N-dimensional Complex Research Articles

On the positivity of the logarithmic cotangent bundle

The aim of this work is to construct examples of pairs whose logarithmic cotangent bundles have strong positivity properties. These examples are constructed from any smooth n-dimensional complex projective varieties by considering the sum of at least n general sufficiently ample hypersurfaces.

On the Distribution of Zero Sets of Holomorphic Functions

Let M be a subharmonic function with Riesz measure ν M in a domain D in the n-dimensional complex Euclidean space ℂ n , and let f be a nonzero function that is holomorphic in D, vanishes on a set Z ⊂ D, and satisfies |f| ⩽ expM on D. Then restrictions on the growth of ν M near the boundary of D imply certain restrictions on the dimensions or the area/volume of Z. We give a quantitative study of this phenomenon in the subharmonic framework.

Regularity of harmonic maps from polyhedra to CAT(1) spaces

We determine regularity results for energy minimizing maps from an n-dimensional Riemannian polyhedral complex X into a CAT(1) space. Provided that the metric on X is Lipschitz regular, we prove Holder regularity with Holder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the $$(n-2)$$ -skeleton, we improve the regularity to locally Lipschitz. Finally, for points $$x \in X^{(k)}$$ with $$k \le n-2$$ , we demonstrate that the Holder exponent depends on geometric and combinatorial data of the link of $$x \in X$$ .

On the Dirichlet problem for a class of singular complex Monge—Ampère equations

We study the Dirichlet problem of the n-dimensional complex Monge—Ampere equation det(u ij ) = F/|z|2α, where 0 < α < n. This equation comes from La Nave—Tian’s continuity approach to the Analytic Minimal Model Program.

Characterization of projective spaces by Seshadri constants

We prove that an n-dimensional complex projective variety is isomorphic to {mathbb {P}}^n if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than n. We also classify complex projective varieties with Seshadri constants equal to n.

Classification of δ(2,n − 2)-ideal Lagrangian submanifolds in n-dimensional complex space forms

It was proven in [4] that every Lagrangian submanifold M of a complex space form M˜n(4c) of constant holomorphic sectional curvature 4c satisfies the following optimal inequality:δ(2,n−2)≤n2(n−2)4(n−1)H2+2(n−2)c, where H2 is the squared mean curvature and δ(2,n−2) is a δ-invariant on M. In this paper we classify Lagrangian submanifolds of complex space forms M˜n(4c), n≥5, which satisfy the equality case of this improved inequality at every point.

Problem of nonoverlapping polycylindrical domains with poles on the boundary of a polydisk

We propose a generalization of the notion of the inner radius to the case of n-dimensional complex space, which allows us to transfer some results known for the complex plane onto $$ {\mathrm{\mathbb{C}}}^n $$ .

Decomposition of L p (∂D a ) space and boundary value of holomorphic functions

This paper deals with two topics mentioned in the title. First, it is proved that function f in L p (∂D a ) can be decomposed into a sum g + h, where D a is an angular domain in the complex plane, g and h are the non-tangential limits of functions in H p (D a ) and $${H^p}\left( {\overline D _a^c} \right)$$ in the sense of L p (D a ), respectively. Second, the sufficient and necessary conditions between boundary values of holomorphic functions and distributions in n-dimensional complex space are obtained.

On irrationality of surfaces in [formula omitted

The degree of irrationality irr(X) of a n-dimensional complex projective variety X is the least degree of a dominant rational map X⇢Pn. It is a well-known fact that given a product X×Pm or a n-dimensional variety Y dominating X, their degrees of irrationality may be smaller than the degree of irrationality of X. In this paper, we focus on smooth surfaces S⊂P3 of degree d≥5, and we prove that irr(S×Pm)=irr(S) for any integer m≥0, whereas irr(Y)<irr(S) occurs for some Y dominating S if and only if S contains a rational curve.

Periods of continuous maps on some compact spaces

The objective of this paper is to provide information on the set of periodic points of a continuous self-map defined in the following compact spaces: (the n-dimensional sphere), (the product space of the n-dimensional with the m-dimensional spheres), (the n-dimensional complex projective space) and (the n-dimensional quaternion projective space). We use as main tool the action of the map on the homology groups of these compact spaces.

On Algebraic Basis of the Algebra of Symmetric Polynomials on lp(Cn)

We consider polynomials on spaces lpCn,1≤p&lt;+∞, of p-summing sequences of n-dimensional complex vectors, which are symmetric with respect to permutations of elements of the sequences, and describe algebraic bases of algebras of continuous symmetric polynomials on lpCn.

Generalizations of the Shadow Problem

We solve the shadow problem in the n-dimensional Euclidean space $$ {\mathbb{N}}^n $$ for a family of sets obtained from any convex domain with nonempty interior with the help of parallel translations and homotheties. Moreover, we determine the number of balls with centers on the sphere sufficient for making a shadow in the n-dimensional complex (hypercomplex) space.

Explicit Serre duality on complex spaces

In this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization, as well as a new analytic proof, of Serre duality on any reduced pure n-dimensional paracompact complex space X. At the core of the paper is the introduction of certain fine sheaves BXn,q of currents on X of bidegree (n,q), such that the Dolbeault complex (BXn,•,∂¯) becomes, in a certain sense, a dualizing complex. In particular, if X is Cohen–Macaulay then (BXn,•,∂¯) is an explicit fine resolution of the Grothendieck dualizing sheaf.

Norm Retrievable Frames and Their Perturbation in Finite Dimensional Complex Hilbert Spaces

ABSTRACTIn this article, we study the property of norm retrievability of spanning vectors in a finite dimensional complex Hilbert space ℋ. Using the set of zero trace operators on ℋ and two sets of self-adjoint operators on ℋ denoted by 𝒮1,0 and 𝒮1,1, we present some equivalent conditions to the norm retrievable frames in ℋ. We will also show that the property of norm retrievability for n-dimensional complex Hilbert space ℋ with n≠2 is stable under enough small perturbation of the frame set only for phase retrievable frames.

A norm equivalence for the mixed norm of Fock type

In this paper, we prove that the mixed norm for an exponential type weighted integral of an entire function is equivalent to the mixed norm of its radial derivative and the distortion function from the weight function in the n-dimensional complex space.

Decomposition of complex hyperbolic isometries by involutions

A k-reflection of the n-dimensional complex hyperbolic space HCn is an element in U(n,1) with negative type eigenvalue λ, |λ|=1, of multiplicity k+1 and positive type eigenvalue 1 of multiplicity n−k. We prove that a holomorphic isometry of HCn is a product of at most four involutions and a complex k-reflection, k≤2. Along the way, we prove that every element in SU(n) is a product of four or five involutions according as n≢2mod4 or n≡2mod4. We also give a short proof of the well-known result that every holomorphic isometry of HCn is a product of two anti-holomorphic involutions.

Inversion of the x-ray transform for complexes of lines in

Consider the problem of inversion of the x-ray transform for n-dimensional complexes of lines in In this article we propose a general method for deriving of explicit inverse formulas. The method is based on geometrical properties of complexes of lines, which are studied in the article. Specifically, we recall notions of admissibility and critical point in the real space. A new criterion of admissibility is obtained. We also introduce a new notion of ω-critical point, which plays crucial role in our method. Some new examples of exact inversion formulas for admissible and non-admissible complexes of lines are given.

Holomorphic Minorants of Plurisubharmonic Functions

Let φ be a plurisubharmonic function on a pseudoconvex domain D in an n-dimensional complex space. We show that there exists a nonzero holomorphic function f on D such that some local mean value of φ with logarithmic additional terms majorizes log|f|. A similar problem is discussed for a locally integrable function on D in terms of balayage of positive measures.

Complex representation growth of finite quasisimple groups of Lie type

Abstract We give upper bounds to the number of n-dimensional irreducible complex representations of finite quasisimple groups belonging to different families of groups of Lie type. The bounds have the form c ns , where c and s are explicit positive constants that both depend on the family in question. From these bounds, we deduce a uniform bound of the form c n to the number of n-dimensional irreducible representations of all finite quasisimple groups of Lie type. Finally, an application of these results to counting conjugacy classes of maximal subgroups of Lie groups is discussed.

English

Let v/ be an entire self-map of the n-dimensional Euclidean complex space n C and u be an entire function on . n C A weighted composition operator induced by v/ with weight u is given by ( )( ) ( ) ( ) ( ) , z v f z u z f uCv / = / for n z C ∈ and f is entire function on . n C In this paper, we study weighted composition operators act between generalized Fock-types spaces. We characterize the boundedness and compactness of these operators act between () 0 , or ∞ φ   p and () 0 , or ∞ ∞   when . 0 ∞ ≤ < p