[Paper Review] On the Structure of Complex Homogeneous Supermanifolds

Elizaveta Vishnyakova|ArXiv.org|Nov 16, 2008

Homotopy and Cohomology in Algebraic Topology3 references3 citations

TL;DR

This paper establishes that every complex homogeneous supermanifold arises as a quotient of a complex Lie supergroup by a closed Lie subsupergroup, generalizing the classical result for complex manifolds. Using sheaf-theoretic methods and differential geometry of supermanifolds, it proves the existence of a $G$-equivariant isomorphism between $G/H$ and any $G$-homogeneous supermanifold, thereby classifying such objects via homogeneous supercoset spaces.

ABSTRACT

For a Lie group $G$ and a closed Lie subgroup $H\subset G$, it is well known that the coset space $G/H$ can be equipped with the structure of a manifold homogeneous under $G$ and that any $G$-homogeneous manifold is isomorphic to one of this kind. An interesting problem is to find an analogue of this result in the case of supermanifolds. In the classical setting, $G$ is a real or a complex Lie group and $G/H$ is a real and, respectively, a complex manifold. Now, if $G$ is a real Lie supergroup and $H\subset G$ is a closed Lie subsupergroup, there is a natural way to consider $G/H$ as a supermanfold. Furthermore, any $G$-homogeneous real supermanifold can be obtained in this way, see \cite{Kostant}. The goal of this paper is to give a proof of this result in the complex case.

Motivation & Objective

• To extend the classical classification of complex homogeneous manifolds as quotients $G/H$ to the supergeometric setting.
• To establish that every complex $G$-homogeneous supermanifold is isomorphic to a supercoset space $G/H$ for a closed Lie subsupergroup $H\subset G$.
• To provide a rigorous sheaf-theoretic and differential-geometric proof of this classification in the complex-analytic supermanifold framework.
• To generalize the real supermanifold result of [1] to the complex case, filling a gap in the theory of superhomogeneous spaces.

Proposed method

• Uses the framework of complex-analytic supermanifolds in the sense of Berest-Roytenberg, with sheaves of superalgebras and morphisms of supermanifolds.
• Applies the theory of Lie supergroups, including multiplication, inverse, and identity morphisms satisfying group axioms in the super setting.
• Constructs a $G$-equivariant isomorphism $\beta: (G/H, \mathcal{O}_{G/H}) \to (M, \mathcal{O}_M)$ via sheaf isomorphisms $\beta_2$ induced by the action $\mu_x$ and the projection $p$.
• Employs the differential of the action $\mu_x$ at the identity, showing surjectivity and nondegeneracy to establish local isomorphisms between superdomains.
• Uses commutative diagrams involving left translations $\widetilde{l}_g$ and the action $\mu$ to prove equivariance of the constructed isomorphism.
• Relies on Theorem 1 (local isomorphism of $\mu|_S$) and properties of the differential $d\mu_x$ to show that $\beta_2$ is well-defined and invertible.

Experimental results

Research questions

• RQ1 Can every complex $G$-homogeneous supermanifold be realized as a quotient $G/H$ for a closed Lie subsupergroup $H\subset G$?
• RQ2 How can the classical classification of complex homogeneous manifolds be extended to the super setting using Lie supergroup actions?
• RQ3 What conditions ensure that the quotient $G/H$ of a complex Lie supergroup by a closed subsupergroup carries a natural supermanifold structure compatible with the group action?
• RQ4 Is there a $G$-equivariant isomorphism between $G/H$ and any $G$-homogeneous supermanifold $M$?
• RQ5 How can the sheaf-theoretic structure of $G/H$ be matched to that of $M$ via the action morphism?

Key findings

• Every complex $G$-homogeneous supermanifold is isomorphic to a supercoset space $G/H$ for a closed Lie subsupergroup $H\subset G$.
• The isomorphism $\beta: (G/H, \mathcal{O}_{G/H}) \to (M, \mathcal{O}_M)$ is explicitly constructed via sheaf isomorphisms $\beta_2$ induced by the action $\mu_x$.
• The constructed isomorphism $\beta$ is $G$-equivariant, meaning the diagram (10) commutes, ensuring compatibility with the group action.
• The differential $d\mu_x$ at the identity is surjective due to transitivity, and its nondegeneracy allows the local isomorphism of $\mu|_S$ to a superdomain in $M$.
• The sheaf $p_2(\mathcal{O}_{G/H})$ coincides with $\mu_{x,2}(\mathcal{O}_M)$ locally, implying global agreement and thus the existence of the inverse sheaf morphism $\beta_2$.
• The proof relies on the commutativity of translation diagrams (13), which ensures that the sheaf structure is preserved under left multiplication and action morphisms.