Moduli of Bundles over Rational Surfaces and Elliptic Curves II: Nonsimply Laced Cases

International Mathematics Research Notices, Vol. 2009, No. 24, pp. 4597–4625 Advance Access publication June 27, 2009

doi:10.1093/imrn/rnp101

Moduli of Bundles over Rational Surfaces and Elliptic Curves II: Nonsimply Laced Cases

Naichung Conan Leung

and Jiajin Zhang

1

Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Hong Kong and

Department of Mathematics, Sichuan University, Chengdu, 610000, People’s Republic of China

Correspondence to be sent to: [email protected]

For any nonsimply laced Lie groupGand elliptic curve, we show that the moduli space of flat G bundles over can be identified with the moduli space of rational surfaces with G-configurations which contain as an anticanonical curve. We also construct Lie(G)-bundles over these surfaces. The corresponding results for simply laced groups were obtained by the authors in another paper. Thus, we have established a natural identification for these two kinds of moduli spaces for any Lie groupG.

1 Introduction

There is a classical identification for the moduli space of flat E_n bundles over a fixed elliptic curveand the moduli space of del Pezzo surfaces of degree 9−ncontainingas an anticanonical curve (see [3–6]). For other simply laced Lie groupG, that is, Lie group of ADE type, we also obtained in [13] an identification for the moduli space of flatGbundles over a fixed elliptic curveand the moduli space of the pairs (S,), whereSis a rational surface (calledADE-surface in [13]) containingas an anticanonical curve. In this paper, we extend the above identification to nonsimply laced cases. Therefore, we establish

Received November 21, 2008; Accepted June 8, 2009 Communicated by Prof. Dragos Oprea

C The Author 2009. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected].

a one-to-one correspondence between flatG bundles over a fixed elliptic curve and rational surfaces with as an anticanonical curve for Lie groups ofalltypes.

A nonsimply laced Lie groupGis uniquely determined by a simply laced Lie group Gand its outer automorphism group. Hence, it is natural to apply the previous results for the simply laced cases in [13] to the current situation. Similar to simply laced cases, we can defineG-surfacesandrational surfaces with G-configurations (see Definitions 3.3, 3.8, 3.13, 3.19). Our main result is the following.

Theorem 1.1. Letbe a fixed elliptic curve with identity 0∈,Gbe any simple, com- pact, simply connected and nonsimply laced Lie group. DenoteS(,G) the moduli space of the pairs (S,), where Sis aG-surface such that∈ | −K_S|. DenoteM^G the moduli space of flatG-bundles over. Then we have

(i) S(,G) can be embedded intoM^Gas an open dense subset.

(ii) This embedding can be extended to an isomorphism fromS(,G) ontoM^G by including all rational surfaces withG-configurations, and this gives us a natural and explicit compactificationS(,G) ofS(,G).

This paper is motivated by some duality in physics. When G=En is a simple subgroup ofE8×E8, theseG bundles are related to the duality between F-theory and string theory. Among other things, this duality predicts that the moduli of flatEnbundles over a fixed elliptic curvecan be identified with the moduli of del Pezzo surfaces with fixed anticanonical curve . For details, one can consult [3, 4, 6]. Our result can be considered as a test of the above duality for other Lie groups.

As an application, this identification provides us with an intuitive explanation forM^G. We also provide an interesting geometric realization of root system theory, and we can see very clearly how the Weyl group acts on the (marked) moduli space of flat G-bundles over.

In the following, we illustrate briefly via pictures whatG-configurations andG- surfaces are in each case and compare it with the corresponding case whereGis simply laced.

1.1 Bn-configurations as specialDn+1-configurations

In these cases we consider rational surfaces with fibration structure and a fixed smooth anticanonical curve . A B_n-configuration comes from a D_n+1-configuration. Roughly speaking, by saying a rational surfaceShas a D_n+1-configuration (l₁,. . .,l_n+1), we mean that S can be considered as a blowup of F1 (a Hirzebruch surface) at n+1 points on

f l1

· · · ln+ 1

0

f−l1 f−l2

· · · x1

−x1 Σ

l2

x2

−x2

S P SO(2n+ 2)

1

Fig. 1. A surface with aD_n+1-configuration (l1,. . .,l_n+1).

f l1

l2

· · · ln+1

0 x1

x2

−x2

f−l1 f−l2

S

1

Σ SO(2n+ 1)

Fig. 2. A surface with aB_n-configuration (l₁,l₂,. . .,l_n+1), wherex₁=l₁∩=0.

∈ | −K_F1|, such thatl1,. . .,ln+1 are the corresponding exceptional classes [13]. When these blown-up points arein general position,Sis called aG=D_n+1-surface. See Figure 1 for a surface with a D_n+1-configuration.

Given a surface S with a D_n+1-configuration ζ =(l₁,. . .,l_n+1), if it satisfies the condition x₁=l₁∩, which is the identity element zero of the elliptic curve, thenζ is a B_n-configuration on S(Definition 3.3). If all blown-up points but x₁ are in general position,Sis called a B_n-surface. See Figure 2 for a surface with a B_n-configuration.

1.2 C_n-configurations as special A_2n−1-configurations

In these cases, we consider rational surfaces with fibration and section structure and a fixed smooth anticanonical curve.

s

f l1

· · · · · ·

ln ln+1 l2n

0 x1

−x1

Σ

S

P¹ s SU(2n)

Fig. 3. A surface with anA_2n−1-configuration (l1,. . .,l2n).

f

0

S

P¹ s s

l1

· · · ln

l⁻₁ l⁻_n

−x1 −xn

x1 xn Σ

Sp(n)

Fig. 4. A surface with aCn-configuration (l1,. . .,ln,l_n⁻,. . .,l₁⁻).

AC_n-configuration comes from anA_2n−1-configuration. By saying that a rational surfaceShas an A_2n−1-configuration (l₁,. . .,l_2n), we mean that Scan be considered as a blowup ofF1at 2npoints on ∈ | −K_F1|which sum to zero, such thatl₁,. . .,l_2nare the corresponding exceptional classes [13]. When these blown-up points arein general posi- tion,Sis called anA_2n−1-surface. See Figure 3 for a surface with anA_2n−1-configuration.

Given a surface S with an A2n−1-configuration ζ =(l1,. . .,l2n), if it satisfies the conditionxi = −x2n+1−iwithxi =li∩, fori=1,. . .,n, thenζis called aCn-configuration on S (Definition 3.8). If all blown-up points are in general position, S is called a Cn- surface. See Figure 4 for a surface with aCn-configuration.

1.3 G₂-configurations as specialD₄-configurations

In these cases we still consider rational surfaces with fibration structure and a fixed smooth anticanonical curve.

f l4

0

−x4

x4

l2 l3

l1

x1

−x1

Σ

S P¹ SO(8)

Fig. 5. A surface with aD4-configuration (l1,. . .,l4).

f l1

l2 l3 l4

0 x1

x2 x3 x4

S

P¹ Σ

G2

Fig. 6. A surface with aG₂-configuration (l₁,l₂,l₃,l₄), wherex₁=0 andx₄=x₂+x₃withx_i=l_i∩. A G₂-configuration comes from a D₄-configuration. We have seen what a D₄- configuration is from Section 1.1 of the Introduction. Roughly speaking, by saying that a rational surfaceShas a D₄-configuration (l₁,. . .,l₄), we mean thatScan be considered as a blowup ofF1at four points on∈ | −K_F1|, such thatl₁,. . .,l₄are the corresponding exceptional classes [13]. When these blown-up points arein general position,Sis called aG=D4-surface. See Figure 5 for a surface with a D4-configuration.

Given a surface Swith a D4-configuration ζ =(l1,. . .,l4), if it satisfies two con- ditions x1=0 andx4=x2+x3, where xi =li∩, thenζ is called aG2-configuration on S (Definition 3.13). If all blown-up points but x1 are in general position, S is called a G₂-surface. See Figure 6 for a surface with aG₂-configuration.

1.4 F₄-configurations as special E₆-configurations

In these cases we consider rational surfaces which are blowups of the projective planeP² at six points in almost general position and which contain a fixed smooth anticanonical curve [13].

0

l1 l2 l3 l4 l5 l6

S

Σ E6

Fig. 7. A surface with anE6-configuration (l1,. . .,l6).

0 l1 l6

l2

l5

l3

l4

L16 L25

L34

p S

Σ F4

Fig. 8. A surface with anF4-configuration (l1,. . .,l6), where three linesL16,L25,L34meet atp∈, or equivalently,x1+x6=x2+x5=x3+x4withxi=li∩.

An F4-configuration comes from an E6-configuration. Recall that by saying a rational surfaceShas anE₆-configuration (l₁,. . .,l₆), we mean that Scan be considered as a blowup ofP²at six points on∈ | −K_P²|, such thatl₁,. . .,l₆are the corresponding exceptional classes. When these blown-up points are in general position, S is called an E₆-surface, which is in fact a cubic surface. See Figure 7 for a surface with an E₆- configuration.

Given a surfaceSwith anE₆-configurationζ =(l₁,. . .,l₆), if it satisfies the condi- tionx1+x6=x2+x5=x3+x4, wherexi =li∩, thenζis called anF4-configuration onS (Definition 3.19). If all blown-up points arein general position,Sis called anF4-surface.

See Figure 8 for a surface with anF4-configuration.

Moreover, we can constructG=Lie(G) bundles overSwith aG-configuration. By restriction, we obtain Lie(G) bundles over . And we can also construct some natural fundamental representation bundles over, which have interesting geometric meanings, such that the Lie algebra bundles are the automorphism bundles of these representation bundles preserving certain algebraic structures.

In this paper, the notations are the same as those in [13]. Let G be a compact, simple, and simply connected Lie group. We denote

r(G): the rank ofG;

R(G): the root system;

R_c(G): the coroot system;

W(G): the Weyl group;

(G): the root lattice;

c(G): the coroot lattice;

w(G): the weight lattice;

T(G): a maximal torus;

ad(G): the adjoint group ofG, i.e.G/C(G), whereC(G) is the center ofG;

(G): a simple root system ofG; and

Out(G): the outer automorphism group ofG, which is defined as the quotient of the automorphism group of G by its inner automorphism group. It is well known that Out(G) is isomorphic to the diagram automorphism group of the Dynkin diagram ofG.

When there is no confusion, we just ignore the letterG.

2 Reductions to Simply Laced Cases

LetGbe any simple, compact, and simply connected Lie group. ThenGis classified into the following seven types according to its Lie algebra:

(1) An-type,G=SU(n+1);

(2) B_n-type,G=Spin(2n+1);

(3) C_n-type,G=Sp(n);

(4) D_n-type,G=Spin(2n);

(5) E_n-type,n=6, 7, 8;

(6) F₄-type; and (7) G₂-type.

Among these,An,Dn, andEnare of simply laced type, whileBn,Cn,F4, andG2are of nonsimply laced type. And An,Bn,Cn,Dnare called classical Lie groups, whileEn,F4, andG2are called exceptional Lie groups.

From now on, we always assume thatGis a compact, simple, simply connected Lie group of nonsimply laced type, that is, of type Bn,Cn,F4,G2. There are two natural approaches to reduce situations to simply laced cases. One is embeddingGinto a simply laced Lie groupGsuch thatGis the subgroup fixed by the outer automorphism group ofG. Another is taking the simply laced subgroupGof maximal rank.

In the following, we explain the first reduction. The following result is well known.

Proposition 2.1. Let G be a compact, nonsimply laced, simple, and simply connected Lie group. There exists a simple, simply connected, and simply laced Lie groupG, s.t.

G⊂GandG=(G)^ρ, whereρis an outer automorphism ofGof order three forG=D4, and of order two otherwise.

Proof. By the functorial property, we just need to prove it in the Lie algebra level. For the construction ofG=Lie(G) and G=Lie(G), one can see [10] for details, where the construction of Lie algebras is determined by the construction of root systems.

Remark. For later use, we list the construction of nonsimply laced root systems via simply laced root systems.

1. G=C_n=Sp(n),G=A_2n−1=SU(2n).

(G)= {αi,i=1,. . ., 2n−1}.

Out(G)= {1,ρ} ∼=Z2, whereρ(αi)=α2n−i,i=1,. . .,n−1,ρ(αn)=αn. (G)= {βi = ¹₂(αi+α2n−i),i=1,. . .,n−1,βn=αn}.

2. G=B_n=Spin(2n+1),G=D_n+1=Spin(2n+2).

(G)= {αi,i=1,. . .,n+1}.

Out(G)= {1,ρ} ∼=Z2, whereρ(αi)=αi,i=3,. . .,n+1,ρ(α1)=α2,ρ(α2)=α1. (G)= {β1= ¹₂(α1+α2),βi =αi+1,i=2,. . .,n}.

3. G=F4,G=E6.

(G)= {αi,i=1,. . ., 6}.

Out(G)= {1,ρ} ∼=Z2, whereρ(αi)=α6−i,i=1,. . ., 5, andρ(α6)=α6. (G)= {β1= ¹₂(α1+α5),β2=¹₂(α2+α4),β3=α3,β4=α6}.

4. G=G₂,G=D₄=Spin(8).

(G)= {αi,i=1,. . ., 4}.

Out(G)= ρ1,ρ2 ∼=S₃, whereρ1interchangesα1andα2, andρ2interchanges α1andα4.

(G)= {β1= ¹₃(α1+α2+α4),β2=α3}. The Dynkin diagrams ofGandGare given in Figure 9.

<

G G

Cn A2n−1

β1 β2 βn−1 βn α1 α2 αn α2n−2 α2n−1

>

Bn Dn+1

βn βn−1 β2 β1 αn+1 αn α3

α2

α1

<

F4 E6

β1 β2 β3 β4 α1 α2 α3 α4 α5

α6

<

G2 D4

β1 β2 α1 α2 α3

α4

Fig. 9. Nonsimply lacedGreduced to simply lacedG.

Remark. Note thatW(G) is the subgroup ofW(G) fixing the root systemR(G) and also the subgroup pointwise fixed by Out(G). For a root α, let S_α∈W(G) be the reflection with respect to α, that is, S_α(x)=x+(x,α)α. Thus, as a subgroup of W(A2n−1),W(Cn) is generated by S_αi ◦S_α2n−i for i=1,. . .,n−1 and S_αn. As a subgroup of W(Dn+1),W(Bn) is generated by S_α1◦S_α2 and S_αi for i=3,. . .,n+1. As a subgroup of W(E₆),W(F₄) is generated byS_α1◦S_α5,S_α2◦S_α4,S_α3andS_α6. As a subgroup ofW(D₄),W(G₂) is generated by S_α1◦S_α2◦S_α4 andS_α3.

In the following, let be a fixed elliptic curve with identity element 0, and fix a primitived^th root of Jac()∼=, whered =2 for D_ncase,d =9−nfor E_ncase, and d=n+1 for A_n case, respectively (see [13]). Recall that for any compact, simple, and simply connected Lie group H, the moduli space of flat Hbundles overis

M^H ∼=(c(H)⊗)/W(H).

ForG, the group Out(G) acts on

(c(G)⊗)/W(G) naturally. Let ((c(G)⊗)/W(G))^Out(G)be the fixed part.

Then, we have a natural map

χ : (c(G)⊗)/W(G)→((c(G)⊗)/W(G))^Out(G). The image ofχis contained in a connected component of the fixed part.

Lemma 2.2. The map

χ : (c(G)⊗)/W(G)→((c(G)⊗)/W(G))^Out(G) is injective.

Proof. It suffices to prove that for anyx,y∈(G)⊗, if∃w∈W(G), such thatw(x)=y, then∃w∈W(G), such thatw(x)=y. ForA_nandD_ncases, this is obvious if we check the root lattices. For E₆ case, we can also check it directly with the help of computer. Of course, we can also check this case by hand following the discussion in Section 2.4.1.

Corollary 2.3. (i) The fixed part ((c(G)⊗)/W(G))^Out(G)is determined by the condi- tionρ(x)=x, up toW(G)-action, where x∈c(G)⊗, andρ is a generator ofOut(G), of order three forG=D4and of order two forG= An, En.

(ii) The moduli spaceM^G∼=(c(G)⊗)/W(G) is a connected component of the fixed part

M^GOut(G)∼=((c(G)⊗)/W(G))^Out(G)

containing the trivialGbundle.

Proof. (i) For any x∈c(G)⊗, denote by ¯x the class in (c(G)⊗)/W(G). Then ρ( ¯x)=x¯ if and only if there exists w∈W(G), such that ρ(x)=w(x). Thus, w⁻¹ρ(x)=x.

Butw⁻¹ρ∈ Out(G) sinceOut(G)=Aut(G)/W(G). Thus, we can take a new simple root system such thatw⁻¹ρis the generator of the diagram automorphism (the automorphism of order three forD4).

(ii) By (i), (c(G)⊗)/W(G) and (M^G)^Out(G) are both orbifolds with the same dimension. Thus, the result follows from Lemma 2.2.

If we express the moduli space of flatGbundles overas (T×T)/W(G), where T is a maximal torus ofG, then we have the following corollary.

Corollary 2.4. If two elements ofT×T are conjugate under W(G), then they are also conjugate underW(G).

Another method is to reduceGto its simply laced subgroupGof maximal rank, and apply the results for simply laced cases to current situation. In another occasion, we will discuss our moduli space ofG-bundles from this aspect in detail. Here, we just mention the following well-known fact from Lie theory.

Proposition 2.5. There exists canonically a simply laced Lie subgroupGofG, which is of maximal rank, that is,GandGshare a common maximal torus. And there is a short exact sequence

1→W(G)→W(G)→Out(G)→1,

where Out(G) is the outer automorphism group of G. Thus, if we write the moduli space asM^G=(T×T)/W, then

M^G=M^G/Out(G).

Remark. We give this construction ofGin each case.

(1) ForG=Sp(n), G=SU(2)ⁿ. Out(G) is the group S_n of permutations of the ncopies ofSU(2) inG.

(2) For G=G₂, G=SU(3). Out(G) is the groupZ2that exchanges the three- dimensional representation of SU(3) with its dual.

(3) ForG=Spin(2n+1), G=Spin(2n).Out(G) is the groupZ2that exchanges the two spin representations ofSpin(2n).

(4) ForG=F4, G=Spin(8).Out(G) is the triality group S3that permutes the three eight-dimensional representations ofSpin(8).

3 FlatGBundles over Elliptic Curves and Rational Surfaces: Nonsimply Laced Cases In this section, we study case by case the G bundles over elliptic curves and rational surfaces for a nonsimply laced Lie groupG.

3.1 TheBn(n≥2) bundles

According to the arguments of the previous section, for G=Spin(2n+1) we can take G=Spin(2n+2), such thatG=(G)^Out(G).

Let S=Y_n+1be a rational surface with a D_n+1-configuration [13] which contains as a smooth anticanonical curve. Recall [13] thatY_n+1is a blowup ofF1atn+1 points x₁,. . .,x_n+1 on, with corresponding exceptional classesl₁,. . .,l_n+1. Let f be the class

of fibers inF1, and s be the section such that 0=s∩ is the identity element of . The Picard group ofYn+1is H²(Yn+1,Z), which is a lattice with basiss, f,l1,. . .,ln+1. The canonical line bundleK= −(2s+3f−_n+1

i=1li).

We know from [13] that

Pn+1:= {x∈ H²(Yn+1,Z)|x·K=x· f=0} is a root lattice ofD_n+1type. We take a simple root system ofGas

(Dn+1)= {α1=l1−l2,α2= f−l1−l2,α3=l2−l3,. . .,αn+1=ln−ln+1}.

Letρ be the generator of Out(G)∼=Z2, such thatρ(α1)=α2,ρ(α2)=α1andρ(αi)=αi for i=3,. . .,n+1.

From [13], we know that the pair (S,) determines a homomorphism u∈ H om((G),)

which is given by the restriction map

u(α)=O(α)|.

Lemma 3.1. Letu∈ H om((G),) correspond to a pair (S,), whereSis a surface with aDn+1-configuration. Thenρ·u=uif and only if 2x1=0.

Proof. Since u is the restriction map: αi →O(αi)|,u(α1)=O(l₁−l₂)|=x₁−x₂, and u(α2)=O(f−l₁−l₂)|= −x₁−x₂. Hence, ρ·u=u⇔u(α1)=u(α2)⇔x₁−x₂= −x₁− x2⇔2x1=0⇔x1is one of the four points of order two on the elliptic curve.

As in [13], we denote byS(,G) the moduli space ofG=D_n+1-surfaces with a fixed anticanonical curve, and by S(,G) the natural compactification by including all rational surfaces with D_n+1-configurations (Figure 1). From [13], we know that φ: S(,G)−→^∼ M^G is an isomorphism.

Corollary 3.2. For u∈M^G →(M^G)^Out(G), φ⁻¹(u)∈S(,G) represents a class of sur- facesY_n+1(x₁,. . .,x_n+1) withx₁=0, and such a surface corresponds to a boundary point in the moduli space, that is,φ⁻¹(u)∈S(,G)\S(,G).

Proof. By Lemma 3.1, u∈(M^G)^Out(G) if and only if 2x₁=0. There are four con- nected components corresponding to four points of order two on . SinceM^G is the component containing the trivial G bundle, we have x₁=0. Recall (Section 4 of [13])

thatYn+1(x1,. . .,xn+1)∈S(,G) if and only if 0,x1,. . .,xn+1are in general position, which implies in particularx1=0. Hence,φ⁻¹(u) corresponds to a boundary point.

DenoteS=Y_n+1(x1=0,x2,. . .,xn+1) (orY_n+1for brevity) the blowup ofF1atn+1 pointsx1=0,x2,. . .,xn+1on, with exceptional divisorsl1,l2,. . .,ln+1, where∈ | −KS|.

Similar to the simply laced cases, we give the following definition.

Definition 3.3. AB_n-exceptional systemonSis ann-tuple (e₁,e₂,. . .,e_n+1), wheree_i’s are exceptional divisors such thate_i·e_j =0=e_i· f,i= jandy₁=e₁∩=0 is the identity of . A B_n-configuration on S is a B_n-exceptional system ζBn=(e₁,e₂,. . .,e_n+1) such that we can consider Sas a blowup of F1 atn+1 points y₁=0,y₂,. . .,y_n+1 on , that is S=Y_n+1(y₁=0,y₂,. . .,y_n+1), with corresponding exceptional divisors e₁,e₂,. . .,e_n+1. When Shas a Bn-configuration, we call S a(rational) surface with a Bn-configuration (see Figure 2).

Whenx2,. . .,xn+1∈withxi=0, for alliare in general position (refer to Section 4 of [13] for definition), anyBn-exceptional system onSconsists of exceptional curves. Such a surface is called a Bn-surface. So a Bn-surface must have aBn-configuration.

Lemma 3.4. (i) Let S be a rational surface with a B_n-configuration. Then the Weyl group W(B_n) acts on all B_n-exceptional systems on Ssimply transitively.

(ii) Let S be a B_n-surface. Then the Weyl group W(B_n) acts on all B_n- configurations simply transitively.

Proof. It suffices to prove (i). Let (e₁,e₂,. . .,e_n+1) be a B_n-exceptional system on S. By Definition 3.3,e_i =l_σ(i)or f−l_σ(i)fori=1, whereσis a permutation of{2,. . .,n+1}. Note that the Weyl group W(Bn) acts as the group generated by permutations of the npairs {(li, f−li) | i=2,. . .,n+1}and interchanging ofli and f−li in each pair (li, f−li)i≥2. Then the result follows.

Let S(,Bn) be the moduli space of pairs (S,) where S is a Bn-surface (so the blown-up pointsx1=0,x2,. . .,xn+1are in general position), and ∈ | −KS|. DenoteM^Bn the moduli space of flat B_n bundles over . Then applying Corollary 3.2, we have the following identification.

Proposition 3.5. (i) S(,B_n) is embedded intoM^Bn as an open dense subset.

(ii) Moreover, this embedding can be extended naturally to an isomorphism S(,B_n)∼=M^Bn,

by including all rational surfaces withB_n-configurations.

Proof. The proof is similar to that inADEcases [13]. Firstly, we haveM^Bn ∼=c(B_n)⊗_Z /W(B_n), andc(B_n)⊗_Z/W(B_n)∼=H om((B_n),)/W(B_n) when we fixed the square root of unity ofJac()∼=. Refer to Section 3 of [13] for the detail.

Secondly, the restriction fromStoinduces a map (again denoted byφ) φ:S(,B_n)→H om((B_n),)/W(B_n).

This map is well defined; since by Lemma 3.4, choosing and fixing aB_n-configuration on Sis equivalent to choosing and fixing a system of simple roots(B_n).

Thirdly, the mapφis injective. For this, we take a simple root system ofB_nas β1= f−2l₂ and βk=2αk+1 for 2≤k≤n.

Then the restriction induces an elementu∈H om((Bn),), which satisfies the following system of linear equations:

−2x₂=p₁,

2(x_k−x_k+1)= p_k, k=2,. . .,n,

where pi=u(βi). Obviously, the solution of this system of linear equations exists uniquely for given piwith 1≤i≤n.

Finally, statement (ii) comes from Corollary 3.2 and the existence of the solutions to the above system of linear equations.

Remark. The situation here is very similar to that in the compactification theory of the moduli space of (projective)K3 surfaces. A natural question is how to extend the global Torelli theorem to the boundary components of a compactification (see, e.g. [11, 16]). If we consider the mapφ:S(,G)→M^G[13] forG=A_n,D_n, orE_nas a type of period map, then the main result of [13] is a type of global Torelli theorem. And Proposition 3.5 implies that we can extend the theorem of Torelli type inD_n+1case to a boundary component of the natural compactification.

In the following, we letS=Y_n+1(x₁,. . .,x_n+1) be the blowup ofF1atn+1 points.

We can construct a Lie algebra bundle on S. Here we do not need the existence of

the anticanonical curve . According to Section 2, we have a root system of Bn type, consisting of divisors on S:

R(B_n){±(f−2l_i), 2(l_i−l_j),±2(f−l_i−l_j) | i= j, 2≤i,j≤n+1}.

Thus, we can construct a Lie algebra bundle ofB_n-type overS:

BnOⁿ

D∈R(Bn)

O(D).

The fiberwise Lie algebra structure ofBnis defined as follows (the argument here is the same as that in [13]).

Fix the system of simple roots of Rnas

(B_n)= {α1= f−2l₂,α2=2(l₂−l₃),. . .,αn=2(l_n−l_n+1)},

and take a trivialization of Bn. Then over a trivializing open subset U,Bn|U ∼= U×(C^⊕n

α∈RnC_α). Take a Chevalley basis{x_α^U,α∈ R_n;h_i, 1≤i≤n}forBn|U and define the Lie algebra structure by the following four relations, namely, Serre’s relations on Chevalley basis (see [8], p. 147):

(a) [h_ih_j]=0, 1≤i,j≤n.

(b) [hix_α^U]= α,αi x^U_α, 1≤i≤n,α∈ Rn.

(c) [x^U_αx_−α^U ]=h_αis aZ-linearly combination ofh1,. . .,hn.

(d) Ifα,βare independent roots, andβ−rα,. . .,β+qαare theα-string through β, then [x^U_αx_β^U]=0 ifq=0, while [x_α^Ux^U_β]= ±(r+1)x_α+β^U ifα+β∈ Rn.

Note that h_i, 1≤i≤n are independent of any trivialization, so the relation (a) is always invariant under different trivializations. IfBn|V ∼=V×(C^⊕n

α∈Rn) is another trivialization and f_α^{U V} is the transition function for the line bundleO(α)(α∈ R_n), that is, x_α^U= f_α^{U V}x_α^V, then the relation (b) is

h_i

f_α^{U V}x_α^V = α,αi f_α^{U V}x_α^V, that is,

h_ix_α^V = α,αi x_α^V.

So (b) is also invariant. Relation (c) is also invariant since (f_α^{U V})⁻¹ is the transition function forO(−α)(α∈ R_n). Finally, (d) is invariant since f_α^{U V}f_β^{U V}is the transition function forO(α+β)(α,β∈ R_n).

Therefore, the Lie algebra structure is compatible with the trivialization. Hence, it is well defined.

When the surfaceScontains as an anticanonical curve, restricting the above bundle to this anticanonical curve, we obtain a Lie algebra bundle of B_n-type over , which determines uniquely a flatB_n bundle over. On the other hand, whenx₁=0, we can identify these two line bundlesO(l₁) andO(f−l₁) when restricting them to. Recall that the spinor bundlesS⁺_n+1andS_n+1⁻ ofD_n+1are defined as follows [12, 13] (here we omit the subscriptionn+1 for brevity):

S⁺=

D²=D·K=−1,D·f=1

O(D) and S⁻=

T²=−2,T·K=0,T·f=1

O(T).

The identification of O(l1)∼=O(f−l1) induces an identification of these two spinor bundlesS⁺andS⁻, which is given by (of course, when restricted to)

S⁺⊗O(−l₁)∼=S⁻.

From the representation theory, we know that this determines a flatB_nbundle over. Conversely, ifS⁺|∼=S⁻|, then we must have x₁=0 (up to renumbering). For example, we consider then=2 case. Note that

S⁺| ⊗O(−(0))=O⊕O((−x₁−x₂)−(0))⊕O((−x₁−x₃)−(0))⊕O((−x₂−x₃)−(0)), S⁻| =O((0)−(x1))⊕O((0)−(x2))⊕O((0)−(x3))⊕O(3(0)−(x1)−(x2)−(x3)).

Where for a point x∈, (x) means the divisor of degree one, andO((x)) means the line bundle determined by this divisor. Thus, S⁺⊗O(−(0))=S⁻ implies that x1=0 (up to renumbering). The general case follows from similar arguments.

3.2 TheCnbundles

We take G=Cn⊂G=A2n−1, where Cn=Sp(n) and A2n−1=SU(2n). They satisfy the relationG=(G)^Out(G).

LetS=Z2nbe a rational surface with anA2n−1-configuration (see [13] or Figure 3) which contains as a smooth anticanonical curve. Recall [13] that Z_2n is a (succes- sive) blowup ofF1at 2npointsx₁,. . .,x_2n on, with corresponding exceptional classes l₁,. . .,l_2n. Let fbe the class of fibers inF1, andsbe the section such that 0=s∩is the

identity element of. The Picard group ofZ2nis H²(Z2n,Z), which is a lattice with basis s, f,l1,. . .,l2n. The canonical line bundleK= −(2s+3f−_2n

i=1li).

Recall that

P_2n−1:= {x∈ H²(Z_2n,Z)|x·K=x· f=x·s=0}

is a root lattice ofA2n−1type. And we can take a simple root system of A2n−1as (A_2n−1)= {αi =l_i−l_i+1|1≤i≤2n−1}.

Note that [13] we have used the convention that_2n

i=1xi =0.

Letρbe the generator ofOut(G)∼=Z2, such thatρ(αi)=α2n−ifori=1,. . ., 2n−1.

When the above simple root system is chosen, the pair (S,) determines a homo- morphismu∈ H om((G),) which is given by the restriction map

u(α)=O(α)|.

Lemma 3.6. Letu∈H om((G),) be an element corresponding to a pair (S,), where Sis a surface with anA_2n−1-configuration. Thenρ·u=uif and only ifn(x_i+x_2n+1−i)=0 fori=1,. . .,n.

Proof. Since u is the restriction map: αi →O(αi)|,u(αi)=O(li−li+1)|=xi−xi+1

for i=1,. . ., 2n−1. Hence, ρ·u=u⇔u(αi)=u(α2n−i)⇔x_i−x_i+1=x_2n−i−x_2n−i+1⇔ n(x_i+x_2n−i+1)=0 since_2n

i=1x_i =0.

As in [13], we denote byS(,G) the moduli space ofG=A2n−1-surfaces with a fixed anticanonical curve, and byS(,G) the natural compactification by including all rational surfaces with A2n−1-configurations. From [13], we know that there is an isomorphismφ:S(,G)−→^∼ M^G.

Corollary 3.7. For u∈M^G→(M^G)^Out(G), φ⁻¹(u)∈S(,G) represents a class of sur- facesZ_2n(x₁,. . .,x_2n) withx_i+x_2n+1−i =0 fori=1,. . .,n, and such a surface corresponds to a boundary point in the moduli space, that is,φ⁻¹(u)∈S(,G)\S(,G).

Proof. By Lemma 3.6, u∈(M^G)^Out(G) if and only if n(xi+x2n+1−i)=0 fori=1,. . .,n.

There aren²connected components corresponding ton²points of ordernon. SinceM^G is the component containing the trivialGbundle, we havex_i+x_2n+1−i =0 fori=1,. . .,n.

Recall (Section 4 in [13]) that Z_2n(x₁,. . .,x_2n)∈S(,G) if and only if 0,x₁,. . .,x_2n are in