terms of local functionals. This expansion satisfies its own form of the Schr\"odinger equation from which the expansion coefficents can be found.…

We introduce relative preresolving subcategories and precoresolving subcategories of an abelian category and define homological dimensions and codimensions relative to these subcategories respectively. We study the properties of these…

Given a prime number $p$, a field $F$ with $\operatorname{char}(F)=p$ and a positive integer $n$, we study the class-preserving modifications of Kato-Milne classes of decomposable differential forms. These modifications demonstrate a…

If the $\ell$-adic cohomology of a projective smooth variety, defined over a $\frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $\ge 1$, then any model over the ring of integers of $K$ has a $k$-rational…