Let $M$ be a positive quaternionic Kähler manifold of dimension $4m$. In earlier papers, Fang and the first author showed that if the symmetry rank is greater than or equal to $[m/2]+3$, then $M$ is isometric to $\mathbf{HP}^{m}$ or $\mathit{Gr}_{2}(\mathbf{C}^{m+2})$. The goal of this paper is to give a more refined classification result for positive quaternionic Kähler manifolds (in particular, of relatively low dimension or with even $m$) whose fourth Betti number equals one. To be precise, we show in this paper that if the symmetry rank of $M$ with $b_{4}(M)=1$ is no less than $[m/2]+2$ for $m \ge 5$, then $M$ is isometric to $\mathbf{HP}^{m}$.

We prove global subelliptic estimates for systems of quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. In a previous work, we pointed out the existence of a particular linear subvector space in the phase space intrinsically associated to their Weyl symbols, called singular space, which rules a number of fairly general properties of non-elliptic quadratic operators. About the subelliptic properties of these operators, we established that quadratic operators with zero singular spaces fulfill global subelliptic estimates with a loss of derivatives depending on certain algebraic properties of the Hamilton maps associated to their Weyl symbols. The purpose of the present work is to prove similar global subelliptic estimates for overdetermined systems of quadratic operators. We establish here a simple criterion for the subellipticity of these systems giving an explicit measure of the loss of derivatives and highlighting the non-trivial interactions played by the different operators composing those systems.

We consider real versions of Brauer's $\mathrm{k}(B)$ conjecture, Olsson's conjecture and Eaton's conjecture. We prove the real version of Eaton's conjecture for $2$-blocks of groups with cyclic defect group and for the principal $2$-blocks of groups with trivial real core. We also characterize $G$-classes, real and rational $G$-classes of the defect group of$B$.

In this paper, we show that the space of vector valued Siegel modular forms of $\Gamma_{0} (N) \subset \mathrm{Sp}(2, \mathbb{Z})$ with respect to the symmetric tensor of degree $2$ has a simple unified structure for $N=2,3,4$. Each structure is similar to the structure of the full modular group.

We discuss constant mean curvature bubbletons in Euclidean 3-space via dressing with simple factors, and prove that single-bubbletons are not embedded.

In this paper, of concern is the singular impulsive functional integral equations subject to nonlocal conditions in a Banach space. Sufficient conditions, ensuring the existence of solutions, are presented. An example is also given to illustrate the applications of the abstract results. Our results essentially extend some existing results in this area.

We introduce in this paper a new resolution graph for an isolated complex plane curve singularity and then calculate the monodromy zeta function and the Alexander polynomial for the singularity in terms of this graph.

Our main goal is to prove the existence of multiple solutions with precise sign information for a Neumann problem driven by the $p$-Laplacian differential operator with a ($p-1$)-superlinear term which does not satisfy the Ambrosetti--Rabinowitz condition. Using minimax methods we show that the problem has five nontrivial smooth solutions, two positive, two negative and the fifth nodal. In the semilinear case ($p = 2$), using Morse theory, we produce a second nodal solution (for a total of six nontrivial smooth solutions).

For a complete hyperbolic three manifold $M$, we consider the representations of $\pi_{1}(M)$ obtained by composing a lift of the holonomy with complex finite dimensional representations of $\mathrm{SL}(2,\mathbf{C})$. We prove a vanishing result for the cohomology of $M$ with coefficients twisted by these representations, using techniques of Matsushima--Murakami. We give some applications to local rigidity.

We describe a class of translation planes whose orders are of the form $q^{n}$, where $n$ is odd and $q$ is an odd prime power $>3$. These planes have the property that a translation complement fixes a triangle and acts transitively on the set of non-vertices of each side. The planes form an odd order analogue to the planes of Kantor--Williams [17] which have even order. The construction of the planes is based on a certain type of Dembowski--Ostrom polynomials.

In this paper we will introduce the new notion of generalized geometric structures defined by systems of closed differential forms. From a cohomological point of view, we develop a unified approach to deformation problems and establish a criterion for unobstructed deformations of the generalized geometric structures. We construct the moduli spaces of the structures with the action of $d$-closed $b$-fields and show that the period map of the moduli space is locally injective under the certain cohomological condition (the local Torelli type theorem). We apply our approach to generalized Calabi--Yau structures and generalized $\mathrm{SU}(n)$-structures and obtain unobstructed deformations of generalized Calabi--Yau structures if the $dd^{\mathcal{J}}$-property is satisfied. We also have unobstructed deformations of generalized $\mathrm{SU}(n)$-structures and show that the period map of the moduli space of generalized $\mathrm{SU}(n)$-structures is locally injective.

We treat Kollár's injectivity theorem from the analytic (or differential geometric) viewpoint. More precisely, we give a curvature condition which implies Kollár type cohomology injectivity theorems. Our main theorem is formulated for a compact Kähler manifold, but the proof uses the space of harmonic forms on a Zariski open set with a suitable complete Kähler metric. We need neither covering tricks, desingularizations, nor Leray's spectral sequence.