Cohomological Invariants Research Articles

On the moment measure conjecture

AbstractThe moment measure conjecture of Białynicki-Birula and Sommese gives a combinatorial characterization of all open substacks of a global quotient stack for a torus action on a normal projective variety that admit a proper good moduli space, in other words it characterizes the invariant open subvarieties that admit a proper quotient. In this article we prove the conjecture for actions on smooth varieties. This gives an instance of stability conditions defined by cohomological invariants that are not given by Chern classes of line bundles.

Witt invariants of Weyl groups

We describe the Witt invariants of a Weyl group over a field k0 by giving generators for the W(k0)-module of Witt invariants, under the assumption that the characteristic of k0 does not divide the order of the group. For the Weyl groups of types Bn, Cn, Dn, and G2, we show that the Witt invariants are generated as a W(k0)-algebra by trace forms and their exterior powers, extending a result due to Serre in type An. Many of our computational methods are applicable to computing Witt invariants of any smooth linear algebraic group over k0, including a technique for lifting module generators from cohomological invariants to Witt invariants.

Reduced resonance schemes and Chen ranks

Abstract The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups.

Reciprocity obstruction to strong approximation over p-adic function fields

Over function fields of p-adic curves, we construct stably rational varieties in the form of homogeneous spaces of SLn with semisimple simply connected stabilizers and we show that strong approximation away from a non-empty set of places fails for such varieties. The construction combines the Lichtenbaum duality and the degree 3 cohomological invariants of the stabilizers. We then establish a reciprocity obstruction which accounts for this failure of strong approximation. We show that this reciprocity obstruction to strong approximation is the only one for counterexamples we constructed, and also for classifying varieties of tori. We also show that this reciprocity obstruction to strong approximation is compatible with known results for tori. At the end, we explain how a similar point of view shows that the reciprocity obstruction to weak approximation is the only one for classifying varieties of tori over p-adic function fields.

On the Computation of the Cohomological Invariants of Bott–Samelson Resolutions of Schubert Varieties

Let X⊆G/B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X\\subseteq G/\\mathcal {B}$$\\end{document} be a Schubert variety in a flag manifold and let π:X~→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi : \ ilde{X} \\rightarrow X$$\\end{document} be a Bott–Samelson resolution of X. In this paper, we prove an effective version of the decomposition theorem for the derived pushforward Rπ∗QX~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R \\pi _{*} \\mathbb {Q}_{\ ilde{X}}$$\\end{document}. As a by-product, we obtain recursive procedure to extract Kazhdan–Lusztig polynomials from the polynomials introduced by Deodhar [7], which does not require prior knowledge of a minimal set. We also observe that any family of equivariant resolutions of Schubert varieties allows to define a new basis in the Hecke algebra and we show a way to compute the transition matrix, from the Kazhdan–Lusztig basis to the new one.

Chern–Simons theory and cohomological invariants of representation varieties

Chern–Simons theory and cohomological invariants of representation varieties

Cohomological and geometric invariants of simple complexes of groups

Cohomological and geometric invariants of simple complexes of groups

Group boundaries for semidirect products with $\mathbb{Z}$

Bestvina's notion of a \mathcal{Z} -structure provides a general framework for group boundaries that includes Gromov boundaries of hyperbolic groups and visual boundaries of CAT(0) groups as special cases. A refinement, known as an E\mathcal{Z} -structure, has proven useful in attacks on the Novikov conjecture and related problems. Characterizations of groups admitting a \mathcal{Z} - or an E\mathcal{Z} -structure are longstanding open problems. In this paper, we examine groups of the form G\rtimes_{\phi}\mathbb{Z} . For example, we show that if G is torsion-free and admits a \mathcal{Z} -structure, then so does every semidirect product of this type. We prove a similar theorem for E\mathcal{Z} -structures, under an additional hypothesis. As applications, we show that all closed 3-manifold groups admit \mathcal{Z} -structures, as do all strongly polycyclic groups and all groups of polynomial growth. In those latter cases, our \mathcal{Z} -boundaries are always spheres. This allows one to make strong conclusions about the group cohomology and end invariants of those groups. In another direction, we expand upon the notion of an E\mathcal{Z} -structure and discuss new applications to the Novikov conjecture.

Coarse Sheaf Cohomology

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0, they see the number of ends of the space. In this paper, a resolution of the constant sheaf via cochains is developed. It serves to be a valuable tool for computing cohomology. In addition, coarse homotopy invariance of coarse cohomology with constant coefficients is established. This property can be used to compute cohomology of Riemannian manifolds. The Higson corona of a proper metric space is shown to reflect sheaves and sheaf cohomology. Thus, we can use topological tools on compact Hausdorff spaces in our computations. In particular, if the asymptotic dimension of a proper metric space is finite, then higher cohomology groups vanish. We compute a few examples. As it turns out, finite abelian groups are best suited as coefficients on finitely generated groups.

Divided powers in the Witt ring of symmetric bilinear forms

The Witt ring of quadratic forms over a field has divided power operations. On the other hand, it follows from Garibaldi-Merkurjev-Serre's work on cohomological invariants that all operations on the Witt ring are linear combinations of exterior powers. We find the explicit formula for the divided powers as a linear combination of exterior powers. The coefficients involve the ``tangent numbers'', related to Bernoulli numbers. The divided powers on the Witt ring give another construction of the divided powers on Milnor K-theory modulo 2.

Product-Type Classes for Vertex Algebra Cohomology of Foliations on Complex Curves

We introduce the vertex algebra cohomology of foliations on complex curves. Generalizing the classical case, the orthogonality condition with respect to a product of elements of the double complexes associated to a grading-restricted vertex algebra matrix elements leads to the construction of cohomology invariants of codimension one foliations.

Minimal Euler characteristics for even-dimensional manifolds with finite fundamental group

Abstract We consider the Euler characteristics $\chi (M)$ of closed, orientable, topological $2n$ -manifolds with $(n-1)$ -connected universal cover and a given fundamental group G of type $F_n$ . We define $q_{2n}(G)$ , a generalised version of the Hausmann-Weinberger invariant [19] for 4–manifolds, as the minimal value of $(-1)^n\chi (M)$ . For all $n\geq 2$ , we establish a strengthened and extended version of their estimates, in terms of explicit cohomological invariants of G. As an application, we obtain new restrictions for nonabelian finite groups arising as fundamental groups of rational homology 4–spheres.

Cousin complex on the complement to the strict normal-crossing divisor in a local essentially smooth scheme over a field

For any $\mathbb{A}^1$-invariant cohomology theory that satisfies Nisnevich excision on the category of smooth schemes over a field $k$ it is proved that the Cousin complex on the complement $U-D$ to the strict normal-crossing divisor $D$ in a local essentially smooth scheme $U$ is acyclic. This claim is also proved for the schemes $(X-D)\times(\mathbb{A}^1_k-Z_0)\times…\times(\mathbb{A}^1_k-Z_l)$, where $Z_0,…,Z_l$ is a finite family of closed subschemes in the affine line over $k$. Bibliography: 32 titles.

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.

Explicit Tamagawa numbers for certain algebraic tori over number fields

Given a number field extension K / k K/k with an intermediate field K + K^+ fixed by a central element of Gal ⁡ ( K / k ) \operatorname {Gal}(K/k) of prime order p p , there exists an algebraic torus over k k whose rational points are elements of K × K^\times sent to k × k^\times by the norm map N K / K + N_{K/K^+} . The goal is to compute the Tamagawa number such a torus explicitly via Ono’s formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when K / k K/k is Galois. Partial results including the numerator of Ono’s formula are given when the extension is not Galois, or more generally when the torus is defined by an étale algebra. We also present tools developed in SageMath for this purpose, allowing us to build and compute the cohomology and explore the local-global principles for such an algebraic torus. Particular attention is given to the case when [ K : K + ] = 2 [K:K^+]=2 and K K is a CM-field. This case corresponds to maximal tori in G S p 2 n \mathrm {GSp}_{2n} , and most examples will be in that setting. This is motivated by the application to abelian varieties over finite fields and the Hasse principle for bilinear forms.

On linear combinations of cohomological invariants of compact complex manifolds

We prove that there are no unexpected universal integral linear relations and congruences between Hodge, Betti and Chern numbers of compact complex manifolds and determine the linear combinations of such numbers which are bimeromorphic or topological invariants. This extends results in the Kähler case by Kotschick and Schreieder. We then develop a framework to tackle the more general questions taking into account ‘all’ cohomological invariants (e.g. the dimensions of the higher pages of the Frölicher spectral sequence, Bott-Chern and Aeppli cohomology). This allows us to reduce the general questions to specific construction problems. We solve these problems in many cases. In particular, we obtain full answers to the general questions concerning universal relations and bimeromorphic invariants in low dimensions.

De Rham’s Theorem for Orlicz Cohomology

We prove that the de Rham $ L^{\phi} $ -cohomology of a Riemannian manifold $ M $ admitting a convenient triangulation $ X $ is isomorphic to the simplicial $ \ell^{\phi} $ -cohomology of $ X $ under some assumptions on the Young function $ \phi $ . This result implies the quasi-isometry invariance of the first cohomology.

Cohomological DT invariants from localization

Given a quiver with potential associated with a toric Calabi–Yau threefold, the numerical Donaldson–Thomas invariants for the moduli space of framed representations can be computed by using toric localization, which reduces the problem to the enumeration of pyramid partitions. We provide a refinement of this localization procedure, which allows to compute cohomological Donaldson–Thomas invariants. Using this approach, we prove a universal formula which gives the BPS invariants of any toric quiver, up to undetermined contributions which are invariant under Poincaré duality. When the toric Calabi–Yau threefold has compact divisors, these self-Poincaré dual contributions have a complicated dependence on the stability parameters, but explicit computations suggest that they drastically simplify for the self-stability condition (also called attractor chamber). We conjecture a universal formula for the attractor invariants, which applies to any toric Calabi–Yau singularity with compact divisors.

On the Bauer–Furuta and Seiberg–Witten invariants of families of 4‐manifolds

We show how the families Seiberg-Witten invariants of a family of smooth $4$-manifolds can be recovered from the families Bauer-Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families Seiberg-Witten invariants. We give a formula for the Steenrod squares of the families Seiberg-Witten invariants leading to a series of mod $2$ relations between these invariants and the Chern classes of the spin$^c$ index bundle of the family. As a result we discover a new aspect of the ordinary Seiberg-Witten invariants of a $4$-manifold $X$: they obstruct the existence of certain families of $4$-manifolds with fibres diffeomorphic to $X$. As a concrete geometric application, we shall detect a non-smoothable family of $K3$ surfaces. Our formalism also leads to a simple new proof of the families wall crossing formula. Lastly, we introduce $K$-theoretic Seiberg-Witten invariants and give a formula expressing the Chern character of the $K$-theoretic Seiberg-Witten invariants in terms of the cohomological Seiberg-Witten invariants. This leads to new divisibility properties of the families Seiberg-Witten invariants.

Groupes de Coxeter finis: involutions et cubes

We review the properties of the finite Coxeter groups which are most useful for applications to cohomological invariants, namely their classes of involutions and their “cubes”, i.e., their abelian subgroups which are generated by reflections. Résumé. Pour l’étude des invariants cohomologiques des groupes de Coxeter finis, il est utile d’avoir une description détaillée de leurs involutions et des sous-groupes abéliens qu’elles engendrent.