Terms Of Cohomology Research Articles

Differential contracting homotopy in higher-spin theory

A new efficient approach to the analysis of nonlinear higher-spin equations, that treats democratically auxiliary spinor variables ZA and integration homotopy parameters in the non-linear vertices of the higher-spin theory, is developed. Being most general, the proposed approach is the same time far simpler than those available so far. In particular, it is free from the necessity to use the Schouten identity. Remarkably, the problem of reconstruction of higher-spin vertices is mapped to certain polyhedra cohomology in terms of homotopy parameters themselves. The new scheme provides a powerful tool for the study of higher-order corrections in higher-spin theory and, in particular, its spin-locality. It is illustrated by the analysis of the lower order vertices, reproducing not only the results obtained previously by the shifted homotopy approach but also projectively-compact vertices with the minimal number of derivatives, that were so far unreachable within that scheme.

The construction and deformation of Novikov H-pseudoalgebras

In this paper, we are devoted to constructing a large number of Novikov H-pseudoalgebras from commutative associative (H-pseudo)algebras with (H-)derivations, infinitesimal H-pseudobialgebras with H-derivations and Novikov (H-pseudo)algebras with Rota-Baxter (H-)operators. Meanwhile, the relationship between Novikov H-pseudoalgebras and Lie H-pseudoalgebras is established. Furthermore, we develop the 1-parameter formal deformation theory of Novikov H-pseudoalgebras and characterize it in terms of cohomology.

K$K$‐Motives and Koszul duality

We construct an ungraded version of Beilinson–Ginzburg–Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of K $K$ -theory. For this, we introduce and study categories DK S ( X ) $\hbox{DK}_\mathcal {S}(X)$ of S $\mathcal {S}$ -constructible K $K$ -motivic sheaves on varieties X $X$ with an affine stratification. We show that there is a natural and geometric functor, called Beilinson realisation, from S $\mathcal {S}$ -constructible mixed sheaves D S m i x ( X ) $\hbox{D}^{mix}_\mathcal {S}(X)$ to DK S ( X ) $\hbox{DK}_\mathcal {S}(X)$ . We then show that Koszul duality intertwines the Betti realisation and Beilinson realisation functors and descends to an equivalence of constructible sheaves and constructible K $K$ -motivic sheaves on Langlands dual flag varieties.

On the computation of Hopf 2-cocycles with an example of diagonal type

AbstractWe present a framework for the computation of the Hopf 2-cocycles involved in the deformations of Nichols algebras over semisimple Hopf algebras. We write down a recurrence formula and investigate the extent of the connection with invariant Hochschild cohomology in terms of exponentials. As an example, we present detailed computations leading to the explicit description of the Hopf 2-cocycles involved in the deformations of a Nichols algebra of Cartan type $A_2$ with $q=-1$ , a.k.a. the positive part of the small quantum group $\mathfrak{u}^+_{\sqrt{-\text{1}}}(\mathfrak{sl}_3)$ . We show that these cocycles are generically pure, that is they are not cohomologous to exponentials of Hochschild 2-cocycles.

On the variational principle in the unfolded dynamics

The interplay between off-shell and on-shell unfolded systems is analyzed. The formulation of invariant constraints that put an off-shell system on shell is developed by adding new variables and derivation in the target space, that extends the original Q-derivation of the unfolded system to a bicomplex. The analogue of the Euler-Lagrange equations in the unfolded dynamics is suggested. The general class of invariant on-shell equation constraints is defined in cohomological terms. The necessary and sufficient condition for the on-shell equation constraints being Euler-Lagrange for some Lagrangian system is proven. The proposed construction is illustrated by the scalar field example.

Cohomologie des courbes analytiques $p$-adiques

Cohomology of affinoids does not behave well; often, this can be remedied by making affinoids overconvergent. In this paper, we focus on dimension 1 and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that cohomology of affinoids (in dimension 1) is not that pathological. From this we deduce a computation of cohomologies of curves without boundary (like the Drinfeld half-plane and its coverings). In particular, we obtain a description of their $p$-adic pro-etale cohomology in terms of de the Rham complex and the Hyodo-Kato cohomology, the later having properties similar to the ones of $\ell$-adic pro-etale cohomology, for $\ell\neq p$.

Abstract intersection theory for zeta-functions: geometric aspects

We give a further elaboration of our previous works on abstract intersection theory for zeta-functions, emphasizing geometric aspects. We review and streamline Weil's proof of the Riemann hypothesis for curves over finite fields and then present it in cohomological terms. Then by using a spectral interpretation of the Riemann zeta-function we construct a model of our abstract intersection theory for the Riemann zeta-function as an analogue of the cohomological version of Weil's classical intersection theory.

Vassiliev invariants for flows via Chern–Simons perturbation theory

The perturbative expansion of Chern–Simons gauge theory leads to invariants of knots and links, the so-called finite type invariants or Vassiliev invariants. It has been proved that at any order in perturbation theory the superposition of certain amplitudes is an invariant of that order. Bott–Taubes integrals on configuration spaces are introduced in the present context to write Feynman diagrams at a given order in perturbation theory in a geometrical and topological framework. One of the consequences of this formalism is that the resulting amplitudes are rewritten in cohomological terms in configuration spaces. This cohomological structure can be used to translate Bott–Taubes integrals into Chern–Simons perturbative amplitudes and vice versa. In this paper, this program is performed up to third order in the coupling constant. This expands some work previously worked out by Thurston. Finally we take advantage of these results to incorporate in the formalism a smooth and divergenceless vector field on the [Formula: see text]-manifold. The Bott–Taubes integrals obtained are used for constructing higher-order average asymptotic Vassiliev invariants extending the work of Komendarczyk and Volić.

Lie 2-algebra moment maps in multisymplectic geometry

Consider a closed non-degenerate 3-form ω with an infinitesimal action of a Lie algebra g. Motivated by the fact that the observables associated to ω form a Lie 2-algebra, we introduce homotopy moment maps defined on a Lie 2-algebra rather than just on the Lie algebra g.We formulate existence criteria and provide a construction for such homotopy moment maps, by characterizing them in terms of cohomology.

Index Formulae for Line Bundle Cohomology on Complex Surfaces

AbstractWe conjecture and prove closed‐form index expressions for the cohomology dimensions of line bundles on del Pezzo and Hirzebruch surfaces. Further, for all compact toric surfaces we provide a simple algorithm which allows expression of any line bundle cohomology in terms of an index. These formulae follow from general theorems we prove for a wider class of surfaces. In particular, we construct a map that takes any effective line bundle to a nef line bundle while preserving the zeroth cohomology dimension. For complex surfaces, these results explain the appearance of piecewise polynomial equations for cohomology and they are a first step towards understanding similar formulae recently obtained for Calabi‐Yau three‐folds.

Immaculate line bundles on toric varieties

We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional sequences, investigating the diagonal property, or the toric Frobenius morphism. In the present paper we focus on line bundles on toric varieties. First, we present a possibility of understanding their cohomology in terms of their (generalized) momentum polytopes. Then we present a method to exhibit the entire locus of immaculate divisors within the class group. This will be applied to the cases of smooth toric varieties of Picard rank two and three and to those being given by splitting fans. The locus of immaculate line bundles contains several linear strata of varying dimensions. We introduce a notion of relative immaculacy with respect to certain contraction morphisms. This notion will be stronger than plain immaculacy and provides an explanation of some of these linear strata.

Characterizing finite length local cohomology in terms of bounds on Koszul cohomology

Let (R,m,κ) be a local ring. We give a characterization of R-modules M whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. In particular, we show that a quasi-unmixed module M is asymptotically Cohen-Macaulay if and only if M is Cohen-Macaulay on the punctured spectrum if and only if sup⁡{ℓ(Hi(f1,…,fd;M))|f1,…,fd=m, i<d}<∞ for d=dim⁡(M)=dim⁡(R).

Crossed Products and Coding Theory

Families of codes, such as group codes, constacyclic, and skew cyclic codes, which were independently suggested, turn out to be special instances of the general family of crossed product codes. Under this framework, the ambient code spaces are classified here with respect to Hamming-isometry in cohomological terms. This classification is demonstrated by two families of examples. We also determine when crossed products belonging to these families are semisimple and, in particular, when they admit only trivial codes.

Relative Čech–Dolbeault homology and applications

We define the relative Dolbeault homology of a complex manifold with currents via a Cech approach, and we prove its equivalence with the relative Cech–Dolbeault cohomology as defined by Suwa (in: Singularities–Niigata–Toyama 2007. Advanced studies in pure mathematics, vol 56. Mathematical Society of Japan, Tokyo, pp 321–340, 2009). This definition is then used to compare the relative Dolbeault cohomology groups of two complex manifolds of the same dimension related by a suitable proper surjective holomorphic map. Finally, an application to blow-ups is considered and a blow-up formula for the Dolbeault cohomology in terms of relative cohomology is presented.

The Poincaré-Shannon Machine: Statistical Physics and Machine Learning Aspects of Information Cohomology

Previous works established that entropy is characterized uniquely as the first cohomology class in a topos and described some of its applications to the unsupervised classification of gene expression modules or cell types. These studies raised important questions regarding the statistical meaning of the resulting cohomology of information and its interpretation or consequences with respect to usual data analysis and statistical physics. This paper aims to present the computational methods of information cohomology and to propose its interpretations in terms of statistical physics and machine learning. In order to further underline the cohomological nature of information functions and chain rules, the computation of the cohomology in low degrees is detailed to show more directly that the k multivariate mutual information () are -coboundaries. The -cocycles condition corresponds to , which generalizes statistical independence to arbitrary degree k. Hence, the cohomology can be interpreted as quantifying the statistical dependences and the obstruction to factorization. I develop the computationally tractable subcase of simplicial information cohomology represented by entropy and information landscapes and their respective paths, allowing investigation of Shannon’s information in the multivariate case without the assumptions of independence or of identically distributed variables. I give an interpretation of this cohomology in terms of phase transitions in a model of k-body interactions, holding both for statistical physics without mean field approximations and for data points. The components define a self-internal energy functional and components define the contribution to a free energy functional (the total correlation) of the k-body interactions. A basic mean field model is developed and computed on genetic data reproducing usual free energy landscapes with phase transition, sustaining the analogy of clustering with condensation. The set of information paths in simplicial structures is in bijection with the symmetric group and random processes, providing a trivial topological expression of the second law of thermodynamics. The local minima of free energy, related to conditional information negativity and conditional independence, characterize a minimum free energy complex. This complex formalizes the minimum free-energy principle in topology, provides a definition of a complex system and characterizes a multiplicity of local minima that quantifies the diversity observed in biology. I give an interpretation of this complex in terms of unsupervised deep learning where the neural network architecture is given by the chain complex and conclude by discussing future supervised applications.

ON THE HUMPHREYS CONJECTURE ON SUPPORT VARIETIES OF TILTING MODULES

Let G be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic p, assumed to be larger than the Coxeter number. The of a G-module M is a certain closed subvariety of the nilpotent cone of G, defined in terms of cohomology for the first Frobenius kernel of G. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for G=SLn in earlier work of the second author. In this paper, we show that for any G, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when p is sufficiently large. We also prove variants of these statements involving relative support varieties.

Euler characteristics and $q$-difference equations

This paper is concerned with linear q-difference equations. Our main result is an explicit formula for the Euler characteristic of the sheaf of analytic solutions attached to any linear algebraic q-difference equation. This formula involves certain invariants attached to the so-called intermediate singularities. As an application, we interpret the index of rigidity recently introduced by Sakai and Yamaguchi in cohomological terms.

Tracing cyclic homology pairings under twisting of graded algebras

We give a description of cyclic cohomology and its pairing with K-groups for 2-cocycle deformation of algebras graded over discrete groups. The proof relies on a realization of monodromy for the Gauss–Manin connection on periodic cyclic cohomology in terms of the cup product action of group cohomology.

De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds

Oeljeklaus-Toma (OT) manifolds are complex non-Kahler manifolds whose construction arises from specific number fields. In this note, we compute their de Rham cohomology in terms of invariants associated to the background number field. This is done by two distinct approaches, one using invariant cohomology and the other one using the Leray-Serre spectral sequence. In addition, we compute also their Morse-Novikov cohomology. As an application, we show that the low degree Chern classes of any complex vector bundle on an OT manifold vanish in the real cohomology. Other applications concern the OT manifolds admitting locally conformally Kahler (LCK) metrics: we show that there is only one possible Lee class of an LCK metric, and we determine all the possible Morse-Novikov classes of an LCK metric, which implies the nondegeneracy of certain Lefschetz maps in cohomology.

On non-Kähler degrees of complex manifolds

Abstract We study cohomological properties of complex manifolds. In particular, under suitable metric conditions, we extend to higher dimensions a result by A. Teleman, which provides an upper bound for the Bott– Chern cohomology in terms of Betti numbers for compact complex surfaces according to the dichotomy b 1 even or odd.