bluuebunny/crossref_metadata_2025 huggingface dataset:
Around the Gysin triangle. II.
Authors: Déglise Frédéricke
Published: January 2023

The notions of orientation and duality are well understood in algebraic topology in the framework of the stable homotopy category. In this work, we follow these lines in algebraic geometry, in the framework of motivic stable homotopy, introduced by F. Morel and V. Voevodsky. We use an axiomatic treatment which allows us to consider both mixed motives and oriented spectra over an arbitrary base scheme. In this context, we introduce the Gysin triangle and prove several formulas extending the traditional panoply of results on algebraic cycles modulo rational equivalence. We also obtain the Gysin morphism of a projective morphism and prove a duality theorem in the (relative) pure case. These constructions involve certain characteristic classes (Chern classes, fundamental classes, cobordism classes) together with their usual properties. They imply statements in motivic cohomology, algebraic K-theory (assuming the base is regular) and “abstract” algebraic cobordism as well as the dual statements in the corresponding homology theories. They apply also to ordinary cohomology theories in algebraic geometry through the notion of a mixed Weil cohomology theory, introduced by D.-C. Cisinski and the author in [CD06], notably rigid cohomology.
A note on the $p$-adic Galois representations attached to Hilbert modular forms
Authors: Skinner Christopher
Published: January 2023

We show that the

p

-adic Galois representations attached to Hilbert modular forms of motivic weight are potentially semistable at all places above

p

and are compatible with the local Langlands correspondence at these places, proving this for those forms not covered by the previous works of T. Saito and of D. Blasius and J. Rogawski.
Homology of the Steinberg variety and Weyl group coinvariants
Authors: Douglass J.M. , Röhrle G.
Published: January 2023

Let

G

be a complex, connected, reductive algebraic group with Weyl group

W

and Steinberg variety

Z

. We show that the graded Borel-Moore homology of

Z

is isomorphic to the smash product of the coinvariant algebra of

W

and the group algebra of

W

.
Vanishing of Hochschild cohomology for affine group schemes and rigidity of homomorphisms between algebraic groups
Authors: Margaux Benedictus
Published: January 2023

Let

k

be an algebraically closed field. If

\mathbf G

is a linearly reductive

k

-group and

\mathbf H

is a smooth algebraic

k

-group, we establish a rigidity property for the set of group homomorphisms

\mathbf G \to \mathbf H

up to the natural action of

\mathbf H(k)

by conjugation. Our main result states that this set remains constant under any base change

K/k

with

K

algebraically closed. This is proven as consequence of a vanishing result for Hochschild cohomology of affine group schemes.
The line bundles on moduli stacks of principal bundles on a curve
Authors: Biswas Indranil , Hoffmann Norbert
Published: January 2023

Let

G

be an affine reductive algebraic group over an algebraically closed field

k

. We determine the Picard group of the moduli stacks of principal

G

-bundles on any smooth projective curve over

k

.
Excess charge for pseudo-relativistic atoms in Hartree-Fock theory
Authors: Dall'acqua Anna , Solovej Jan Philip
Published: January 2023

We prove within the Hartree-Fock theory of pseudo-relativistic atoms that the maximal negative ionization charge and the ionization energy of an atom remain bounded independently of the nuclear charge

Z

and the fine structure constant

\alpha

as long as

Z\alpha

is bounded.
On the solutions of quadratic Diophantine equations
Authors: Yoshinaga Takashi
Published: January 2023

We determine a finite set of representatives of the set of local solutions in a maximal lattice modulo the stabilizer of the lattice in question for a quadratic Diophantine equation. Our study is based on the works of Shimura on quadratic forms, especially citeSh3 and citeSh4. Indeed, as an application of the result, we present a criterion (in both global and local cases) of the maximality of the lattice of

(11.6\textrm{a} )

in citeSh3. This gives an answer to the question

(11.6\textrm{a} )

. As one more global application, we investigate primitive solutions contained in a maximal lattice for the sums of squares on each vector space of dimension

4, 6, 8

, or

10

over the field of rational numbers.
Semigroup properties for the second fundamental form
Authors: Wang Feng-Yu
Published: January 2023

Let

M

be a compact Riemannian manifold with boundary

\partial M

and

L= \delta+Z

for a

C^1

-vector field

Z

on

M

. Several equivalent statements, including the gradient and Poincaré/log-Sobolev type inequalities of the Neumann semigroup generated by

L

, are presented for lower bound conditions on the curvature of

L

and the second fundamental form of

\partial M

. The main result not only generalizes the corresponding known ones on manifolds without boundary, but also clarifies the role of the second fundamental form in the analysis of the Neumann semigroup. Moreover, the Lévy-Gromov isoperimetric inequality is also studied on manifolds with boundary.
Duality and integrability on contact Fano manifolds
Authors: Buczynski Jaroslaw
Published: January 2023

We address the problem of classification of contact Fano manifolds. It is conjectured that every such manifold is necessarily homogeneous. We prove that the Killing form, the Lie algebra grading and parts of the Lie bracket can be read from geometry of an arbitrary contact manifold. Minimal rational curves on contact manifolds (or contact lines) and their chains are the essential ingredients for our constructions.
Fuss-Catalan numbers in noncommutative probability
Authors: Mlotkowski Wojciech
Published: January 2023

We prove that if

p,r\in{R}, p\ge1

and

0le rle p

then the Fuss-Catalan sequence

\binom{mp+r}m\frac{r}{mp+r}

is positive definite. We study the family of the corresponding probability measures

\mu(p,r)

on

{R}

from the point of view of noncommutative probability. For example, we prove that if

0le 2rle p

and

r+1le p

then

\mu(p,r)

is

\boxplus

-infinitely divisible. As a by-product, we show that the sequence

\frac{m^m}{m!}

is positive definite and the corresponding probability measure is

\boxtimes

-infinitely divisible.
Asymptotic behavior of word metrics on Coxeter groups.
Authors: Noskov G.A.
Published: January 2023

We study the geometry of tessellation defined by the walls in the Moussong complex

\mathcal M_W

of a Coxeter group

W

. It is proved that geodesics in

\mathcal M_W

can be approximated by geodesic galleries of the tessellation. A formula for the translation length of an element of

W

is given. We prove that the restriction of the word metric on the

W

to any free abelian subgroup

A

is Hausdorff equivalent to a regular norm on

A.
Modules homotopiques
Authors: Déglise F.
Published: January 2023

Based on previous works, we compare over a perfect field

k

the category of homotopy invariant sheaves with transfers introduced by V. Voevodsky and the category of cycle modules introduced by M. Rost: the former is a full subcategory of the latter. Using the recent construction by D.C. Cisinski and the author of a non effective version

DM(k)

of the category of motivic complexes, we show that cycle modules form the heart of a natural t-structure on

DM(k)

, generalizing the homotopy t-structure on motivic complexes.
On the cuspidal divisor class group of a Drinfeld modular curve
Authors: Gekeler Ernst-Ulrich
Published: January 2023

The theory of theta functions for arithmetic groups

\Gamma

that act on the Drinfeld upper half-plane is extended to allow degenerate parameters. This is used to investigate the cuspidal divisor class groups of Drinfeld modular curves. These groups are finite for congruence subgroups

\Gamma

and may be described through the corresponding quotients of the Bruhat-Tits tree by

\Gamma

. The description given is fairly explicit, notably in the most important special case of Hecke congruence subgroups

\Gamma

over a polynomial ring.
The Hilbert-Chow morphism and the incidence divisor
Authors: Ross Joseph
Published: January 2023

For a smooth projective variety

P

of dimension

n

, we construct a Cartier divisor supported on the incidence locus in the product of Chow varieties

\mathscr{C}_a (P) \times \mathscr{C}_{n -a - 1}(P)

. There is a natural definition of the corresponding line bundle on a product of Hilbert schemes, and we show this bundle descends to the Chow varieties. This answers a question posed by Barry Mazur.
Irreducible modules over the Virasoro algebra
Authors: Lü Rencai , Guo Xiangqian , Zhao Kaiming
Published: January 2023

In this paper, we construct two different classes of Virasoro modules from twisting Harish-Chandra modules over the twisted Heisenberg-Virasoro algebra by an automorphism of the twisted Heisenberg-Virasoro algebra. Weight modules in the first class are some irreducible highest weight modules over the twisted Heisenberg-Virasoro algebra. The non-weight modules in the first class are irreducible Whittaker modules over the Virasoro algebra. We obtain concrete bases for all irreducible Whittaker modules (instead of a quotient of modules). This generalizes known results on Whittaker modules. The second class of modules are non-weight modules which are not Whittaker modules. We determine the irreducibility and isomorphism classes of these modules.
The equivariant cohomology of isotropy actions on symmetric spaces
Authors: Goertsches Oliver
Published: January 2023

We show that for every symmetric space

G/K

of compact type with

K

connected, the

K

-action on

G/K

by left translations is equivariantly formal.
Decompositions of motives of generalized Severi-Brauer varieties
Authors: Zhykhovich Maksim
Published: January 2023

Let

p

be a positive prime number and

X

be a Severi-Brauer variety of a central division algebra

D

of degree

p^n

, with

n\geq 1

. We describe all shifts of the motive of

X

in the complete motivic decomposition of a variety

Y

, which splits over the function field of

X

and satisfies the nilpotence principle. In particular, we prove the motivic decomposability of generalized Severi-Brauer varieties

X(p^m,D)

of right ideals in

D

of reduced dimension

p^m, m=0,1,ldots,n-1

, except the cases

p=2, m=1

and

m=0 (for any prime p)

, where motivic indecomposability was proven by Nikita Karpenko.
The $C^\ast$-algebra of an affine map on the 3-torus
Authors: Andersen Kasper K.S. , Thomsen Klaus
Published: January 2023

We study the

C^*

-algebra of an affine map on a compact abelian group and give necessary and sufficient conditions for strong transitivity when the group is a torus. The structure of the

C^*

-algebra is completely determined for all strongly transitive affine maps on a torus of dimension one, two or three.
Modularity of the Consani-Scholten quintic. With an appendix by José Burgos Gil and Ariel Pacetti
Authors: Dieulefait Luis , Pacetti Ariel , Schütt Matthias
Published: January 2023

We prove that the Consani-Scholten quintic, a Calabi-Yau threefold over

\mathbb Q

, is Hilbert modular. For this, we refine several techniques known from the context of modular forms. Most notably, we extend the Faltings-Serre-Livné method to induced four-dimensional Galois representations over

\mathbb Q

. We also need a Sturm bound for Hilbert modular forms; this is developed in an appendix by José Burgos Gil and the second author.
Lifting tropical intersections
Authors: Osserman Brian , Payne Sam
Published: January 2023

We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension. We also prove a similar result for intersections inside an ambient subvariety of the torus, when the tropicalizations meet inside a facet of multiplicity 1. The proofs require not only the geometry of compactified tropicalizations of subvarieties of toric varieties, but also new results about the geometry of finite type schemes over non-noetherian valuation rings of rank 1. In particular, we prove subadditivity of codimension and a principle of continuity for intersections in smooth schemes over such rings, generalizing well-known theorems over regular local rings. An appendix on the topology of finite type morphisms may also be of independent interest.
Factorial cluster algebras
Authors: Geiß Christof , Leclerc Bernard , Schröer Jan
Published: January 2023

We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an application we give a criterion for a cluster algebra to be a factorial algebra. This can be used to construct cluster algebras, which are isomorphic to polynomial rings. We also study various kinds of upper bounds for cluster algebras, and we prove that factorial cluster algebras coincide with their upper bounds.
On 14-dimensional quadratic forms in $I^3$, 8-dimensional forms in $I^2$, and the common value property
Authors: Hoffmann Detlef W. , Tignol Jean-Pierre
Published: January 2023

Let

F

be a field of characteristic

\neq 2

. We define certain properties

D(n), n\in\{ 2,4,8,14\}

, of

F

as follows :

F

has property

D(14)

if each quadratic form

\varphi\in I^3F

of dimension

14

is similar to the difference of the pure parts of two 3-fold Pfister forms;

F

has property

D(8)

if each form

\varphi\in I^2F

of dimension 8 whose Clifford invariant can be represented by a biquaternion algebra is isometric to the orthogonal sum of two forms similar to 2-fold Pfister forms;

F

has property

D(4)

if any two 4-dimensional forms over

F

of the same determinant which become isometric over some quadratic extension always have (up to similarity) a common binary subform;

F

has property

D(2)

if for any two binary forms over

F

and for any quadratic extension

E/F

we have that if the two binary forms represent over

E

a common nonzero element, then they represent over

E

a common nonzero element in

F

. Property

D(2)

has been studied earlier by Leep, Shapiro, Wadsworth and the second author. In particular, fields where

D(2)

does not hold have been known to exist.
On the generalized semi-relativistic Schrödinger-Poisson system in $\mathbb{R}^n$
Authors: Salem Walid Abou , Chen Thomas , Vougalter Vitali
Published: January 2023

The Cauchy problem for the semi-relativistic Schrödinger-Poisson system of equations is studied in

R^n, n \ge 1

, for a wide class of nonlocal interactions. Furthermore, the asymptotic behavior of the solution as the mass tends to infinity is rigorously discussed, and compared with solutions to the non-relativistic Schrödinger-Poisson system.
Metrics on states from actions of compact groups
Authors: Rieffel Marc A.
Published: January 2023

Let a compact Lie group act ergodically on a unital

C^*

-algebra

A

. We consider several ways of using this structure to define metrics on the state space of

A

. These ways involve length functions, norms on the Lie algebra, and Dirac operators. The main thrust is to verify that the corresponding metric topologies on the state space agree with the weak-* topology.
Elliptic curves on some homogeneous spaces
Authors: Pasquier B. , Perrin N.
Published: January 2023

Let

X

be a minuscule homogeneous space, an odd-dimensional quadric, or an adjoint homogenous space of type different from

A

and

G_2

. Le

C

be an elliptic curve. In this paper, we prove that for

d

large enough, the scheme of degree

d

morphisms from

C

to

X

is irreducible, giving an explicit lower bound for

d

which is optimal in many cases.
P-jets of finite algebras. I: $p$-divisible groups
Authors: Buium Alexandru
Published: January 2023

p

-jets of finite flat maps of schemes are generally neither finite nor flat. However, for

p

-isogenies, and in particular for

p

-divisible groups, this pathology tends to disappear “in the limit”. We illustrate this in the case of

G_m

, elliptic curves, and formal groups of finite height.
P-jets of finite algebras. II: $p$-typical Witt rings
Authors: Buium Alexandru
Published: January 2023

We study the structure of the

p

-jet spaces of the

p

-typical Witt rings of the

p

-adics. We also study the

p

-jets of the comonad map. These data can be viewed as an arithmetic analogue, for the ring

\mathbb Z

, of the Lie groupoid of the line and hence as an infinitesimal version of the Galois group of

\mathbb Q

over “

\mathbb F_1

”.
On Artin representations and nearly ordinary Hecke algebras over totally real fields
Authors: Sasaki Shu
Published: January 2023

We prove many new cases of the strong Artin conjecture for two-dimensional, totally odd, insoluble (icosahedral) representations

\mathrm{Gal}(\overline{F}/F)\rightarrow GL_2(\mathbf{C})

of the absolute Galois group of a totally real field

F

.
Burniat surfaces. III: Deformations of automorphisms and extended Burniat surfaces
Authors: Bauer I. , Catanese F.
Published: January 2023

We continue our investigation of the connected components of the moduli space of surfaces of general type containing the Burniat surfaces, correcting a mistake in part II. We define the family of extended Burniat surfaces with

K_S^2 = 4

, resp. 3, and prove that they are a deformation of the family of nodal Burniat surfaces with

K_S^2 = 4

, resp. 3. We show that the extended Burniat surfaces together with the nodal Burniat surfaces with

K_S^2=4

form a connected component of the moduli space. We prove that the extended Burniat surfaces together with the nodal Burniat surfaces with

K_S^2=3

form an irreducible open set in the moduli space. Finally we point out an interesting pathology of the moduli space of surfaces of general type given together with a group of automorphisms

G

. In fact, we show that for the minimal model

S

of a nodal Burniat surface

(G = (\mathbb Z/2 \mathbb Z)^2)

we have

\operatorname{Def}(S,G) \neq \operatorname{Def}(S)

, whereas for the canonical model

X

it holds

\operatorname{Def}(X,G) =\operatorname{Def}(X)

. All deformations of

S

have a

G

-action, but there are different deformation types for the pairs

(S,G)

of the minimal models

S

together with the

G

-action, while the pairs

(X,G)

have a unique deformation type.
Evidence for a generalization of Gieseker's conjecture on stratified bundles in positive characteristic
Authors: Kindler Lars
Published: January 2023

Let

X

be a smooth, connected, projective variety over an algebraically closed field of positive characteristic. In [Gie75], Gieseker conjectured that every stratified bundle (i.e. every

\mathcal O_X

-coherent

\mathscr{D}_{X/k}

-module) on

X

is trivial, if and only if

\pi_1^{\mathrm{ét}}(X)=0

. This was proven by Esnault–Mehta, [EM10]. Building on the classical situation over the complex numbers, we present and motivate a generalization of Gieseker's conjecture, using the notion of regular singular stratified bundles developed in the author's thesis and [Kin12a]. In the main part of this article we establish some important special cases of this generalization; most notably we prove that for not necessarily proper

X, \pi_1^{\mathrm{tame}}(X)=0

implies that there are no nontrivial regular singular stratified bundles with abelian monodromy.
Which moments of a logarithmic derivative imply quasiinvariance
Authors: Scheutzow Michael , von Weizsäcker Heinrich
Published: January 2023

In many special contexts quasiinvariance of a measure under a one-parameter group of transformations has been established. A remarkable classical general result of A.V. Skorokhod citeSkorokhod74 states that a measure

\mu

on a Hilbert space is quasiinvariant in a given direction if it has a logarithmic derivative

\beta

in this direction such that

e^{a|\beta|}

is

\mu

-integrable for some

a > 0

. In this note we use the techniques of citeSmolyanov-Weizsaecker93 to extend this result to general one-parameter families of measures and moreover we give a complete characterization of all functions

\psi:[0,\infty) \rightarrow [0,\infty)

for which the integrability of

\psi(|\beta|)

implies quasiinvariance of

\mu

. If

\psi

is convex then a necessary and sufficient condition is that

\log \psi(x)/{x^2}

is not integrable at

\infty

.
Exact criterion for global existence and blow up to a degenerate Keller-Segel system
Authors: Chen Li , Wang Jinhuan
Published: January 2023

A degenerate Keller-Segel system with diffusion exponent

m

with

\frac{2n}{n+2}<m<2-\frac{2}n

in multi dimension is studied. An exact criterion for global existence and blow up of solution is obtained. The estimates on

L^{\frac{2n}{n+2}}

norm of the solution play important roles in our analysis. These estimates are closely related to the optimal constant in the Hardy- Littlewood- Sobolev inequality. In the case of initial free energy less than a universal constant which depends on the inverse of total mass, there exists a constant such that if the

L^{\frac{2n}{n+2}}

norm of initial data is less than this constant, then the weak solution exists globally; if the

L^{\frac{2n}{n+2}}

norm of initial data is larger than the same constant, then the solution must blow-up in finite time. Our result shows that the total mass, which plays the deterministic role in two dimension case, might not be an appropriate criterion for existence and blow up discussion in multi-dimension, while the

L^{\frac{2n}{n+2}}

norm of the initial data and the relation between initial free energy and initial mass are more important.
An equivariant Lefschetz fixed-point formula for correspondences
Authors: Dell'Ambrogio Ivo , Emerson Heath , Meyer Ralf
Published: January 2023

We compute the trace of an endomorphism in equivariant bivariant K-theory for a compact group (G) in several ways: geometrically using geometric correspondences, algebraically using localisation, and as a Hattori–Stallings trace. This results in an equivariant version of the classical Lefschetz fixed-point theorem, which applies to arbitrary equivariant correspondences, not just maps. textitWe dedicate this article to Tamaz Kandelaki, who was a coauthor in an earlier version of this article, and passed away in 2012. We will remember him for his warm character and his perseverance in doing mathematics in difficult circumstances.
On quadratic Diophantine equations in four variables and orders associated with lattices
Authors: Murata Manabu
Published: January 2023

This paper treats certain lattices in ternary quadratic spaces, which are obtained from the data of a non-zero element and a maximal lattice in a quaternary space. Each class in the genus of such a lattice with respect to the special orthogonal group corresponds to an isomorphism class in the genus of an order associated with the lattice in a quaternion algebra. Using this result together with the principle of Shimura, we show that the number of classes of the primitive solutions of a quadratic Diophantine equation in four variables coincides with the type number of the order under suitable conditions on the given element. We illustrate this result by explicit examples.
The first $L^2$-Betti number and approximation in arbitrary characteristic
Authors: Ershov Mikhail , Lück Wolfgang
Published: January 2023

Let

G

be a finitely generated group and

G = G_0 \supseteq G_1 \supseteq G_2 \supseteq \cdots

a descending chain of finite index normal subgroups of

G

. Given a field

K

, we consider the sequence

\frac{b_1(G_i;K)}{[G:G_i]}

of normalized first Betti numbers of

G_i

with coefficients in

K

, which we call a

K

-approximation for

b_1^{(2)}(G)

, the first

L^2

-Betti number of

G

. In this paper we address the questions of when

\mathbb Q

-approximation and

\mathbb F_p

-approximation have a limit, when these limits coincide, when they are independent of the sequence

(G_i)

and how they are related to

b_1^{(2)}(G)

. In particular, we prove the inequality

\lim_{i\to\infty} \frac{b_1(G_i;\mathbb F_p)}{[G:G_i]}\geq b_1^{(2)}(G)

under the assumptions that

\cap G_i={1}

and each

G/G_i

is a finite

p

-group.
The Jordan-Hölder series of the locally analytic Steinberg representation
Authors: Orlik Sascha , Schraen Benjamin
Published: January 2023

We determine the composition factors of a Jordan-Hölder series including multiplicities of the locally analytic Steinberg representation. For this purpose, we prove the acyclicity of the evaluated locally analytic Tits complex giving rise to the Steinberg representation. Further we describe some analogue of the Jacquet functor applied to the irreducible principal series representation constructed in citeOS2.
Corrigendum to: “Andreotti-Mayer loci and the Schottky problem” [cf. Documenta Math. 13 (2008) 398–440]
Authors: Ciliberto Ciro , van der Geer Gerard
Published: January 2023

The main result of [2] (C. Ciliberto, G. van der Geer: Andreotti-Mayer Loci and the Schottky problem. Documenta Math. 13 (2008), 389–440) is based on a proposition whose proof is incorrect. We therefore give an alternative proof of the main result of [2].
Szpiro's small points conjecture for cyclic covers
Authors: Javanpeykar Ariyan , von Känel Rafael
Published: January 2023

Let

X

be a smooth, projective and geometrically connected curve of genus at least two, defined over a number field. In 1984, Szpiro conjectured that

X

has a «small point». In this paper we prove that if

X

is a cyclic cover of prime degree of the projective line, then

X

has infinitely many «small points». In particular, we establish the first cases of Szpiro's small points conjecture, including the genus two case and the hyperelliptic case. The proofs use Arakelov theory for arithmetic surfaces and the theory of logarithmic forms.
Chern classes of fibered products of surfaces
Authors: Teicher Mina
Published: January 2023

In this paper we introduce a formula to compute Chern classes of fibered products of algebraic surfaces. For

f: X\to \mathbb{CP}^2

a generic projection of an algebraic surface, we define

X_k

for

k\le n(n=\deg f)

to be the closure of

k

products of

X

over

f

minus the big diagonal. For

k=n

(or

n-1), X_k

is called the full Galois cover of

f

w.r.t. full symmetric group. We give a formula for

c_1^2

and

c_2

of

X_k.

For

k=n

the formulas were already known. We apply the formula in two examples where we manage to obtain a surface with a high slope of

c_1^2/c_2.

We pose conjectures concerning the spin structure of fibered products of Veronese surfaces and their fundamental groups.
On the structure of Witt–Burnside rings attached to pro-$p$ groups
Authors: Miller Lance Edward
Published: January 2023

The

p

-typical Witt vectors are a ubiquitous object in algebra and number theory. They arise as a functorial construction that takes perfect fields

k

of prime characteristic

p > 0

to

p

-adically complete discrete valuation rings of characteristic 0 with residue field

k

and are universal in that sense. A. Dress and C. Siebeneicher generalized this construction by producing a functor

\mathbf W_G

attached to any profinite group

G

. The

p

-typical Witt vectors arise as those attached to the

p

-adic integers. Here we examine the ring structure of

\mathbf W_G(k)

for several examples of pro-

p

groups

G

and fields

k

of characteristic

p

. We will show that the structure is surprisingly more complicated than the

p

-typical case.
Log homogeneous compactifications of some classical groups
Authors: Huruguen Mathieu
Published: January 2023

We generalize in positive characteristics some results of Bien and Brion on log homogeneous compactifications of a homogeneous space under the action of a connected reductive group. We also construct an explicit smooth log homogeneous compactification of the general linear group by successive blow-ups starting from a grassmannian. By taking fixed points of certain involutions on this compactification, we obtain smooth log homogeneous compactifications of the special orthogonal and the symplectic groups.
The variation of the monodromy group in families of stratified bundles in positive characteristic
Authors: Battiston Giulia
Published: January 2023

In this article we study smooth families of stratified bundles in positive characteristic and the variation of their monodromy group. Our aim is, in particular, to strengthen the weak form of the positive equicharacteristic

p

-curvature conjecture stated and proved by Esnault and Langer in citeEL:pcur. The main result is that if the ground field is uncountable then the strong form holds. In the case where the ground field is countable we provide positive and negative answers to possible generalizations.
Rokhlin dimension: obstructions and permanence properties
Authors: Hirshberg Ilan , Phillips N.Christopher
Published: January 2023

This paper is a further study of finite Rokhlin dimension for actions of finite groups and the integers on

C^*

-algebras, introduced by the first author, Winter, and Zacharias. We extend the definition of finite Rokhlin dimension to the nonunital case. This definition behaves well with respect to extensions, and is sufficient to establish permanence of finite nuclear dimension and

\mathcal Z

-absorption. We establish

K

-theoretic obstructions to the existence of actions of finite groups with finite Rokhlin dimension (in the commuting tower version). In particular, we show that there are no actions of any nontrivial finite group on the Jiang-Su algebra or on the Cuntz algebra

\mathcal O_{\infty}

with finite Rokhlin dimension in this sense.
Linear Koszul duality and Fourier transform for convolution algebras
Authors: Mirković Ivan , Riche Simon
Published: January 2023

In this paper we prove that the
linear Koszul duality
isomorphism for convolution algebras in

K

-homology of [MR3] and the
Fourier transform
isomorphism for convolution algebras in Borel–Moore homology of [EM] are related by the Chern character. So, Koszul duality appears as a categorical upgrade of Fourier transform of constructible sheaves. This result explains the connection between the categorification of the Iwahori–Matsumoto involution for graded affine Hecke algebras in [EM] and for ordinary affine Hecke algebras in [MR3].
Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case
Authors: Bischoff Marcel , Kawahigashi Yasuyuki , Longo Roberto
Published: January 2023

Let

\mathcal A

be a completely rational local Möbius covariant net on

S^1

, which describes a set of chiral observables. We show that local Möbius covariant nets

\mathcal B_2

on 2D Minkowski space which contains

\mathcal A

as chiral left-right symmetry are in one-to-one correspondence with Morita equivalence classes of Q-systems in the unitary modular tensor category

\mathrm{DHR}(\mathcal A)

. The Möbius covariant boundary conditions with symmetry

\mathcal A

of such a net

\mathcal B_2

are given by the Q-systems in the Morita equivalence class or by simple objects in the module category modulo automorphisms of the dual category. We generalize to reducible boundary conditions. To establish this result we define the notion of Morita equivalence for Q-systems (special symmetric

\ast

-Frobenius algebra objects) and non-degenerately braided subfactors. We prove a conjecture by Kong and Runkel, namely that Rehren's construction (generalized Longo–Rehren construction,

\alpha

-induction construction) coincides with the categorical full center. This gives a new view and new results for the study of braided subfactors.
Arithmetic quotients of the complex ball and a conjecture of Lang
Authors: Dimitrov Mladen , Ramakrishnan Dinakar
Published: January 2023

We prove that various arithmetic quotients of the unit ball in

C^{n}

are Mordellic, in the sense that they have only finitely many rational points over any finitely generated field extension of Q. In the previously known case of compact hyperbolic complex surfaces, we give a new proof using their Albanese in conjunction with some key results of Faltings, but without appealing to the Shafarevich conjecture. In higher dimension, our methods allow us to solve an alternative of Ullmo and Yafaev. Our strongest result uses in addition Rogawski's theory and establishes the Mordellicity of the Baily-Borel compactifications of Picard modular surfaces of some precise levels related to the discriminant of the imaginary quadratic fields.
On the smoothness of normalisers, the subalgebra structure of modular Lie algebras, and the cohomology of small representations
Authors: Herpel Sebastian , Stewart David I.
Published: January 2023

We provide results on the smoothness of normalisers in connected reductive algebraic groups

G

over fields

k

of positive characteristic

p

. Specifically we we give bounds on

p

which guarantee that normalisers of subalgebras of

\mathfrak{g}

in

G

are smooth, i.e.so that the Lie algebras of these normalisers coincide with the infinitesimal normalisers. One of our main tools is to exploit cohomology vanishing of small dimensional modules. Along the way, we obtain complete reducibility results for small dimensional modules in the spirit of similar results due to Jantzen, Guralnick, Serre and Bendel–Nakano–Pillen.
Operator-valued monotone convolution semigroups and an extension of the Bercovici-Pata bijection.
Authors: Anshelevich Michael , Williams John D.
Published: January 2023

In a 1999 paper, Bercovici and Pata showed that a natural bijection between the classically, free and Boolean infinitely divisible measures held at the level of limit theorems of triangular arrays. This result was extended to include monotone convolution by the authors in [Ans-Williams-Chernoff]. In recent years, operator-valued versions of free, Boolean and monotone probability have also been developed. Belinschi, Popa and Vinnikov showed that the Bercovici-Pata bijection holds for the operator-valued versions of free and Boolean probability. In this article, we extend the bijection to include monotone probability theory even in the operator-valued case. To prove this result, we develop the general theory of composition semigroups of non-commutative functions and largely recapture Berkson and Porta's classical results on composition semigroups of complex functions in operator-valued setting. As a byproduct, we deduce that operator-valued monotonically infinitely divisible distributions belong to monotone convolution semigroups. Finally, in the appendix, we extend the result of the second author on the classification of Cauchy transforms for non-commutative distributions to the Cauchy transforms associated to more general completely positive maps.
Difference scheme for the Vlasov-Manev system
Authors: Illner R. , Rjasanow S.
Published: January 2023

We develop and test a finite difference scheme for the Vlasov-Manev Equation in one space and one velocity dimension. The Manev correction to the Newtonian potential produces visible qualitative differences in the behaviour of stellar systems; the most notable effect observed in this paper is a stabilisation of the separate identities of two Maxwellian concentrations at different locations.
On quadratic forms of height 3 and degree $\leq 2$
Authors: Laghribi Ahmed
Published: January 2023

Let

F

be a commutative field of characteristic not 2. In this paper, we give some results on the classification of

F

-quadratic forms of height 3 and degree

\leq 2

.
Liberté et accumulation
Authors: Peyre Emmanuel
Published: January 2023

Le principe de Batyrev et Manin et ses variantes donnent une interprétation conjecturale précise pour le terme dominant du nombre de points de hauteur bornée d'une variété algébrique dont l'opposé du faisceau canonique est suffisamment positif. Comme l'a clairement montré le contre-exemple de Batyrev et Tschinkel la mise en œuvre de ce principe nécessite l'exclusion de domaines d'accumulation qui sont le plus souvent déterminés en procédant par récurrence sur la dimension de la variété. Cette méthode ne donne cependant pas de critère direct permettant de dire si un point rationnel donné doit être exclu ou pas. L'ambition de cet article est de définir une mesure de la liberté d'un point rationnel de sorte que les points d'une liberté suffisante se répartissent effectivement de manière uniforme sur la variété, c'est-à-dire qu'ils soient distribués sur l'espace adélique associé à la variété conformément à la mesure de distribution adélique introduite dans un article antérieur de l'auteur. De ce point de vue, les points assez libres devraient être ceux qui respectent le principe de Batyrev et Manin.
Graded Frobenius Cluster Categories
Authors: Grabowski Jan E. , Pressland Matthew
Published: January 2023

Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi-Yau triangulated categories with a choice of cluster-tilting object. The grading on the category corresponds to a grading on the cluster algebra without coefficients categorified by the cluster category and hence knowledge of one of these structures can help us study the other. In this work, we extend the above to certain Frobenius categories that categorify cluster algebras with coefficients. We interpret the grading K-theoretically and prove similar results to the triangulated case, in particular obtaining that degrees are additive on exact sequences. We show that the categories of Buan, Iyama, Reiten and Scott, some of which were used by Geiß, Leclerc and Schröer to categorify cells in partial flag varieties, and those of Jensen, King and Su, categorifying Grassmannians, are examples of graded Frobenius cluster categories.
On Commutative Nonarchimedean Banach Fields
Authors: Kedlaya Kiran S.
Published: January 2023

We study the problem of whether a commutative nonarchimedean Banach ring which is algebraically a field can be topologized by a multiplicative norm. This can fail in general, but it holds for uniform Banach rings under some mild extra conditions. Notably, any perfectoid ring whose underlying ring is a field is a perfectoid field.
On the Homotopy Exact Sequence for Log Algebraic Fundamental Groups
Authors: Di Proietto Valentina , Shiho Atsushi
Published: January 2023

We construct a log algebraic version of the homotopy sequence for a normal crossing log variety over a log point of characteristic zero and prove some exactness properties of it. Our proofs are purely algebraic.
On the automorphism group of a complex sphere
Authors: Brunella Marco
Published: January 2023

Let

X

be a compact complex threefold with the integral homology of

{\bf S}^6

and let

Aut(X)

be its holomorphic automorphism group. By [HKP] and [CDP] the dimension of

Aut(X)

is at most 2. We prove that

Aut(X)

cannot be isomorphic to the complex affine group.
The Motivic Hopf Map Solves the Homotopy Limit Problem for $K$-Theory
Authors: Röndigs Oliver , Spitzweck Markus , Østvær Paul Arne
Published: January 2023

We solve affirmatively the homotopy limit problem for (effective)

K

-theory over fields of finite virtual cohomological dimension. Our solution uses the motivic slice filtration and the first motivic Hopf map.
The Log Term in the Bergman and Szegő Kernels in Strictly Pseudoconvex Domains in $\mathbb C^2$
Authors: Ebenfelt Peter
Published: January 2023

In this paper, we consider bounded strictly pseudoconvex domains

D\subset \mathbb C^2

with smooth boundary

M=M^3:=\partial D

, and the asymptotic expansion of the Bergman kernel on the diagonal

K_B\sim \frac{\phi_{B}}{\rho^{n+1}}+\psi_B\log\rho,

where

\rho>0

is a Fefferman defining equation for

D

. The Ramadanov Conjecture states that if the log term

\psi_B

vanishes to infinite order on

M

, then

M

is locally spherical. In

\mathbb C^2

, the validity of this conjecture is known and follows from work of Boutet de Monvel, Burns, and Graham; indeed, it suffices that

\psi_B=O(\rho^2)

locally on

M

to conclude that

M

is locally spherical. On the other hand, it is also known that the boundary values alone of the log term

b\psi_B:=(\psi_B)|_M

on

M

does not determine the CR geometry of

M

locally; e.g., the vanishing of

b\psi_B

on an open subset of

M

does not imply that

M

is locally spherical there. The main result in this paper, however, is that if

D\subset \mathbb C^2

is assumed to have transverse symmetry, then the global vanishing of

b\psi_B

on

M

implies that

M

is locally spherical. A similar result is proved for the Szegő kernel.
Einstein metrics and stability for flat connections on compact Hermitian manifolds, and a correspondence with Higgs operators in the surface case
Authors: Lübke M.
Published: January 2023

A flat complex vector bundle

(E,D)

on a compact Riemannian manifold

(X,g)

is stable (resp. polystable) in the sense of Corlette [C] if it has no

D

-invariant subbundle (resp. if it is the

D

-invariant direct sum of stable subbundles). It has been shown in [C] that the polystability of

(E,D)

in this sense is equivalent to the existence of a so-called harmonic metric in

E

. In this paper we consider flat complex vector bundles on compact Hermitian manifolds

(X,g)

. We propose new notions of

g

-(poly-)stability of such bundles, and of

g

-Einstein metrics in them; these notions coincide with (poly-)stability and harmonicity in the sense of Corlette if

g

is a Kähler metric, but are different in general. Our main result is that the

g

-polystability in our sense is equivalent to the existence of a

g

-Hermitian-Einstein metric. Our notion of a

g

-Einstein metric in a flat bundle is motivated by a correspondence between flat bundles and Higgs bundles over compact surfaces, analogous to the correspondence in the case of Kähler manifolds [S1], [S2], [S3].
$p$-Adic Deformation of Motivic Chow Groups
Authors: Langer Andreas
Published: January 2023

For a smooth projective scheme

Y

over

W(k)

we consider an element in the motivic Chow group of the reduction

Y_{m}

over the truncated Witt ring

W_{m}(k)

and give a “Hodge” criterion – using the crystalline cycle class in relative crystalline cohomology – for the element to lift to the continuous Chow group of the associated

p

-adic formal scheme

Y_\bullet

. The result extends previous work of Bloch–Esnault–Kerz on the

p

-adic variational Hodge conjecture to a relative setting. In the course of the proof we derive two new results on the relative de Rham–Witt complex and its Nygaard filtration, and work with a relative version of syntomic complexes to define relative motivic complexes for a smooth lifting of

Y_{m}

over the ind-scheme

\mathrm{Spec}\,W_{\bullet}(W_{m}(k))

.
Functoriality Properties of the Dual Group
Authors: Knop Friedrich
Published: January 2023

Let

G

be a connected reductive group. Previously, it was shown that for any

G

-variety

X

one can define the dual group

G^\vee_X

which admits a natural homomorphism with finite kernel to the Langlands dual group

G^\vee

of

G

. Here, we prove that the dual group is functorial in the following sense: if there is a dominant

G

-morphism

X\to Y

or an injective

G

-morphism

Y\to X

then there is a unique homomorphism with finite kernel

G^\vee_Y\to G^\vee_X

which is compatible with the homomorphisms to

G^\vee

.
What are cumulants?
Authors: Mattner Lutz
Published: January 2023

Let

\mathcal P

be the set of all probability measures on

\mathbb R

possessing moments of every order. Consider

\mathcal P

as a semigroup with respect to convolution. After topologizing

\mathcal P

in a natural way, we determine all continuous homomorphisms of

\mathcal P

into the unit circle and, as a corollary, those into the real line. The latter are precisely the finite linear combinations of cumulants, and from these all the former are obtained via multiplication by

i

and exponentiation.
Rational Curves on Fibered Calabi-Yau Manifolds
Authors: Diverio Simone , Fontanari Claudio , Martinelli Diletta
Published: January 2023

We show that a smooth projective complex manifold of dimension greater than two endowed with an elliptic fiber space structure and with finite fundamental group always contains a rational curve, provided its canonical bundle is relatively trivial. As an application of this result, we prove that any Calabi-Yau manifold that admits a fibration onto a curve whose general fiber is an abelian variety always contains a rational curve.
Some properties of the symmetric enveloping algebra of a subfactor, with applications to amenability and property $T$
Authors: Popa Sorin
Published: January 2023

We undertake here a more detailed study of the structure and basic properties of the symmetric enveloping algebra

M\boxtimes_{e_N} M^{\mathrm{op}}

associated to a subfactor

N\subset M

, as introduced in [Po5]. We prove a number of results relating the amenability properties of the standard invariant of

N\subset M,{\mathcal G}_{N,M}

, its graph

\Gamma_{N,M}

and the inclusion

M\vee M^{\mathrm{op}} \subset M\boxtimes_{e_N}M^{\mathrm{op}}

, notably showing that

M\boxtimes_{e_N}M^{\mathrm{op}}

is amenable relative to its subalgebra

M\vee M^{\mathrm{op}}

iff

\Gamma_{N,M}

(or equivalently

{\mathcal G}_{N,M})

is amenable, i.e.,

\|\Gamma_{N,M}\|^2=[M:N]

. We then prove that the hyperfiniteness of

M\boxtimes_{e_N}M^{\mathrm{op}}

is equivalent to

M

being hyperfinite and

\Gamma_{N,M}

being amenable. We derive from this a hereditarity property for the amenability of graphs of subfactors showing that if an inclusion of factors

Q\subset P

is embedded into an inclusion of hyperfinite factors

N\subset M

with amenable graph, then its graph

\Gamma_{Q,P}

follows amenable as well. Finally, we use the symmetric enveloping algebra to introduce a notion of property T for inclusions

N\subset M

, by requiring

M\boxtimes_{e_N}M^{\mathrm{op}}

to have the property T relative to

M\vee M^{\mathrm{op}}

. We prove that this property doesn't in fact depend on the inclusion

N\subset M

but only on its standard invariant

\mathcal G_{N,M}

, thus defining a notion of property T for abstract standard lattices

\mathcal G

.
Parabolic Conformally Symplectic Structures. III: Invariant Differential Operators and Complexes
Authors: Čap Andreas , Salač Tomáš
Published: January 2023

This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study of these structures, the second part focused on the case that the underlying structure is conformally symplectic (PCS-structures). In that case, we obtained a close relation to parabolic contact structures via a concept of parabolic contactification. It was also shown that special symplectic connections (and thus all connections of exotic symplectic holonomy) arise as the canonical connection of such a structure.

In this last part, we use parabolic contactifications and constructions related to Bernstein-Gelfand-Gelfand (BGG) sequences for parabolic contact structures, to construct sequences of differential operators naturally associated to a PCS-structure. In particular, this gives rise to a large family of complexes of differential operators associated to a special symplectic connection. In some cases, large families of complexes for more general instances of PCS-structures are obtained.