Complex supermanifold and Berezinian conventions

Write a complex superalgebra as a unital associative ${\mathbb{Z}}_2$-graded algebra $A=A_0\oplus A_1$ satisfying $ab=(-1{)}^{\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }}ba$ for homogeneous $a,b$. A locally ringed superspace is a topological space $X$ with a sheaf ${\mathcal{O}}_X$ of such algebras whose stalk at each point has a unique maximal homogeneous ideal. A morphism $(f,f^{\mathrm{\#}}):(X,{\mathcal{O}}_X)\to (Y,{\mathcal{O}}_Y)$ consists of a continuous map $f:X\to Y$ and a parity-preserving unital sheaf morphism $f^{\mathrm{\#}}:{\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$ whose stalk maps ${\mathcal{O}}_{Y,f(x)}\to {\mathcal{O}}_{X,x}$ are local: the inverse image of the maximal homogeneous ideal of ${\mathcal{O}}_{X,x}$ is the maximal homogeneous ideal of ${\mathcal{O}}_{Y,f(x)}$. An isomorphism is a morphism with a two-sided inverse.

For $m\ge 1$ and open $U\subseteq {\mathbb{C}}^m$, let ${\mathcal{O}}_U(V)$ be the algebra of functions $f:V\to \mathbb{C}$ that are complex Fréchet differentiable at every point of the open set $V\subseteq U$. For $m=0$, ${\mathbb{C}}^0$ is a point; on that point ${\mathcal{O}}_U(U)=\mathbb{C}$, the algebra of all functions from the point to $\mathbb{C}$. For integers $m,n\ge 0$, the complex superdomain $U^{m\mathrm{\mid }n}$ has underlying topological space $U$ and sheaf ${\mathcal{O}}_{U^{m\mathrm{\mid }n}}(V)={\mathcal{O}}_U(V){\otimes }_{\mathbb{C}}{\mathrm{\Lambda }}_{\mathbb{C}}({\theta }_1,\dots ,{\theta }_n),$ where the generators ${\theta }_i$ are odd, so ${\theta }_i{\theta }_j=-{\theta }_j{\theta }_i$ and ${\theta }_i^2=0$. The complex supermanifold of dimension $m\mathrm{\mid }n$ is a Hausdorff, second-countable locally ringed superspace locally isomorphic to these superdomains by the morphisms just defined. Its reduced space $S_{\mathrm{r}\mathrm{e}\mathrm{d}}$ is the ringed space obtained by quotienting ${\mathcal{O}}_S$ by the ideal locally generated by odd sections; on $U^{m\mathrm{\mid }n}$ this is $(U,{\mathcal{O}}_U)$.

Use left odd derivatives. A homogeneous derivation $X$ of parity $\mathrm{\mid }X\mathrm{\mid }\in \{0,1\}$ satisfies the graded Leibniz rule $X(fg)=X(f)g+(-1{)}^{\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }f\mathrm{\mid }}fX(g)$. The tangent sheaf $T_S$ is the locally free sheaf of such derivations. A distribution of rank $r\mathrm{\mid }s$ is a locally free direct summand of $T_S$ with even rank $r$ and odd rank $s$.

For an even invertible coordinate Jacobian written in blocks $J=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$, with $A,D$ even square blocks and $B,C$ odd blocks, its Berezinian is $\mathrm{Ber}(J)=\det (A-B{D}^{-1}C)\mathrm{/}\det D$. The Berezinian line is the line bundle whose transition functions are these Berezinians. All uses below have invertible $D$ blocks; the alternative formula using invertible $A$ is equivalent on the common domain.

Witten, Notes on Super Riemann Surfaces and Their Moduli, §2.1

Wang–Yin, On the Equivalence between SRS and PCO Formulations, §§2.1, 4.1

This is local complex-analytic supergeometry. It does not assert that a global supermoduli space, a compactification, an integration cycle, or a physical functional measure exists.