Related papers: Simplicial Abel-Jacobi maps and reciprocity laws
In this article, we study the Gross--Kudla--Schoen diagonal cycle on the triple product of Shimura curves at a place of good reduction and prove an unramified arithmetic level raising theorem for the cohomology of this triple product. We…
We discuss several approaches to motivic complexes and explicit constructions of the regulator maps from the motivic complexes to Deligne complexes.
We show that the image of the Abel-Jacobi map admits functorially a model over the field of definition, with the property that the Abel-Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the…
The goal of this paper is to study non-$\mathbb{A}^1$-invariant motivic cohomology, recently defined by Elmanto, Morrow, and the first-named author, for smooth schemes over possibly non-discrete valuation rings. We establish that the cycle…
We generalize a construction of Barthel-Brasselet-Fieseler-Gabber-Kaup in the setting of complex varieties to the setting of finite type, complex algebraic stacks. Given two such stacks $\mathcal{X},\mathcal{Y}$ with affine stabilizers, and…
A classical result of A. Connes asserts that the Frechet algebra of smooth functions on a smooth compact manifold X provides, by a purely algebraic procedure, the de Rham cohomology of X. Namely the procedure uses Hochschild and cyclic…
De Rham cohomology with spacelike compact and timelike compact supports has recently been noticed to be of importance for understanding the structure of classical and quantum Maxwell theory on curved spacetimes. Similarly causally…
We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex $\chH$ over free associative algebra or path algebra of a quiver. The $\chH$ complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau…
Let $\mathcal{E}(X)$ be the group of homotopy classes of self homotopy equivalences for a connected CW complex $X$. We observe two classes of maps $\mathcal{E}$-maps and co-$\mathcal{E}$-maps. They are defined as the maps $X\to Y$ that…
We develop an extension of the usual theory of formal group laws where the base ring is not required to be commutative and where the formal variables need neither be central nor have to commute with each other. We show that this is the…
Let $\Pi$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $\lambda$ of the field of algebraic numbers which is prime to p, consider the $\lambda$-adic pro-semisimple completion of $\Pi$ as an object…
Our main goal is to give a sense of recent developments in the (stable) rationality problem from the point of view of unramified cohomology and 0-cycles as well as derived categories and semiorthogonal decompositions, and how these…
We construct several pairings in Hopf-cyclic cohomology of (co)module (co)algebras with arbitrary coefficients. The key ideas instrumental in constructing these pairings are the derived functor interpretation of Hopf-cyclic and equivariant…
Starting from a very general trace-form entropy, we introduce a pair of algebraic structures endowed by a generalized sum and a generalized product. These algebras form, respectively, two Abelian fields in the realm of the complex numbers…
The pairings between the cyclic cohomologies and the K-theories of separable $C^\ast$-algebras supply topological invariants that often relate to physical response coefficients of materials. Using three numerical simulations, we exemplify…
This paper presents a proof of reciprocity laws for the Parshin symbol and for two new local symbols, defined here, which we call 4-function local symbols. The reciprocity laws for the Parshin symbol are proven using a new method - via…
This informal note collects key results and open problems on the (co)homology of the Deligne-Mumford moduli spaces of real marked rational curves. The open problems are both of topological nature, aiming to investigate the (co)homology of…
We extend Kato explicit reciprocity law, in the version written by Scholl, for a modular curve to a product of two modular curves. By embedding the product of two modular curves in the Siegel threefold, we deduce an explicit reciprocity law…
We relate Fourier transforms between compactified Jacobians over the moduli space of stable curves to logarithmic Abel-Jacobi theory. As an application, we compute the pushforward of divisor monomials on compactified Jacobians in terms of…
We introduce an extension of the standard cohomology which is characterised by maps that fail to be classical cocycles by products of simpler maps. The construction is motivated by the study of Manin's noncommutative modular symbols and of…