arXiv subjects

Saad Varsaie

Publications and source records attributed to Saad Varsaie.

10 recordsLinked to original sources

arXiv2025

Euler-Poincare characteristic pair of orientable supermanifolds

The Euler-Poincare characteristic, or Euler characteristic in short, is a fundamental topological invariant of compact manifolds that plays a crucial role in a variety of geometric and topological situations. From this point of view, we tried to expand on this important concept in supergeometry. In this article, we introduce the Euler-Poincare characteristic pair in the supergeometry. In the final section, we examine transversality in the category of $Π$-symmetric supermanifolds.

math.DG↗

arXiv2025

Supertransversality and $Π$-symmetric supermanifolds

The main objective of this article is to extend the concept of transversality to supergeometry. Transversality has two important properties in the classical case, namely " stability" and " genericity", which we show in the following that in the category of smooth supermanifolds, supertransversality has stable property. By extending Sard's theorem to supergeometry, genericity property is proved. In the final section, we examine transversality in the category of $Π$-symmetric supermanifolds. The theory presented here is a step towards an extension of the concept of Euler-Poincaré characteristic to supermanifolds.

math.DG↗

arXiv2023

Homotopy Classification of Super Vector Bundles and Universality

This study first provides a brief overview of the structure of typical Grassmann manifolds. Then a new type of supergrassmannians is construced using an odd involution in a super ringed space and by gluing superdomains together. Next, constructing a Gauss morphism of a super vector bundle, some properties of this morphism is discussed. By this, we generalize one of the main theorems of homotopy classification for vector bundles in supergeometry. Afterwards, a similar structure is introduced in the state of infinite-dimensional. Here our tools mainly include multilinear algebra of Grassmann algebras, the direct limit of the base spaces and the inverse limit of the structure sheaf of ringed spaces. We show that the resulting super vector bundle is a universal member of its category.

math.DG↗

arXiv2022

Extension of local positive definite $\mathbb{Z}_{2}^{n}$-superfunctions

By defining a concept of boundedness for $ \mathbb{Z}_2^n $-superfunction, we show that each local positive definite $ \mathbb{Z}_2^n $-superfunctions, which is bounded in some sense, on a $ \mathbb{Z}_2^n $-Lie supergroup has a positive definite extension to all of the $\mathbb{Z}_{2}^{n}$-Lie supergroup.

math.DG↗

arXiv2020

Analytic Approach To a Generalization of Chern Classes in Supergeometry

Some cohomology elements, called $ν$ classes, as a supergeneralization of universal Chern classes, are introduced for canonical super line bundles over $ν$ projective spaces, a novel supergeometric generalization of projective spaces. It is shown that these classes may be described by analytic representatives of elements of generalized de Rham cohomology.

math.DG↗

arXiv2019

On the Construction of $\mathbb{Z}_2^n-$Supergrassmannians as Homogeneous $\mathbb{Z}_2^n-$Superspaces

In this paper, we construct the $\mathbb Z_{2}^{n}-$supergrassmannians by gluing of the $\mathbb Z_{2}^{n}-$superdomains and give an explicit description of the action of the $\mathbb Z_{2}^{n}-$super Lie group $GL(\overrightarrow{\textbf{m}})$ on the $\mathbb Z_{2}^{n}-$supergrassmannian $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ in the functor of points language. In particular, we give a concrete proof of the transitively of this action, and the gluing of the local charts of the supergrassmannian.

math.DG↗

arXiv2019

The Super Lie Groups Associated to Odd Involutions

A new generalization of Grassmannians in supergeometry, called $ν-$Grassmannians, are constructed by gluing $ν-$domains. By a $ν-$domain, we mean a superdomain with an odd involution say $ν$ on its structure sheaf, as morphism of modules. Then we show that $ν-$Grassmannians are homogeneous superspaces. In addition, in the last section, a supergroup associated to the odd involution $ν$ is introduced.

math.DG↗

arXiv2018

Universal Super Vector Bundles

A new generalization of Grassmannians, called ν-grassmannians, and a canonical super vector bundle over this new space, say Γ, are introduced. Then, constructing a Gauss supermap of a super vector bundle, the universal property of Γ is discussed. Finally, we generalize one of the main theorems of homotopy classification for vector bundles in supergeometry.

math.DG↗

arXiv2018

A Novel Supergeometric Generalization of Grassmannians

A new generalization of Grassmannians to supergeometry, different from the well known supergrassmannian, is introduced. These are constructed by gluing a finite number of copies of a ν\- domain, i.e. a superdomain with an odd involution, say ν\, on their structure sheaf considered as a sheaf of C^\infty_{R^m}-modules.

math.DG↗

arXiv2018

Supergrassmannians as Homogeneous Superspaces

A homogeneous space is a manifold on which a Lie group acts transitively. Super generalization of this concept is also studied in [2] and [4]. In this paper we explicitly show that super Lie group GL(m|n) acts transitively on supergrassmannian G_{k|l}(m|n). In this regard, by using functor of point approach, this action is constructed by gluing local actions.

math.DG↗