Defining supermanifolds by equations

Question

I would like to in what sense supermanifolds may be defined by systems of equations in the ordinary flat superspace. I am particularly interested in the approach to supergeometry via ringed spaces.

Remark: situation is quite clear for me in the case of affine super schemes. In this case subscheme of the spectrum $\mathrm{Spec}(\mathcal A)$ of the supercommutative superalgebra $\mathcal A$ which may be described as the zero locus of an ideal $J \subseteq \mathcal A$ is simply the spectrum of the supercommutative superalgebra $\frac{\mathcal A}{J}$. We have a canonical morphism $\mathrm{Spec} \left (\frac{\mathcal A}{J} \right ) \to \mathrm{Spec}(\mathcal A)$. This construction doesn't seem to easily generalize to supermanifolds.

Answer 1

Recall that a super ringed space is a pair $X= (|X|, \mathcal{O}_X)$ consisting of a topological space $|X|$ and a sheaf $\mathcal{O}_X$ of supercommutative algebras over an algebraically closed field $k$.

Supermanifolds are glued together from local models, usually $\mathbb{C}^{m|n}$ or $\mathbb{R}^{m|n}$ which are both super ringed space. This is analogous to superschemes being locally modeled by affine superschemes.

The superspace $\mathbb{C}^{m|n} = (\mathbb{C}^m, \mathcal{O}_{\mathbb{C}^{m|n}})$ is a super ringed space with a sheaf of supercommutative algebras, $$ \mathcal{O}_{\mathbb{C}^{m|n}}: U \to \operatorname{Hol}(U) \otimes \bigwedge \theta_i$$ where $\operatorname{Hol}(U)$ is the ordinary algebra of holomorphic functions on an open subset $U$ and $\theta_i$ are a choice of $n$ odd indeterminates. Clearly, $\operatorname{Hol}(U) \otimes \bigwedge \theta_i$ is a supercommutative algebra.

In this case, one may view the odd indeterminates $\theta_i$ as generators of a rank $n$ vector bundle $V$ over the ordinary manifold $\mathbb{C}^m$. Any such vector bundle is locally free and thus we know exactly how the $\theta_i$ glue together. You will sometimes see $\operatorname{Hol}(U) \otimes \bigwedge \theta_i$ denoted simply by $\bigwedge V^*$.

You can now think of supermanifolds as being cut out locally by the vanishing sets of (super) functions $f \in \operatorname{Hol}(U) \otimes \bigwedge \theta_i$.