geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
superalgebra and (synthetic ) supergeometry
There are two different things that one might mean by a “complex supermanifold”, and the term is in fact used for two different notions in the literature (the terminology is a mess!):
In the first sense, a complex supermanifold generalizes the notion of a smooth manifold with its sheaf of smooth complex-valued functions, just as an ordinary supermanifold is a generalization of an ordinary manifold with its sheaf of smooth real-valued functions. However, considering ordinary smooth manifolds as ringed spaces with either their sheaves of real or complex smooth functions gives two equivalent categories, whereas this is not true in the case of real and complex supermanifolds; the corresponding functor is neither essentially surjective nor fully faithful. (For $X$ a complex supermanifold in this sense, the underlying reduced manifold $X_{red}$ is not a complex manifold but just a smooth manifold regarded as a ringed space with structure sheaf taken to be the sheaf of $\mathbb{C}$-valued smooth functions on the ordinary real manifold.)
In the second sense, a complex supermanifold is a super(complex manifold), a super-version of complex manifold.
A complex supermanifold is a ringed space $X = (|X|, O_X)$ such that
the structure sheaf $O_X$ a sheaf of commutative complex super algebras
locally $O_X$ is isomorphic to $C^\infty(\mathbb{R}^d) \otimes_{\mathbb{C}} \wedge^\bullet \mathbb{C}^\delta$
Write cSDiff? for the category of complex supermanifolds.
Example The functor $\Pi : \{real vector bundles\} \to SDiff$ has a complex analogue $\Pi : \{complex vector bundles\} \to cSDiff$.
Let $E \to X$ be a complex vector bundle of rank $\delta$. This gives rise to the complex supermanifold $\Pi E$, in the same way as a real vector bundle gives rise to a real supermanifold: the structure sheaf is given by sections of the exterior algebra of the dual of $E$.
Remark
$C^\infty(X) := O_X(X)$ does not in general have a $\mathbb{C}$-antilinear involution $\bar{-} : C^\infty(X) \to C^\infty(X)$ but there does exist a canonical complex conjugation on the quotient $C^\infty(X)$ by the ideal of nilpotent sections, which is $C^\infty(X_{red}; \mathbb{C})$. So on a complex supermanifold we have complex conjugation only on the reduced manifold.
As for ordinary supermanifolds (and with same proof as in the real case) we have the following two statements:
Theorem
Every complex supermanifold is isomorphic to one of the form $\Pi E$.
$cSDiff(X,Y) \simeq ComplexSuperAlg(C^\infty(Y), C^\infty(X)).$
Remark It turns out that a $\mathbb{C}$-super algebra homomorphism $\phi : C^\infty(Y) \to C^\infty(X)$ automatically satisfies $\phi_{red}(\overline{f_{red}}) = \overline{\phi_{red}(f_{red})}$.
Define the complex supermanifold $\mathbb{R}_{cs}^{d|\delta}$ as $\mathbb{R}^d$ with structure sheaf $U\mapsto C^\infty(U) \otimes_{\mathbb{C}} \wedge^\bullet \mathbb{C}^\delta$.
Then for $S$ an arbitrary complex supermanifold we have
Example
For
we shall write