Advances in Mathematics最新文献

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Maximality of correspondence representations

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110097

Boris Bilich

In this paper, we fully characterize maximal representations of a C*-correspondence, thereby strengthening several earlier results. We demonstrate the maximality criteria through diverse examples. We also describe the noncommutative Choquet boundary and provide additional counterexamples to Arveson's hyperrigidity conjecture following the counterexample recently found by the author and Dor-On. Furthermore, we identify several classes of correspondences for which the hyperrigidity conjecture holds.

引用次数: 0

Flag versions of quiver Grassmannians for Dynkin quivers have no odd cohomology

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110078

Ruslan Maksimau

We prove the conjecture that flag versions of quiver Grassmannians (also known as Lusztig's fibers) for Dynkin quivers (types A, D, E) have no odd cohomology groups over an arbitrary ring. Moreover, for types A and D we prove that these varieties have affine pavings. We also show that to prove the same statement for type E, it is enough to check this for indecomposable representations.

We also give a flag version of the result of Cerulli Irelli-Esposito-Franzen-Reineke on rigid representations: we prove that flag versions of quiver Grassmannians for rigid representations have a diagonal decomposition. In particular, they have no odd cohomology groups.

引用次数: 0

Bosonization of Feigin-Odesskii Poisson varieties

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110096

Zheng Hua , Alexander Polishchuk

The derived moduli stack of complexes of vector bundles on a Gorenstein Calabi-Yau curve admits a 0-shifted Poisson structure. Projective spaces with Feigin-Odesskii Poisson brackets are examples of such moduli spaces over complex elliptic curves [6], [7]. By generalizing several results in our previous work [10], [11], [12] we construct a collection of auxiliary Poisson varieties equipped with Poisson morphisms to Feigin-Odesskii varieties. We call them bosonizations of Feigin-Odesskii varieties. These spaces appear as special cases of the moduli spaces of chains, which we introduce. We show that the moduli space of chains admits a shifted Poisson structure when the base is a Calabi-Yau variety of an arbitrary dimension. Using bosonization spaces mapping to the zero loci of the Feigin-Odesskii varieties, we show that the Feigin-Odesskii Poisson brackets on projective spaces (associated with stable bundles of arbitrary rank on elliptic curves) admit no infinitesimal symmetries. We also derive explicit formulas for the Poisson brackets on the bosonizations of the Feigin-Odesskii varieties associated with line bundles in a simplest nontrivial case.

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引用次数: 0

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

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