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\begin{document}

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\section*{complex supermanifold}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

\hypertarget{supergeometry}{}\paragraph*{{Supergeometry}}\label{supergeometry}

[[!include supergeometry - contents]]

\hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{complex}{complex}\dotfill \pageref*{complex} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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!):

\begin{enumerate}%
\item 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 \textbf{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.)


\item In the second sense, a complex supermanifold is a super(complex manifold), a super-version of [[complex manifold]].



\end{enumerate}
\hypertarget{complex}{}\subsection*{{complex}}\label{complex}

A \textbf{complex supermanifold} is a [[ringed space]] $X = (|X|, O_X)$ such that

\begin{itemize}%
\item the [[structure sheaf]] $O_X$ a [[sheaf]] of commutative complex [[super algebra]]s


\item locally $O_X$ is [[isomorphism|isomorphic]] to $C^\infty(\mathbb{R}^d) \otimes_{\mathbb{C}} \wedge^\bullet \mathbb{C}^\delta$



\end{itemize}
Write [[cSDiff]] for the [[category]] of complex supermanifolds.

\textbf{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 [[supermanifold|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 $E$.

\textbf{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 [[supermanifold]]s (and with same proof as in the real case) we have the following two statements:

\textbf{Theorem}

\begin{enumerate}%
\item Every complex supermanifold is isomorphic to one of the form $\Pi E$.


\item $cSDiff(X,Y) \simeq  ComplexSuperAlg(C^\infty(Y), C^\infty(X)).$



\end{enumerate}
\textbf{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

\begin{displaymath}
\mathbb{R}_{cs}^{d|\delta}(S)
  =
   cSDiff(S, \mathbb{R}_{cs}^{d|\delta})
  =
  \{
    (x_1, \cdots, x_d, \theta_1, \cdots, \theta_{\delta})|
    x_i \in C^\infty(S)^{ev} , 
    \theta_j \in C^\infty(S)^{odd};
    with x_i real in that \overline{(x_i)_{red}} = (x_i)_{red}
  \}
\end{displaymath}
\textbf{Example}

For

\begin{displaymath}
\mathbb{R}^{2|1}(S)
  =
  \{
    (x,y,\theta) |
    x,y \in C^\infty(S)^{ev}, 
    \theta \in C^\infty(S)^{odd}; x,y real
  \}
\end{displaymath}
we shall write

\begin{displaymath}
\simeq \{
     (z,\bar z, \theta)
     | z, \bar z \in C^\infty(S)^{ev}; \overline{z_{red}}=(\overline z )_{red}
  \}.
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[super Riemann surface]]

\end{itemize}
[[!redirects complex supermanifolds]]



\end{document}