arXiv subjects

Masoud Sabzevari

Publications and source records attributed to Masoud Sabzevari.

At least 19 recordsLinked to original sources

arXiv2026

Holomorphic normal forms of six-dimensional totally nondegenerate CR manifolds in C^5

The class of six-dimensional totally nondegenerate CR submanifolds in $\mathbb C^5$ contains an infinite number of models parameterized by nonzero pairs $(a, b)\in\mathbb C\times\mathbb R$ appearing in their defining equations. Employing the equivariant moving frame method, we construct normal forms of this class. The applied normalizations yield into fourteen biholomorphically inequivalent subclasses, each with its own normal form. For each subclass, we determine the Lie algebra of infinitesimal CR automorphisms. Finally, using the method of involution, we prove the convergence of the constructed normal forms.

math.CV↗

arXiv2025

Convergence of Normal Form Power Series for Infinite-Dimensional Lie Pseudo-Group Actions

We prove the convergence of normal form power series for suitably nonsingular analytic submanifolds under a broad class of infinite-dimensional Lie pseudo-group actions. Our theorem is illustrated by a number of examples, and includes, as a particular case, Chern and Moser's celebrated convergence theorem for normal forms of real hypersurfaces. The construction of normal forms relies on the equivariant moving frame method, while the convergence proof is based on the realization that the normal form can be recovered as part of the solution to an initial value problem for an involutive system of differential equations, whose analyticity is guaranteed by the Cartan-K\"ahler Theorem.

math-ph↗

arXiv2023

Complete normal forms for real hypersurfaces in $\mathbb C^3$ at 2-nondegenerate points of Levi non-uniform rank zero

We construct complete normal forms for $5$-dimensional real hypersurfaces in $\mathbb C^3$ which are $2$-nondegenerate and also of Levi non-uniform rank zero at the origin point ${\bf p} =0$. The latter condition means that the rank of the Levi form vanishes at ${\bf p}$ but not identically in a neighborhood of it. The mentioned hypersurfaces are the only finitely nondegenerate real hypersurfaces in $\mathbb C^3$ for which their complete normal forms were absent in the literature. As a byproduct, we also treat the underlying biholomorphic equivalence problem between the hypersurfaces. Our primary approach in constructing the desired complete normal forms is to utilize the techniques derived in the theory of equivariant moving frames. It notably offers the advantage of systematic and symbolic manipulation of the associated computations.

math.DG↗

arXiv2022

Normal forms, moving frames, and differential invariants for nondegenerate hypersurfaces in C^2

We use the method of equivariant moving frames to revisit the problem of normal forms and equivalence of nondegenerate real hypersurfaces M \subset C^2 under the pseudo-group action of holomorphic transformations. The moving frame recurrence formulae allow us to systematically and algorithmically recover the results of Chern and Moser for hypersurfaces that are either non-umbilic at a point p \in M or umbilic in an open neighborhood of it. In the former case, the coefficients of the normal form expansion, when expressed as functions of the jet of the hypersurface at the point, provide a complete system of functionally independent differential invariants that can be used to solve the equivalence problem. We prove that under a suitable genericity condition, the entire algebra of differential invariants for such hypersurfaces can be generated, through the operators of invariant differentiation, by a single real differential invariant of order 7. We then apply moving frames to construct new convergent normal forms for nondegenerate real hypersurfaces at singularly umbilic points, namely those umbilic points where the hypersurface is not identically umbilic around them.

math.DG↗

arXiv2020

Convergent normal forms for five dimensional totally nondegenerate CR manifolds in C^4

Applying the equivariant moving frames method, we construct convergent normal forms for real-analytic 5-dimensional totally nondegenerate CR submanifolds of C^4. These CR manifolds are divided into several biholomorphically inequivalent subclasses, each of which has its own complete normal form. Moreover it is shown that, biholomorphically, Beloshapka's cubic model is the unique member of this class with the maximum possible dimension seven of the corresponding algebra of infinitesimal CR automorphisms. Our results are also useful in the study of biholomorphic equivalence problem between CR manifolds, in question.

math.CV↗

arXiv2019

On the geometric order of totally nondegenerate CR manifolds

A CR manifold $M$, with CR distribution $\mathcal D^{10}\subset T^\mathbb C M$, is called {\it totally nondegenerate of depth $μ$} if: (a) the complex tangent space $T^\mathbb C M$ is generated by all complex vector fields that might be determined by iterated Lie brackets between at most $μ$ fields in $\mathcal D^{10} + \overline{\mathcal D^{10}}$; (b) for each integer $2 \leq k \leq μ-1$, the families of all vector fields that might be determined by iterated Lie brackets between at most $k$ fields in $\mathcal D^{10} + \overline{\mathcal D^{10}}$ generate regular complex distributions; (c) the ranks of the distributions in (b) have the {\it maximal values} that can be obtained amongst all CR manifolds of the same CR dimension and satisfying (a) and (b) -- this maximality property is the {\it total nondegeneracy} condition. In this paper, we prove that, for any Tanaka symbol $\frak m = \frak m^{-μ}+ \ldots + \frak m^{-1}$ of a totally nondegenerate CR manifold of depth $μ\geq 4$, the full Tanaka prolongation of $\frak m$ has trivial subspaces of degree $k \geq 1$, i.e. it has the form $\frak m^{-μ}+ \ldots + \frak m^{-1} + \frak g^0$. This result has various consequences. For instance it implies that any (local) CR automorphism of a regular totally nondegenerate CR manifold is uniquely determined by its first order jet at a fixed point of the manifold. It also gives a complete proof of a conjecture by Beloshapka on the group of automorphisms of homogeneous totally nondegenerate CR manifolds.

math.CV↗

arXiv2018

On the maximum conjecture

We verify the maximum conjecture on the rigidity of totally nondegenerate model CR manifolds in the following two cases: (i) for all models of CR dimension one (ii) for the so-called full-models, namely those in which their associated symbol algebras are free CR. In particular, we discover that in each arbitrary CR dimension and length >= 3, there exists at least one totally nondegenerate model, enjoying this conjecture. Our proofs rely upon some recent results in the Tanaka theory of transitive prolongation of fundamental algebras.

math.CV↗

arXiv2017

Biholomorphic equivalence to totally nondegenerate model CR manifolds and Beloshapka's maximum conjecture

Applying Elie Cartan's classical method, we show that the biholomorphic equivalence problem to a totally nondegenerate Beloshapka's model of CR dimension one and codimension $k> 1$, whence of real dimension $2+k$, is reducible to some absolute parallelism, namely to an {e}-structure on a certain prolonged manifold of real dimension either $3+k$ or $4+k$. The proof relies on the weight analysis of the structure equations associated with the mentioned problem of equivalence. Thanks to the achieved results, we prove Beloshapka's maximum conjecture about the rigidity of his CR models of certain lengths equal or greater than three: "CR automorphism Lie groups of these models do not contain any nonlinear map, preserving the origin". Here, we mainly deal with CR models of the fixed CR dimension one though the results seem generalizable by means of certain analogous proofs.

math.DG↗

arXiv2017

The rigidity of totally nondegenerate model CR manifolds

In this paper, we prove that every real analytic totally nondegenerate model CR manifold of length >= 3 has rigidity. This result was actually conjectured before by Valerii Beloshapka as the so-called "maximum conjecture". It follows that the transformation Lie group of all CR automorphisms associated with each of the mentioned models does not include any nonlinear map.

math.DG↗

arXiv2016

Totally nondegenerate models and standard manifolds in CR dimension one

It is shown that two Levi-Tanaka and infinitesimal CR automorphism algebras, associated with a totally nondegenerate model of CR dimension one are isomorphic. As a result, the model surfaces are maximally homogeneous and standard. This gives an affirmative answer in CR dimension one to a certain question formulated by Beloshapka.

math.DG↗

arXiv2014

Moduli spaces of model real submanifolds: two alternative approaches

Instead of the invariant theory approach employed by Beloshaoka and Mamai for constructing the moduli spaces of Beloshapka's universal CR-models, we consider two alternative approaches borrowed from the theories of equivalence problem and Lie symmetries, each of them having its own advantages. Also the moduli space M(1,4) associated to the class of universal CR-models of CR-dimension 1 and codimension 4 is computed by means of the presented methods.

math.CV↗

arXiv2014

Canonical Cartan Connections on Maximally Minimal Generic Submanifolds M^5 in C^4

On a real analytic 5-dimensional CR-generic submanifold M^5 in C^4 of codimension 3, hence of CR dimension 1, which enjoys the generically satisfied nondegeneracy condition that Lie brackets up to length 3 of T^{1,0}M generate CTM, a canonical Cartan connection is constructed after reduction to a certain partially explicit e-structure of the concerned local biholomorphic equivalence problem. More advanced explorations of the incoming differential invariants due to the first and to the third authors already appeared in January 2014, hence the purpose is to show, while studying this specific Class III-1 of 5-dimensional CR structures, why and how the construction of Cartan geometries usually provides less information than a complete ramified discussion of potentially normalizable essential torsion coefficients.

math.CV↗

arXiv2014

Lie algebras of infinitesimal CR-automorphisms of finite type, holomorphically nondegenerate, weighted homogeneous CR-generic submanifolds of C^N

We consider the significant class of holomorphically nondegenerate CR manifolds of finite type that are represented by some weighted homogeneous polynomials and we derive some useful features which enable us to set up a fast effective algorithm to compute their Lie algebras of infinitesimal CR-automorphisms. This algorithm mainly relies upon a natural gradation of the sought Lie algebras, and it also consists in treating separately the related graded components. While some other methods are based on constructing and solving an associated pde systems which become time consuming as soon as the number of variables increases, the new method presented here is based on plain techniques of linear algebra. Furthermore, it benefits from a divide-and-conquer strategy to break down the computations into some simpler sub-computations. Moreover, we consider the new and effective concept of comprehensive Gröbner systems which provides us some powerful tools to treat the computations in the parametric cases. The designed algorithm is also implemented in the Maple software.

math.DG↗

arXiv2014

Cartan equivalences for 5-dimensional CR-manifolds in C^4 belonging to General Class III_1

We reduce to various absolute parallelisms, namely to certain {e}-structures on manifolds of dimensions 7, 6, 5, the biholomorphic equivalence problem or the intrinsic CR equivalence problem for generic submanifolds M^5 in C^4 of CR dimension 1 and of codimension 3 that are maximally minimal and are geometry-preserving deformations of one natural cubic model of Beloshapka, somewhere else called the General Class III-1 of 5-dimensional CR manifolds. Some inspiration links exist with the treatment of the General Class II previously done in 2007 by Beloshapka, Ezhov, Schmalz, and also with the classification of nilpotent Lie algebras due to Goze, Khakimdjanov, Remm.

math.CV↗

arXiv2014

Cartan equivalences for Levi-nondegenerate hypersurfaces M^3 in C^2 belonging to General Class I

We develope in great computational details the classical Cartan equivalence problem for Levi-nondegenerate C^6-smooth real hypersurfaces M^3 in C^2, performing all calculations effectively in terms of a (local) graphing function φ. In particular, we present explicitly the unique (complex) essential invariant J of the problem. Its expansion in terms of the 3-variables function φincorporates millions of differential monomials, while, when φis assumed to depend only on 2 variables (rigid case), J writes out in two lines (7 monomials).

math.DG↗

arXiv2013

Equivalences of 5-dimensional CR manifolds (II): General classes I, II, III-1, III-2, IV-1, IV-2

For later use in subsequent upcoming arxiv.org prepublications, basic foundational material on local, smooth or real analytic, CR-generic submanifolds of complex Euclidean spaces is developed from scratch, with strong emphasis on the interplay between extrinsic and intrinsic aspects, a constructive option that commands to perform computational syntheses in coordinates. Mainly, one finds a self-contained proof of the existence of precisely six general classes: I, II, III-1, III-2, IV-1, IV-2 of nondegenerate general CR manifolds up to dimension 5, class III-2 being unobserved untill now.

math.CV↗

arXiv2012

Applications of differential algebra for computing Lie algebras of infinitesimal CR-automorphisms

We perform detailed computations of Lie algebras of infinitesimal CR-automorphisms associated to three specific model real analytic CR-generic submanifolds in C^9 by employing differential algebra computer tools -- mostly within the Maple package DifferentialAlgebra -- in order to automate the handling of the arising highly complex linear systems of pde's. Before treating these new examples which prolong previous works of Beloshapka, of Shananina and of Mamai, we provide general formulas for the explicitation of the concerned pde systems that are valid in arbitrary codimension k >= 1 and in any CR dimension n >= 1. Also, we show how Ritt's reduction algorithm can be adapted to the case under interest, where the concerned pde systems admit so-called complex conjugations.

math.RA↗

arXiv2012

Modules Satisfying the Prime Radical Condition and a Sheaf Construction for Modules I

The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring $R$, called $\mathbb{P}$-radical modules (modules $M$ satisfying the prime radical condition "$(\sqrt[p]{\cal{P}M}:M)={\cal{P}}$" for every prime ideal ${\cal{P}}\supseteq {\rm Ann}(M)$, where $\sqrt[p]{\cal{P}M}$ is the intersection of all prime submodules of $M$ containing ${\cal{P}}M$). This class contains the family of primeful modules properly. This yields that over any ring all free modules and all finitely generated modules lie in the class of $\mathbb{P}$-radical modules. Also, we show that if $R$ is a domain (or a Noetherian ring), then all projective modules are $\mathbb{P}$-radical. In particular, if $R$ is an Artinian ring, then all $R$-modules are $\mathbb{P}$-radical and the converse is also true when $R$ is a Noetherian ring. Also an $R$-module $M$ is called $\mathbb{M}$-radical if $(\sqrt[p]{\cal{M}M}:M)={\cal{M}}$; for every maximal ideal ${\cal{M}}\supseteq {\rm Ann}(M)$. We show that the two concepts $\mathbb{P}$-radical and $\mathbb{M}$-radical are equivalent for all $R$-modules if and only if $R$ is a Hilbert ring. Semisimple $\mathbb{P}$-radical ($\mathbb{M}$-radical) modules are also characterized. In Part II we shall continue the study of this construction, and as an application, we show that the sheaf theory of spectrum of $\mathbb{P}$-radical modules (with the Zariski topology) resembles to that of rings.

math.AC↗