International Mathematics Research Notices

Ornea, Liviu; Verbitsky, Misha

doi: 10.1093/imrn/rnaf014pmid: N/A

A complex Hermitian $n$-manifold $(M,I, \omega )$ is called locally conformally Kähler (LCK) if $d\omega =\theta \wedge \omega $, where $\theta $ is a closed 1-form, balanced if $\omega ^{n-1}$ is closed, and SKT if $dId\omega =0$. We conjecture that any compact complex manifold admitting two of these three types of Hermitian forms (balanced, SKT, LCK) also admits a Kähler metric, and prove partial results towards this conjecture. We conjecture that the (1,1)-form $-d(I\theta )$ is Bott–Chern homologous to a positive (1,1)-current. This conjecture implies that $(M,I)$ does not admit a balanced Hermitian metric. We verify this conjecture for all known classes of LCK manifolds.

Lamzouri, Youness

doi: 10.1093/imrn/rnaf006pmid: N/A

We study the distribution functions of several classical error terms in analytic number theory, focusing on the remainder term in the Dirichlet divisor problem $\Delta (x)$. We first bound the discrepancy between the distribution function of $\Delta (x)$ and that of a corresponding probabilistic random model, improving results of Heath-Brown and Lau. We then determine the shape of its large deviations in a certain uniform range, which we believe to be the limit of our method, given our current knowledge about the linear relations among the $\sqrt{n}$ for square-free positive integers $n$. Finally, we obtain similar results for the error terms in the Gauss circle problem and in the second moment of the Riemann zeta function on the critical line.

Basmajian, Ara; Brisson, Jade; Hassannezhad, Asma; Métras, Antoine

doi: 10.1093/imrn/rnaf001pmid: N/A

We consider the Steklov eigenvalue problem on a compact pinched negatively curved manifold $M$ of dimension at least three with totally geodesic boundaries. We obtain a geometric lower bound for the first nonzero Steklov eigenvalue in terms of the total volume of $M$ and the volume of its boundary. We provide examples illustrating the necessity of these geometric quantities in the lower bound. Our result can be seen as a counterpart of the lower bound for the first nonzero Laplace eigenvalue on closed pinched negatively curved manifolds of dimension at least three proved by Schoen in 1982. The proof is composed of certain key elements. We provide a uniform lower bound for the first eigenvalue of the Steklov–Dirichlet problem on a neighborhood of the boundary of $M$ and show that it provides an obstruction to having a small first nonzero Steklov eigenvalue. As another key element of the proof, we give a tubular neighborhood theorem for totally geodesic hypersurfaces in a pinched negatively curved manifold. We give an explicit dependence for the width function in terms of the volume of the boundary and the pinching constant.

Andreasson, Rolf; Hultgren, Jakob; Jonsson, Mattias; Mazzon, Enrica; McCleerey, Nicholas

doi: 10.1093/imrn/rnaf013pmid: N/A

Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge–Ampère operator on the boundary of a simplex. This can be formulated in terms of optimal transport, and we consider, more generally, the problem of optimal transport between symmetric probability measures on the boundary of a simplex and of the dual simplex. For suitably regular measures, we obtain regularity properties of the transport map, and of its convex potential. To do so, we exploit boundary regularity results for optimal transport maps by Caffarelli, together with the symmetries of the simplex.

Coron, Basile

doi: 10.1093/imrn/rnaf009pmid: N/A

We introduce a new type of operad-like structure called a ${\mathcal{P}}$-operad, which depends on the choice of some collection of posets ${\mathcal{P}}$, and which is governed by chains in posets of ${\mathcal{P}}$. We introduce several examples of such structures that are related to classical poset theoretic notions such as poset homology, Cohen–Macaulayness, and lexicographic shellability. We then show that ${\mathcal{P}}$-operads form a satisfactory framework to categorify Kazhdan–Lusztig polynomials of geometric lattices and their $P$-kernel. In particular, this leads to a new proof of the positivity of the coefficients of Kazhdan–Lusztig polynomials of geometric lattices.

Naryshkin, Petr; Petrakos, Spyridon

doi: 10.1093/imrn/rnaf016pmid: N/A

We introduce the property of having good subgroups for actions of countable discrete groups on compact metrizable spaces and show that it implies comparison when the acting group is amenable. As a consequence, free actions on finite-dimensional spaces of many notable amenable groups of dynamical origin are almost finite. For instance, this applies to topological full groups of Cantor minimal systems and the Basilica group. In particular, minimal such actions give rise to classifiable crossed products.

Choe, Junho

doi: 10.1093/imrn/rnaf017pmid: N/A

McCullough and Peeva found sequences of counterexamples to the Eisenbud–Goto conjecture on the Castelnuovo–Mumford regularity by using Rees-like algebras, where entries of each sequence have increasing dimensions and codimensions. In this paper we suggest another method to construct counterexamples to the conjecture with any fixed dimension $n\geq 3$ and any fixed codimension $e\geq 2$. Our strategy is an unprojection process and utilizes the possible complexity of homogeneous ideals with three generators. Furthermore, our counterexamples exhibit how singularities affect the Castelnuovo–Mumford regularity.

Bauer, Wolfram; Laaroussi, Abdellah; Tarama, Daisuke

doi: 10.1093/imrn/rnaf002pmid: N/A

Four subriemannian (SR) structures over the Euclidean sphere $\mathbb{S}^{7}$ are considered in accordance to the previous literature. The defining bracket generating distribution is chosen as the horizontal space in the Hopf fibration, the quaternionic Hopf fibration, or spanned by a suitable number of canonical vector fields. In all cases the induced SR geodesic flow on $T^{*}\mathbb{S}^{7}$ is studied. Adapting a method by A. Thimm in [36], a maximal set of functionally independent and Poisson commuting first integrals are constructed, including the corresponding SR Hamiltonian. As a result, the complete integrability in the sense of Liouville is proved for the SR geodesic flow. It is observed that these first integrals arise as the symbols of commuting second-order differential operators one of them being a (not necessarily intrinsic) sublaplacian. On the way one explicitly derives the Lie algebras of all SR isometry groups intersected with $O(8)$.

Baluyot, Siegfred; Turnage-Butterbaugh, Caroline L

doi: 10.1093/imrn/rnaf010pmid: N/A

Consider the family of Dirichlet $L$-functions of all even primitive characters of conductor at most $Q$, where $Q$ is a parameter tending to infinity. For $X=Q^{\eta }$ with $1<\eta <2$, we examine Dirichlet polynomials of length $X$ with coefficients those of the Dirichlet series of a product of an arbitrary (finite) number of shifted $L$-functions from the family. Assuming the Generalized Lindelöf Hypothesis for Dirichlet $L$-functions, we prove an asymptotic formula for averages of these Dirichlet polynomials. Our result agrees with the prediction of the recipe of Conrey, Farmer, Keating, Rubinstein, and Snaith for these averages. One may view our result as evidence for the “one-swap” terms in the recipe prediction for the general $2k$th moment of the family of Dirichlet $L$-functions.

Leng, James; Sawhney, Mehtaab

doi: 10.1093/imrn/rnae294pmid: N/A

Let $g$ be sufficiently large, $b\in \{0,\ldots ,g-1\}$, and $\mathcal{S}_{b}$ be the set of integers with no digit equal to $b$ in their base $g$ expansion. We prove that every sufficiently large odd integer $N$ can be written as $p_{1} + p_{2} + p_{3}$ where $p_{i}$ are prime and $p_{i}\in \mathcal{S}_{b}$.