International Journal of Mathematics

We classify finite groups acting by birational transformations of a nontrivial Severi–Brauer surface over a field of characteristic zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field $𝕂$ of characteristic zero that has no $𝕂$-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.

We consider and resolve the gap problem for almost quaternion-Hermitian structures, i.e. we determine the maximal and submaximal symmetry dimensions, both for Lie algebras and Lie groups, in the class of almost quaternion-Hermitian manifolds. We classify all structures with such symmetry dimensions. Geometric properties of the submaximally symmetric spaces are studied, in particular, we identify locally conformally quaternion-Kähler structures as well as quaternion-Kähler with torsion.

In this paper, suppose $F:{ℝ}^N\to [0,+\infty )$ be a convex function of class $C^2({ℝ}^N\setminus \{0\})$ which is even and positively homogeneous of degree 1. We establish the Lions type concentration-compactness principle of singular Trudinger–Moser Inequalities involving $N$-Finsler–Laplacian operator. Let $\mathrm{\Omega }\subset {ℝ}^N(N\ge 2)$ be a smooth bounded domain. $\{u_n\}\subset W_0^{1,N}(\mathrm{\Omega })$ be a sequence such that anisotropic Dirichlet norm${\int }_{\mathrm{\Omega }}{F}^N(\nabla u_n)dx=1$, $u_n\rightharpoonup u\not\equiv 0$ weakly in $W_0^{1,N}(\mathrm{\Omega })$. Denote

Let $𝔾$ be a three-dimensional Lie group with a bi-invariant metric. Consider $\mathrm{\Omega }\subset 𝔾$ a strictly convex domain in $𝔾$. We prove that if $\mathrm{\Sigma }\subset \mathrm{\Omega }$ is a stable CMC free-boundary surface in $\mathrm{\Omega }$ then $\mathrm{\Sigma }$ has genus either 0 or 1, and at most three boundary components. This result was proved by Nunes [I. Nunes, On stable constant mean curvature surfaces with free-boundary, Math. Z. 287(1–2) (2017) 73–479] for the case where $𝔾={ℝ}^3$ and by R. Souam for the case where $𝔾={𝕊}^3$ and $\mathrm{\Omega }$ is a geodesic ball with radius $r<\frac{\pi }{2}$, excluding the possibility of $\mathrm{\Sigma }$ having three boundary components. Besides ${ℝ}^3$ and ${𝕊}^3$, our result also apply to the spaces ${𝕊}^1\times {𝕊}^1\times {𝕊}^1$, ${𝕊}^1\times {ℝ}^2$, ${𝕊}^1\times {𝕊}^1\times ℝ$ and $\mathrm{\text{SO}}(3)$. When $𝔾={𝕊}^3$ and $\mathrm{\Omega }$ is a geodesic ball with radius $r<\frac{\pi }{2}$, we obtain that if $\mathrm{\Sigma }$ is stable then $\mathrm{\Sigma }$ is a totally umbilical disc. In order to prove those results, we use an extended stability inequality and a modified Hersch type balancing argument to get a better control on the genus and on the number of connected components of the boundary of the surfaces.

We investigate when the tangent bundle of a projective manifold has a nontrivial first-order (or positive-dimensional) deformation. This leads to a new conjectural characterization of the complex projective space.

We consider integrable analytic deformations of codimension one holomorphic foliations near an initially singular point. Such deformations are of two possible types. The first type is given by an analytic family ${\{{\mathrm{\Omega }}^t\}}_{t\in D}$ of integrable one-forms ${\mathrm{\Omega }}^t$ defined in a neighborhood $U\subset {ℂ}^n$ of the initial singular point, and parametrized by the disc $D\subset ℂ$. The initial foliation is defined by ${\mathrm{\Omega }}^0$. The second type, more restrictive, is given by an integrable holomorphic one-form $\mathrm{\Omega }(x,t)$ defined in the product $U\times D\subset {ℂ}^n\times ℂ$. Then, the initial foliation is defined by the slice restriction $\mathrm{\Omega }(x,0)$. In the first part of this work, we study the case where the starting foliation has a holomorphic first integral, i.e. it is given by $df=0$ for some germ of holomorphic function $f\in {{𝒪}}_n$ at the origin $0\in {ℂ}^n,n\ge 3$. We assume that the germ $f$ is irreducible and that the typical fiber of $f$ is simply-connected. This is the case if outside of a dimension $\le n-3$ analytic subset $Y\subset {ℂ}^n$, the analytic hypersurface $X_f:(f=0)$ has only normal crossings singularities. We then prove that, if cod sing $\mathrm{\Omega }(x,0)\ge 2$ then the (germ of the) developing foliation given by $\mathrm{\Omega }(x,t)=0$ also exhibits a holomorphic first integral. For the general case, i.e. cod sing $\mathrm{\Omega }(x,0)\ge 1$, we obtain a dimension two normal form for the developing foliation. In the second part of the paper, we consider analytic deformations ${\{{{ℱ}}_t\}}_{t\in ℂ,0}$, of a local pencil ${{ℱ}}_0:\frac{f}{g}=\mathrm{\text{constant}}$, for $f,g\in {{𝒪}}_n$. For dimension $n=2,$ we consider $f=x,g=y$. For dimension $n\ge 3,$ we assume some generic geometric conditions on $f$ and $g$. In both cases, we prove: (i) in the case of an analytic deformation there is a multiform formal first integral of type $\widehat{F}=\frac{f^{1+\widehat{\lambda }(t)}}{g^{1+\widehat{\mu }(t)}}{e}^{Ĥ(x,y,t)}$ with some properties; (ii) in the case of an integrable deformation there is a meromorphic first integration of the form $M=\frac{f}{g}{e}^{P(t)+H(x,y,t)}$ with some additional properties, provided that for $n=2$ the axes remain invariant for the foliations ${{ℱ}}_t$.

We study holographic operators associated with Rankin–Cohen brackets which are symmetry breaking operators for the restriction of tensor products of holomorphic discrete series of the universal covering of $S{L}_2(ℝ)$. Furthermore, we investigate a geometrical interpretation of these operators and their relations to classical Jacobi polynomials.

It has been recently discovered that in smooth unfoldings of maps with a rank-one homoclinic tangency there are codimension two laminations of maps with infinitely many sinks. Indeed, these laminations, called Newhouse laminations, occur also in the holomorphic context. In the space of polynomials of ${ℂ}^2$, with bounded degree, there are Newhouse laminations.

In this paper, we investigate a connection between convolution products for quiver Hecke algebras and tensor products for quantum groups. We give a categorification of the natural projection ${\pi }_{\lambda ,\mu }:V_{𝔸}{(\lambda )}^{\vee }{\otimes }_{𝔸}{V}_{𝔸}{(\mu )}^{\vee }\twoheadrightarrow V_{𝔸}{(\lambda +\mu )}^{\vee }$ sending the tensor product of the highest weight vectors to the highest weight vector in terms of convolution products. When the quiver Hecke algebra is symmetric and the base field is of characteristic $0$, we obtain a positivity condition on some coefficients associated with the projection ${\pi }_{\lambda ,\mu }$ and the upper global basis, and prove several results related to the crystal bases. We then apply our results to finite type $A$ using the homogeneous simple modules ${{𝒮}}^T$ indexed by one-column tableaux $T$.