N-dimensional Complex Research Articles

Symmetries of Fano varieties

Abstract Prokhorov and Shramov proved that the BAB conjecture, which Birkar later proved, implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of finite semi-simple groups (i.e., those with no nontrivial normal abelian subgroups) acting faithfully on 𝑛-dimensional complex Fano varieties, and this bound only depends on 𝑛. We investigate the geometric consequences of an action by a certain semi-simple group: the symmetric group. We give an effective upper bound for the maximal symmetric group action on an 𝑛-dimensional Fano variety. For certain classes of varieties – toric varieties and Fano weighted complete intersections – we obtain optimal upper bounds. Finally, we draw a connection between large symmetric actions and boundedness of varieties, by showing that the maximally symmetric Fano fourfolds form a bounded family. Along the way, we also show analogues of some of our results for Calabi–Yau varieties and log terminal singularities.

On a B-field transform of generalized complex structures over complex tori

Let (Xn,Xˇn) be a mirror pair of an n-dimensional complex torus Xn and its mirror partner Xˇn. Then, by SYZ transform, we can construct a holomorphic line bundle with an integrable connection from each pair of a Lagrangian section of Xˇn→Rn/Zn and a unitary local system along it, and those holomorphic line bundles with integrable connections form a dg-category DGXn. In this paper, we focus on a certain B-field transform of the generalized complex structure induced from the complex structure on Xn, and interpret it as the deformation XGn of Xn by a flat gerbe G. Moreover, we construct the deformation of DGXn associated to the deformation from Xn to XGn, and also discuss the homological mirror symmetry between XGn and its mirror partner on the object level.

Multidimensional Model of Information Struggle with Impulse Perturbation in Terms of Levy Approximation

The focus of this research was on building a decision support system for a model that characterizes the conflict interaction of n-dimensional complex systems with non-trivial internal structures. The interpretation of the new model was focused on information warfare as the impact of rare events that quickly change certain perceptions of a large number of people. Consequently, the support for various ideas experiences stochastic jumps, a phenomenon observable through a non-classical Levy approximation scheme. The essence of our decision support system lies in its ability to navigate the complex dynamics of conflict interaction among multifaceted systems. Through the utilization of advanced modeling techniques, our aim is to illuminate the complicated interplay of factors influencing information warfare and its cascading effects on societal perceptions and behaviors. Key components of our decision support system encompass model development, simulation capabilities, data integration, and visualization tools. The significance of our work lies in its potential to inform policy formulation, conflict resolution strategies, and societal resilience in the face of information warfare. By providing decision-makers with actionable intelligence and foresight into emerging threats and opportunities, our decision support system serves as a valuable tool for navigating the complexities of modern conflict dynamics. In conclusion, developing a decision support system for modeling conflict interaction in complex systems represents an essential step toward enhancing our understanding of information warfare and its consequences. Through interdisciplinary collaboration and innovative modeling techniques, we aim to provide stakeholders with the insights and capabilities needed to navigate the developing landscape of conflict and ensure the stability and resilience of society.

A PACKAGE OF PROCEDURES AND FUNCTIONS FOR CONSTRUCTIONS AND INVERSION OF ANALYTIC MAPPINGS WITH UNIT JACOBIAN

The set of polynomial mappings of n-dimensional complex space into itself the Jacobian matrix of which has a constant nonzero determinant is known to be very vast in any dimension that exceeds one. The well-known Jacobian conjecture states that any such mapping is polynomially invertible. Even though the computation of the determinant of the Jacobian matrix is very well supported in modern computer algebra systems, the algorithmic inversion of a polynomial mapping is still a problem of considerable computational complexity. In this paper, we present a Mathematica package JC that can be used for construction and inversion of polynomial mappings and more general analytic mappings with the unit determinant of the Jacobian matrix. The package includes functions that allow one to algorithmically construct these mappings for a given dimension of the space of variables and a given degree of mapping components. The package, together with a library of datasets for testing it and results of computational experiments, is available for free public use at https://www.researchgate.net/publication/358409332_JC_Package_and_Datasets.

Codebook Training for Trellis-Based Hierarchical Grassmannian Classification

We consider classification of points on a complex-valued Grassmann manifold of m-dimensional subspaces within the n-dimensional complex Euclidean space. We introduce a trellis-based hierarchical classification network, which is based on an orthogonal product decomposition of the orthogonal basis representing the m-dimensional subspace. Exploiting the similarity of the proposed trellis classifier with a neural network, we propose stochastic gradient-based training techniques. We apply the proposed methods to two important applications in wireless communication, namely Grassmannian channel state information quantization in multiple-input multiple-output communications and non-coherent Grassmannian multi-resolution transmission.

Face numbers of barycentric subdivisions of cubical complexes

The h-polynomial of the barycentric subdivision of any n-dimensional cubical complex with nonnegative cubical h-vector is shown to have only real roots and to be interlaced by the Eulerian polynomial of type Bn. This result applies to barycentric subdivisions of shellable cubical complexes and, in particular, to barycentric subdivisions of cubical convex polytopes and answers affirmatively a question of Brenti, Mohammadi and Welker.

Equivalence of Neighborhoods of Embedded Compact Complex Manifolds and Higher Codimension Foliations

We consider an embedded n-dimensional compact complex manifold in \(n+d\) dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert’s formal principle program. We will give conditions ensuring that a neighborhood of \(C_n\) in \(M_{n+d}\) is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold’s result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in \(M_{n+d}\) having \(C_n\) as a compact leaf, extending Ueda’s theory to the high codimension case. Both problems appear as a kind of linearization problems involving small divisors condition arising from solutions to their cohomological equations.

Connecting Homomorphism and Separating Cycles

We discuss the construction of a long semi-exact Mayer–Vietoris sequence for the homology of any finite union of open subspaces. This sequence is used to obtain topological conditions of representation of the integral of a meromorphic n-form on an n-dimensional complex manifold in terms of Grothendieck residues. For such a representation of the integral to exist, it is necessary that the cycle of integration separates the set of polar hypersurfaces of the form. The separation condition in a number of cases turns out to be a sufficient condition for representing the integral as a sum of residues. Earlier, when describing such cases (in the works of Tsikh, Yuzhakov, Ulvert, etc.), the key was the condition that the manifold be Stein. The main result of this article is the relaxation of this condition

Spectral density for the Schrödinger operator with magnetic field in the unit complex ball: solutions of evolutionary equations and applications to special functions

The Schwartz kernel of the spectral density for the Schrodinger operator with magnetic field in the n-dimensional complex ball is given. As applications, we compute the heat, resolvent and the wave kernels. Moreover, the resolvent and wave kernels are used to establish two new formulas for the Gauss-hypergeometric function.

Remarks on the homological mirror symmetry for tori

Let us consider an n-dimensional complex torus TJ=T2n≔ℂn∕2π(Zn⊕TZn). Here, T is a complex matrix of order n whose imaginary part is positive definite. In particular, when we consider the case n=1, the complexified symplectic form of a mirror partner of TJ=T2 is defined by using −1T or T. However, if we assume n≥2 and that T is a singular matrix, we cannot define a mirror partner of TJ=T2n as a natural generalization of the case n=1 to the higher dimensional case. In this paper, we propose a way to avoid this problem, and discuss the homological mirror symmetry.

RUANG PROYEKTIF KOMPLEKS 〖CP〗^n ADALAH MANIFOLD KOMPLEKS

<p>The purpose of this paper to determine the complex projective space as a complex manifold is to calculate the cohomology of the coherent sheaves of . The research method in this paper is to construct an -dimensional complex projective space, namely and then the n-dimensional complex projective space, namely , is a complex manifold. The result of this research is the -dimensional complex projective space, namely is a complex and compact manifold.</p>

A Note on <i>m</i>-Möbius Transformations

Lie groups of bi-Mobius transformations are known and their actions on non orientable n-dimensional complex manifolds have been studied. In this paper, m-Mobius transformations are introduced and similar problems as those related to bi-Mobius transformations are tackled. In particular, it is shown that the subgroup generated by every m-Mobius transformation is a discrete group. Cyclic subgroups are exhibited. Vector valued m-Mobius transformations are also studied.

Lie Groups Actions on Non Orientable <i>n</i>-Dimensional Complex Manifolds

Analytic atlases on can be easily defined making it an n-dimensional complex manifold. Then with the help of bi-Möbius transformations in complex coordinates Abelian groups are constructed making this manifold a Lie group. Actions of Lie groups on differentiable manifolds are well known and serve different purposes. We have introduced in previous works actions of Lie groups on non orientable Klein surfaces. The purpose of this work is to extend those studies to non orientable n-dimensional complex manifolds. Such manifolds are obtained by factorizing with the two elements group of a fixed point free antianalytic involution of . Involutions h(z) of this kind are obtained linearly by composing special Möbius transformations of the planes with the mapping . A convenient partition of is performed which helps defining an internal operation on and finally actions of the previously defined Lie groups on the non orientable manifold are displayed.

Distortion results for a certain subclass of biholomorphic mappings in ℂ n

Let be the space of n-dimensional complex variables and be the unit polydisc in . We obtain the distortion theorems of the Fréchet-derivative type and the Jacobi-determinant type for a certain subclass of normalized biholomorphic mappings defined on . Also, the distortion theorem of Jacobi-determinant type for the corresponding subclass defined on the unit ball in with arbitrary norm is established. Our results allow each component of complex vectors to have different dimensions, which extends severl previous works being closely related to some subclasses of starlike mappings.

Compact Hermitian Symmetric Spaces, Coadjoint Orbits, and the Dynamical Stability of the Ricci Flow

Using a stability criterion due to Kröncke, we show, providing $${n\ne 2k}$$ , the Kähler–Einstein metric on the Grassmannian $$Gr_{k}(\mathbb {C}^{n})$$ of complex k-planes in an n-dimensional complex vector space is dynamically unstable as a fixed point of the Ricci flow. This generalises the recent results of Kröncke and Knopf–Sesum on the instability of the Fubini–Study metric on $$\mathbb {CP}^{n}$$ for $$n>1$$ . The key to the proof is using the description of Grassmannians as certain coadjoint orbits of SU(n). We are also able to prove that Kröncke’s method will not work on any of the other compact, irreducible, Hermitian symmetric spaces.

Doubling chains on complements of algebraic hypersurfaces

A doubling chart on an n-dimensional complex manifold Y is a univalent analytic mapping ψ: B1 → Y of the unit ball in ℂn, which is extendable to the (say) four times larger concentric ball of B1. A doubling covering of a compact set G in Y is its covering with images of doubling charts on Y. A doubling chain is a series of doubling charts with non-empty subsequent intersections. Doubling coverings (and doubling chains) provide, essentially, a conformally invariant version of Whitney’s ball coverings of a domain W ⊂ ℝn, introduced in [19] (compare [11]). We study doubling chains in the complement Y = ℂnH of a complex algebraic hypersurface H of degree d in ℂn, and provide information on their length and other properties. Our main result is that any two points v1, v2 in a distance δ from H can be joined via a doubling chain in the complement Y = ℂnH of length at most c1$$\log \left( {\frac{{{c_2}}}{\delta }} \right)$$ with explicit constants c1, c2 depending only on n and d. As a consequence, we obtain an upper bound on the Kobayashi distance in Y, and an upper bound for the constant in a doubling inequality for regular algebraic functions on Y. We also provide the corresponding lower bounds for the length of the doubling chains, through the doubling constant of specific functions on Y.

English

We study a certain operator of multiplication by monomials in the weighted Bergman space both in the unit disk of the complex plane and in the polydisk of the n-dimensional complex plane. Characterization of the commutant of such operators is given.

On the number and boundedness of log minimal models of general type

We show that the number of marked minimal models of an n-dimensional smooth complex projective variety of general type can be bounded in terms of its volume, and, if n=3, also in terms of its Betti numbers. For an n-dimensional projective klt pair (X,D) with $K_X+D$ big, we show more generally that the number of its weak log canonical models can be bounded in terms of the coefficients of D and the volume of $K_X+D$. We further show that all n-dimensional projective klt pairs (X,D), such that $K_X+D$ is big and nef of fixed volume and such that the coefficients of D are contained in a given DCC set, form a bounded family. It follows that in any dimension, minimal models of general type and bounded volume form a bounded family.

The construction problem for Hodge numbers modulo an integer

For any positive integer m and any dimension n, we show that any n-dimensional Hodge diamond with values in Z/mZ is attained by the Hodge numbers of an n-dimensional smooth complex projective variety. As a corollary, there are no polynomial relations among the Hodge numbers of n-dimensional smooth complex projective varieties besides the ones induced by the Hodge symmetries, which answers a question raised by Koll\'ar in 2012.

Fredholm Toeplitz operators with VMO symbols and the duality of generalized Fock spaces with small exponents

AbstractWe characterize Fredholmness of Toeplitz operators acting on generalized Fock spaces of the n-dimensional complex space for symbols in the space of vanishing mean oscillation VMO. Our results extend the recent characterizations for Toeplitz operators on standard weighted Fock spaces to the setting of generalized weight functions and also allow for unbounded symbols in VMO for the first time. Another novelty is the treatment of small exponents 0 < p < 1, which to our knowledge has not been seen previously in the study of the Fredholm properties of Toeplitz operators on any function spaces. We accomplish this by describing the dual of the generalized Fock spaces with small exponents.