Related papers: De Rham logarithmic classes and Tate conjecture

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real…

For a smooth irreducible affine algebraic variety we study a class of gauge modules admitting compatible actions of both the algebra $A$ of functions and the Lie algebra $\mathcal{V}$ of vector fields on the variety. We prove that a gauge…

We write down a new "logarithmic" quasicoherent category $\operatorname{Qcoh}_{log}(U, X, D)$ attached to a smooth open algebraic variety $U$ with toroidal compactification $X$ and boundary divisor $D$. This is a (large) symmetric monoidal…

If $V$ is a smooth projective variety defined over a local field $K$ with finite residue field, so that its \'etale cohomology over the algebraic closure $\bar{K}$ is supported in codimension 1, then the mod $p$ reduction of a projective…

Given a smooth proper dg-algebra $A$, a perfect dg $A$-module $M$, and an endomorphism $f$ of $M$, we define the Hochschild class of the pair $(M,f)$ with values in the Hochschild homology of $A$. Our main result is a Riemann-Roch type…

We prove a big monodromy result for a smooth family of complex algebraic surfaces of general type, with invariants p_g=q=1 and K^2=3, that has been introduced by Catanese and Ciliberto. This is accomplished via a careful study of…

We define a version of a derived chiral De Rham complex over a locally complete intersection, thereby "chiralizing" a result by Illusie and Bhatt. A similar construction attaches to a graded ring a dg vertex algebra, which we prove to be…

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as…

A finite-dimensional algebra $A$ over an algebraically closed field $K$ is called periodic if it is periodic under the action of the syzygy operator in the category of $A-A-$ bimodules. The periodic algebras are self-injective and occur…

The Deligne conjecture (many times a theorem) endows Hochschild cochains of a linear category with the structure of an $E_2$-algebra, that is, of an algebra over the little 2-disks operad. In this paper, we prove the cyclic Deligne…

We show that the $\ell$-adic Tate conjecture for divisors on smooth proper varieties over finitely generated fields of positive characteristic follows from the $\ell$-adic Tate conjecture for divisors on smooth projective surfaces over…

In this paper, we provide an upgrade of Deligne's geometric class field theory for tamely ramified Galois groups using logarithmic geometry. In particular, we define a framed logarithmic Picard space, and show that a logarithmic…

We investigate a strong version of the integral Tate conjecture for 1-cycles on the product of a curve and a surface over a finite field, under the assumption that the surface is geometrically $CH_0$-trivial. By this we mean that over any…

We introduce the "sharp" (universal) extension of a 1-motive (with additive factors and torsion) over a field of characteristic zero. We define the "sharp de Rham realization" by passing to the Lie-algebra. Over the complex numbers we then…

In order to have cohomological operations for de Rham p-adic cohomology with coefficients as manageable as possible, the main purpose of this paper is to solve intrinsically and from a cohomological point of view the lifting problem of…

Along the lines of Hodge and Tate conjectures, Beilinson conjectured that in the qth cohomology all the weight 2q Hodge cycles of a smooth complex variety and all the weight 2q Tate cycles of a smooth variety over a finitely generated field…

The natural logarithm can be represented by an infinite series that converges for all positive real values of the variable, and which makes concavity patently obvious. Concavity of the natural logarithm is known to imply, among other…

We develop a theory of Prym varieties and cubic threefolds over fields of characteristic $2$. As an application, we prove that smooth cubic threefolds are non-rational over an arbitrary field and solve a conjecture of Deligne regarding…

Let $K$ be a field which is complete for a discrete valuation. We prove a logarithmic version of the N\'eron-Ogg-Shafarevich criterion: if $A$ is an abelian variety over $K$ which is cohomologically tame, then $A$ has good reduction in the…

Tame abstract elementary classes are a broad nonelementary framework for model theory that encompasses several examples of interest. In recent years, progress toward developing a classification theory for them have been made. Abstract…