Sheaves with a Real Structure

Introduction

An involution of a space $X$ is a continuous map $\sigma:X\to X$ with $\sigma^2=\mathrm{id}$. A sheaf with a real structure is a sheaf $\mathcal{F}$ of complex vector spaces on $X$ together with a conjugate-linear involution $\sigma_{\mathcal{F}}$ covering $\sigma$: over an open set $U$ it gives an isomorphism $\sigma_{\mathcal{F}}:\mathcal{F}(U)\to\mathcal{F}(\sigma U)$ with $\sigma_{\mathcal{F}}(zs)=\bar z\,\sigma_{\mathcal{F}}(s)$ for a complex scalar $z$, compatible with the restrictions and with $\sigma_{\mathcal{F}}^2=\mathrm{id}$. The pair $(\mathcal{F},\sigma_{\mathcal{F}})$ is the sheaf-theoretic form of the real structure of linear algebra: the fixed sections are the real sections, the fixed locus $X^{\sigma}$ of the involution is the real locus, and the conjugate-linear involution is the descent datum by which the sheaf is recovered from its real part. The real structure is the involution on the elements of the coefficient sheaf, and it is the two-element-group case of the equivariant sheaves of Equivariant Sheaves and Descent, with the coefficient maps required to be conjugate-linear rather than complex-linear.

This article fixes the notion, the fixed part and the real points, and the descent along the involution. The descent identifies the sheaves with a real structure on $X$ with the equivariant sheaves for the two-element group generated by $\sigma$; for a free involution this is the descent to the quotient $X/\langle\sigma\rangle$, and when $X$ is a complexification, so that $X/\langle\sigma\rangle=X^{\sigma}$, the real-structured sheaves are the sheaves on the real form. The article is the second of the involutive layer of the category; the general group action and the descent are Equivariant Sheaves and Descent, the involution of the structure sheaf is The Involution on the Structure Sheaf, the cohomology of an involutive sheaf is Cohomology with an Involution, and the Galois case over a field extension is Galois Descent for Sheaves.

The article uses no analysis and no geometry: the involution is a homeomorphism of order two, the complexification is the tensor product with $\mathbb{C}$ over $\mathbb{R}$, and no holomorphic function, no derivative, no measure and no norm occurs. Throughout, $X$ is a space with an involution $\sigma$, the quotient by the group $\langle\sigma\rangle\cong\mathbb{Z}/2$ is written $p:X\to X/\langle\sigma\rangle$, the fixed locus is $X^{\sigma}=\{x:\sigma x=x\}$, the sheaves are sheaves of complex vector spaces (or of $\mathcal{O}_X$-modules for a structure sheaf with an involution), and $\sigma_{\mathcal{F}}$ denotes the conjugate-linear involution of a sheaf $\mathcal{F}$.

Involutions of a Space

Definition. An involution of $X$ is a continuous map $\sigma:X\to X$ with $\sigma^2=\mathrm{id}$; it generates an action of $\mathbb{Z}/2$ on $X$ by $e\cdot x=x$ and $\tau\cdot x=\sigma x$, and the fixed locus is $X^{\sigma}=\{x:\sigma x=x\}$, a closed subset of $X$. The orbit of $x$ is $\{x,\sigma x\}$, of one point if $x\in X^{\sigma}$ and of two points otherwise, and the quotient $p:X\to X/\langle\sigma\rangle$ is the orbit map.

Proposition. The involution is free if and only if $X^{\sigma}=\varnothing$; in that case the quotient map is a double cover, and for a free properly discontinuous involution the quotient $X/\langle\sigma\rangle$ and the covering $p$ are those of Transformation Groups and The Fundamental Group and Covering Spaces.

Example (complex conjugation). The complex numbers carry the conjugation $z\mapsto\bar z$, an involution with the real numbers as its fixed locus; on $\mathbb{C}^n$ the conjugation acts coordinatewise, with fixed locus $\mathbb{R}^n$, and on the complement of the fixed locus the quotient is the real projective space $\mathbb{RP}^{2n-1}$. The example is the model of every real structure: the bar is the involution and the fixed locus is the real part.

Example (the complexification). For a real vector space $V$, the complexification $V_{\mathbb{C}}=V\otimes_{\mathbb{R}}\mathbb{C}$ carries the conjugation $v\otimes z\mapsto v\otimes\bar z$, whose fixed locus is $V$; the example shows that a space may be the complexification of its own fixed locus, and this is the case in which the real structure is called a real form.

Sheaves with a Real Structure

Definition. Let $\sigma$ be an involution of $X$. A sheaf with a real structure, or an involutive sheaf, is a sheaf $\mathcal{F}$ of complex vector spaces on $X$ together with, for every $g\in\langle\sigma\rangle$, a conjugate-linear isomorphism

$$ \sigma_{\mathcal{F}}^{\tau}:\mathcal{F}\longrightarrow \sigma_*\mathcal{F},\qquad \sigma_{\mathcal{F}}^{e}=\mathrm{id}, $$

satisfying the cocycle $\sigma_{\mathcal{F}}^{\tau}\circ\sigma_*(\sigma_{\mathcal{F}}^{\tau})=\mathrm{id}$; equivalently, a family of conjugate-linear isomorphisms $\mathcal{F}(U)\to\mathcal{F}(\sigma U)$ compatible with the restrictions and with $\sigma_{\mathcal{F}}^2=\mathrm{id}$. A morphism of sheaves with a real structure is a complex-linear morphism $\psi:\mathcal{F}\to\mathcal{G}$ commuting with the involutions, $\sigma_{\mathcal{G}}\circ\sigma_*(\psi)=\psi\circ\sigma_{\mathcal{F}}$; the category is written $\mathrm{Sh}^{\mathbb{R}}(X,\sigma)$.

Remark (real structure and equivariant structure). A real structure is exactly an equivariant structure for the two-element group generated by $\sigma$ in which the coefficient maps are conjugate-linear; the only difference from Equivariant Sheaves and Descent is the presence of the bar on the scalars. Forgetting the conjugate-linearity gives a $\mathbb{Z}/2$-equivariant sheaf of complex vector spaces; the real structures on a fixed underlying equivariant sheaf are classified by the choices of the antilinear involutions, and they need not exist.

Proposition (the involution on the stalks). The real structure induces, for every $x\in X$, a conjugate-linear involution $\sigma_{\mathcal{F},x}:\mathcal{F}_x\to\mathcal{F}_{\sigma x}$ with $\sigma_{\mathcal{F},\sigma x}\circ\sigma_{\mathcal{F},x}=\mathrm{id}$; over a fixed point $x\in X^{\sigma}$ the involution $\sigma_{\mathcal{F},x}$ is a conjugate-linear involution of the stalk $\mathcal{F}_x$, and the fixed germs are the real germs at $x$.

Proof. The conjugate-linear isomorphism between the sections over $U$ and over $\sigma U$ passes to the colimit of the germs; over a fixed point the two colimits coincide and the involution is a conjugate-linear map of the stalk to itself with square one.

Proposition (the category is abelian). The category $\mathrm{Sh}^{\mathbb{R}}(X,\sigma)$ is abelian, and it is equivalent to the category of $\mathbb{Z}/2$-equivariant sheaves of real vector spaces on the quotient when the involution is free; kernels and cokernels are computed on the underlying sheaves with the induced real structure.

Proof. The constructions of Equivariant Sheaves and Descent apply to the equivariant structure, and the conjugate-linearity is preserved by the formation of kernels and cokernels because the bar is an involution of the coefficient field; the equivalence with the quotient is the descent theorem for the two-element group.

The Fixed Part and the Real Points

Definition. The fixed sheaf of an involutive sheaf $(\mathcal{F},\sigma_{\mathcal{F}})$ is the subsheaf $\mathcal{F}^{\sigma}$ of real sections, the sections $s$ with $\sigma_{\mathcal{F}}(s)=s$ wherever the involution is defined; for a $\sigma$-invariant open set $U$ it is the fixed subspace $\mathcal{F}(U)^{\sigma}$ of the conjugate-linear involution $\sigma_{\mathcal{F}}(U)$, whose elements are the real sections over $U$. The real points of $X$ are the points of $X^{\sigma}$, and the restriction $\mathcal{F}|_{X^{\sigma}}$ with the involutions on the stalks is the real part of the sheaf.

Proposition (real sections form a module over the real functions). Over a $\sigma$-invariant open set $U$ the real sections $\mathcal{F}(U)^{\sigma}$ form a module over the fixed functions $\mathcal{O}_X(U)^{\sigma}=\{a:\bar a=a\}$, that is, over the real-valued functions; for the constant sheaf $\underline{\mathbb{C}}$ with the conjugation, the real sections are the real-valued locally constant functions.

Proof. If $s$ is real and $a$ is real-valued then $as$ is real because $\sigma_{\mathcal{F}}(as)=\bar a\,\sigma_{\mathcal{F}}(s)=as$; the module axioms are those of the complex sections. For the constant sheaf the computation is the same with $z$ in place of $s$.

Proposition (the real part at a fixed point). For $x\in X^{\sigma}$ the stalk of the fixed sheaf at $x$ is the fixed subspace $(\mathcal{F}_x)^{\sigma}$ of the conjugate-linear involution of the stalk, and the real dimension of $(\mathcal{F}_x)^{\sigma}$ over $\mathbb{R}$ equals the complex dimension of $\mathcal{F}_x$; the fixed sheaf is therefore a sheaf of real vector spaces on the real locus.

Proof. The stalk of the fixed sheaf at a fixed point is the direct limit of the fixed subspaces, which is the fixed subspace of the direct limit because the involution is conjugate-linear and an involution; the dimension statement is the standard fact that an antilinear involution of a complex vector space of dimension $n$ has a real form of dimension $n$.

Example (the conjugation constant sheaf). On $X=\mathbb{C}$ with the conjugation, the constant sheaf $\underline{\mathbb{C}}$ with the real structure $\sigma_{\mathcal{F}}(s)=\bar s$ has fixed sheaf $\underline{\mathbb{R}}$ on the fixed locus $\mathbb{R}$, extended by zero off it; the real sections over a conjugation-invariant open set are the real-valued locally constant functions. The example is the sheaf-theoretic form of the real form of a complex vector space.

Descent Along the Involution

Theorem (descent). Let $\sigma$ be an involution of $X$. The sheaves with a real structure on $X$ are exactly the $\mathbb{Z}/2$-equivariant sheaves on $X$ whose coefficient maps are conjugate-linear, and the forgetful functor to the $\mathbb{Z}/2$-equivariant sheaves of complex vector spaces is faithful. The fixed sheaf is the invariants functor and the descent along the quotient,

$$ p_*^{\sigma}\mathcal{F}(V)=\mathcal{F}\bigl(p^{-1}V\bigr)^{\sigma},\qquad V\subseteq X/\langle\sigma\rangle, $$

is right adjoint to the pullback $p^*$ along the orbit map. For a free and properly discontinuous involution the pullback and the descent are inverse equivalences

$$ \mathrm{Sh}\bigl(X/\langle\sigma\rangle\bigr)\simeq\mathrm{Sh}^{\mathbb{R}}(X,\sigma), $$

realised by the conjugate-linear equivariant structures.

Proof. The statements are those of Equivariant Sheaves and Descent for the two-element group, with the coefficient maps conjugate-linear; the descent theorem for a free properly discontinuous action applies verbatim to the underlying equivariant sheaves, the bar being the identity on the constants.

Corollary (the real form). Suppose $X$ is the complexification of a space $Y$, so that $X=Y_{\mathbb{C}}=Y\otimes_{\mathbb{R}}\mathbb{C}$ with the conjugation involution, $X^{\sigma}=Y$, and $X/\langle\sigma\rangle=Y$. Then the descent is an equivalence between the sheaves with a real structure on $X$ and the sheaves of real vector spaces on $Y$,

$$ \mathrm{Sh}^{\mathbb{R}}\bigl(Y_{\mathbb{C}},\sigma\bigr)\simeq\mathrm{Sh}_{\mathbb{R}}(Y), $$

the functor in one direction being $(-)^{\sigma}$ and in the other the complexification $\mathcal{G}\mapsto\mathcal{G}\otimes_{\mathbb{R}}\underline{\mathbb{C}}$; the real structure is the real form of the complex sheaf.

Proof. The complexification is the pullback along $Y\to Y_{\mathbb{C}}$ followed by the extension of scalars, and it carries the real structure given by the conjugation on the second factor; the fixed sheaf and the complexification are inverse by the descent theorem, since the quotient map identifies $X/\langle\sigma\rangle$ with $Y$.

Proposition (cohomology of a real structure). The real structure acts on the cohomology, $\sigma_{\mathcal{F}}^*:H^i(X,\mathcal{F})\to H^i(X,\sigma_*\mathcal{F})$, and for a $\sigma$-invariant situation it gives a conjugate-linear involution of $H^i(X,\mathcal{F})$; the fixed classes are the real classes, and the descent identifies the cohomology of the real form with the fixed part under the appropriate hypotheses. The functoriality and the fixed part of the action are the subject of Cohomology with an Involution.

Proof. The conjugate-linear isomorphism of coefficients induces a conjugate-linear map on cohomology by the functoriality of $H^i$, and the involution property passes because of the cocycle; the identification with the real form is the descent of the previous theorem applied to an acyclic resolution.

Worked Cases

The Real Structure of a Constant Sheaf

For $X=\mathbb{C}^n$ with the coordinatewise conjugation and $\mathcal{F}=\underline{\mathbb{C}}$, the real structure is $s\mapsto\bar s$; the fixed sheaf is $\underline{\mathbb{R}}$ on the real locus $\mathbb{R}^n$, the real structure is the typical one, and the descended sheaf on the quotient $\mathbb{C}^n/\langle\sigma\rangle$ restricts on the real locus to the sheaf of real locally constant functions. The cohomology $H^i(\mathbb{C}^n,\underline{\mathbb{C}})$ is the complexification of $H^i(\mathbb{R}^n,\underline{\mathbb{R}})$ in the sense of the corollary: the fixed classes of the conjugation are the real classes.

The Structure Sheaf of a Complexification

For $X=Y_{\mathbb{C}}$ the structure sheaf $\mathcal{O}_X$ carries the real structure induced by the conjugation of the coefficients, $\sigma_{\mathcal{O}}(a)=\bar a$ in the complexified coordinates; the fixed sheaf is the sheaf of real-valued functions on $Y$ extended to $Y_{\mathbb{C}}$, and the descent recovers the real structure sheaf of $Y$. The example is the geometric case of the corollary and the entry point to The Involution on the Structure Sheaf.

The Free Involution and the Double Cover

For a free involution of $X$ the quotient is a double cover and the descent is the one of Equivariant Sheaves and Descent for the two-element group; a sheaf with a real structure on $X$ is the same thing as a sheaf on the quotient, the real structure being the descent datum. When the quotient map is a covering of a space that is the real locus of a complexification, the two descriptions of the descent — through the quotient and through the real form — coincide.

Summary

An involution of a space is a homeomorphism of order two, with fixed locus the real points and orbit map the quotient by the two-element group. A sheaf with a real structure is a sheaf of complex vector spaces with a conjugate-linear involution covering the involution of the space; the real structure is the involution on the elements of the coefficient sheaf, and it is the conjugate-linear case of the equivariant sheaves of the previous article. The fixed sheaf consists of the real sections, a module over the real-valued functions, with stalk at a fixed point the real form of the stalk; the real part of the sheaf is its restriction to the real locus, and the real structure acts on the cohomology by a conjugate-linear involution whose fixed classes are the real classes.

The descent identifies the sheaves with a real structure on $X$ with the equivariant sheaves for the two-element group, and for a free and properly discontinuous involution with the sheaves on the quotient $X/\langle\sigma\rangle$. When $X$ is the complexification of a real space $Y$, so that the quotient is the fixed locus, the descent is an equivalence between the sheaves with a real structure on $X$ and the sheaves on the real form $Y$: the real structure is the real form of the complex sheaf, and the complexification and the fixed part are inverse functors.

Summary of Notation

Further Reading

• Glen E. Bredon, Sheaf Theory (Springer, second edition, 1997), for equivariant sheaves and the fixed sheaf under a group action of order two.
• Alexander Grothendieck, Théorie des topos et cohomologie étale des schémas (SGA 4) (Springer Lecture Notes in Mathematics 269, 270, 305, 1972–1973), for the conjugate-linear descent and the real forms of sheaves.
• Max-Albert Knus, Alexander Merkurjev, Markus Rost and Jean-Pierre Tignol, The Book of Involutions, Colloquium Publications 44 (American Mathematical Society, 1998), for the descent of the linear structures along an involution and the real forms.
• Michael F. Atiyah, "K-theory and reality", Quarterly Journal of Mathematics (2) 17 (1966), 367–386, for the real structures on vector bundles and the descent to the real locus, cited for the geometric case.
• Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic (Springer, 1992), for the equivariant sheaves and the descent in the topos-theoretic form.
• Jean-Pierre Serre, Galois Cohomology (Springer, 1997), for the descent of the linear structures and the comparison with the Galois case.