Degree Cohomology Research Articles

Unmixed torsions and Hilbert coefficients of d-sequences

In this paper, we define an unmixed torsion associated with a certain cohomological degree. We establish a connection between the Hilbert coefficients and the unmixed torsions of the module with respect to a parameter ideal generated by a d-sequence. As an application, we characterize the vanishing of the Hilbert coefficients through the depth of the module.

On the v-Picard Group of Stein Spaces

Abstract We study the image of the Hodge–Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of closed differential forms. We show that it indeed always contains such forms but the quotient can be non-trivial: it contains a slightly mysterious $\mathbf{Z}_{p}$-module that maps, via the Bloch–Kato exponential map, to integral classes in the pro-étale cohomology. This quotient is already non-trivial for open unit disks of dimension strictly greater than $1$.

Mixed Hodge structures and Siegel operators

Abstract In this paper, we study mixed Hodge structures on the cohomology of locally symmetric varieties and give an application to modular forms. After proving vanishing of some Hodge numbers, we focus on the weight filtration on the last Hodge subspace of the middle degree cohomology. We prove that the weight filtration coincides with the corank filtration on the space of modular forms of canonical weight defined by the Siegel operators, and calculate the graded quotients. As an application, we deduce surjectivity of the total Siegel operators in many cases, and identify an obstruction space in the remaining case.

Formal deformations, cohomology theory and L∞[1]-structures for differential Lie algebras of arbitrary weight

We introduced a cohomology theory for differential Lie algebras of arbitrary weight which generalised a previous work of Jiang and Sheng. The underlying L∞[1]-structure on the cochain complex is also determined via a generalised version of higher derived brackets. The equivalence between L∞[1]-structures for absolute and relative differential Lie algebras is established. Formal deformations and abelian extensions are interpreted by using lower degree cohomology groups. Also we introduce the homotopy differential Lie algebras.

On deformations of Rota–Baxter algebra morphisms

Rota–Baxter algebras are important in probability, combinatorics, associative Yang–Baxter equation and splitting of algebras. This paper studies the formal deformations of Rota–Baxter algebra morphisms. As a consequence, we develop a cohomology theory of Rota–Baxter algebra morphisms to interpret the lower degree cohomology groups as formal deformations. Finally, we prove the cohomology comparison theorem of Rota–Baxter algebra morphisms, i.e. the cohomology of a morphism of Rota–Baxter algebras is isomorphic to the cohomology of an auxiliary Rota–Baxter algebra.

Rota–Baxter family Ω-associative conformal algebras and their cohomology theory

In this paper, we first propose the concept of Rota–Baxter family Ω-associative conformal algebras, then we study the cohomology theory of Rota–Baxter family Ω-associative conformal algebras of any weight and justify it by interpreting the lower degree cohomology groups as formal deformations.

Floyd’s manifold is a conjugation space

We prove that there is an action of the cyclic group $\mathbf{C}_2$ on the $10$-dimensional Floyd manifold which turns it into a conjugation manifold. The submanifold of fixed points is the $5$-dimensional Floyd manifold, whose cohomology is isomorphic to that of the large one, scaled down by dividing the cohomological degree by a factor two.

Stable cohomology of the universal degree d hypersurface in ℙn

Stable cohomology of the universal degree d hypersurface in ℙn

On the base locus of linear systems of general double points

Fix integers n ≥ 1, d ≥ 4 and x>0 such that (n+1)(x-1) +Bin(n+2, 2) ≤ Bin(n+d, n). Take a general S ⊂ Pn such that #S=x and let B denote the scheme-theoretic base locus of |I2s(d)|, where 2S is the union of the double points with S as their reduction. Then 2S is the union of the connected components of B containing at least one point of S. We prove this theorem proving that a general union of x-1 double points and one triple point has no higher cohomology in degree d.

Mixed Hodge Structure on Local Cohomology with Support in Determinantal Varieties

Abstract We employ the inductive structure of determinantal varieties to calculate the mixed Hodge module structure of local cohomology modules with determinantal support. We show that the weight of a simple composition factor is uniquely determined by its support and cohomological degree. As a consequence, we obtain the equivariant structure of the Hodge filtration on each local cohomology module. Finally, as an application, we provide a formula for the generation level of the Hodge filtration on these modules.

Intersection theory and Chern classes in Bott–Chern cohomology

In this article, we investigate an axiomatic approach introduced by Grivaux for the study of rational Bott-Chern cohomology, and use it in that context to define Chern classes of coherent sheaves. This method also allows us to derive a Riemann-Roch-Grothendieck formula for a projective morphism between smooth complex compact manifolds. In the general case of complex spaces, the Poincar\'e and Dolbeault-Grothendieck lemmas are not always valid. For this reason, and to simplify the exposition, we only consider non singular complex spaces. The appendix presents a calculation of integral Bott-Chern cohomology in top degree for a connected compact manifold.

Deformations and cohomology theory of Rota-Baxter 3-Lie algebras of arbitrary weights

A cohomology theory of Rota-Baxter 3-Lie algebras of arbitrary weights is introduced. Formal deformations, abelian extensions, skeletal Rota-Baxter 3-Lie 2-algebras and crossed modules of Rota-Baxter 3-Lie algebras are interpreted by using lower degree cohomology groups.

Monodromic model for Khovanov–Rozansky homology

Abstract We describe a new geometric model for the Hochschild cohomology of Soergel bimodules based on the monodromic Hecke category studied earlier by the first author and Yun. Moreover, we identify the objects representing individual Hochschild cohomology groups (for the zero and the top degree cohomology this reduces to an earlier result of Gorsky, Hogancamp, Mellit and Nakagane). These objects turn out to be closely related to explicit character sheaves corresponding to exterior powers of the reflection representation of the Weyl group. Applying the described functors to the images of braids in the Hecke category of type A we obtain a geometric description for Khovanov–Rozansky knot homology, essentially different from the one considered earlier by Webster and Williamson.

On Kodaira fibrations with invariant cohomology

A Kodaira fibration is a compact, complex surface admitting a holomorphic submersion onto a complex curve, such that the fibers have nonconstant moduli. We consider Kodaira fibrations X with nontrivial invariant rational cohomology in degree 1, proving that if the dimension of the holomorphic invariants is 1 or 2, then X admits a branch covering over a product of curves inducing an isomorphism on rational cohomology in degree 1. We also study the class of Kodaira fibrations possessing a holomorphic section, and demonstrate that having a section imposes no restriction on possible monodromies.

Deformation Principle and André motives of Projective Hyperkähler Manifolds

Abstract Let $X_1$ and $X_2$ be deformation equivalent projective hyperkähler manifolds. We prove that the André motive of $X_1$ is abelian if and only if the André motive of $X_2$ is abelian. Applying this to manifolds of $\mbox {K3}^{[n]}$, generalized Kummer and OG6 deformation types, we deduce that their André motives are abelian. As a consequence, we prove that all Hodge classes in arbitrary degree on such manifolds are absolute. We discuss applications to the Mumford–Tate conjecture, showing in particular that it holds for even degree cohomology of such manifolds.

Integral p-adic Hodge theory of formal schemes in low ramification

We prove that for any proper smooth formal scheme $\frak X$ over $\mathcal O_K$, where $\mathcal O_K$ is the ring of integers in a complete discretely valued nonarchimedean extension $K$ of $\mathbb Q_p$ with perfect residue field $k$ and ramification degree $e$, the $i$-th Breuil-Kisin cohomology group and its Hodge-Tate specialization admit nice decompositions when $ie<p-1$. Thanks to the comparison theorems in the recent works of Bhatt, Morrow and Scholze, we can then get an integral comparison theorem for formal schemes when the cohomological degree $i$ satisfies $ie<p-1$, which generalizes the case of schemes under the condition $(i+1)e<p-1$ proven by Fontaine-Messing and Caruso.

Eisenstein series and the top degree cohomology of arithmetic subgroups of SL n /ℚ

Abstract The cohomology H * ⁢ ( Γ , E ) {H^{*}(\Gamma,E)} of a torsion-free arithmetic subgroup Γ of the special linear ℚ {\mathbb{Q}} -group 𝖦 = SL n {\mathsf{G}={\mathrm{SL}}_{n}} may be interpreted in terms of the automorphic spectrum of Γ. Within this framework, there is a decomposition of the cohomology into the cuspidal cohomology and the Eisenstein cohomology. The latter space is decomposed according to the classes { 𝖯 } {\{\mathsf{P}\}} of associate proper parabolic ℚ {\mathbb{Q}} -subgroups of 𝖦 {\mathsf{G}} . Each summand H { P } * ⁢ ( Γ , E ) {H^{*}_{\mathrm{\{P\}}}(\Gamma,E)} is built up by Eisenstein series (or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in { 𝖯 } {\{\mathsf{P}\}} . The cohomology H * ⁢ ( Γ , E ) {H^{*}(\Gamma,E)} vanishes above the degree given by the cohomological dimension cd ⁢ ( Γ ) = 1 2 ⁢ n ⁢ ( n - 1 ) {\mathrm{cd}(\Gamma)=\frac{1}{2}n(n-1)} . We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes { 𝖯 } {\{\mathsf{P}\}} for which the corresponding summand H { 𝖯 } cd ⁢ ( Γ ) ⁢ ( Γ , E ) {H^{\mathrm{cd}(\Gamma)}_{\mathrm{\{\mathsf{P}\}}}(\Gamma,E)} vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span H { 𝖰 } cd ⁢ ( Γ ) ⁢ ( Γ , ℂ ) {H^{\mathrm{cd}(\Gamma)}_{\mathrm{\{\mathsf{Q}\}}}(\Gamma,\mathbb{C})} . Finally, in the case of a principal congruence subgroup Γ ⁢ ( q ) {\Gamma(q)} , q = p ν &gt; 5 {q=p^{\nu}&gt;5} , p ≥ 3 {p\geq 3} a prime, we give lower bounds for the size of these spaces. In addition, for certain associate classes { 𝖰 } {\{\mathsf{Q}\}} , there is a precise formula for their dimension.

On the top-dimensional cohomology groups of congruence subgroups of SL(n, ℤ)

Let $\Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(\mathbb{Z})$. Borel-Serre proved that the cohomology of $\Gamma_n(p)$ vanishes above degree $\binom{n}{2}$. We study the cohomology in this top degree $\binom{n}{2}$. Let $\mathcal{T}_n(\mathbb{Q})$ denote the Tits building of $\text{SL}_n(\mathbb{Q})$. Lee-Szczarba conjectured that $H^{\binom{n}{2}}(\Gamma_n(p))$ is isomorphic to $\widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/\Gamma_n(p))$ and proved that this holds for $p=3$. We partially prove and partially disprove this conjecture by showing that a natural map $H^{\binom{n}{2}}(\Gamma_n(p)) \rightarrow \widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/\Gamma_n(p))$ is always surjective, but is only injective for $p \leq 5$. In particular, we completely calculate $H^{\binom{n}{2}}(\Gamma_n(5))$ and improve known lower bounds for the ranks of $H^{\binom{n}{2}}(\Gamma_n(p))$ for $p \geq 5$.

The Gerstenhaber structure on the Hochschild cohomology of a class of special biserial algebras

We determine the Gerstenhaber structure on the Hochschild cohomology ring of a class of self-injective special biserial algebras. Each of these algebras is presented as a quotient of the path algebra of a certain quiver. In degree one, we show that the cohomology is isomorphic, as a Lie algebra, to a direct sum of copies of a subquotient of the Virasoro algebra. These copies share Virasoro degree 0 and commute otherwise. Finally, we describe the cohomology in degree n as a module over this Lie algebra by providing its decomposition as a direct sum of indecomposable modules.

Rigid-analytic varieties with projective reduction violating Hodge symmetry

We construct examples of smooth proper rigid-analytic varieties admitting formal models with projective special fibers and violating Hodge symmetry for cohomology in degrees ${\geq }3$. This answers negatively the question raised by Hansen and Li.