bluuebunny/crossref_metadata_2025 huggingface dataset:
Minimax principles, Hardy-Dirac inequalities, and operator cores for two and three dimensional Coulomb-Dirac operators
Authors: Müller David
Published: January 2023

For

n\in{2,3}

we prove minimax characterisations of eigenvalues in the gap of the

n

dimensional Dirac operator with an potential, which may have a Coulomb singularity with a coupling constant up to the critical value

1/(4-n)

. This result implies a so-called Hardy-Dirac inequality, which can be used to define a distinguished self-adjoint extension of the Coulomb-Dirac operator defined on

C_{0}^{\infty}({R}^{n}\setminus{0};{C}^{2(n-1)})

, as long as the coupling constant does not exceed

1/(4-n)

. We also find an explicit description of an operator core of this operator.
On the uniqueness of the injective $\mathrm{III}_{1}$ factor
Authors: Haagerup Uffe
Published: January 2023

We give a new proof of a theorem due to Alain Connes, that an injective factor

N

of type

\textup{III}_1

with separable predual and with trivial bicentralizer is isomorphic to the Araki–Woods type

\textup{III}_1

factor

R

_infty. This, combined with the author's solution to the bicentralizer problem for injective

\textup{III}_1

factors provides a new proof of the theorem that up to

*

-isomorphism, there exists a unique injective factor of type

\textup{III}_1

on a separable Hilbert space.
Cohomological Invariants for $G$-Galois Algebras and Self-Dual Normal Bases
Authors: Bayer-Fluckiger E. , Parimala R.
Published: January 2023

We define degree two cohomological invariants for

G

-Galois algebras over fields of characteristic not 2, and use them to give necessary conditions for the existence of a self-dual normal basis. In some cases (for instance, when the field has cohomological dimension

\leq 2

) we show that these conditions are also sufficient.
Concordance Invariance of Levine-Tristram Signatures of Links
Authors: Nagel Matthias , Powell Mark
Published: January 2023

We determine for which complex numbers on the unit circle the Levine-Tristram signature and the nullity give rise to link concordance invariants.
Algebraic Subellipticity and Dominability of Blow-Ups of Affine Spaces
Authors: Lárusson Finnur , Truong Tuyen Trung
Published: January 2023

Little is known about the behaviour of the Oka property of a complex manifold with respect to blowing up a submanifold. A manifold is of Class

\mathcal A

if it is the complement of an algebraic subvariety of codimension at least 2 in an algebraic manifold that is Zariski-locally isomorphic to

\Bbb C^{n}

. A manifold of Class

\mathcal A

is algebraically subelliptic and hence Oka, and a manifold of Class

\mathcal A

blown up at finitely many points is of Class

\mathcal A

. Our main result is that a manifold of Class

\mathcal A

blown up along an arbitrary algebraic submanifold (not necessarily connected) is algebraically subelliptic. For algebraic manifolds in general, we prove that strong algebraic dominability, a weakening of algebraic subellipticity, is preserved by an arbitrary blow-up with a smooth centre. We use the main result to confirm a prediction of Forster's famous conjecture that every open Riemann surface may be properly holomorphically embedded into

\Bbb C^{2}

.
Morita Theory for Hopf Algebroids, Principal Bibundles, and Weak Equivalences
Authors: El Kaoutit Laiachi , Kowalzig Niels
Published: January 2023

We show that two flat commutative Hopf algebroids are Morita equivalent if and only if they are weakly equivalent and if and only if there exists a principal bibundle connecting them. This gives a positive answer to a conjecture due to Hovey and Strickland. We also prove that principal (left) bundles lead to a bicategory together with a 2-functor from flat Hopf algebroids to trivial principal bundles. This turns out to be the universal solution for 2-functors which send weak equivalences to invertible 1-cells. Our approach can be seen as an algebraic counterpart to Lie groupoid Morita theory.
On the Center-Valued Atiyah Conjecture for $L^2$-Betti Numbers
Authors: Knebusch Anselm , Linnell Peter , Schick Thomas
Published: January 2023

The so-called Atiyah conjecture states that the

\mathcal N(G)

-dimensions of the

L^2

-homology modules of finite free

G

-CW-complexes belong to a certain set of rational numbers, depending on the finite subgroups of

G

. In this article we extend this conjecture to a statement for the center-valued dimensions. We show that the conjecture is equivalent to a precise description of the structure as a semisimple Artinian ring of the division closure

D(\Bbb{Q}[G])

of

\Bbb{Q}[G]

in the ring of affiliated operators. We prove the conjecture for all groups in Linnell's class

\frak{C}

, containing in particular free-by-elementary amenable groups. The center-valued Atiyah conjecture states that the center-valued

L^2

-Betti numbers of finite free

G

-CW-complexes are contained in a certain discrete subset of the center of

\Bbb{C}[G]

, the one generated as an additive group by the center-valued traces of all projections in

\Bbb{C}[H]

, where

H

runs through the finite subgroups of

G

. Finally, we use the approximation theorem of Knebusch [15] for the center-valued

L^2

-Betti numbers to extend the result to many groups which are residually in

\frak{C}

, in particular for finite extensions of products of free groups and of pure braid groups.
Hecke Algebra Isomorphisms and Adelic Points on Algebraic Groups
Authors: Cornelissen Gunther , Karemaker Valentijn
Published: January 2023

Let

G

denote a linear algebraic group over

\bold{Q}

and

K

and

L

two number fields. We establish conditions on the group

G

, related to the structure of its Borel groups, under which the existence of a group isomorphism

G(\bold{A}_{K,f}) \cong G(\bold{A}_{L,f})

over the finite adeles implies that

K

and

L

have isomorphic adele rings. Furthermore, if

G

satisfies these conditions,

K

or

L

is a Galois extension of

\bold{Q}

, and

G(\bold{A}_{K,f}) \cong G(\bold{A}_{L,f})

, then

K

and

L

are isomorphic as fields. We use this result to show that if for two number fields

K

and

L

that are Galois over

\bold{Q}

, the finite Hecke algebras for

\mathrm{GL}(n)

(for fixed

n \geq 2

) are isomorphic by an isometry for the

L^1

-norm, then the fields

K

and

L

are isomorphic. This can be viewed as an analogue in the theory of automorphic representations of the theorem of Neukirch that the absolute Galois group of a number field determines the field, if it is Galois over

\bold{Q}

.
Equality of Two Non-Logarithmic Ramification Filtrations of Abelianized Galois Group in Positive Characteristic
Authors: Yatagawa Yuri
Published: January 2023

We prove the equality of two non-logarithmic ramification filtrations defined by Matsuda and Abbes-Saito of the abelianized absolute Galois group of a complete discrete valuation field in positive characteristic. We compute the refined Swan conductor and the characteristic form of a character of the fundamental group of a smooth separated scheme over a perfect field of positive characteristic by using sheaves of Witt vectors.
Representations of $^\ast$-Algebras by Unbounded Operators: C$^\ast$-Hulls, Local-Global Principle, and Induction
Authors: Meyer Ralf
Published: January 2023

We define a C

^\ast

-hull for a

^\ast

-algebra, given a notion of integrability for its representations on Hilbert modules. We establish a local-global principle which, in many cases, characterises integrable representations on Hilbert modules through the integrable representations on Hilbert spaces. The induction theorem constructs a C

^\ast

-hull for a certain class of integrable representations of a graded

^\ast

-algebra, given a C

^\ast

-hull for its unit fibre.
$p$-Adic Interpolation of Automorphic Periods for GL$_2$
Authors: Van Order Jeanine
Published: January 2023

We give a new and representation theoretic construction of

p

-adic interpolation series for central values of self-dual Rankin-Selberg

L

-functions for GL

_2

in dihedral towers of CM fields, using expressions of these central values as automorphic periods. The main novelty of this construction, apart from the level of generality in which it works, is that it is completely local. We give the construction here for a cuspidal automorphic representation of GL

_2

over a totally real field corresponding to a

\mathfrak{p}

-ordinary Hilbert modular forms of parallel weight two and trivial character, although a similar approach can be taken in any setting where the underlying GL

_2

-representation can be chosen to take values in a discrete valuation ring. A certain choice of vectors allows us to establish a precise interpolation formula thanks to theorems of Martin-Whitehouse and File-Martin-Pitale. Such interpolation formulae had been conjectured by Bertolini-Darmon in antecedent works. Our construction also gives a conceptual framework for the nonvanishing theorems of Cornut-Vatsal in that it describes the underlying theta elements. To highlight this latter point, we describe how the construction extends in the parallel weight two setting to give a

p

-adic interpolation series for central derivative values when the root number is generically equal to

-1

, in which case the formula of Yuan-Zhang-Zhang can be used to give an interpolation formula in terms of heights of CM points on quaternionic Shimura curves.
Pseudo-Differential Operators, Wigner Transform and Weyl Systems on Type I Locally Compact Groups
Authors: Mantoiu Marius , Ruzhansky Michael
Published: January 2023

Let

\mathsf{G}

be a unimodular type I second countable locally compact group and let

\hat{\mathsf{G}}

be its unitary dual. We introduce and study a global pseudo-differential calculus for operator-valued symbols defined on

\mathsf{G}\times\hat{\mathsf{G}}

, and its relations to suitably defined Wigner transforms and Weyl systems. We also unveil its connections with crossed products

C^\ast

-algebras associated to certain

C^\ast

-dynamical systems, and apply it to the spectral analysis of covariant families of operators. Applications are given to nilpotent Lie groups, in which case we relate quantizations with operator-valued and scalar-valued symbols.
Cohomological Support and the Geometric Join
Authors: Dao Hailong , Sanders William T.
Published: January 2023

Let

M,N

be finitely generated modules over a local complete intersection

R

. Assume that all the modules Tor

^R_i(M,N)

are zero for

i>0

. We prove that the cohomological support of

M\otimes_R N

(in the sense of Avramov-Buchweitz) is equal to the geometric join of the cohomological supports of

M,N

. This result gives a new connection between two active areas or research, and immediately produces several interesting corollaries. Naturally, it also raises many intriguing new questions about the homological properties of modules over a complete intersection, some of which are investigated in this work.
Basic Operations on Supertropical Quadratic Forms
Authors: Izhakian Zur , Knebusch Manfred
Published: January 2023

In the case that a module

V

over a (commutative) supertropical semiring

R

is free, the

R

-module Quad(

V

) of all quadratic forms on

V

is almost never a free module. Nevertheless, Quad(

V

) has two free submodules, the module QL(

V

) of quasilinear forms with base

\frak{D}_0

and the module Rig(

V

) of rigid forms with base

\frak{H}_0

, such that Quad(

V

)=QL(

V

)+Rig(

V

) and QL(

V

)

\cap

Rig(

V

) =

\{0\}

. In this paper we study endomorphisms of Quad(

V

) for which each submodule

Rq

with

q \in \frak{D}_0\cup\frak{H}_0

is invariant; these basic endomorphisms are determined by coefficients in

R

and do not depend on the base of

V

. We aim for a description of all basic endomorphisms of Quad(

V

), or more generally of its submodules spanned by subsets of

\frak{D}_0\cup\frak{H}_0

. But, due to complexity issues, this naive goal is highly nontrivial for an arbitrary supertropical semiring

R

. Our main stress is therefore on results valid under only mild conditions on

R

, while a complete solution is provided for the case that

R

is a tangible supersemifield.
The Magnetic Laplacian Acting on Discrete Cusps
Authors: Golénia Sylvain , Truc Françoise
Published: January 2023

We introduce the notion of discrete cusp for a weighted graph. In this context, we prove that the form-domain of the magnetic Laplacian and that of the non-magnetic Laplacian can be different. We establish the emptiness of the essential spectrum and compute the asymptotic of eigenvalues for the magnetic Laplacian.
A minimax principle for eigenvalues in spectral gaps: Dirac operators with Coulomb potentials.
Authors: Griesemer Marcel , Lewis Roger T. , Siedentop Heinz
Published: January 2023

We prove the minimax principle for eigenvalues in spectral gaps introduced in citeGriesemerSiedentop1999 based on an alternative set of hypotheses. In the case of the Dirac operator these new assumptions allow for potentials with Coulomb singularites.
On the Milnor Monodromy of the Exceptional Reflection Arrangement of Type $G_{31}$
Authors: Dimca Alexandru , Sticlaru Gabriel
Published: January 2023

Combining recent results by A. D. Mačinic et al. [Doc. Math. (Bielefeld) 22, 135–150 (2017; Zbl 1358.14020)] with a spectral sequence and computer aided computations, we determine the monodromy action on

H^1(F,\Bbb C)

, where

F

denotes the Milnor fiber of the hyperplane arrangement associated to the exceptional irreducible complex reflection group

G_{31}

. This completes the description given by the first author of such monodromy operators for all the other irreducible complex reflection groups.
On the Rank of Universal Quadratic Forms over Real Quadratic Fields
Authors: Blomer Valentin , Kala Vítězslav
Published: January 2023

We study the minimal number of variables required by a totally positive definite diagonal universal quadratic form over a real quadratic field

\Bbb Q(\sqrt{D})

and obtain lower and upper bounds for it in terms of certain sums of coefficients of the associated continued fraction. We also estimate such sums in terms of

D

and establish a link between continued fraction expansions and special values of

L

-functions in the spirit of Kronecker's limit formula.
Wide Subcategories are Semistable
Authors: Yurikusa Toshiya
Published: January 2023

For an arbitrary finite dimensional algebra

\Lambda

, we prove that any wide subcategory of

\mathsf{mod} \Lambda

satisfying a certain finiteness condition is

\theta

-semistable for some stability condition

\theta

. More generally, we show that wide subcategories of

\mathsf{mod} \Lambda

associated with two-term presilting complexes of

\Lambda

are semistable. This provides a complement for Ingalls-Thomas-type bijections for finite dimensional algebras.
Hochschild Cohomology of Polynomial Representations of $\mathrm{GL}_2$
Authors: Miemietz Vanessa , Turner Will
Published: January 2023

We compute the Hochschild cohomology algebras of Ringel-self-dual blocks of polynomial representations of

\mathrm{GL}_2

over an algebraically closed field of characteristic

p>2

, that is, of any block whose number of simple modules is a power of

p

. These algebras are finite-dimensional and we provide an explicit description of their bases and multiplications.
Euler-Like Vector Fields, Deformation Spaces and Manifolds with Filtered Structure
Authors: Haj Saeedi Sadegh Ahmad Reza , Higson Nigel
Published: January 2023

Let

M

be a smooth submanifold of a smooth manifold

V

. Bursztyn, Lima and Meinrenken defined a concept of Euler-like vector field on

V

associated to the embedding of

M

into

V

, and proved that there is a bijection between germs of tubular neighborhoods of

M

and germs of Euler-like vector fields. We shall present a new view of this result by characterizing Euler-like vector fields algebraically and examining their relation to the deformation to the normal cone from algebraic geometry. Then we shall extend our algebraic point of view to smooth manifolds that are equipped with Lie filtrations, and define deformations to the normal cone and Euler-like vector fields in that context. Our algebraic construction of the deformation to the normal cone gives a new approach to Connes' tangent groupoid and its generalizations to filtered manifolds. In addition, Euler-like vector fields give rise to preferred coordinate systems on filtered manifolds.
A Cycle Class Map from Chow Groups with Modulus to Relative $K$-Theory
Authors: Binda Federico
Published: January 2023

Let

\overline{X}

be a smooth quasi-projective

d

-dimensional variety over a field

k

and let

D

be an effective, non-reduced, Cartier divisor on it such that its support is strict normal crossing. In this note, we construct cycle class maps from (a variant of) the higher Chow group with modulus of the pair

(\overline{X};D)

in the range

(d+n, n)

to the relative

K

-groups

K_n(\overline{X}; D)

for every

n\geq 0

.
Lattices in Crystalline Representations and Kisin Modules Associated with Iterate Extensions
Authors: Ozeki Yoshiyasu
Published: January 2023

Cais and Liu extended the theory of Kisin modules and crystalline representations to allow more general coefficient fields and lifts of Frobenius. Based on their theory, we classify lattices in crystalline representations by Kisin modules with additional structures under a Cais-Liu's setting. Furthermore, we give a geometric interpretation of Kisin modules of height one in terms of Dieudonné crystals of

p

-divisible groups, and show a full faithfulness theorem for a restriction functor on torsion crystalline representations.
Spectral Asymptotics for the Schrödinger Operator on the Line with Spreading and Oscillating Potentials
Authors: Duchêne Vincent , Raymond Nicolas
Published: January 2023

This study is devoted to the asymptotic spectral analysis of multiscale Schrödinger operators with oscillating and decaying electric potentials. Different regimes, related to scaling considerations, are distinguished. By means of a normal form filtrating most of the oscillations, a reduction to a non-oscillating effective Hamiltonian is performed.
Some Non-Special Cubic Fourfolds
Authors: Addington Nicolas , Auel Asher
Published: January 2023

In [20], Ranestad and Voisin showed, quite surprisingly, that the divisor in the moduli space of cubic fourfolds consisting of cubics “apolar to a Veronese surface” is not a Noether–Lefschetz divisor. We give an independent proof of this by exhibiting an explicit cubic fourfold

X

in the divisor and using point counting methods over finite fields to show

X

is Noether–Lefschetz general. We also show that two other divisors considered in [20] are not Noether–Lefschetz divisors.
Numerical Dimension and Locally Ample Curves
Authors: Lau Chung-Ching
Published: January 2023

In the paper [the author, “Fujita vanishing theorems for

q

-ample divisors and applications on subvarieties with nef normal bundle”, Preprint, arXiv:1609.07797], was shown that the restriction of a pseudoeffective divisor

D

to a so-called nef subvariety

Y

(e.g.

Y

is lci in

X

and has nef normal bundle) is pseudoeffective. Assuming the normal bundle is ample and that

D|_Y

is not big, we prove that the numerical dimension of

D

is bounded above by that of its restriction, i.e.

\kappa_{\sigma}(D)\leq \kappa_{\sigma}(D|_Y)

. The main motivation is to study the cycle classes of “positive” curves: we show that the cycle class of a curve with ample normal bundle lies in the interior of the cone of curves, and the cycle class of an ample curve lies in the interior of the cone of movable curves. We do not impose any condition on the singularities on the curve or the ambient variety. For locally complete intersection curves in a smooth projective variety, this is the main result of J. C. Ottem [J. Eur. Math. Soc. (JEMS) 18, No. 11, 2459–2468 (2016; Zbl 1362.14007)]. The main tool in this paper is the theory of

q

-ample divisors.
Hermitian Forms and Systems of Quadratic Forms
Authors: Nokhodkar Amir Hossein
Published: January 2023

We associate to every symmetric (antisymmetric) Hermitian form a system of quadratic forms over the base field which determines its isotropy and metabolicity behaviour. It is shown that two even Hermitian forms are isometric if and only if their associated systems are equivalent. As an application, it is also shown that an anisotropic symmetric Hermitian form over a quaternion division algebra in characteristic two remains anisotropic over all odd degree extensions of the ground field.
Motivic Cohomology of Fat Points in Milnor Range
Authors: Park Jinhyun , Ünver Sinan
Published: January 2023

We introduce a new algebraic-cycle model for the motivic cohomology theory of truncated polynomials

k[t]/(t^m)

in one variable. This approach uses ideas from the deformation theory and non-archimedean analysis, and is distinct from the approaches via cycles with modulus. We compute the groups in the Milnor range when the base field is of characteristic 0, and prove that they give the Milnor

K

-groups of

k[t]/(t^m)

, whose relative part is the sum of the absolute Kähler differential forms.
Prolongations of $t$-Motives and Algebraic Independence of Periods
Authors: Maurischat Andreas
Published: January 2023

In this article we show that the coordinates of a period lattice generator of the

n

-th tensor power of the Carlitz module are algebraically independent, if

n

is prime to the characteristic. The main part of the paper, however, is devoted to a general construction for

t

-motives which we call prolongation, and which gives the necessary background for our proof of the algebraic independence. Another incredient is a theorem which shows hypertranscendence for the Anderson-Thakur function

\omega(t)

, i.e. that

\omega(t)

and all its hyperderivatives with respect to

t

are algebraically independent.
Dimension Theory of the Moduli Space of Twisted $K$-Differentials
Authors: Schmitt Johannes
Published: January 2023

In this note we extend the dimension theory for the spaces

\widetilde{\mathcal{H}}_g^k(\mu)

of twisted

k

-differentials defined by Farkas and Pandharipande in [G. Farkas and R. Pandharipande, J. Inst. Math. Jussieu 17, No. 3, 615–672 (2018; Zbl 06868654)] to the case

k>1

. In particular, we show that the intersection

\mathcal{H}_g^k(\mu)=\widetilde{\mathcal{H}}_g^k(\mu) \cap \mathcal{M}_{g,n}

is a union of smooth components of the expected dimensions for all

k\geq 0

. We also extend a conjectural formula from [Zbl 06868654] for a weighted fundamental class of

\widetilde{\mathcal{H}}_g^k(\mu)

and provide evidence in low genus. If true, this conjecture gives a recursive way to compute the cycle class

[\overline{\mathcal{H}}_g^k(\mu)]

of the closure of

\mathcal{H}_g^k(\mu)

for

k\geq 1,\mu

arbitrary.
$p$-Adic Fourier Theory of Differentiable Functions
Authors: Nagel Enno
Published: January 2023

Let

\mathbf K

be a finite extension of

\Bbb{Q}_p

of degree

d

and

\mathcal{O}_{\mathbf{K}}

its ring of integers; let

\Bbb{C}_p

be the completed algebraic closure of

\Bbb{Q}_p

. The Fourier polynomials

P_n:\mathcal{O}_{\mathbf{K}}\to\Bbb{C}_p

show that the topological algebra of all locally analytic distributions

\mu:\mathcal{C}^{\mathrm{la}}(\mathcal{O}_{\mathbf{K}},\Bbb{C}_p)\to\Bbb{C}_p

is, by

\mu\mapsto\sum\mu(P_n) X^n

, isomorphic to that of all power series in

\Bbb{C}_p[[X]]

that converge on the open unit disc of

\Bbb{C}_p

. Given a real number

r\geq d

, we determine the power series that correspond under this isomorphism to all distributions

\mu:\mathcal{C}^r(\mathcal{O}_{\mathbf{K}},\Bbb{C}_p)\to\Bbb{C}_p

that extend to all

r

-times differentiable functions (as arisen in the

p

-adic Langlands program): A function

f:\mathcal{O}_{\mathbf{K}}\to\mathcal{C}_p

is

r

-times differentiable if and only if

f(x)=\Sigma a_nP_n(x)

with

|a_n|n^{r/d}\to 0

as

n\to\infty

.
Cohomological Invariants of Genus Three Hyperelliptic Curves
Authors: Pirisi Roberto
Published: January 2023

We compute the cohomological invariants with coefficients in

\mathbb{Z}/p\mathbb{Z}

of the stack

\mathscr{H}_3

of hyperelliptic curves of genus

3

over an algebraically closed field.
Bivariant Theories in Motivic Stable Homotopy
Authors: Déglise Frédéric
Published: January 2023

The purpose of this work is to study the notion of bivariant theory introduced by Fulton and MacPherson in the context of motivic stable homotopy theory, and more generally in the broader framework of the Grothendieck six functors formalism. We introduce several kinds of bivariant theories associated with a suitable ring spectrum, and we construct a system of orientations (called fundamental classes) for global complete intersection morphisms between arbitrary schemes. These fundamental classes satisfy all the expected properties from classical intersection theory and lead to Gysin morphisms, Riemann-Roch formulas as well asduality statements, valid for general schemes, including singular ones and without need of a base field. Applications are numerous, ranging from classical theories (Betti homology) to generalized theories (algebraic K-theory, algebraic cobordism) and more abstractly to étale sheaves (torsion and

\ell

-adic) and mixed motives.
The Motivic Cofiber of $\tau$
Authors: Gheorghe Bogdan
Published: January 2023

Consider the Tate twist

\tau \in H^{0,1}(S^{0,0})

in the mod 2 cohomology of the motivic sphere. After 2-completion, the motivic Adams spectral sequence realizes this element as a map

\tau:S^{0,-1} \to S^{0,0}

, with cofiber

C\tau

. We show that this motivic 2-cell complex can be endowed with a unique

E_\infty

ring structure. Moreover, this promotes the known isomorphism

\pi_{\ast,\ast} C\tau \cong \mathrm{Ext}^{\ast,\ast}_{BP_{\ast}BP}(BP_{\ast},BP_{\ast})

to an isomorphism of rings which also preserves higher products. We then consider the closed symmetric monoidal category of

C\tau

-modules (

_{C\tau}

Mod,

-\wedge_{C\tau}-)

which lives in the kernel of Betti realization. Given a motivic spectrum

X

, the

C\tau

-induced spectrum

X \wedge C\tau

is usually better behaved and easier to understand than

X

itself. We specifically illustrate this concept in the examples of the mod 2 Eilenberg-Maclane spectrum

H\Bbb F_2

, the mod 2 Moore spectrum

S^{0,0}/2

and the connective hermitian

K

-theory spectrum

kq

.
Proper Holomorphic Mappings onto Symmetric Products of a Riemann Surface
Authors: Bharali Gautam , Biswas Indranil , Divakaran Divakaran , Janardhanan Jaikrishnan
Published: January 2023

We show that the structure of proper holomorphic maps between the

n

-fold symmetric products,

n\geq 2

, of a pair of non-compact Riemann surfaces

X

and

Y

, provided these are reasonably nice, is very rigid. Specifically, any such map is determined by a proper holomorphic map of

X

onto

Y

. This extends existing results concerning bounded planar domains, and is a non-compact analogue of a phenomenon observed in symmetric products of compact Riemann surfaces. Along the way, we also provide a condition for the complete hyperbolicity of all

n

-fold symmetric products of a non-compact Riemann surface.
Weil-Étale Cohomology and Zeta-Values of Proper Regular Arithmetic Schemes
Authors: Flach Matthias , Morin Baptiste
Published: January 2023

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.
On Courant's Nodal Domain Property for Linear Combinations of Eigenfunctions. I
Authors: Bérard Pierre H. , Helffer Bernard
Published: January 2023

According to Courant's theorem, an eigenfunction associated with the

n

-th eigenvalue

\lambda_n

has at most

n

nodal domains. A footnote in the book of Courant and Hilbert, states that the same assertion is true for any linear combination of eigenfunctions associated with eigenvalues less than or equal to

\lambda_n

. We call this assertion the Extended Courant Property. In this paper, we propose new, simple and explicit examples for which the extended Courant property is false: convex domains in

\mathbb{R}^n

(hypercube and equilateral triangle), domains with cracks in

\mathbb{R}^2

, on the round sphere

\mathbb{S}^2

, and on a flat torus

\mathbb{T}^2

. We also give numerical evidence that the extended Courant property is false for the equilateral triangle with rounded corners, and for the regular hexagon.
Big Cohen-Macaulay Modules, Morphisms of Perfect Complexes, and Intersection Theorems in Local Algebra
Authors: Avramov Luchezar L. , Iyengar Srikanth B. , Neeman Amnon
Published: January 2023

There is a well known link from the first topic in the title to the third one. In this paper we thread that link through the second topic. The central result is a criterion for the tensor nilpotence of morphisms of perfect complexes over commutative noetherian rings, in terms of a numerical invariant of the complexes known as their level. Applications to local rings include a strengthening of the Improved New Intersection Theorem, short direct proofs of several results equivalent to it, and lower bounds on the ranks of the modules in every finite free complex that admits a structure of differential graded module over the Koszul complex on some system of parameters.
Rost Nilpotence and Free Theories
Authors: Gille Stefan , Vishik Alexander
Published: January 2023

We introduce coherent cohomology theories

\underline{{h}}

and prove that if such a theory is moreover generically constant then the Rost nilpotence principle holds for projective homogeneous varieties in the category of

\underline{{h}}

-motives. Examples of such theories are algebraic cobordism and its descendants the free theories.
Matrix Factorizations and Curves in $\mathbb{P}^4$
Authors: Schreyer Frank-Olaf , Tanturri Fabio
Published: January 2023

Let

C

be a curve in

\mathbb{P}^4

and

X

be a hypersurface containing it. We show how it is possible to construct a matrix factorization on

X

from the pair

(C,X)

and, conversely, how a matrix factorization on

X

leads to curves lying on

X

. We use this correspondence to prove the unirationality of the Hurwitz space

\mathcal{H}_{12,8}

and the uniruledness of the Brill-Noether space

\mathcal{W}^1_{13,9}

. Several unirational families of curves of genus

16 \leq g \leq 20

in

\mathbb{P}^4

are also exhibited.
The $L^2$-Torsion Polytope of Amenable Groups
Authors: Funke Florian
Published: January 2023

We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the

L^2

-torsion polytope among

G

-CW-complexes for these groups. As another application we prove that the

L^2

-torsion polytope of an amenable group vanishes provided that it contains a non-abelian elementary amenable normal subgroup.
The Minimal Exact Crossed Product
Authors: Buss Alcides , Echterhoff Siegfried , Willett Rufus
Published: January 2023

Given a locally compact group

G

, we study the smallest exact crossed-product functor

(A,G,\alpha)\mapsto A\rtimes_{\mathcal E} G

on the category of

G

-

C^*

-dynamical systems. As an outcome, we show that the smallest exact crossed-product functor is automatically Morita compatible, and hence coincides with the functor

\rtimes_{\mathcal E}

as introduced by P. Baum et al. in their reformulation of the Baum–Connes conjecture [Ann.

K

-Theory 1, No. 2, 155–208 (2016; Zbl 1331.46064)]. We show that the corresponding group algebra

C_{\mathcal E}^*(G)

always coincides with the reduced group algebra, thus showing that the new formulation of the Baum–Connes conjecture coincides with the classical one in the case of trivial coefficients.
Erratum for “Fano Threefolds with 2-Torus Action” [Cf. Documenta Mathematica, vol. 19 (2014), p. 905–940]
Authors: Süß Hendrik
Published: January 2023

The list of Fano threefolds given in Theorem 1.1 [the author, ibid. 19, 905–940 (2014; Zbl 1341.14021] wrongly contained an element of the family 3.8.
Permanence properties of $C^*$-exact groups
Authors: Kirchberg Eberhard , Wassermann Simon
Published: January 2023

It is shown that the class of exact groups, as defined in a previous paper, is closed under various operations, such as passing to a closed subgroup and taking extensions. Taken together, these results imply, in particular, that all almost-connected locally compact groups are exact. The proofs of the permanence properties use a result relating the exactness of sequences of maps in which corresponding algebras are strongly Morita equivalent. The statement of this result is based on a notion of reduced twisted crossed product for covariant systems which are twisted in the sense of Green. The theory of these reduced twisted crossed products and the proof of the exactness result are given in the first part of the paper.
A Step Towards Twist Conjecture
Authors: Huang Jingyin , Przytycki Piotr
Published: January 2023

Under the assumption that a defining graph of a Coxeter group admits only twists in

\mathbb{Z}_2

and is of type FC, we prove Mühlherr's Twist Conjecture.
Restrictions of Eisenstein Series and Rankin-Selberg Convolution
Authors: Keaton Rodney , Pitale Ameya
Published: January 2023

In a 2005 paper, Yang constructed families of Hilbert Eisenstein series, which when restricted to the diagonal are conjectured to span the underlying space of elliptic modular forms. One approach to these conjectures is to show the non-vanishing of an inner product of elliptic eigenforms with the restrictions of Eisenstein series. In this paper, we compute this inner product locally by using explicit values of new vectors in the Waldspurger model.
Localization for Gapped Dirac Hamiltonians with Random Perturbations: Application to Graphene Antidot Lattices
Authors: Barbaroux Jean-Marie , Cornean Horia D. , Zalczer Sylvain
Published: January 2023

In this paper we study random perturbations of first order elliptic operators with periodic potentials. We are mostly interested in Hamiltonians modeling graphene antidot lattices with impurities. The unperturbed operator

H_0 := D_S + V_0

is the sum of a Dirac-like operator

D_S

plus a periodic matrix-valued potential

V_0

, and is assumed to have an open gap. The random potential

V_\omega

is of Anderson-type with independent, identically distributed coupling constants and moving centers, with absolutely continuous probability distributions. We prove band edge localization, namely that there exists an interval of energies in the unperturbed gap where the almost sure spectrum of the family

H_\omega := H_0 +V_\omega

is dense pure point, with exponentially decaying eigenfunctions, that give rise to dynamical localization.
Weighted Khovanov-Lauda-Rouquier Algebras
Authors: Webster Ben
Published: January 2023

In this paper, we define a generalization of Khovanov–Lauda–Rouquier algebras which we call
weighted Khovanov–Lauda–Rouquier algebras
. We show that these algebras carry many of the same structures as the original Khovanov–Lauda–Rouquier algebras, including induction and restriction functors which induce a twisted bialgebra structure on their Grothendieck groups. We also define natural
steadied quotients
of these algebras, which in an important special cases give categorical actions of an associated Lie algebra. These include the algebras categorifying tensor products and Fock spaces defined by the author and
C. Stroppel
[
B. Webster
, Mem. Am. Math. Soc. 1191, iii-vi, 146 p. (2017; Zbl 07000045), p. 141, and
C. Stroppel
and
B. Webster
, “Quiver Schur algebras and

q

-Fock space”, Preprint, ]. For symmetric Cartan matrices, weighted KLR algebras also have a natural geometric interpretation as convolution algebras, generalizing that for the original KLR algebras by
M. Varagnolo
and
E. Vasserot
[J. Reine Angew. Math. 659, 67–100 (2011; Zbl 1229.17019)]; this result has positivity consequences important in the theory of crystal bases. In this case, we can also relate the Grothendieck group and its bialgebra structure to the Hall algebra of the associated quiver.
Metrics on state spaces
Authors: Rieffel Marc A.
Published: January 2023

In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly non-commutative compact spaces are usually not determined by the restriction of the metric they define on the state space, to the extreme points of the state space. We characterize the Lipschitz norms which are determined by their metric on the whole state space as being those which are lower semicontinuous. We show that their domain of Lipschitz elements can be enlarged so as to form a dual Banach space, which generalizes the situation for ordinary Lipschitz seminorms. We give a characterization of the metrics on state spaces which come from Lipschitz seminorms. The natural (broader) setting for these results is provided by the “function spaces” of Kadison. A variety of methods for constructing Lipschitz seminorms is indicated.
The Homotopy Groups of the Simplicial Mapping Space between Algebras
Authors: Rodríguez Cirone Emanuel Darío
Published: January 2023

Let

\ell

be a commutative ring with unit. To every pair of

\ell

-algebras

A

and

B

one can associate a simplicial set

\text{Hom}(A,B^\Delta)

so that

\pi_0\text{Hom}(A,B^\Delta)

equals the set of polynomial homotopy classes of morphisms from

A

to

B

. We prove that

\pi_n\text{Hom}(A,B^\Delta)

is the set of homotopy classes of morphisms from

A

to

B^{\mathfrak{S}_n}_\bullet

, where

B^{\mathfrak{S}_n}_\bullet

is the ind-algebra of polynomials on the

n

-dimensional cube with coefficients in

B

vanishing at the boundary of the cube. This is a generalization to arbitrary dimensions of a theorem of Cortiñas-Thom, which addresses the cases

n\leq 1

. As an application we give a simplified proof of a theorem of Garkusha that computes the homotopy groups of his matrix-unstable algebraic

KK

-theory space in terms of polynomial homotopy classes of morphisms.
Good Lattices of Algebraic Connections (with an Appendix by Claude Sabbah)
Authors: Esnault Hélène , Sabbah Claude
Published: January 2023

We construct a logarithmic model of connections on smooth quasi-projective

n

-dimensional geometrically irreducible varieties defined over an algebraically closed field of characteristic

0

. It consists of a good compactification of the variety together with

(n+1)

lattices on it which are stabilized by log differential operators, and compute algebraically de Rham cohomology. The construction is derived from the existence of good Deligne-Malgrange lattices, a theorem of Kedlaya and Mochizuki which consists first in eliminating the turning points. Moreover, we show that a logarithmic model obtained in this way, called a good model, yields a formula predicted by Michael Groechenig, computing the class of the characteristic variety of the underlying

\mathcal{D}

-module in the

K

-theory group of the variety.
On Zhang's Semipositive Metrics
Authors: Gubler Walter , Martin Florent
Published: January 2023

Zhang introduced semipositive metrics on a line bundle of a proper variety. In this paper, we generalize such metrics for a line bundle

L

of a paracompact strictly

K

-analytic space

X

over any non-archimedean field

K

. We prove various properties in this setting such as density of piecewise

\mathbb{Q}

-linear metrics in the space of continuous metrics on

L

. If

X

is proper scheme, then we show that algebraic, formal and piecewise linear metrics are the same. Our main result is that on a proper scheme

X

over an arbitrary non-archimedean field

K

, the set of semipositive model metrics is closed with respect to pointwise convergence generalizing a result from Boucksom, Favre and Jonsson where

K

was assumed to be discretely valued with residue characteristic

0

.
A GIT Construction of Degenerations of Hilbert Schemes of Points
Authors: Gulbrandsen Martin G. , Halle Lars H. , Hulek Klaus
Published: January 2023

We present a Geometric Invariant Theory (GIT) construction which allows us to construct good projective degenerations of Hilbert schemes of points for simple degenerations. A comparison with the construction of Li and Wu shows that our GIT stack and the stack they construct are isomorphic, as are the associated coarse moduli schemes. Our construction is sufficiently explicit to obtain good control over the geometry of the singular fibres. We illustrate this by giving a concrete description of degenerations of degree

n

Hilbert schemes of a simple degeneration with two components.
On the Non Commutative Iwasawa Main Conjecture for Abelian Varieties over Function Fields
Authors: Trihan Fabien , Vauclair David
Published: January 2023

We establish the Iwasawa main conjecture for semistable abelian varieties over a function field of characteristic

p

under certain restrictive assumptions. Namely we consider

p

-torsion free

p

-adic Lie extensions of the base field which contain the constant

\mathbb{Z}_p

-extension and are everywhere unramified. Under the usual

\mu=0

hypothesis, we give a proof which mainly relies on the interpretation of the Selmer complex in terms of

p

-adic cohomology [F. Trihan, D. Vauclair, A comparison theorem for semi abelian schemes over a smooth curve, preprint arXiv:1505.02942, 2015] together with the trace formulas of
J.-Y. Etesse
and
B. Le Stum
[Math. Ann. 296, No. 3, 557–576 (1993; Zbl 0789.14015)].
Multiplicativity of the Double Ramification Cycle
Authors: Holmes David , Pixton Aaron , Schmitt Johannes
Published: January 2023

The double ramification cycle satisfies a basic multiplicative relation

\mathrm{DRC}_a \cdot \mathrm{DRC}_b = \mathrm{DRC}_a \cdot \mathrm{DRC}_{a + b}

over the locus of compact-type curves, but this relation fails in the Chow ring of the moduli space of stable curves. We restore this relation over the moduli space of stable curves by introducing an extension of the double ramification cycle to the small

\mathrm{b}

-Chow ring (the colimit of the Chow rings of all smooth blowups of the moduli space). We use this to give evidence for the conjectured equality between the (twisted) double ramification cycle and a cycle

P_g^{d,k}(A)

described by the second author in [
F. Janda
et al., Publ. Math., Inst. Hautes Étud. Sci. 125, 221–266 (2017; Zbl 1370.14029)].
Higher Zigzag Algebras
Authors: Grant Joseph
Published: January 2023

Given a Koszul algebra of finite global dimension we define its higher zigzag algebra as a twisted trivial extension of the Koszul dual. If our original algebra is the path algebra of a tree-type quiver, this construction recovers the zigzag algebras of Huerfano-Khovanov. We study examples of higher zigzag algebras coming from Iyama's type A higher representation finite algebras, give their presentations by quivers and relations, and describe relations between spherical twists acting on their derived categories. We connect this to the McKay correspondence in higher dimensions: if

G

is a finite abelian subgroup of

SL_{d+1}

then these relations occur between spherical twists for

G

-equivariant sheaves on affine

(d+1)

-space.
Equivariant $A$-Theory
Authors: Malkiewich Cary , Merling Mona
Published: January 2023

We give a new construction of the equivariant

K

-theory of group actions [
C. Barwick
, “Spectral Mackey functors and equivariant algebraic

K

-theory (I)”, Adv. Math. 304, 646–727 (2017; Zbl 1348.18020) and
C. Barwick
et al., “Spectral Mackey functors and equivariant algebraic

K

-theory (II)”, Preprint (2015); ], producing an infinite loop

G

-space for each Waldhausen category with

G

-action, for a finite group

G

. On the category

R(X)

of retractive spaces over a

G

-space

X

, this produces an equivariant lift of Waldhausen's functor

A(X)

, and we show that the

H

-fixed points are the bivariant

A

-theory of the fibration

X_{hH}\to BH

. We then use the framework of spectral Mackey functors to produce a second equivariant refinement

A_G(X)

whose fixed points have tom Dieck type splittings. We expect this second definition to be suitable for an equivariant generalization of the parametrized

h

-cobordism theorem.
Completeness: When Enough is Enough
Authors: Diener Hannes , Hendtlass Matthew
Published: January 2023

We investigate the notion of a complete enough metric space that, while classically vacuous, in a constructive setting allows for the generalisation of many theorems to a much wider class of spaces. In doing so, this notion also brings the known body of constructive results significantly closer to that of classical mathematics. Most prominently, we generalise the Kreisel-Lacome-Shoenfield Theorem/Tseytin's Theorem on the continuity of functions in recursive mathematics.
The Mean Curvature of Transverse Kähler Foliations
Authors: Jung Seoung Dal , Richardson Ken
Published: January 2023

We study properties of the mean curvature one-form and its holomorphic and antiholomorphic cousins on a transverse Kähler foliation. If the mean curvature of the foliation is automorphic, then there are some restrictions on basic cohomology similar to that on Kähler manifolds, such as the requirement that the odd basic Betti numbers must be even. However, the full Hodge diamond structure does not apply to basic Dolbeault cohomology unless the foliation is taut.
On the Boundary and Intersection Motives of Genus 2 Hilbert-Siegel Varieties
Authors: Cavicchi Mattia
Published: January 2023

We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties

S_K

corresponding to the group

\mathrm{G}\mathrm{Sp}_{4,F}

over a totally real field

F

, along with the relative Chow motives

^\lambda\mathcal{V}

of abelian type over

S_K

obtained from irreducible representations

V_\lambda

of

\mathrm{G}\mathrm{Sp}_{4,F}

. We analyse the weight filtration on the degeneration of such motives at the boundary of the Baily-Borel compactification and we find a criterion on the highest weight

\lambda

, potentially generalisable to other families of Shimura varieties, which characterizes the absence of the
middle weights
0 and 1 in the corresponding degeneration. Thanks to Wildeshaus' theory, the absence of these weights allows us to construct Hecke-equivariant Chow motives over

\mathbb{Q}

, whose realizations equal interior (or intersection) cohomology of

S_K

with

V_{\lambda}

-coefficients. We give applications to the construction of homological motives associated to automorphic representations.
The Syntomic Realization of the Elliptic Polylogarithm via the Poincaré Bundle
Authors: Sprang Johannes
Published: January 2023

We give an explicit description of the syntomic elliptic polylogarithm on the universal elliptic curve over the ordinary locus of the modular curve in terms of certain

p

-adic analytic moment functions associated to Katz' two-variable

p

-adic Eisenstein measure. The present work generalizes previous results of Bannai-Kobayashi-Tsuji and Bannai-Kings on the syntomic Eisenstein classes.