nLab
complex supermanifold

Context

Complex geometry

Supergeometry

Manifolds and cobordisms

Contents

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!):

  1. 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 XX a complex supermanifold in this sense, the underlying reduced manifold X redX_{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.)

  2. In the second sense, a complex supermanifold is a super(complex manifold), a super-version of complex manifold.

complex

A complex supermanifold is a ringed space X=(X,O X)X = (|X|, O_X) such that

  • the structure sheaf O XO_X a sheaf of commutative complex super algebras

  • locally O XO_X is isomorphic to C ( d) δ C^\infty(\mathbb{R}^d) \otimes_{\mathbb{C}} \wedge^\bullet \mathbb{C}^\delta

Write cSDiff? for the category of complex supermanifolds.

Example The functor Π:{realvectorbundles}SDiff \Pi : \{real vector bundles\} \to SDiff has a complex analogue Π:{complexvectorbundles}cSDiff \Pi : \{complex vector bundles\} \to cSDiff .

Let EXE \to X be a complex vector bundle of rank δ\delta. This gives rise to the complex supermanifold ΠE\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 EE.

Remark

C (X):=O X(X)C^\infty(X) := O_X(X) does not in general have a \mathbb{C}-antilinear involution ¯:C (X)C (X)\bar{-} : C^\infty(X) \to C^\infty(X) but there does exist a canonical complex conjugation on the quotient C (X)C^\infty(X) by the ideal of nilpotent sections, which is C (X red;)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

  1. Every complex supermanifold is isomorphic to one of the form ΠE\Pi E.

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

Remark It turns out that a \mathbb{C}-super algebra homomorphism ϕ:C (Y)C (X) \phi : C^\infty(Y) \to C^\infty(X) automatically satisfies ϕ red(f red¯)=ϕ red(f red)¯\phi_{red}(\overline{f_{red}}) = \overline{\phi_{red}(f_{red})}.

Define the complex supermanifold cs dδ\mathbb{R}_{cs}^{d|\delta} as d\mathbb{R}^d with structure sheaf UC (U) δU\mapsto C^\infty(U) \otimes_{\mathbb{C}} \wedge^\bullet \mathbb{C}^\delta.

Then for SS an arbitrary complex supermanifold we have

cs dδ(S)=cSDiff(S, cs dδ)={(x 1,,x d,θ 1,,θ δ)x iC (S) ev,θ jC (S) odd;withx irealinthat(x i) red¯=(x i) red} \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} \}

Example

For

21(S)={(x,y,θ)x,yC (S) ev,θC (S) odd;x,yreal} \mathbb{R}^{2|1}(S) = \{ (x,y,\theta) | x,y \in C^\infty(S)^{ev}, \theta \in C^\infty(S)^{odd}; x,y real \}

we shall write

{(z,z¯,θ)z,z¯C (S) ev;z red¯=(z¯) red}. \simeq \{ (z,\bar z, \theta) | z, \bar z \in C^\infty(S)^{ev}; \overline{z_{red}}=(\overline z )_{red} \}.

Revised on November 6, 2012 19:00:03 by Urs Schreiber (131.174.188.92)