Cohomology Of Complex Research Articles

On the Mixed Hodge Structures of the Intersection Cohomology Stalks of Complex Hypersurfaces

We consider a hypersurface in \mathbb C^n with an isolated singular point at the origin, and study the mixed Hodge structure of the stalk of its intersection cohomology complex at the origin. In particular, we express the dimension of each graded piece of the weight filtration of this mixed Hodge structure in terms of the numbers of Jordan blocks in the Milnor monodromy.

Explicit decomposition theorem for special Schubert varieties

Abstract We provide an explicit canonical description of the perverse cohomology sheaves and of the primitive perverse cohomology complexes for the non-small resolution π : 𝒮 ~ → 𝒮 {\pi:\tilde{\mathcal{S}}\to\mathcal{S}} of a special Schubert variety 𝒮 {\mathcal{S}} . For such a resolution, we also discuss a way to obtain an explicit splitting of R ⁢ π ⁢ ℚ 𝒮 ~ * {R\pi{{}_{*}}\mathbb{Q}_{\widetilde{{\mathcal{S}}}}} , in the derived category, by means of Gysin morphisms and cohomology extensions.

The BV formalism: Theory and application to a matrix model

We review the BV formalism in the context of [Formula: see text]-dimensional gauge theories. For a gauge theory [Formula: see text] with an affine configuration space [Formula: see text], we describe an algorithm to construct a corresponding extended theory [Formula: see text], obtained by introducing ghost and anti-ghost fields, with [Formula: see text] a solution of the classical master equation in [Formula: see text]. This construction is the first step to define the (gauge-fixed) BRST cohomology complex associated to [Formula: see text], which encodes many interesting information on the initial gauge theory [Formula: see text]. The second part of this article is devoted to the application of this method to a matrix model endowed with a [Formula: see text]-gauge symmetry, explicitly determining the corresponding [Formula: see text] and the general solution [Formula: see text] of the classical master equation for the model.

Gorenstein cohomology of N-complexes

Let [Formula: see text] and [Formula: see text] be [Formula: see text]-complexes with [Formula: see text] an integer such that [Formula: see text] has finite Gorenstein projective dimension and [Formula: see text] has finite Gorenstein injective dimension. We define the [Formula: see text]th Gorenstein cohomology groups [Formula: see text] [Formula: see text] via a strict Gorenstein precover [Formula: see text] of [Formula: see text] and a strict Gorenstein preenvelope [Formula: see text] of [Formula: see text]. Using Gaussian binomial coefficients we show that there exists an isomorphism [Formula: see text] which extends the balance result of Liu [Relative cohomology of complexes. J. Algebra 502 (2018) 79–97] to the [Formula: see text]-complex case.

An operadic approach to vertex algebra and Poisson vertex algebra cohomology

We translate the construction of the chiral operad by Beilinson and Drinfeld to the purely algebraic language of vertex algebras. Consequently, the general construction of a cohomology complex associated to a linear operad produces the vertex algebra cohomology complex. Likewise, the associated graded of the chiral operad leads to the classical operad, which produces a Poisson vertex algebra cohomology complex. The latter is closely related to the variational Poisson cohomology studied by two of the authors.

Relative cohomology of complexes based on cotorsion pairs

Let [Formula: see text] be an associative ring with identity. The purpose of this paper is to establish relative cohomology theories based on cotorsion pairs in the setting of unbounded complexes of modules over [Formula: see text]. Let [Formula: see text] be a complete hereditary cotorsion pair in [Formula: see text]-Mod. Then [Formula: see text] and [Formula: see text] are complete hereditary cotorsion pairs in the category of [Formula: see text]-complexes. For any complexes [Formula: see text] and [Formula: see text] and any [Formula: see text], we define the [Formula: see text]th relative cohomology groups [Formula: see text] and [Formula: see text] by special [Formula: see text]-precovers of [Formula: see text] and by special [Formula: see text]-preenvelopes of [Formula: see text], respectively. They are common generalizations of absolute cohomology groups and Gorenstein cohomology groups of complexes. Some induced exact sequences concerning relative cohomology groups are considered. It is also shown that the relative cohomology functor of complexes we considered is balanced.

Balance for relative cohomology of complexes

Let [Formula: see text] be an arbitrary ring. We use a strict [Formula: see text]-resolution [Formula: see text] of a complex [Formula: see text] with finite [Formula: see text]-projective dimension, where [Formula: see text] denotes a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits an injective cogenerator [Formula: see text], to define the [Formula: see text]th relative cohomology functor [Formula: see text] as [Formula: see text]. If a complex [Formula: see text] has finite [Formula: see text]-injective dimension, then one can use a dual argument to define a notion of a relative cohomology functor [Formula: see text], where [Formula: see text] is a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits a projective generator. Under several orthogonal conditions, we show that there exists an isomorphism [Formula: see text] of relative cohomology groups for each [Formula: see text]. This result simultaneously encompasses a balance result of Holm on Gorenstein projective and injective modules, a balance result of Sather-Wagstaff, Sharif and White on Gorenstein projective and injective modules with respect to semidualizing modules, and a balance result of Liu on Gorenstein projective and injective complexes. In particular, as an application of this result, we extend the above balance result of Sather-Wagstaff, Sharif and White to the setting of complexes.

Note on weight-monodromy conjecture for $p$-adically uniformized varieties

We prove the weight-monodromy conjecture for varieties which are p-adically uniformized by a product of the Drinfeld upper half spaces. It is an easy consequence of Dat’s work on the cohomology complex of the Drinfeld upper half space.

Parallels between Moduli of Quiver Representations and Vector Bundles over Curves

This is a review article exploring similarities between moduli of quiver representations and moduli of vector bundles over a smooth projective curve. After describing the basic properties of these moduli problems and constructions of their moduli spaces via geometric invariant theory and symplectic reduction, we introduce their hyperk\"ahler analogues: moduli spaces of representations of a doubled quiver satisfying certain relations imposed by a moment map and moduli spaces of Higgs bundles. Finally, we survey a surprising link between the counts of absolutely indecomposable objects over finite fields and the Betti cohomology of these (complex) hyperk\"ahler moduli spaces due to work of Crawley-Boevey and Van den Bergh and Hausel, Letellier and Rodriguez-Villegas in the quiver setting, and work of Schiffmann in the bundle setting.

Topological comparison theorems for Bredon motivic cohomology

We prove equivariant versions of the Beilinson-Lichtenbaum conjecture for Bredon motivic cohomology of smooth complex and real varieties with an action of the group of order two. This identifies equivariant motivic and topological invariants in a large range of degrees.

Supersymmetry and cohomology of graph complexes

This is preprint HAL-00429963 (2009). I describe a combinatorial construction of the cohomology classes in compactified moduli spaces of curves $\widehat{Z}_{I}\in H^{*}(\bar{\mathcal{M}}_{g,n})$ starting from the following data: an odd derivation $I$, whose square is non-zero in general, $I^{2}\neq 0$, acting on a $\mathbb{Z}/2\mathbb{Z}$-graded associative algebra with odd scalar product. The constructed cocycles were first described in the theorem 2 in the author's paper "Noncommmutative Batalin-Vilkovisky geometry and Matrix integrals". Comptes Rendus Mathematique, 348, pp. 359-362, arXiv:0912.5484 , preprint HAL-00102085 (09/2006). By the theorem 3 from loc.cit. the family of the cohomology classes obtained in the case of the algebra $Q(N)$ and the derivation $I=\left[\Lambda,\cdot\right]$ coincided with the generating function of products of $\psi-$classes. This was the first nontrivial computation of categorical Gromov-Witten invariants of higher genus. The result matched with the mirror symmetry prediction, i.e. with the classical (non-categorical) Gromov-Witten descendent invariants of a point for all genus. As a byproduct of that computation a new combinatorial formula for products of $\psi$-classes $\psi_{i}=c_{1}(T_{p_{i}}^{*})$ in the cohomology $H^{*}(\bar{\mathcal{M}}_{g,n})$ is written out.

Integrable 3D Statistical Models on Six-Valent Graphs

The paper is devoted to the study of a special statistical model on graphs with vertices of degrees 6 and 1. We show that this model is invariant with respect to certain Roseman moves if one regards the graph as the singular point set of the diagram of a 2-knot. Our approach is based on the properties of the tetrahedron cohomology complex.

(0,2) versions of exotic (2,2) GLSMs

In this paper we extend work on exotic two-dimensional (2,[Formula: see text]2) supersymmetric gauged linear sigma models (GLSMs) in which, for example, geometries arise via nonperturbative effects, to (0,[Formula: see text]2) theories, and in so doing find some novel (0,[Formula: see text]2) GLSM phenomena. For one example, we describe examples in which bundles are constructed physically as cohomologies of short complexes involving torsion sheaves, a novel effect not previously seen in (0,[Formula: see text]2) GLSMs. We also describe examples related by RG flow in which the physical realizations of the bundles are related by quasi-isomorphism, analogous to the physical realization of quasi-isomorphisms in D-branes and derived categories, but novel in (0,[Formula: see text]2) GLSMs. Finally, we also discuss (0,[Formula: see text]2) deformations in various duality frames of other examples.

Tate Cohomology for Complexes with Finite Gorenstein AC-Injective Dimension

In this paper, we introduce and study a notion of Gorenstein AC-injective dimension for complexes of left modules over associative rings. We show first that the class of complexes with finite Gorenstein AC-injective dimension is exactly the class of complexes admitting a complete $$\mathcal {AC}$$ -coresolution. Then the interaction between the corresponding relative and Tate cohomologies of complexes is given. Finally, the relationships between Gorenstein AC-injective dimensions and injective dimensions for complexes are given.

Cohomology and deformations of n-Lie algebra morphisms

The study of $n$-Lie algebras which are natural generalization of Lie algebras is motivated by Nambu Mechanics and recent developments in String Theory and M-branes. The purpose of this paper is to define cohomology complexes and study deformation theory of $n$-Lie algebra morphisms. We discuss infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide various examples.

Existence and uniqueness of weak homotopy moment maps

We show that the classical results on the existence and uniqueness of moment maps in symplectic geometry generalize directly to weak homotopy moment maps in multisymplectic geometry. In particular, we show that their existence and uniqueness is governed by a Lie algebra cohomology complex which reduces to the Chevalley–Eilenberg complex in the symplectic setup.

Hom-Lie-Rinehart algebras

ABSTRACTWe introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterization of low dimensional cohomology spaces in terms of the group of automorphisms of certain abelian extension and the equivalence classes of those abelian extensions in the category of hom-Lie-Rinehart algebras, respectively. We also construct a canonical example of hom-Lie-Rinehart algebra associated to a given Poisson algebra and an automorphism.

Relative cohomology of complexes

Let R be an arbitrary ring and C a complex with finite Gorenstein projective dimension (that is, the supremum of Gorenstein projective dimension of all R-modules in C is finite). For each complex D, we define the nth relative cohomology group ExtGPn(C,D) by the equality ExtGPn(C,D)=HnHom(G,D), where G⟶C is a strict Gorenstein projective precover of C. If D is a complex with finite Gorenstein injective dimension (that is, the supremum of Gorenstein injective dimension of all R-modules in D is finite), then one can use a dual argument to define a notion of relative cohomology group ExtGIn(C,D). We show that if C is a complex with finite Gorenstein projective dimension and D a complex with finite Gorenstein injective dimension, then for each n∈Z there exists an isomorphism ExtGPn(C,D)≅ExtGIn(C,D). This shows that the relative cohomology functor of complexes is balanced. Some induced exact sequences concerning relative cohomology groups are considered.

Weil-Étale Cohomology and Zeta-Values of Proper Regular Arithmetic Schemes

We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme \mathcal{X} at any integer n in terms of Weil-étale cohomology complexes. This extends work of S. Lichtenbaum [Compos. Math. 141, No. 3, 689–702 (2005; Zbl 1073.14024)] and T. Geisser [Math. Ann. 330, No. 4, 665–692 (2004; Zbl 1069.14021)] for \mathcal{X} of characteristic p , of S. Lichtenbaum [Ann. Math. (2) 170, No. 2, 657–683 (2009; Zbl 1278.14029)] for \mathcal{X}=\mathrm{Spec}(\mathcal{O}_F) and n=0 where F is a number field, and of the second author for arbitrary \mathcal{X} and n=0 [B. Morin, Duke Math. J. 163, No. 7, 1263–1336 (2014; Zbl 06303878)]. We show that our conjecture is compatible with the Tamagawa number conjecture of S. Bloch and K. Kato [Prog. Math. 86, 333–400 (1990; Zbl 0768.14001)], and J.-M. Fontaine and B. Perrin-Riou [Proc. Symp. Pure Math. 55, 599–706 (1994; Zbl 0821.14013)] if \mathcal{X} is smooth over \mathrm{Spec}(\mathcal{O}_F) , and hence that it holds in cases where the Tamagawa number conjecture is known.

Morphisms Cohomology and Deformations of Hom-algebras

The purpose of this paper is to define cohomology complexes and study deformation theory of Hom-associative algebra morphisms and Hom-Lie algebra morphisms. We discuss infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide various examples.