Advances in Mathematics最新文献

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Monochromatic products and sums in 2-colorings of N

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110095

Matt Bowen

We show that any 2-coloring of

N

contains infinitely many monochromatic sets of the form

{x, y, x y, x + y}

, and more generally monochromatic sets of the form

{x_{i}, \prod x_{i}, \sum x_{i} : i \leq n}

for any

n \in N

. Along the way we prove a monochromatic products of sums theorem that extends Hindman's theorem and a colorful variant of this result that holds in any ‘balanced’ coloring.

引用次数: 0

Mirror of orbifold singularities in the Hitchin fibration: The case (SLn,PGLn)

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110076

Yongbin Ruan , Cheng Shu

We study the geometry of singular

{SL}_{n}

-Hitchin fibres over the elliptic locus. We show that orbifold singularities appear in the

{PGL}_{n}

-moduli space

M_{C}^{e l l} ({PGL}_{n})

exactly when the

{SL}_{n}

side

M_{C}^{e l l} ({SL}_{n})

has a reducible Hitchin fibre. Our main theorem shows that the Fourier-Mukai transform of a skyscraper sheaf supported at an orbifold singularity in

M_{C}^{e l l} ({PGL}_{n})

satisfies a version of the fractional Hecke eigenproperty, as conjectured by Frenkel and Witten.

引用次数: 0

On the codimension of permanental varieties

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110079

A. Boralevi , E. Carlini , M. Michałek , E. Ventura

In this article, we study permanental varieties, i.e., varieties defined by the vanishing of permanents of fixed size of a generic matrix. Permanents and their varieties play an important, and sometimes poorly understood, role in combinatorics. However, there are essentially no geometric results about them in the literature, in very sharp contrast to the well-behaved and ubiquitous case of determinants and minors. Motivated by the study of the singular locus of the permanental hypersurface, we focus on the codimension of these varieties. We introduce a

C^{⁎}

-action on matrices and prove a number of results. In particular, we improve a lower bound on the codimension of the aforementioned singular locus established by von zur Gathen in 1987.

引用次数: 0

Border subrank via a generalised Hilbert-Mumford criterion

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110077

Benjamin Biaggi , Chia-Yu Chang , Jan Draisma , Filip Rupniewski

We show that the border subrank of a sufficiently general tensor in

{(C^{n})}^{\otimes d}

O (n^{1 / (d - 1)})

for

n \to \infty

. Since this matches the growth rate

Θ (n^{1 / (d - 1)})

for the generic (non-border) subrank recently established by Derksen-Makam-Zuiddam, we find that the generic border subrank has the same growth rate. In our proof, we use a generalisation of the Hilbert-Mumford criterion that we believe will be of independent interest.

引用次数: 0

Non-optimal levels of some reducible mod p modular representations

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110074

Shaunak V. Deo

Let

p \geq 5

be a prime, N be an integer not divisible by p,

{\bar{ρ}}_{0}

be a reducible, odd and semi-simple representation of

G_{Q, N p}

of dimension 2 and

{ℓ_{1}, \dots, ℓ_{r}}

be a set of primes not dividing Np. After assuming that a certain Selmer group has dimension at most 1, we find sufficient conditions for the existence of a cuspidal eigenform f of level

N \prod_{i = 1}^{r} ℓ_{i}

and appropriate weight lifting

{\bar{ρ}}_{0}

such that f is new at every

ℓ_{i}

. Moreover, suppose

p | ℓ_{i_{0}} + 1

for some

1 \leq i_{0} \leq r

. Then, after assuming that a certain Selmer group vanishes, we find sufficient conditions for the existence of a cuspidal eigenform of level

N ℓ_{i_{0}}^{2} \prod_{j \neq i_{0}} ℓ_{j}

and appropriate weight which is new at every

ℓ_{i}

and which lifts

{\bar{ρ}}_{0}

. As a consequence, we prove a conjecture of Billerey–Menares in many cases.

引用次数: 0

On the existence of weighted-cscK metrics

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-31 DOI: 10.1016/j.aim.2025.110125

Jiyuan Han , Yaxiong Liu

In this paper, we prove that on a smooth Kähler manifold, the

G

-coercivity of the weighted Mabuchi functional implies the existence of the

(v, w)

-weighted-cscK (extremal) metric with v log-concave (firstly studied in [33]), e.g., extremal metrics, Kähler–Ricci solitons, μ-cscK metrics.

引用次数: 0

Stability of homomorphisms, coverings and cocycles II: Examples, applications and open problems

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-30 DOI: 10.1016/j.aim.2025.110117

Michael Chapman , Alexander Lubotzky

Coboundary expansion (with

F_{2}

coefficients), and variations on it, have been the focus of intensive research in the last two decades. It was used to study random complexes, property testing, and above all Gromov's topological overlapping property.

In part I of this paper, we extended the notion of coboundary expansion (and its variations) to cochains with permutation coefficients, equipped with the normalized Hamming distance. We showed that this gives a unified language for studying covering stability of complexes, as well as stability of group homomorphisms — a topic that drew a lot of attention in recent years.

In this part, we extend the theory to the permutation coefficients setting. This gives some new results, even for

F_{2}

coefficients, opens several new directions of research, and suggests a pattern to proving the existence of non-sofic groups. Along the way, we solve the dimension 2 case of a problem of Gromov, exhibiting a family of bounded degree coboundary expanders with

F_{2}

coefficients.

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引用次数: 0

Sliced skein algebras and geometric Kauffman bracket

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-30 DOI: 10.1016/j.aim.2025.110118

Charles D. Frohman , Joanna Kania-Bartoszynska , Thang T.Q. Lê

The sliced skein algebra of a closed surface of genus g with m punctures,

S = Σ_{g, m}

, is the quotient of the Kauffman bracket skein algebra

S_{ξ} (S)

corresponding to fixing the scalar values of its peripheral curves. We show that the sliced skein algebra of a finite type surface is a domain if the ground ring is a domain. When the quantum parameter ξ is a root of unity we calculate the center of the sliced skein algebra and its PI-degree. Among applications we show that any smooth point of a sliced character variety is an Azumaya point of the skein algebra

S_{ξ} (S)

For any

S L_{2} (C)

-representation ρ of the fundamental group of an oriented connected 3-manifold M and a root of unity ξ with the order of

ξ^{2}

odd, we introduce the ρ-reduced skein module

S_{ξ, ρ} (M)

. We show that

S_{ξ, ρ} (M)

has dimension 1 when M is closed and ρ is irreducible. We also show that if ρ is irreducible the ρ-reduced skein module of a handlebody, as a module over the skein algebra of its boundary, is simple and has the dimension equal to the PI-degree of the skein algebra of its boundary.

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引用次数: 0

The stabilizer bitorsors of the module and algebra harmonic coproducts are equal

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-28 DOI: 10.1016/j.aim.2025.110128

Benjamin Enriquez , Hidekazu Furusho

In earlier work, we constructed a pair of “Betti” and “de Rham” Hopf algebras and a pair of module-coalgebras over this pair, as well as the bitorsors related to both structures (which will be called the “module” and “algebra” stabilizer bitorsors). We showed that Racinet's torsor constructed out of the double shuffle and regularization relations between multiple zeta values is essentially equal to the “module” stabilizer bitorsor, and that the latter is contained in the “algebra” stabilizer bitorsor. In this paper, we show the equality of the “algebra” and “module” stabilizer bitorsors. We reduce the proof to showing the equality of the associated “algebra” and “module” graded Lie algebras. The argument for showing this equality involves the relation of the “algebra” Lie algebra with the kernel of a linear map, the expression of this linear map as a composition of three linear maps, the relation of one of them with the “module” Lie algebra and the computation of the kernel of the other one by discrete topology arguments.

引用次数: 0

Revisiting the moduli space of 8 points on P1

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-28 DOI: 10.1016/j.aim.2025.110126

Klaus Hulek , Yota Maeda

The moduli space of 8 points on

P^{1}

, a so-called ancestral Deligne-Mostow space, is, by work of Kondō, also a moduli space of K3 surfaces. We prove that the Deligne-Mostow isomorphism does not lift to a morphism between the Kirwan blow-up of the GIT quotient and the unique toroidal compactification of the corresponding ball quotient. Moreover, we show that these spaces are not K-equivalent, even though they are natural blow-ups at the unique cusps and have the same cohomology. This is analogous to the work of Casalaina-Martin-Grushevsky-Hulek-Laza on the moduli space of cubic surfaces. The moduli spaces of ordinary stable maps, that is the Fulton-MacPherson compactification of the configuration space of points on

P^{1}

, play an important role in the proof. We further relate our computations to new developments in the minimal model program and the recent work of Odaka. We briefly discuss other cases of moduli space of points on

P^{1}

where a similar behaviour can be observed, hinting at a more general, but not yet fully understood phenomenon.

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引用次数: 0

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