A Torelli theorem for moduli spaces of principal bundles on curves defined over $\mathbb R$

TL;DR

This paper proves a Torelli-type theorem showing that the isomorphism class of the moduli space of principal G-bundles on a real algebraic curve uniquely determines the curve itself, extending classical Torelli results.

Contribution

It establishes a Torelli theorem for moduli spaces of principal bundles over real algebraic curves, a significant extension of classical results to the real setting.

Findings

The moduli space's isomorphism class determines the curve uniquely.

The result applies to nonabelian groups with one simple factor.

It extends Torelli theorems to real algebraic geometry.

Abstract

Let $X$ be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let $G$ be a connected reductive affine algebraic group, defined over $\mathbb R$, such that $G$ is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal $G$--bundles on $X$ determine uniquely the isomorphism class of $X$.

Equations

$$\det :{\mathcal{M}}_X(r,d)⟶{\text{Pic}}^d(X)$$

$${\mathcal{N}}_X(r,\mathcal{L}):={\det }^{-1}(\mathcal{L})\subset {\mathcal{M}}_X(r,d).$$

$$\sigma :X_{\mathbb{C}}⟶X_{\mathbb{C}}.$$

$$\begin{matrix}E&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}&\sigma^{\ast}\overline{E}\\ \Big{\downarrow}&&\Big{\downarrow}\\ X&\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}&X\end{matrix}$$

$${\sigma }_{\mathcal{N}}:{\mathcal{N}}_{X_{\mathbb{C}}}(r,{\mathcal{L}}_{\mathbb{C}})⟶{\mathcal{N}}_{X_{\mathbb{C}}}(r,{\mathcal{L}}_{\mathbb{C}})$$

$${\tau }_{\mathcal{N}}=f^{-1}{\sigma }_{\mathcal{N}}f.$$

$$\Psi :\text{Aut}({\mathcal{N}}_{X_{\mathbb{C}}}(r,L_{\mathbb{C}}))⟶\text{Aut}(X_{\mathbb{C}}),h⟼H^{-1}.$$

$$f(V)=A\otimes {\phi }^{\ast }V,$$

$$f^{-1}(V)=(({\phi }^{-1}{)}^{\ast }{A}^{-1})\otimes ({\phi }^{-1}{)}^{\ast }V.$$

$$V⟼{\tau }^{\ast }\overline{(({\phi }^{-1}{)}^{{}^{\ast }}{A}^{-1})\otimes ({\phi }^{-1}{)}^{{}^{\ast }}(({\sigma }^{\ast }\overline{A})\otimes ({\sigma }^{\ast }\overline{{\phi }^{\ast }V}))}$$

$$=B\otimes {\tau }^{\ast }({\phi }^{-1}{)}^{\ast }{\sigma }^{\ast }{\phi }^{\ast }V=B\otimes (\phi \circ \sigma \circ {\phi }^{-1}\circ \tau {)}^{\ast }V,$$

$$\eta :=\Psi ({\tau }_{\mathcal{N}}^{-1}\circ f^{-1}\circ {\sigma }_{\mathcal{N}}\circ f)=\phi \circ \sigma \circ {\phi }^{-1}\circ \tau .$$

$$\tau =\phi \circ \sigma \circ {\phi }^{-1}.$$

$$f(V)=A\otimes {\phi }^{\ast }{V}^{\vee },$$

$$f^{-1}(V)=(({\phi }^{-1}{)}^{\ast }A)\otimes ({\phi }^{-1}{)}^{\ast }{V}^{\vee }.$$

$$V⟼{\tau }^{\ast }\overline{(({\phi }^{-1}{)}^{{}^{\ast }}A)\otimes (({\phi }^{-1}{)}^{{}^{\ast }}(({\sigma }^{\ast }\overline{A})\otimes ({\sigma }^{\ast }\overline{{\phi }^{\ast }{V}^{\vee }}){)}^{\vee })}$$

$$=B\otimes {\tau }^{\ast }({\phi }^{-1}{)}^{\ast }{\sigma }^{\ast }{\phi }^{\ast }V=B\otimes (\phi \circ \sigma \circ {\phi }^{-1}\circ \tau {)}^{\ast }V,$$

$$\Psi ({\tau }_{\mathcal{N}}^{-1}\circ f^{-1}\circ {\sigma }_{\mathcal{N}}\circ f)=\phi \circ \sigma \circ {\phi }^{-1}\circ \tau .$$

$${\sigma }_G:G_{\mathbb{C}}⟶G_{\mathbb{C}}$$

$$\overline{E}=E({\sigma }_G)=E{\times }^{{\sigma }_{{}_G}}{G}_{\mathbb{C}}⟶X_{\mathbb{C}}$$

$${\sigma }_{\mathcal{M}}:{\mathcal{M}}_{X_{\mathbb{C}}}(G_{\mathbb{C}})⟶{\mathcal{M}}_{X_{\mathbb{C}}}(G_{\mathbb{C}}),E⟼{\sigma }^{\ast }\overline{E}.$$

$$y^{-1}Ly\subset y^{-1}Py={\sigma }_G(P)$$

$$L^{\prime }=[L,L]$$

$${\pi }_1(L^{\prime })⟶{\pi }_1(G_{\mathbb{C}})$$

$${\pi }_2(G_{\mathbb{C}})⟶{\pi }_2(G_{\mathbb{C}}/L^{\prime })⟶{\pi }_1(L^{\prime })⟶{\pi }_1(G_{\mathbb{C}})$$

$${\pi }_2(G_{\mathbb{C}}/L^{\prime })=0.$$

$${\pi }_2(P/L^{\prime })⟶{\pi }_2(G_{\mathbb{C}}/L^{\prime })⟶{\pi }_2(G_{\mathbb{C}}/P)⟶{\pi }_1(G_{\mathbb{C}})$$

$${\pi }_2(P/L^{\prime })⟶{\pi }_2(G_{\mathbb{C}}/L^{\prime })$$

$${\pi }_2(L/L^{\prime })⟶{\pi }_2(P/L^{\prime })⟶{\pi }_2(P/L)$$

$$L/L^{\prime }⟶P/L^{\prime }⟶P/L.$$

Full text

1. Introduction

Let $X$ be a geometrically irreducible smooth projective curve defined over the field of real numbers, of genus $g$, with $g\,\geq\,3$. Let $\mathcal{L}\,\in\,\text{Pic}^{d}(X)$ be a point defined over $\mathbb{R}$. We note that $\mathcal{L}$ need not correspond to a line bundle over $X$. For example, the unique $\mathbb{R}$–point of $\text{Pic}^{1}$ of the anisotropic conic does not correspond to a line bundle over the anisotropic conic. Let ${\mathcal{N}}_{X}(r,\mathcal{L})$ denote the moduli space of semistable vector bundles on $X$ of rank $r$ and determinant $\mathcal{L}$, where $r\,\geq\,2$.

We prove that the isomorphism class of the variety ${\mathcal{N}}_{X}(r,\mathcal{L})$ uniquely determines the isomorphism class of the real curve $X$ (Theorem 2.1).

When the base field is complex numbers, this was proved in [MN] for rank two, and in [Tj], [KP, p. 229, Theorem E] for general $r$ and $d$.

Let ${{X_{\mathbb{C}}}}$ be the complexification of $X$. Let $G_{\mathbb{C}}$ be a connected reductive affine algebraic group defined over $\mathbb{C}$, and let $G$ be a real form of $G_{\mathbb{C}}$. We assume that $G_{\mathbb{C}}$ is nonabelian and it has exactly one simple factor. The antiholomorphic involution of $G_{\mathbb{C}}$ corresponding to $G$ will be denoted by $\sigma_{G}$. Let $\mathcal{M}_{{X_{\mathbb{C}}}}(G_{\mathbb{C}})$ denote the moduli space of topologically trivial semistable principal $G_{\mathbb{C}}$–bundles on ${X_{\mathbb{C}}}$. The variety $\mathcal{M}_{{X_{\mathbb{C}}}}(G_{\mathbb{C}})$ is the complexification of the component of the moduli space of principal $G$–bundles on $X$ that contains the trivial $G$–bundle. The involution $\sigma_{G}$ and the antiholomorphic involution of ${X_{\mathbb{C}}}$ together produce the antiholomorphic involution $\sigma_{\mathcal{M}}$ of $\mathcal{M}_{{X_{\mathbb{C}}}}(G_{\mathbb{C}})$.

We prove that the isomorphism class of the real variety $(\mathcal{M}_{{X_{\mathbb{C}}}}(G_{\mathbb{C}})\,,\sigma_{\mathcal{M}})$ uniquely determines the isomorphism class of $X$ (Theorem 3.4).

The proof of Theorem 3.4 crucially uses a result of [BHo] which says that the isomorphism class of $\mathcal{M}_{{X_{\mathbb{C}}}}(G_{\mathbb{C}})$ uniquely determines the isomorphism class of ${X_{\mathbb{C}}}$.

2. Moduli spaces of vector bundles

Let $X$ be a geometrically irreducible smooth projective curve defined over $\mathbb{R}$. Let $g$ denote the genus of $X$. We will assume that $g\,\geq\,3$. For any $d\,\in\,\mathbb{Z}$ and any integer $r\,\geq\,2$, let ${\mathcal{M}}_{X}(r,d)$ be the moduli space of semistable vector bundles on $X$ of rank $r$ and degree $d$; see [BHu], [BGH], [BHH], [Sc1], [Sc2], [Sc3] for moduli spaces of bundles over $X$. Let

[TABLE]

be the morphism defined by $E\,\longmapsto\,\bigwedge^{r}E$. Take any $\mathbb{R}$–point $\mathcal{L}\,\in\,\text{Pic}^{d}(X)$. Define

[TABLE]

This ${\mathcal{N}}_{X}(r,\mathcal{L})$ is a geometrically irreducible normal projective variety defined over $\mathbb{R}$, of dimension $(r^{2}-1)(g-1)$.

Let ${X_{\mathbb{C}}}\,:=\,X_{\mathbb{C}}\,=\,X\times_{\mathbb{R}}\mathbb{C}$ be the complex projective curve obtained from $X$ by extending the base field to $\mathbb{C}$. Let $\mathcal{L}_{\mathbb{C}}\,\in\,\text{Pic}^{d}({X_{\mathbb{C}}})$ be the pull-back of $\mathcal{L}$ to ${X_{\mathbb{C}}}$ by the natural morphism $\xi\,:\,{X_{\mathbb{C}}}\,\longrightarrow\,X$. The nontrivial element of the Galois group $\text{Gal}(\xi)\,=\,\text{Gal}({\mathbb{C}}/{\mathbb{R}})\,=\,{\mathbb{Z}}/2\mathbb{Z}$ produces an antiholomorphic involution

[TABLE]

The conjugate vector bundle of a holomorphic vector bundle $E$ on ${X_{\mathbb{C}}}$ will be denoted by $\overline{E}$. We recall that the underlying real vector bundle for $\overline{E}$ is identified with that of $E$, while the multiplication on $\overline{E}$ by any $c\,\in\,\mathbb{C}$ coincides with the multiplication by $\overline{c}$ on $E$. The $C^{\infty}$ vector bundle $\sigma^{\ast}\overline{E}$ has a natural holomorphic structure which is uniquely determined by the condition that the natural $\mathbb{R}$–linear identification of it with $E$ is anti-holomorphic. Note that we have a commutative diagram

[TABLE]

It is easy to see that $E$ is semistable (respectively, stable) if and only if $\sigma^{\ast}\overline{E}$ is semistable (respectively, stable). Similarly, $E$ is polystable if and only if $\sigma^{\ast}\overline{E}$ is polystable.

The above hypothesis that $\mathcal{L}\,\in\,\text{Pic}^{d}(X)$ means that the line bundle $\mathcal{L}_{\mathbb{C}}$ is holomorphically isomorphic to the line bundle $\sigma^{\ast}\overline{\mathcal{L}_{\mathbb{C}}}$.

Let ${\mathcal{N}}_{{X_{\mathbb{C}}}}(r,\mathcal{L}_{\mathbb{C}})$ be the moduli space of semistable vector bundles on ${X_{\mathbb{C}}}$ of rank $r$ and determinant $\mathcal{L}_{\mathbb{C}}$. The complex variety ${\mathcal{N}}_{{X_{\mathbb{C}}}}(r,\mathcal{L}_{\mathbb{C}})$ coincides with the complexification ${{\mathcal{N}}_{X}(r,\mathcal{L})\times_{\mathbb{R}}\mathbb{C}}$ of $\mathcal{N}_{X}(r,\mathcal{L})$; the resulting antiholomorphic involution

[TABLE]

sends a vector bundle $E$ on ${X_{\mathbb{C}}}$ to the vector bundle $\sigma^{\ast}\overline{E}$.

Theorem 2.1**.**

The isomorphism class of the $\mathbb{R}$–variety $\mathcal{N}_{X}(r,\mathcal{L})$ uniquely determines the isomorphism class of the real curve $X$.

Proof.

First note that the isomorphism class of the complex variety $\mathcal{N}_{{X_{\mathbb{C}}}}(r,\mathcal{L}_{\mathbb{C}})$ uniquely determines the complex curve ${X_{\mathbb{C}}}$ [Tj], [KP, p. 229, Theorem E]. We have to prove that the antiholomorphic involution $\sigma_{\mathcal{N}}$ determines $\sigma$.

Let $\tau$ be an antiholomorphic involution of ${X_{\mathbb{C}}}$ such that the involution $E\,\longmapsto\,\tau^{\ast}\overline{E}$ preserves $\mathcal{N}_{{X_{\mathbb{C}}}}(r,\mathcal{L}_{\mathbb{C}})$. The resulting antiholomorphic involution of ${\mathcal{N}}_{{X_{\mathbb{C}}}}(r,\mathcal{L}_{\mathbb{C}})$ will be denoted by $\tau_{\mathcal{N}}$. The two real varieties $(\mathcal{N}_{{X_{\mathbb{C}}}}(r,\mathcal{L}_{\mathbb{C}})\,,\tau_{\mathcal{N}})$ and $(\mathcal{N}_{{X_{\mathbb{C}}}}(r,\mathcal{L}_{\mathbb{C}})\,,\sigma_{\mathcal{N}})$ are isomorphic if and only if there exists a complex algebraic automorphism $f$ of ${\mathcal{N}}_{{X_{\mathbb{C}}}}(r,\mathcal{L}_{\mathbb{C}})$ such that

[TABLE]

Assume that the two real varieties $(\mathcal{N}_{{X_{\mathbb{C}}}}(r,\mathcal{L}_{\mathbb{C}})\,,\tau_{\mathcal{N}})$ and $(\mathcal{N}_{{X_{\mathbb{C}}}}(r,\mathcal{L}_{\mathbb{C}})\,,\sigma_{\mathcal{N}})$ are isomorphic. Fix an automorphism $f$ of $\mathcal{N}_{{X_{\mathbb{C}}}}(r,\mathcal{L}_{\mathbb{C}})$ satisfying (2.2).

The dual a vector bundle $E$ will be denoted by $E^{\vee}$; the dual of a line bundle $\nu$ will also be denoted by $\nu^{-1}$.

Take any algebraic automorphism $h$ of ${\mathcal{N}}_{{X_{\mathbb{C}}}}(r,L_{\mathbb{C}})$. It follows from [KP, p. 228, Theorem B] and [KP, p. 228, remark 0.1] that $h$ is either of the form $E\,\longmapsto\,H^{*}E\otimes\nu$ or $E\,\longmapsto\,H^{*}E^{\vee}\otimes\nu_{1}$, where $H$ is an automorphism of ${X_{\mathbb{C}}}$ uniquely determined by $h$ while $\nu$ a line bundle on ${X_{\mathbb{C}}}$ with $\nu^{\otimes r}\,=\,{\mathcal{O}}_{X}$ and $\nu_{1}$ a line bundle on ${X_{\mathbb{C}}}$ with $\nu^{\otimes r}_{1}\,=\,L^{\otimes 2}_{\mathbb{C}}$; it should be clarified both $\nu$ and $\nu_{1}$ are independent of $E$. Therefore, we get a map

[TABLE]

It is straight-forward to check that $\Psi$ is a homomorphism of groups.

We will denote $\Psi(f)\,\in\,\text{Aut}({X_{\mathbb{C}}})$ by $\varphi$, where $\Psi$ is defined in (2.3) and $f$ is the automorphism in (2.2). First assume that

[TABLE]

where $A$ is a line bundle on ${X_{\mathbb{C}}}$. Therefore, we have

[TABLE]

Hence the automorphism $\tau^{-1}_{\mathcal{N}}\circ f^{-1}\circ\sigma_{\mathcal{N}}\circ f$ of ${\mathcal{N}}_{{X_{\mathbb{C}}}}(r,L_{\mathbb{C}})$ is the morphism defined by

[TABLE]

[TABLE]

where $B$ is a line bundle which does not depend on $V$. This implies that

[TABLE]

Now from (2.2) we conclude that $\eta\,=\,\text{Id}_{{X_{\mathbb{C}}}}$. So from (2.4) we have

[TABLE]

Therefore, $\varphi$ produces an isomorphism between the two curves $({{X_{\mathbb{C}}}}\,,\sigma)$ and $({{X_{\mathbb{C}}}}\,,\tau)$.

Next assume that

[TABLE]

where $A$ is a line bundle on ${X_{\mathbb{C}}}$. Then

[TABLE]

Therefore, the automorphism $\tau^{-1}_{\mathcal{N}}\circ f^{-1}\circ\sigma_{\mathcal{N}}\circ f$ of ${\mathcal{N}}_{{X_{\mathbb{C}}}}(r,L_{\mathbb{C}})$ is the morphism defined by

[TABLE]

[TABLE]

where $B$ is a line bundle which does not depend on $V$. This implies that

[TABLE]

Hence, as before, $\tau\,=\,\varphi\circ\sigma\circ\varphi^{-1}$. This completes the proof of the theorem. ∎

3. Moduli spaces of principal bundles

Let $G_{\mathbb{C}}$ be a connected nonabelian reductive group over $\mathbb{C}$ with only one simple factor and let

[TABLE]

be an antiholomorphic automorphism of order two. We denote by $G$ the real form of $G_{\mathbb{C}}$ corresponding to $\sigma_{G}$.

Let $\mathcal{M}_{{X_{\mathbb{C}}}}(G_{\mathbb{C}})$ denote the moduli space of topologically trivial semistable principal $G_{\mathbb{C}}$–bundles on ${X_{\mathbb{C}}}$. It is an irreducible normal projective variety defined over $\mathbb{C}$. For any holomorphic principal $G_{\mathbb{C}}$–bundle $E$ on ${X_{\mathbb{C}}}$, let

[TABLE]

be the $C^{\infty}$ principal $G_{\mathbb{C}}$–bundle obtained by twisting the action of $G_{\mathbb{C}}$ using the homomorphism $\sigma_{G}$. So the total space of $\overline{E}$ is identified with that of $E$, but the action of any $y\,\in\,G_{\mathbb{C}}$ on $\overline{E}$ is the action of $\sigma_{G}(y)$ on $E$ in terms of the identification of $E$ with ${\overline{E}}$. The pullback $\sigma^{\ast}\overline{E}$ has a holomorphic structure uniquely determined by the condition that the above identification between the total spaces of $E$ and $\sigma^{\ast}\overline{E}$ is anti-holomorphic; since the total spaces of $\overline{E}$ and $\sigma^{\ast}\overline{E}$ are naturally identified, the above identification between the total spaces of $E$ and $\overline{E}$ produces an identification of the total spaces of $E$ and $\sigma^{\ast}\overline{E}$. The complex projective variety $\mathcal{M}_{{X_{\mathbb{C}}}}(G_{\mathbb{C}})$ carries a real structure associated to the antiholomorphic involution

[TABLE]

Let $\mathcal{M}_{X}(G)$ denote the variety over $\mathbb{R}$ defined by the above pair $(\mathcal{M}_{G_{\mathbb{C}}}({X_{\mathbb{C}}})\,,\sigma_{\mathcal{M}})$.

A Zariski closed connected subgroup $P\,\subset\,G_{\mathbb{C}}$ is called a parabolic subgroup if $G_{C}/P$ is a complete variety. A Levi subgroup of $P$ is a maximal connected reductive subgroup of $P$ containing a maximal torus. Any two Levi subgroups of $P$ are conjugate by some element of $P$. A proper parabolic subgroup $P\,\subset\,G_{\mathbb{C}}$ is called maximal if there is no proper parabolic subgroup of $G_{\mathbb{C}}$ containing $P$.

Lemma 3.1**.**

There exists a maximal parabolic subgroup $P\,\subset\,G_{\mathbb{C}}$ and a Levi subgroup $L\,\subset\,P$, such that the two subgroups $\sigma_{G}(L)$ and $L$ are conjugate by some element of $G_{\mathbb{C}}$.

Proof.

For any parabolic subgroup $P\,\subset\,G_{\mathbb{C}}$, the image $\sigma_{G}(P)$ is also a parabolic subgroup of $G_{\mathbb{C}}$. Since $\sigma_{G}(y^{-1}Py)\,=\,\sigma_{G}(y)^{-1}\sigma_{G}(P)\sigma_{G}(y)$, we get a self-map of the conjugacy classes of parabolic subgroups of $G_{\mathbb{C}}$ that sends the conjugacy class of any $P$ to the conjugacy class of $\sigma_{G}(P)$. Therefore, the involution $\sigma_{G}$ also acts on the Dynkin diagram $D$ of $G_{\mathbb{C}}$ as an involution $\tau$. Examining the Dynkin diagrams we observe that an involution of the Dynkin diagram of $G_{\mathbb{C}}$ must have a fixed point unless $G_{\mathbb{C}}$ is of type $A_{n}$ for $n$ even.

If $G_{\mathbb{C}}$ is not of type $A_{n}$, let $P$ be a maximal parabolic subgroup corresponding to a vertex of $D$ fixed by the above constructed involution $\tau$. Then $P$ and $\sigma_{G}(P)$ are conjugate in $G_{\mathbb{C}}$. Let $y\,\in\,G_{\mathbb{C}}$ be such that $\sigma_{G}(P)\,=\,y^{-1}Py$. Then for any Levi subgroup $L$ of $P$,

[TABLE]

is a Levi subgroup of $\sigma_{G}(P)$.

If $G_{\mathbb{C}}$ is of type $A_{n}$, then $\sigma_{G}(L)$ and $L$ are conjugate for every Levi subgroup of every maximal parabolic subgroup of $G_{\mathbb{C}}$. It is enough to check this for $G_{\mathbb{C}}\,=\,\mathrm{SL}(n+1,\mathbb{C})$, in which case this is obvious. ∎

Remark 3.2**.**

We can be more precise as follows. The two subgroups $\sigma_{G}(L)$ and $L$ are conjugate for every Levi subgroup of every maximal parabolic subgroup of $G_{\mathbb{C}}$ unless $G_{\mathbb{C}}$ is of type $D_{n}$ (with $n\,\geq\,4$) or $E_{6}$.

Lemma 3.3**.**

Let $L$ be any Levi subgroup of a parabolic subgroup $P$ of $G_{\mathbb{C}}$, and let

[TABLE]

be its derived subgroup. Then the homomorphism

[TABLE]

induced by the inclusion $L^{\prime}\,\hookrightarrow\,G_{\mathbb{C}}$ is injective.

Proof.

Consider the fibration $L^{\prime}\,\longrightarrow\,G_{\mathbb{C}}\,\longrightarrow\,G_{\mathbb{C}}/L^{\prime}$. Let

[TABLE]

be the long exact sequence of homotopy groups associated to it. From (3.2) we conclude that the homomorphism in (3.1) is injective if

[TABLE]

Since $\pi_{2}(G_{\mathbb{C}})\,=\,0$ and $\pi_{1}(L^{\prime})$ is a finite group (recall that $L^{\prime}$ is semisimple), from (3.2) it follows that $\pi_{2}(G_{\mathbb{C}}/L^{\prime})$ is a finite group.

Now consider the fibration $P/L^{\prime}\,\longrightarrow\,G_{\mathbb{C}}/L^{\prime}\,\longrightarrow\,G_{\mathbb{C}}/P$. Let

[TABLE]

be the long exact sequence of homotopy groups associated to it. Since $G_{\mathbb{C}}/P$ is simply connected, the second homotopy group $\pi_{2}(G_{\mathbb{C}}/P)$ is isomorphic to $H_{2}(G_{\mathbb{C}}/P,\,\mathbb{Z})$, which is a free abelian group. Therefore, there is no nonzero homomorphism from the finite group $\pi_{2}(G_{\mathbb{C}}/L^{\prime})$ to $\pi_{2}(G_{\mathbb{C}}/P)$. Hence, the homomorphism

[TABLE]

in (3.4) is surjective.

Finally, consider the long exact sequence of homotopy groups

[TABLE]

associated to the fibration

[TABLE]

Since $P/L$ is diffeomorphic to the unipotent radical of $P$, which is contractible, we have $\pi_{2}(P/L)\,=\,0$. Also, $\pi_{2}(L/L^{\prime})\,=\,0$ because $L/L^{\prime}$ is a Lie group. Hence from (3.6) it follows that $\pi_{2}(P/L^{\prime})\,=\,0$. This implies that (3.3) holds because the homomorphism in (3.5) is surjective. ∎

Theorem 3.4**.**

The real variety $\mathcal{M}_{X}(G)$ uniquely determines the real curve $X$.

Proof.

We already know that the complex variety $\mathcal{M}_{{X_{\mathbb{C}}}}(G_{\mathbb{C}})\,=\,\mathcal{M}_{X}(G)\times_{\mathbb{R}}\mathbb{C}$ determines the complex curve ${X_{\mathbb{C}}}$ [BHo].

Let $\mathrm{Sing}(\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}}))$ denote the singular locus of the variety $\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}})$. Recall from [BHo] that the strictly semi–stable locus

[TABLE]

is the Zariski closure of the set of closed points $[E]\,\in\,\mathrm{Sing}(\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}}))$ with the property that every Euclidean neighborhood $U$ of $[E]$ contains an open neighborhood $U^{\prime}\,\ni\,[E]$ such that $U^{\prime}\setminus(U^{\prime}\bigcap\mathrm{Sing}(\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}})))$ is connected and simply connected. Moreover, this closed subset $\Delta_{G}$ is the union of irreducible components corresponding to the conjugacy classes of Levi subgroups of maximal parabolic subgroups of $G_{\mathbb{C}}$. More precisely, the decomposition of $\Delta_{G}$ into irreducible components is the union

[TABLE]

where $L$ ranges over conjugacy classes of Levi subgroups of maximal parabolic subgroups of $G_{\mathbb{C}}$, and $M_{L}$ is the image of the morphism $\mathcal{M}_{X_{\mathbb{C}}}(L)\,\longrightarrow\,\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}})$ given by the inclusion of $L$ in $G_{\mathbb{C}}$. This can be deduced from [BHo, Proposition 3.1] and Lemma 3.3. Indeed, every closed point in $\Delta_{G}$ is defined by a principal $G_{\mathbb{C}}$–bundle $E$ admitting a reduction of structure group $E_{L}$ to a Levi subgroup $L$ of a maximal parabolic subgroup of $G_{\mathbb{C}}$. Moreover, this $L$–bundle $E_{L}$ is semistable and its topological type $\delta\,\in\,\pi_{1}(L)$ is torsion, which means that $\delta$ belongs to $\pi_{1}([L,L])$; this is because $\pi_{1}(L/[L,L])$ is free abelian. Now, since $E$ is topologically trivial, $\delta$ must be trivial (follows from Lemma 3.3), i.e., $[E]\,\in\,\mathcal{M}_{{X_{\mathbb{C}}}}(G_{\mathbb{C}})$ belongs to $M_{L}$.

Moreover, $M_{L}$ is never empty, since there always exist semi-stable principal $L$–bundles which are topologically trivial, for example the trivial holomorphic principal $L$–bundle. The fact that the subvarieties $M_{L_{1}}$ and $M_{L_{2}}$ of $\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}})$ are distinct when $L_{1}$ and $L_{2}$ are not conjugate by some element of $G_{\mathbb{C}}$ is contained in the last part of the proof of [BHo, Proposition 3.1].

The antiholomorphic involution $\sigma_{\mathcal{M}}$ maps the strictly semi–stable locus $\Delta_{G}$ into itself, permuting its irreducible components. It follows from Lemma 3.1 that there exists at least one component $M_{L}$ which is fixed by $\sigma_{\mathcal{M}}$. The restriction of $\sigma_{\mathcal{M}}$ to this component $M_{L}$ has at least one fixed point, namely the closed point corresponding to the trivial bundle.

We now proceed as in the proof of [BHo, Theorem 4.1] to recover the involution $\sigma$ defining the real curve $X$.

First, one can assume $G_{\mathbb{C}}$ to be semi–simple. To see this, let $Z^{0}_{G_{\mathbb{C}}}$ be the connected component of the center of $G_{\mathbb{C}}$ containing the identity element. Let us denote by $G^{\prime}$ the quotient $G_{\mathbb{C}}/Z^{0}_{G_{\mathbb{C}}}$ of $G_{\mathbb{C}}$. Note that $\sigma_{G}$ preserves $Z^{0}_{G_{\mathbb{C}}}$, so it produces a real structure on the quotient $G^{\prime}$. The canonical line bundle of $\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}})$ (respectively, $\mathcal{M}_{X_{\mathbb{C}}}(G^{\prime})$) will be denoted by $\omega_{\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}})}$ (respectively, $\omega_{\mathcal{M}_{X_{\mathbb{C}}}(G^{\prime})}$). We note that $\omega_{\mathcal{M}_{X_{\mathbb{C}}}(G^{\prime})}$ pulls back to $\omega_{\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}})}$ under the morphism $\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}})\,\longrightarrow\,\mathcal{M}_{X_{\mathbb{C}}}(G^{\prime})$ given by the quotient map $G_{\mathbb{C}}\,\longrightarrow\,G^{\prime}$. There exists an integer $m$ such that the pluri–anti–canonical system $|-m\omega_{\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}})}|$ factors into the natural map $\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}})\,\longrightarrow\,\mathcal{M}_{X_{\mathbb{C}}}(G^{\prime})$ followed by the embedding

[TABLE]

Since the dualizing sheaves are defined over the reals, we have real structures on $|-m\omega_{\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}})}|^{\ast}$ and $|-m\omega_{\mathcal{M}_{X_{\mathbb{C}}}(G^{\prime})}|^{\ast}$. All the maps above are defined over $\mathbb{R}$. Therefore, it is enough to prove the theorem for $G^{\prime}$.

So let us assume that $G_{\mathbb{C}}$ is semi–simple. We have seen above that $\Delta_{G}\,\subset\,\mathcal{M}_{X_{\mathbb{C}}}(G_{\mathbb{C}})$ contains at least one irreducible component fixed by $\sigma_{\mathcal{M}}$, which is equal to the variety $M_{L}$ associated to a Levi subgroup $L$ of a maximal parabolic subgroup of $G_{\mathbb{C}}$. Let

[TABLE]

be the normalization of $M_{L}$, and let $\sigma_{L}$ be the antiholomorphic involution of $M_{L}$. Since normalization commutes with the base change of field of definition, the variety $\widetilde{M}_{L}$ is also defined over $\mathbb{R}$, and the morphism $\alpha$ is also defined over $\mathbb{R}$. Hence the antiholomorphic involution $\sigma_{L}$ of $M_{L}$ lifts to $\widetilde{M}_{L}$. Moreover, $\widetilde{M}_{L}$ is isomorphic to the quotient $\mathcal{M}_{X_{\mathbb{C}}}(L)/\Gamma_{L}$, where $\Gamma_{L}$ is the image of $N_{G_{\mathbb{C}}}(L)$ in $\mathrm{Out}(L)$, which is either trivial or ${\mathbb{Z}}/2{\mathbb{Z}}$ (see [BHo]), and this quotient map is compatible with the real structures on $\mathcal{M}_{X_{\mathbb{C}}}(L)$ and $\widetilde{M}_{L}$.

Let $Z^{0}_{L}$ be the connected component of the center of $L$ containing the identity element. Let us denote by $L^{\prime}$ the quotient $L/Z_{L}^{0}$. Then the above group $\Gamma_{L}$ also acts on $\mathcal{M}_{X_{\mathbb{C}}}(L^{\prime})$, and the morphism (defined over the real numbers)

[TABLE]

can be recovered from the second tensor power of the canonical line bundle on the smooth locus of $\widetilde{M}_{L}$. Indeed, this second tensor power extends to a line bundle on the whole variety, and a sufficiently negative power of it gives the morphism $\theta$ (see [BHo]).

Let $\beta\,:\,\mathcal{M}_{X_{\mathbb{C}}}(L^{\prime})\,\longrightarrow\,\mathcal{M}_{X_{\mathbb{C}}}(L^{\prime})/\Gamma_{L}$ be the quotient map. Consider $\theta$ in (3.7). For any point $y\,\in\,\mathcal{M}_{X_{\mathbb{C}}}(L^{\prime})/\Gamma_{L}$, the fiber $\theta^{-1}(y)$ is $J_{X_{\mathbb{C}}}$ (respectively, $J_{X_{\mathbb{C}}}/({\mathbb{Z}}/2{\mathbb{Z}})$) if $\#\beta^{-1}(y)\,=\,2$ (respectively, $\#\beta^{-1}(y)\,=\,1$).

Now take any smooth point

[TABLE]

fixed by the antiholomorphic involution. As noted above, the fiber $\theta^{-1}(y)$ is isomorphic to either $J_{X_{\mathbb{C}}}$ or the singular Kummer variety $J_{X_{\mathbb{C}}}/({\mathbb{Z}}/2{\mathbb{Z}})$. The real structure on $\theta^{-1}(y)$ induced by that of $\widetilde{M}_{L}$ comes from the real structure on the Jacobian associated to the curve. So in both cases we recover the Jacobian variety together with its natural real structure: when $\theta^{-1}(y)$ is isomorphic to $J_{X_{\mathbb{C}}}/({\mathbb{Z}}/2{\mathbb{Z}})$, then $J_{X}$ is obtained from the two–sheeted cover of the smooth locus of the Kummer variety defined by the unique maximal torsion–free subgroup in its fundamental group. The antiholomorphic involution can be lifted to this cover, and this lift extends to $J_{X_{\mathbb{C}}}$ because its construction is over $\mathbb{R}$.

Finally, the class of the canonical principal polarization on $J_{X_{\mathbb{C}}}$ is determined as in [BHo]. Now the theorem follows from the real analog of Torelli theorem [GH, Theorem 9.4]. ∎

Acknowledgements

We thank the referee for helpful comments. The first author acknowledges support of a J. C. Bose Fellowship.