Vanishing Result Research Articles

Local homology, Koszul homology and Serre classes

Given a Serre class $\mathcal {S}$ of modules, we compare the containment of Koszul homology, Ext modules, Tor modules, local homology and local cohomology in $\mathcal {S}$ up to a given bound $s \geq 0$. As applications, we give a full characterization of Noetherian local homology modules. Furthermore, we establish a comprehensive vanishing result which readily leads to formerly known descriptions of the numerical invariants' width and depth in terms of Koszul homology, local homology and local cohomology. In addition, we recover a few renowned vanishing criteria scattered throughout the literature.

Quantitative Representation Stability Over Linear Groups

We introduce a technique for proving quantitative representation stability theorems for sequences of representations of certain finite linear groups over a field of characteristic zero. In particular, we prove a vanishing result for higher syzygies of VIC- and SI-modules, which can be thought of as a weaker version of a regularity theorem of Church-Ellenberg in the context of FI-modules. We apply these techniques to the rational homology of congruence subgroups of mapping class groups and congruence subgroups of automorphism groups of free groups. This partially resolves a question raised by Church and Putman-Sam. We also prove new homological stability results for mapping class groups and automorphism groups of free groups with twisted coefficients.

Relative geometric assembly and mapping cones, part I: the geometric model and applications

Inspired by an analytic construction of Chang, Weinberger and Yu, we define an assembly map in relative geometric $K$-homology. The properties of the geometric assembly map are studied using a variety of index theoretic tools (e.g., the localized index and higher Atiyah-Patodi-Singer index theory). As an application we obtain a vanishing result in the context of manifolds with boundary and positive scalar curvature; this result is also inspired and connected to work of Chang, Weinberger and Yu. Furthermore, we use results of Wahl to show that rational injectivity of the relative assembly map implies homotopy invariance of the relative higher signatures of a manifold with boundary.

A vanishing result for the supersymmetric nonlinear sigma model in higher dimensions

We prove a vanishing result for critical points of the supersymmetric nonlinear sigma model on complete non-compact Riemannian manifolds of positive Ricci curvature that admit an Euclidean type Sobolev inequality, assuming that the dimension of the domain is bigger than two and that a certain energy is sufficiently small.

The Spectral Gaps of Generalized Flag Complexes and a Geometric Hall-type Theorem

Abstract Let $X$ be a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$. Let $L_{k}$ denote the $k$-Laplacian acting on real $k$-cochains of $X$ and let $\mu _{k}(X)$ denote its minimal eigenvalue. We study the connection between the spectral gaps $\mu _{k}(X)$ for $k\geq d$ and $\mu _{d-1}(X)$. In particular, we establish the following vanishing result: if $\mu _{d-1}(X)>\big(1-\binom{k+1}{d}^{-1}\big)n$, then $\tilde{H}^{j}\left (X;{\mathbb{R}}\right )=0$ for all $d-1\leq j \leq k$. As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Martínez-Sandoval, and Montejano for general position sets in matroids.

On the Lee classes of locally conformally symplectic complex surfaces

We prove that the deRham cohomology classes of Lee forms of locally conformally symplectic structures taming the complex structure of a compact complex surface S with first Betti number equal to 1 is either a non-empty open subset of H 1 dR (S, R), or a single point. In the latter case, we show that S must be biholomorphic to a blow-up of an Inoue-Bombieri surface. Similarly, the deRham co-homology classes of Lee forms of locally conformally Kahler structures of a compact complex surface S with first Betti number equal to 1 is either a non-empty open subset of H 1 dR (S, R), a single point or the empty set. We give a characterization of Enoki surfaces in terms of the existence of a special foliation, and obtain a vanishing result for the Lichnerowicz-Novikov cohomology groups on the class VII compact complex surfaces with infinite cyclic fundamental group.

An algorithm for the normal bundle of rational monomial curves

We give a method to calculate the cohomology of the twisted normal bundle over a smoth rational curve. From this method we derive a vanishing result and an algorithm for calculating the splitting type of the bundle for any rational monomial curve.

On the length of global integrals for

In this paper we study global integrals defined on the group GL_n({mathbf A}). We prove a vanishing result for certain integrals which involve Speh representations and certain Eisenstein series, and which satisfies a certain dimension equation. This is part of a general Conjecture which states that a global nonzero integral which satisfies a dimension equation involves at most three nontrivial representations.

Morse–Floer theory for superquadratic Dirac equations, II: construction and computation of Morse–Floer homology

We give a construction and computation of Morse–Floer homology for a class of superquadratic Dirac functionals defined on the set of spinors on a compact spin manifold. For the superquadratic functionals, we show that the Morse–Floer homology is well defined for generic choice of metric on the set of spinors and its isomorphism class is independent of the choice of such a generic metric. Moreover, we show that it is also independent of the choice of superquadratic Dirac functional. Therefore, it defines an invariant for the set of spinors which we call the superquadratic-Dirac–Morse–Floer homology. We prove a vanishing result for this homology. As an application, we give existence and multiplicity results for a class of superquadratic Dirac equations on compact spin manifolds.

P-harmonic [formula omitted]-forms on Riemannian manifolds with a weighted Poincaré inequality

Given a Riemannian manifold with a weighted Poincaré inequality, in this paper, we will show some vanishing type theorems for p-harmonic ℓ-forms on such a manifold. We also prove a vanishing result on submanifolds in Euclidean space with flat normal bundle. Our results can be considered as generalizations of the work of Lam, Li–Wang, Lin, and Vieira (see Lam (2008), Li and Wang (2001), Lin (2015), Vieira (2016)). Moreover, we also prove a vanishing and splitting type theorem for p-harmonic functions on manifolds with Spin (9) holonomy provided a (p,p,λ)-Sobolev type inequality which can be considered as a general Poincaré inequality holds true.

String theory in polar coordinates and the vanishing of the one-loop Rindler entropy

We analyze the string spectrum of flat space in polar coordinates, following the small curvature limit of the $SL(2,\mathbb{R})/U(1)$ cigar CFT. We first analyze the partition function of the cigar itself, making some clarifications of the structure of the spectrum that have escaped attention up to this point. The superstring spectrum (type 0 and type II) is shown to exhibit an involution symmetry, that survives the small curvature limit. We classify all marginal states in polar coordinates for type II superstrings, with emphasis on their links and their superconformal structure. This classification is confirmed by an explicit large $\tau_2$ analysis of the partition function. Next we compare three approaches towards the type II genus one entropy in Rindler space: using a sum-over-fields strategy, using a Melvin model approach and finally using a saddle point method on the cigar partition function. In each case we highlight possible obstructions and motivate that the correct procedures yield a vanishing result: $S=0$. We finally discuss how the QFT UV divergences of the fields in the spectrum disappear when computing the free energy and entropy using Euclidean techniques.

Vanishing of Rabinowitz Floer homology on negative line bundles

Following [Fra08, AF14] we construct Rabinowitz Floer homology for negative line bundles over symplectic manifolds and prove a vanishing result. In [Rit14] Ritter showed that symplectic homology of these spaces does not vanish, in general. Thus, the theorem $\mathrm{SH}=0\Leftrightarrow\mathrm{RFH}=0$, [Rit13], does not extend beyond the symplectically aspherical situation. We give a conjectural explanation in terms of the Cieliebak-Frauenfelder-Oancea long exact sequence [CFO10].

Simple endotrivial modules for linear, unitary and exceptional groups

Motivated by a recent result of Robinson showing that simple endotrivial modules essentially come from quasi-simple groups we classify such modules for finite special linear and unitary groups as well as for exceptional groups of Lie type. Our main tool is a lifting result for endotrivial modules obtained in a previous paper which allows us to apply character theoretic methods. As one application we prove that the $$\ell $$ -rank of quasi-simple groups possessing a faithful simple endotrivial module is at most 2. As a second application we complete the proof that principal blocks of finite simple groups cannot have Loewy length 4, thus answering a question of Koshitani, Kulshammer and Sambale. Our results also imply a vanishing result for irreducible characters of special linear and unitary groups.

Finite volume effects on the extraction of form factors at zero momentum

Hadronic matrix elements that depend on momentum are required for numerous phenomenological applications. Probing the low-momentum regime is often problematic for lattice QCD computations on account of the restriction to periodic momentum modes. Recently a novel method has been proposed to compute matrix elements at zero momentum, for which straightforward evaluation of the matrix elements would otherwise yield a vanishing result. We clarify an assumption underlying this method, and thereby establish the theoretical framework required to address the associated finite volume effects. Using the pion electromagnetic form factor as an example, we show how the charge radius and two higher moments can be calculated at zero momentum transfer, and determine the corresponding finite volume effects. These computations are performed using chiral perturbation theory to account for modified infrared physics, and can be generalized to ascertain finite volume effects for other hadronic matrix elements extracted at zero momentum.

On vanishing theorems for Higgs bundles

We introduce the notion of Hermitian Higgs bundle as a natural generalization of the notion of Hermitian vector bundle and we study some vanishing theorems concerning Hermitian Higgs bundles when the base manifold is a compact complex manifold. We show that a first vanishing result, proved for these objects when the base manifold was Kähler, also holds when the manifold is compact complex. From this fact and some basic properties of Hermitian Higgs bundles, we conclude several results. In particular we show that, in analogy to the classical case, there are vanishing theorems for invariant sections of tensor products of Higgs bundles. Then, we prove that a Higgs bundle admits no nonzero invariant sections if there is a condition of negativity on the greatest eigenvalue of the Hitchin–Simpson mean curvature. Finally, we prove that the invariant sections of certain tensor products of a weak Hermitian–Yang–Mills Higgs bundle are all parallel in the classical sense.

On the cohomology of weakly almost periodic group representations

We initiate a study of cohomological aspects of weakly almost periodic group representations on Banach spaces, in particular, isometric representations on reflexive Banach spaces. Using the Ryll–Nardzewski fixed point theorem, we prove a vanishing result for the restriction map (with respect to a subgroup) in the reduced cohomology of weakly periodic representations. Combined with the Alaoglu–Birkhoff decomposition theorem, this generalizes and complements theorems on continuous group cohomology by several authors.

Periodicity in the stable representation theory of crystallographic groups

Deformation K -theory associates to each discrete group G a spectrum built from spaces of finite dimensional unitary representations of G . In all known examples, this spectrum is 2-periodic above the rational cohomological dimension of G (minus 2), in the sense that T. Lawson's Bott map is an isomorphism on homotopy in these dimensions. We establish a periodicity theorem for crystallographic subgroups of the isometries of k -dimensional Euclidean space. For a certain subclass of torsion-free crystallographic groups, we prove a vanishing result for the homotopy groups of the stable moduli space of representations, and we provide examples relating these homotopy groups to the cohomology of G . These results are established as corollaries of the fact that for each , the one-point compactification of the moduli space of irreducible n -dimensional representations of G is a CW complex of dimension at most k . This is proven using real algebraic geometry and projective representation theory.

Finite decomposition complexity and the integral Novikov conjecture for higher algebraic K-theory

Abstract Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded geometry metric spaces with finite decomposition complexity. This leads to a proof of the integral K-theoretic Novikov conjecture, regarding split injectivity of the K-theoretic assembly map, for groups with finite decomposition complexity and finite CW models for their classifying spaces. By work of Guentner, Tessera, and Yu, this includes all (geometrically finite) linear groups.

Dirac operators and symmetries of quasitoric manifolds

We establish a vanishing result for indices of certain twisted Dirac operators on $\text{Spin}^c$-manifolds with non-abelian Lie-group actions. We apply this result to study non-abelian symmetries of quasitoric manifolds. We give upper bounds for the degree of symmetry of these manifolds.

On the extension problem and the nil groups of rings of finite global dimension

A vanishing result is obtained in respect of nil groups of rings of finite global dimension. Also a connection is established with the extension problem.