Related papers: Bounded Connected Components of Polynomial Lemnisc…
A polynomial lemniscate is a curve in the complex plane defined by $\{z \in \mathbb{C}:|p(z)|=t\}$. Erd\"os, Herzog, and Piranian posed the extremal problem of determining the maximum length of a lemniscate $\Lambda=\{ z \in…
Given a complex manifold containing a relatively compact $Z(q)$ domain, we give sufficient geometric conditions on the domain so that its $L^2$-cohomology in degree $(p,q)$ (known to be finite dimensional) vanishes. The condition consists…
Determining the Jordan canonical form of the tensor product of Jordan blocks has many applications including to the representation theory of algebraic groups, and to tilting modules. Although there are several algorithms for computing this…
We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold and Lefschetz singularities. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson…
We determine the isomorphism classes of Jordan algebras in dimension two over the field of real numbers. Using techniques of non-standard analysis we study the properties of the variety of Jordan algebras, and also the contractions among…
For a linear algebraic group $G$ over $\bf Q$, we consider the period domains $D$ classifying $G$-mixed Hodge structures, and construct the extended period domains $D_{\mathrm{BS}}$, $D_{\mathrm{SL}(2)}$, and $\Gamma \backslash D_{\Sigma}$.…
We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge theoretic conditions, the cohomology ring of the complement of the hypersurface functorially…
Different topics on the differential geometry of the complex Grassmann manifold are surveyed in relation to the coherent states. A calculation of the tangent conjugate locus and conjugate locus in the complex Grassmann manifold is…
We establish a criterion for local boundedness and hence normality of a family $\F$ of analytic functions on a domain $D$ in the complex plane whose corresponding family of derivatives is locally bounded. Furthermore we investigate the…
Let G be a bounded Jordan domain in the complex plane with piecewise analytic boundary. We present theoretical estimates and numerical evidence for certain phenomena, regarding the application of the Bergman kernel method with algebraic and…
In this article we prove that the elliptic, hyperbolic and nilpotent (or unipotent) additive (or multiplicative) Jordan components of an endomorphism $X$ (or an isomorphism $g$) of a finite dimensional vector space are given by polynomials…
We consider the planar unit disk $\mathbb D$ as the reference configuration and a Jordan domain $\mathbb Y$ as the deformed configuration, and study the problem of extending a given boundary homeomorphism $\varphi \colon \partial \mathbb D…
We prove several boundedness results for log Fano pairs with certain K-stability. In particular, we prove that K-semistable log Fano pairs of Maeda type form a log bounded family. We also compute K-semistable domains for some examples.
We shall give a variational formula of the full Bergman kernels associated to a family of smoothly bounded strongly pseudoconvex domains. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the…
We prove some basic theorems concerning lemniscate configurations in an Euclidean space of dimension $ n \geq 3$. Lemniscates are defined as follows. Given m points $w_j $ in $\mathbb R^n$, consider the function $F(x)$ which is the product…
In \cite{kamz} the author proved that every quasiconformal harmonic mapping between two Jordan domains with $C^{1,\alpha}$, $0<\alpha\le 1$, boundary is bi-Lipschitz, providing that the domain is convex. In this paper we avoid the…
We study Serrin's overdetermined boundary value problems in bounded domains on weighted Riemannian manifolds. When the closure of the domain is compact, we establish a rigidity result that characterizes both the solution and the geometry of…
We discuss Poisson structures on a weighted polynomial algebra $A:=\Bbbk[x, y, z]$ defined by a homogeneous element $\Omega\in A$, called a potential. We start with classifying potentials $\Omega$ of degree deg$(x)+$deg$(y)+$deg$(z)$ with…
We show that the family of semi log canonical pairs with ample log canonical class and with fixed volume is bounded.
Let R be a noetherian connected graded domain of Gelfand-Kirillov dimension 3 over an uncountable algebraically closed field. Suppose that the graded quotient ring of R is a skew-Laurent ring over a field; we say that R is a birationally…