International Journal of Mathematics

In this note, we show that for a closed almost-Kähler manifold $(X,J)$ with the almost complex structure $J$ satisfies $dimker{P}_J=b_2-1$ the space of de Rham harmonic forms is contained in the space of symplectic-Bott–Chern harmonic forms. In particular, suppose that $X$ is four-dimensional, if the self-dual Betti number $b_2^{+}=1$, then we prove that the second non-HLC degree measures the gap between the de Rham and the symplectic-Bott–Chern harmonic forms.

We consider the classical pretzel knots $P(a_1,a_2,a_3)$, where $a_1,a_2,a_3$ are positive odd integers. By using continuous paths of elliptic ${\mathrm{\text{SL}}}_2(ℝ)$-representations, we show that (i) the 3-manifold obtained by $\frac{m}{l}$-surgery on $P(a_1,a_2,a_3)$ has left orderable fundamental group if $\frac{m}{l}<1$ and (ii) the $n\mathrm{\text{th}}$-cyclic branched cover of $P(a_1,a_2,a_3)$ has left orderable fundamental group if $n>2\pi /arccos(1-2/(1+a_1{a}_2+a_2{a}_3+a_3{a}_1))$.

In this paper, we consider a $p$-Kirchhoff problem with Dirichlet boundary problem in a bounded domain. Under suitable conditions, we get a priori estimates for positive solutions to an auxiliary problem by the well-known blow-up argument. As an application, a existence result for positive solutions is proved by the topological degree theory.

In this paper, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows — generalizing a result of Ransford–White for the spectral unit ball — that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider $2$-point and $3$-point Pick–Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary conditions for the existence of a holomorphic interpolant for these problems. In particular, we shall observe that these results are generalizations of the results provided by Bharali and Chandel related to these problems.

In Finsler geometry, the projective Ricci curvature is an important projective invariant. In this paper, we investigate the projective Ricci curvature of a class of general $(\alpha ,\beta )$-metrics satisfying a certain condition, which is invariant under the change of volume form. Moreover, we construct a class of new nontrivial examples on such Finsler metrics.

We collect some results on endomorphisms on projective varieties related to the Kawaguchi–Silverman conjecture. We discuss certain conditions on automorphism groups of projective varieties and positivity conditions on leading real eigendivisors of self-morphisms. We prove Kawaguchi–Silverman conjecture for endomorphisms on projective bundles on a smooth Fano variety of Picard number one. In the last section, we discuss endomorphisms and augmented base loci of their eigendivisors.

We investigate the deformations of pairs $(X,L)$, where $L$ is a line bundle on a smooth projective variety $X$, defined over an algebraically closed field $𝕂$ of characteristic 0. In particular, we prove that the DG-Lie algebra controlling the deformations of the pair $(X,L)$ is homotopy abelian whenever $X$ has trivial canonical bundle, and so these deformations are unobstructed.

In this paper, we study characterizations of Griffiths semi-positivity through $L^2$-estimates of the $\bar{\partial }$-equation and $L^2$-extension theorems for symmetric powers of a holomorphic vector bundle. We also investigate several versions of the converse of the Demailly–Skoda theorem.