Cohomology Of Sheaves Research Articles

Cohomologie analytique des arrangements d’hyperplans

In this article, we study the cohomology of some analytic sheaves on the complementary in the projective space of a suitable infinite collection of hyperplane like the Drinfel'd symetric space. In particular, the sheaf of invertible functions on these rigid spaces has no cohomology in degree greater or equal to $1$. This proves the vanishing of the Picard goup and the methods used give a convenient description of the global invertible functions.

Reciprocity sheaves and logarithmic motives

We connect two developments that aim to extend Voevodsky's theory of motives over a field in such a way as to encompass non-$\mathbf {A}^1$-invariant phenomena. One is theory of reciprocity sheaves introduced by Kahn, Saito and Yamazaki. The other is theory of the triangulated category $\operatorname {\mathbf {logDM}}^{{\operatorname {eff}}}$ of logarithmic motives launched by Binda, Park and Østvær. We prove that the Nisnevich cohomology of reciprocity sheaves is representable in $\operatorname {\mathbf {logDM}}^{{\operatorname {eff}}}$.

Degree 2 cohomological invariants of linear algebraic groups

The paper deals with the cohomological invariants of smooth and connected linear algebraic groups over an arbitrary field. More precisely, we study degree $2$ invariants with coefficients $\mathbb{Q}/\mathbb{Z}(1)$, that is invariants taking values in the Brauer group. Our main tool is the \'etale cohomology of sheaves on simplicial schemes. We get a description of these invariants for \emph{every} smooth and connected linear groups, in particular for non reductive groups over an imperfect field.

On the cohomology of reciprocity sheaves

AbstractIn this paper, we show the existence of an action of Chow correspondences on the cohomology of reciprocity sheaves. In order to do so, we prove a number of structural results, such as a projective bundle formula, a blow-up formula, a Gysin sequence and the existence of proper pushforward. In this way, we recover and generalise analogous statements for the cohomology of Hodge sheaves and Hodge-Witt sheaves.We give several applications of the general theory to problems which have been classically studied. Among these applications, we construct new birational invariants of smooth projective varieties and obstructions to the existence of zero cycles of degree 1 from the cohomology of reciprocity sheaves.

A note on Lie algebra cohomology

Given a finite dimensional Lie algebra $L$ let $I$ be the augmentation ideal in the universal enveloping algebra $U(L)$. We study the conditions on $L$ under which the $Ext$-groups $Ext (k,k)$ for the trivial $L$-module $k$ are the same when computed in the category of all $U(L)$-modules or in the category of $I$-torsion $U(L)$-modules. We also prove that the Rees algebra $\oplus _{n\geq 0}I^n$ is Noetherian if and only if $L$ is nilpotent. An application to cohomology of equivariant sheaves is given.

Cohomology of Lie Groupoid Modules and the Generalized van Est Map

Abstract The van Est map is a map from Lie groupoid cohomology (with respect to a sheaf taking values in a representation) to Lie algebroid cohomology. We generalize the van Est map to allow for more general sheaves, namely to sheaves of sections taking values in a (smooth or holomorphic) $G$-module, where $G$-modules are structures, which differentiate to representations. Many geometric structures involving Lie groupoids and stacks are classified by the cohomology of sheaves taking values in $G$-modules and not in representations, including $S^1$-groupoid extensions and equivariant gerbes. Examples of such sheaves are $\mathcal{O}^*$ and $\mathcal{O}^*(*D)\,,$ where the latter is the sheaf of invertible meromorphic functions with poles along a divisor $D\,.$ We show that there is an infinitesimal description of $G$-modules and a corresponding Lie algebroid cohomology. We then define a generalized van Est map relating these Lie groupoid and Lie algebroid cohomologies and study its kernel and image. Applications include the integration of several infinitesimal geometric structures, including Lie algebroid extensions, Lie algebroid actions on gerbes, and certain Lie $\infty $-algebroids.

RUANG PROYEKTIF KOMPLEKS 〖CP〗^n ADALAH MANIFOLD KOMPLEKS

<p>The purpose of this paper to determine the complex projective space as a complex manifold is to calculate the cohomology of the coherent sheaves of . The research method in this paper is to construct an -dimensional complex projective space, namely and then the n-dimensional complex projective space, namely , is a complex manifold. The result of this research is the -dimensional complex projective space, namely is a complex and compact manifold.</p>

Displaying the cohomology of toric line bundles

There is a standard approach to calculate the cohomology of torus-invariant sheaves on a toric variety via the simplicial cohomology of the associated subsets of the space of 1-parameter subgroups of the torus. For a line bundle represented by a formal difference of polyhedra in the character space , [1] contains a simpler formula for the cohomology of , replacing by the set-theoretic difference . Here, we provide a short and direct proof of this formula.

Calabi–Yau generalized complete intersections and aspects of cohomology of sheaves

We consider generalized complete intersection manifolds in the product space of projective spaces and work out useful aspects pertaining to the cohomology of sheaves over them. First, we present and prove a vanishing theorem on the cohomology groups of sheaves for subvarieties of the ambient product space of projective spaces. We then prove a birational equivalence between configuration matrices of complete intersection Calabi–Yau manifolds. We also present a formula of the genus of curves in generalized complete intersection manifolds. Some of these curves arise as the fixed point locus of certain symmetry group action on the generalized complete intersection Calabi–Yau manifolds. We also make a blowing-up along curves by which one can generate new Calabi–Yau manifolds. Moreover, an approach on spectral sequences is used to compute Hodge numbers of generalized complete intersection Calabi–Yau manifolds and the genus of curves therein.

Groupe mirabolique, stratification de Newton raffinée et cohomologie des espaces de Lubin-Tate

In my 2009 paper at Inventiones, we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle $\Psi$ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf cohomology of both Harris-Taylor perverse sheaves and those of $\Psi$. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.

On One Property of Bounded Complexes of Discrete $$\mathbb{F}_p[\pi]$$ -modules

The aim of this paper is to prove the following assertion: let π be a profinite group and K* be a bounded complex of discret $$\mathbb{F}_p[\pi]$$ -modules. Suppose that Hi(K*) are finite Abelian groups. Then, there exists a quasi-isomorphism L* → K*, where L* is a bounded complex of discrete $$\mathbb{F}_p[\pi]$$ -modules such that all Li are finite Abelian groups. This is an analog for discrete $$\mathbb{F}_p[\pi]$$ -modules of the wellknown lemma on bounded complexes of A-modules (e.g., concentrated in nonnegative degrees), where A is a Noetherian ring, which states that any such complex is quasi-isomorphic to a complex of finitely generated A-modules, that are free with a possible exception of the module lying in degree 0. This lemma plays a key role in the proof of the base-change theorem for cohomology of coherent sheaves on Noetherian schemes, which, in turn, can be used to prove the Grothendieck theorem on the behavior of dimensions of cohomology groups of a family of vector bundles over a flat family of varieties.

Morse cohomology estimates for jet differential operators

We provide detailed holomorphic Morse estimates for the cohomology of sheaves of jet differentials and their dual sheaves. These estimates apply on arbitrary directed varieties, and a special attention has been given to the analysis of the singular situation. As a consequence, we obtain existence results for global jet differentials and global differential operators under positivity conditions for the canonical or anticanonical sheaf of the directed structure.

Projective superspaces in practice

We study the supergeometry of complex projective superspaces $\mathbb{P}^{n|m}$. First, we provide formulas for the cohomology of invertible sheaves of the form $\mathcal{O}_{\mathbb{P}^{n|m}} (\ell)$, that are pull-back of ordinary invertible sheaves on the reduced variety $\mathbb{P}^n$. Next, by studying the even Picard group $\mbox{Pic}_0 (\mathbb{P}^{n|m})$, classifying invertible sheaves of rank $1|0$, we show that the sheaves $\mathcal{O}_{\mathbb {P}^{n|m}} (\ell)$ are not the only invertible sheaves on $\mathbb{P}^{n|m}$, but there are also new genuinely supersymmetric invertible sheaves that are unipotent elements in the even Picard group. We study the $\Pi$-Picard group $\mbox{Pic}_\Pi (\mathbb{P}^{n|m})$, classifying $\Pi$-invertible sheaves of rank $1|1$, proving that there are also non-split $\Pi$-invertible sheaves on supercurves $\mathbb{P}^{1|m}$. Further, we investigate infinitesimal automorphisms and first order deformations of $\mathbb{P}^{n|m}$, by studying the cohomology of the tangent sheaf using a supersymmetric generalisation of the Euler exact sequence. A special special attention is paid to the meaningful case of supercurves $\mathbb{P}^{1|m}$ and of Calabi-Yau's $\mathbb{P}^{n|n+1}$. Last, with an eye to applications to physics, we show in full detail how to endow $\mathbb{P}^{1|2}$ with the structure of $\mathcal{N}=2$ super Riemann surface and we obtain its SUSY-preserving infinitesimal automorphisms from first principles, that prove to be the Lie superalgebra $\mathfrak{osp} (2|2)$. A particular effort has been devoted to keep the exposition as concrete and explicit as possible.

A geometric construction of colored HOMFLYPT homology

The aim of this paper is two-fold. First, we give a fully geometric description of the HOMFLYPT homology of Khovanov-Rozansky. Our method is to construct this invariant in terms of the cohomology of various sheaves on certain algebraic groups, in the same spirit as the authors' previous work on Soergel bimodules. All the differentials and gradings which appear in the construction of HOMFLYPT homology are given a geometric interpretation. In fact, with only minor modifications, we can extend this construction to give a categorification of the colored HOMFLYPT polynomial, colored HOMFLYPT homology. We show that it is in fact a knot invariant categorifying the colored HOMFLYPT polynomial and that this coincides with the categorification proposed by Mackaay, Stosic and Vaz.

Adams operations and Galois structure

We present a new method for determining the Galois module structure of the cohomology of coherent sheaves on varieties over the integers with a tame action of a finite group. This uses a novel Adams-Riemann-Roch type theorem obtained by combining the Kunneth formula with localization in equivariant K-theory and classical results about cyclotomic fields. As an application, we show two conjectures of Chinburg-Pappas-Taylor, in the case of curves.

Cohomology of exact categories and (non-)additive sheaves

We use (non-)additive sheaves to introduce an (absolute) notion of Hochschild cohomology for exact categories as Ext's in a suitable bisheaf category. We compare our approach to various definitions present in the literature.

On stratified Morse theory: from topology to constructible sheaves

Stratied Morse theory is the generalization of usual Morse theory to functions on stratied spaces. There are versions for the topological type, homotopy type or (co)homology. A standard reference is the book of Goresky-MacPherson which primarily treats the topolog- ical type. Corresponding results about the homotopy type or cohomology may be expected to be consequences but in fact usually one needs some extra information, in particular in the case of cohomology of constructible sheaves, as we will see in this paper.

Reflexive differential forms on singular spaces. Geometry and cohomology

Abstract Based on a recent extension theorem for reflexive differential forms, that is, regular differential forms defined on the smooth locus of a possibly singular variety, we study the geometry and cohomology of sheaves of reflexive differentials. First, we generalise the extension theorem to holomorphic forms on locally algebraic complex spaces. We investigate the (non-)existence of reflexive pluri-differentials on singular rationally connected varieties, using a semistability analysis with respect to movable curve classes. The necessary foundational material concerning this stability notion is developed in an appendix to the paper. Moreover, we prove that Kodaira–Akizuki–Nakano vanishing for sheaves of reflexive differentials holds in certain extreme cases, and that it fails in general. Finally, topological and Hodge-theoretic properties of reflexive differentials are explored.

Formula omitted]-theory of log-schemes II: Log-syntomic [formula omitted]-theory

We prove that suitably truncated topological log-syntomic-étale homotopy K-theory of proper semistable schemes in mixed characteristic surjects onto the Kummer log-étale p-adic K-theory of the generic fiber. As a corollary we get that the log-syntomic-étale cohomology is a direct factor of the log-syntomic-étale cohomology of homotopy K-theory sheaves. The proofs use p-adic Hodge theory computations of p-adic nearby cycles.

The geometry of Markov traces

We give a geometric interpretation of the Jones-Ocneanu trace on the Hecke algebra using the equivariant cohomology of sheaves on SLn. This construction makes sense for all simple groups, so we obtain a generalization of the Jones-Ocneanu trace to Hecke algebras of other types. We give a geometric expansion of this trace in terms of the irreducible characters of the Hecke algebra and conclude that it agrees with a trace defined independently by Gomi. Based on our proof, we also prove that certain simple perverse sheaves on a reductive algebraic group G are equivariantly formal for the conjugation action of a Borel B, or equivalently, that the Hochschild homology of any Soergel bimodule is free, as the authors had previously conjectured. This construction is also closely tied to knot homology. This interpretation of the Jones-Ocneanu trace is a more elementary manifestation of the geometric construction of HOMFLYPT homology given by the authors.