Hypercomplex Structures on Group Manifolds

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Jan 5, 1999 - generates hypercomplex automorphism, i.e. automorphisms which are holomorphic with respect to each complex structure of the hypercom-.
Hypercomplex Structures on Group Manifolds  Henrik Pedersen

y

Yat Sun Poon

z

January 5, 1999

Abstract We study deformations of hypercomplex structures on com-

pact Lie groups. Our calculation is through the complex deformation theory of the associated twistor spaces. In general, we nd complete parameter spaces of hypercomplex structures associated to compact semisimple Lie groups. In particular, we discover the complete moduli space of hypercomplex structures on the product of Hopf surfaces.

Keywords: Lie Groups, Homogeneous, Hopf Surfaces, Hypercomplex, Deformations. AMS Subject Classi cation: Primary 32G05. Secondary 53C10, 32L25, 32G20.

Introduction A manifold is said to have a hypercomplex structure if there exist three integrable complex structures fI1 ; I2 ; I3 g satisfying the identities of pure quaternions: I1 I2 = ?I2 I1 = I3 . Hypercomplex manifolds interest physicists as they arise in the theory of N = 4 super-symmetric models with Wess-Zumino terms. In this content, Spindel, Sevrin, Troot and Van Proeyen discovered left-invariant hypercomplex structures on compact Lie groups [18]. When Joyce constructed homogeneous hypercomplex and homogeneous quaternionic manifolds, he independently discovered these hypercomplex structures [11].  A project supported by the Institut des Hautes E tudes Scienti ques. Joint

IHES/Odense preprint. y Address: Institut for Matematik og Datalogi, Odense Universitet, Campusvej 55, Odense M, DK-5230, Denmark. E-mail: [email protected]. z Address: Department of Mathematics, University of California at Riverside, Riverside, CA 92521, U.S.A.. E-mail: [email protected].

1

Joyce's construction of homogeneous hypercomplex structures has its roots in the work of Borel [1], Samelson [17] and Wang [19] about homogeneous complex manifolds. It is also tied to Wolf's construction of symmetric quaternionic Kahler manifolds [20]. Suppose that G is a compact semi-simple Lie group of rank r. Joyce and Spindel et al. found that the group T 2n?r  G has a left-invariant hypercomplex structure, where n is the number of commuting sp(1) subalgebras in the Lie algebra g of G. Here T 2n?r is the (2n ? r)-dimensional compact Abelian group. In Section 1, we explain how the hypercomplex structures are de ned. In this paper, we investigate deformations of these hypercomplex structures. Deformations of homogeneous complex manifolds were studied by Griths [8]. An important ingredient in [8] is a relation between the homogeneous spaces and ag manifolds. Applying twistor theory [14] [16], we establish a link between the twistor space of T 2n?r  G and ag manifolds. The twistor space associated to the hypercomplex manifold T 2n?r  G is a complex manifold W encompassing all complex structures of the given hypercomplex structure. There is a holomorphic map p from the twistor space W to CP1 so that the pre-images of points in CP1 are the complex structures of T 2n?r  G [15] [16]. It is known that deformations of hypercomplex structures are equivalent to deformations of the holomorphic map p [15]. The rst step to describe the deformation theory is a computation of the relevant cohomology groups on the twistor spaces. It is done via Leray's spectral sequence established by the link between the twistor spaces and ag manifolds. The computation is possible due to the vanishing theorems of Borel-Hirzebruch and Bott [2] [3]. From this computation, we nd that the obstruction space to deformations is non-trivial. Therefore, in Section 4 we complete our calculation of the deformations through Kuranishi theory. The result is: Theorem Suppose G is a compact semi-simple Lie group of rank r. Then the local moduli at a generic deformation of the left-invariant hypercomplex structure on T 2n?r  G is a smooth manifold of dimension n(n + r). The identity component of the group of hypercomplex symmetries of a generic deformation is the Abelian group T 2n . With more details, this theorem is stated as Theorem 3 in Section 4. 2

In the computation of the Kuranishi family, it is seen that the in nitesimal deformations of hypercomplex structures on T 2n?r  G are entirely described by the in nitesimal deformations of an Abelian variety contained in the twistor space. Inspired by the theory of Abelian varieties, we construct a family of hypercomplex structures on (S3 S1 )n . This family is complete due to the theorem above. In fact, we nd the global moduli space of hypercomplex structures on (S3  S1 )n . To be precise, we obtain Theorem The quotient space ((R=Z)n2  GL(n; R))=(Zn2  GL(n; Z)) is a complete moduli space for hypercomplex structures on the product manifold (S3  S1 )n . This is Theorem 4 proved in Section 5.1. The action of the group Zn2  GL(n; Z) on the space (R=Z)n2  GL(n; R) is given by (26) and Lemma 14.

1 Hypercomplex Structures on Lie Groups Suppose G is a compact semi-simple Lie group. Let U be a maximal torus. The space G=U has a left-invariant homogeneous complex structure [1, x29]. The (1; 0)-forms at the identity coset is the space of positive roots with respect to the Cartan subalgebra uC . The complex structures at the other points of the quotient space are obtained by left translations. The Joyce construction is similar. It starts with the Wolf decomposition of the Lie algebra gC [20]. Choose a system of ordered roots with respect to uC . Let 1 be a maximal positive root, and h1 the dual space of 1 . Let @1 be the sp(1)-subalgebra of g such that its complexi cation is isomorphic to h1  g 1  g? 1 where g 1 and g? 1 are the root spaces for 1 and ? 1 respectively. Let b1 be the centralizer of @1 . Then there is a vector subspace f1 composed of root spaces such that g = b1  @1  f1 . If b1 is not Abelian, Joyce applies this decomposition to it. By inductively searching for sp(1) subalgebras, he nds the following [11, Lemma 4.1]. Lemma 1 The Lie algebra g of a compact Lie group G decomposes as g = b nj=1 @j nj=1 fj ; (1) with the following properties. (1) b is Abelian and @j is isomorphic to sp(1). (2) b nj=1 @j contains u. (3) Set b0 = g, bn = b and bk = 3

b nj=k+1 @j

nj=k+1 fj . Then [bk ; @j ] = 0 for k  j . (4) [@l ; fl ]  fl. (5)

The adjoint representation of @l on fl is reducible to a direct sum of the irreducible 2-dimensional representations of sp(1).

Note that this theorem is true for the Lie algebra of any compact Lie group. We refer to this decomposition of the algebra g as the Joyce decomposition. In this decomposition n is the number of commuting sp(1) subalgebras, and the rank of the algebra g is equal to n + dim b.

Proposition 1 The Joyce decompositions of compact simple Lie algebras are

Table: For `  1

n dim b

a2`

` `

a2`?1

`

b`

`

c`

d2`

` 2`

`?1 0 0

0

d2`+1

2` 1

e6

4 2

e7

7 0

e8

8 0

f4

4 0

g2

2 0

Proof: We derive the decompositions through the following recursive relations. a` c` e6 f4

= u(1)  a`?2  @1  f1 : b` = b`?2  @2  @1  f1 : = c`?1  @1  f1 : d` = d`?2  @2  @1  f1 : = a 5  @ 1  f 1 : e 7 = d 6  @ 1  f 1 : e 8 = e 7  @1  f 1 : = c 3  @ 1  f 1 : g 2 = @ 2  @1  f 1 : q: e: d:

Let G be a compact semi-simple Lie group. Then

(2n ? r)u(1)  g  = Rn nj=1 @j nj=1 fj :

(2)

At the tangent space of the identity element of T 2n?r  G, i.e. the Lie algebra (2n ? r)u(1)  g, a hypercomplex structure fI1 ; I2 ; I3 g is de ned as follows. Let fe1 ; : : : ; en g be the standard basis for Rn . Choose isomorphisms j from sp(1), the real vector space of imaginary quaternions, to @j . It gives a real linear identi cation from the quaternions H to hej i  @j . De ne the action of Ia on fj by Ia (v) = [v; j (a )] where 1 = i; 2 = j; 3 = k. The complex structures fI1 ; I2 ; I3 g at the other points of the group T 2n?r  G are obtained by left translations. Due to Proposition 1, Joyce's construction for compact simple Lie groups yields the following lists of left-invariant hypercomplex manifolds. 4

Proposition 2 The following groups have homogeneous hypercomplex

structures.

SU(2`+1); T 1 SU(2`); T ` SO(2`+1); T ` Sp(`); T 2` SO(4`);

T 2`?1 SO(4`+2); T 2 E6 ; T 7 E7 ; T 8 E8 ; T 4 F4 ; T 2 G2 : This list of hypercomplex manifolds is known to Spindel et al. [18, Table 1].

1.1 Hypercomplex Structures on SU(2)  U(1)

The hypercomplex structures on the Hopf manifold S3  S1 are classi ed by Boyer and Kato [4] [12]. Given any complex number such that 0 < j j < 1, the left multiplication by on the right module H generates an integer group ? acting on H . We choose spherical coordinates (q; ) so that H is identi ed to the product space S3  R+ . For = re2i , the action of n is (e2in q; rn ). Since left multiplication commutes with right multiplication of the quaternions, the quotient space (S3  R+ )=? is hypercomplex. The map (q; ) 7! (e?2i lnlnr q; e2i lnln r ) from S3  R+ to S3  S1 factors through the quotient space (S3  R+ )=?, and de nes a di eomorphism from (S3  R+)=? to the Hopf manifold S3  S1. Therefore, these hypercomplex structures on S3  S1 are parameterized by the punctured disk f 2 C : 0 < j j < 1g: Now we treat the manifold S3  S1 as the Lie group SU(2)  U(1). The group multiplication on SU(2)  U(1) is (q1 ; e2iln 1 )  (q2 ; e2iln 2 ) = (q1 q2 ; e2i(ln 1 +ln 2 ) ). Therefore, it is covered by the multiplication on S3  R+ de ned by (q1 ; 1 )  (q2 ; 2 ) = (q1 q2 ; 1 2 ). It follows that the left-invariant hypercomplex structures on the Lie group SU(2)  U(1) is parametrized by the space f?r : 0 < r < 1g [ fr : 0 < r < 1g.

1.2 Hypercomplex Structures on SO(3)  U(1)

Given the action of Z  Z2 on S3  R+ generated by above and (q; ) = (?q; ), we consider the quotient (S3  R+ )=(?  hi). As the adjoint representation Ad of SU(2) on its real algebra is a double covering from SU(2) onto SO(3), the map  ln 

ln 

(q; ) 7! (Ad(e?2i ln r q); e2i ln r )

(3)

determines a di eomorphism from (S3  R+ )=(? hi) to SO(3)  U(1). 5

Since the hypercomplex structures de ned by ( ; ) and (? ; ) are equivalent, they are parametrized by the space fre2i : 0 < r < 1; ?=2    =2g, and the left invariant hypercomplex structures on SO(3)  U(1) are parametrized by the interval fr : 0  r  1g.

1.3 Hypercomplex Structures on U(2)

Although the group U(2) is not semi-simple, we include its description as a hypercomplex manifold with left-invariant hypercomplex structure for later exposition. Considering S3 as the space of unit quaternions fq = (w1 + jw2 ) 2 H : jw1 j2 + jw2 j2 = 1g, and U(1) as the space of unit complex numbers, we de ne a di eomorphism from S3  S1 to the group U(2) by





 

i w1 ; ?w2 : (q; ei ) 7! e0 10  w 2 ; w1

(4)

The multiplication for U(2) in the coordinates (q; ei ) is given by 



(q1 ; ei1 )  (q2 ; ei2 ) = (e?i 22 q1 ei 22 q2 ; ei(1 +2 ) ): The multiplication on S3  R+ is (q1 ; 1 )  (q2 ; 2 ) = (e?i ln 2 q1 ei ln 2 q2 ; 1 2 ): Note that the left multiplication is again hypercomplex. For any = re2i , the action generated by (q; ) = (e2i q; r) commutes with left multiplication if and only if 2 + ln r is an integer. Thus the space of U(2)-left invariant hypercomplex structures is f?re?i ln r : 0 < r < 1g [ fre?i ln r : 0 < r < 1g.

2 The Twistor Spaces of T 2n?r  G In this section, we describe the twistor space of the hypercomplex structure on T 2n?r  G and other related objects. We refer the reader to [15] for general twistor theory on hypercomplex manifolds. We focus on the algebraic nature of the twistor space of T 2n?r  G. By construction, the left action of the group T 2n?r  G on itself generates hypercomplex automorphism, i.e. automorphisms which are holomorphic with respect to each complex structure of the hypercomplex structure. However, there is a subgroup in T 2n?r  G whose right 6

action is also hypercomplex. When b is not trivial, let B be the real Abelian subgroup of G whose algebra is b. As b is in the commutator of all @j , the adjoint action of b preserves the hypercomplex structure on the components hej i  @j , for 1  j  n. Since each fj is a direct sum of root spaces, and b is contained in the Cartan subalgebra, the adjoint action of b also leaves the hypercomplex structure on the component fj invariant. Since the hypercomplex structure is left invariant, it follows that the right action of the group B is hypercomplex.

Lemma 2 The algebra of hyper-holomorphic vector elds on T 2n?r  G contains the direct sum of the Lie algebras g and a, where a is the algebra of right-invariant vector elds generated by T 2n?r  B .

There is also a distinguished subgroup of right multiplication preserving the collection of complex structures although it does not preserve any individual one.

Lemma 3 Let 4 : sp(1) ! g be de ned by 4(a ) = Pnj=1 j (a ), for 1  a  3. Let 4 also represent the corresponding group homomor-

phism. Then for any complex structure Iv = v1 I1 + v2 I2 + v3 I3 = (v1 ; v2 ; v3 ) in the given hypercomplex structure, and any A in Sp(1),

Ad4(A) Iv Ad4(A)?1 = IAdA v :

(5)

In particular, the right action of the subgroup 4(Sp(1)) is a group of quaternionic transformations on T 2n?r  G. Proof: The algebra of endomorphisms on the manifold T 2n?r  G generated by I1 ; I2 and I3 is isomorphic to sp(1). We have IAdA v = AdA (Iv ). It follows that the in nitesimal version of (5) is [ad 4(); Iv ] = [; Iv ], for any element  in sp(1). As this identity is linear in both the variable v and the variable , it suces to verify that

[ad 4(a ); Ia ] = 0

and

[ad 4(a ); Ib ] = 2Ic ;

(6)

when (abc) is an even permutation of (123). We verify the second identity, and leave the veri cation of the rst identity to the reader. For any element w in the Lie algebra (2n ? r)u(1)C  gC , [ad 4(a ); Ib ]w =

X

j

f[j (a ); Ib w] ? Ib [j (a ); w]g: 7

(7)

Recall that the Lie algebra for T 2n?r  G has a decomposition (2). Suppose that w is contained in hek i  @k for some k. The identity (7) is reduced to [k (a ); Ib w] ? Ib [k (a ); w] = [k (a ); wk (b )] ? [k (a ); w]k (b ) = 2wk (c ) = 2Ic w: Suppose that w is contained in fl for some l. Using the Jacobi identity, then the identity (7) is reduced to [l (a ); Ib w] ? Ib [l (a ); w] = [l (a ); [w; l (b )]] ? [[l (a ); w]; l (b )] = [w; [l (a ); l (b )]] = 2[w; l (c )] = 2Ic w: The veri cation of the second identity in (6) is now completed. q. e. d. Let us choose the map j so that the dual j of j (1 ) is a positive root, and j (2 )  ij (3 ) spans the root spaces g j . As observed by Joyce [11, Equation (10) and (11)] the (1; 0)-forms at the identity element with respect to I1 on the complement of the Abelian part is precisely the direct sums of all positive root spaces. Since the Abelian part is the algebra of T 2n?r  U , we have

Lemma 4 With respect to the complex structure I1 , there is a holomorphic map from T 2n?r  G onto the ag manifold G=U . Through Lemma 3, we see that the above lemma can be applied to each complex structure of the hypercomplex structure. We shall explore this observation through twistor theory. Let W be the twistor space of T 2n?r  G. Since the Obata connection trivializes W as a ber bundle over T 2n?r  G, the space W is smoothly the product of T 2n?r  G and the 2-sphere S 2 . To describe the complex structure on W , we recall that the 2-sphere S 2 parameterizes the complex structures of the hypercomplex structure on T 2n?r  G. If v = (v1 ; v2 ; v3) is a unit vector in R3 , we denote the complex structure v1 I1 + v2 I2 + v3 I3 by Iv . Let  = (1; 0; 0). Consider S 2 as the quotient Sp(1)= U(1) with U(1) the isotropy at . Let Jv be the standard complex structure at v on S 2 de ned by the orientation fI1 ; I2 ; I3 g. Let (1; e) be the identity element in the group T 2n?r  G. The complex structure I at the point (1; e; v) in T 2n?r  G  S 2 is the direct sum I(1;e;v) = (Iv ; Jv ). Note that Iv is a complex structure for T 2n?r  G at (1; e). Let A 2 Sp(1) such that A = v. De ne the action A of A on 8

T 2n?r  G  S 2 by A(t; g; w) = (t; 4(A)g4(A)?1 ; Aw). By Lemma 3, (Iv ; Jv ) = dA  (I1 ; J ). Since the complex structure at (t; g; v) is obtained by left translations of the complex structure I(1;e;v) by the group element (t; g), we extend the A-action to an action of T 2n?r  G  Sp(1) on T 2n?r  G  S 2 by A(s; h; A; t; g; B) = (st; h4(A)g4(A)?1 ; AB): (8) It shows that the group of transformations is a semi-direct product of T 2n?r  G with Sp(1). We denote this group by T 2n?r  G ./ Sp(1). We have just seen that the complex structure on the twistor space is homogeneous with respect to this group. Thus, we have Lemma2n?5r The twistor space W is the homogeneous complex manifold G./Sp(1) : W = T U(1)

In the semi-direct product, we are interested in several subgroups. Recall that U is a maximal torus in G. Let E be the product subgroup T 2n?r  U in T 2n?r  G. Note that T 2n?r  B is a subgroup of E such that dim E = 2 dim(T 2n?r  B ). The map from Sp(1) to T 2n?r  G ./ Sp(1) de ned by (A) = (1; 4(A)?1 ; A) is a group homomorphism. The subgroup (Sp(1)) commutes with the subgroup E . It follows that the subgroup generated by E and (Sp(1)) is a direct product. Abusing notation, we denote this product by E ./ Sp(1). Considering U(1) as a subgroup in E ./ Sp(1) through the embedding in Sp(1), we have a homogeneous manifold F = E./USp(1) (1) : Lemma 6 There is a locally trivial holomorphic bration  from W to the ag manifold Z = G=U with F as ber. Proof: We de ne a map  from T 2n?r  G ./ Sp(1) to G by (t; g; A) = g4(A): (9) It is a group homomorphism. Since this map sends the U(1)-subgroup of the Sp(1)-factor into the maximal torus U , this map descends to a map  from the homogeneous space W onto Z . Naturally, the map  intertwines the left action of the groups through the homomorphism . We denote the U (1)-left coset of (t; g; A) 2 T 2n?r  G ./ Sp(1) by [t; g; A], and the U -left coset of g 2 G by [g]. We claim that d[1;e;I]  I[1;e;I] = I[e]  d[1;e;I] : (10) 9

To verify this claim, let h0 be the complexi cation of the algebra of U(1). Consider the decomposition sp(1)C = h0  g+  g? where g+ is the positive root space with respect to h0 . Then the complexi ed tangent space of W at [1; e; I] is (2n?r)u(1)C gC g+ g? = ((2n?r)u(1)bnj=1 @j nj=1 fj )C g+ g?: (11) The Abelian part (2n ? r)u(1)C  uC is the complexi cation of the algebra t of the group E . Since it is in the kernel of d and it is invariant of I1 , the identity (10) holds on tC . It is observed in [11, Equation (10) and (11)] that the (1; 0)-forms in gC with respect to I1 is the direct sum of all positive root spaces, and the (0; 1)-forms in gC is the negative root spaces. This decomposition coincides with the type decomposition on the tangent space of Z at the identity coset. Therefore the identity (10) holds on the complement of the Abelian part in gC. Finally, we consider vectors in g+  g? . Since ([1; e; A]) = [4(A)], the restriction of d to g+  g? is the map 4 on the Lie algebra. By construction, 4 maps g+ to (2n ? r)u(1)C  bC nj=1 g j and g? to (2n ? r)u(1)C  bC nj=1 g? j . Since (2n ? r)u(1)C  bC is in the kernel, the quotient of the image of g+ and g? are contained in nj=1g j and nj=1g? j respectively. Since j , for 1  j  n, is a positive root, the restriction of d to g+  g? preserves type decomposition. The identity (10) is veri ed. It follows that the map  is holomorphic at the identity coset. Since it intertwines the left multiplications, and the complex structures on W and Z are both homogeneous, the map  is holomorphic. Given the de nition of  through (9), it is not hard to see that the ber of  is the homogeneous space F . To prove that the map  is locally trivial, we recall that the space of positive roots  >0 g with the exponential map forms a holomorphic coordinate chart for the identity coset on the ag manifold Z = G=U . Any point in the subgroup E ./ Sp(1) is given by the product of an element (t; u; I) in T 2n?r U and an element (A) = (1; 4(A)?1 ; A) in (Sp(1)). It is (t; u4(A)?1 ; A). We denote its U(1)-coset by its square bracket. Now we consider the map  from  >0 g  F to W de ned by

(v; [t; u4(A)?1 ; A]) = [t; exp(v)u4(A)?1 ; A];

where v is an element in  >0 g . This map is holomorphic as it is holomorphic on both factors. Since the map  sends (v; [t; u4(A)?1 ; A]) 10

to [exp(v)],  is a trivialization of the map . Since  intertwines with left actions, the map  is a locally trivial bration everywhere. q. e. d. Next we consider the map p from W onto Sp(1)= U(1) = CP1 de ned by p([t; g; A]) = [A]: It is clear that this map intertwines the left actions. It follows that the map p is holomorphic. This map is precisely the parameterization of the complex structures of the given hypercomplex structure on T 2n?r  G. Let S be the inverse image of the identity coset. We have seen in Lemma 4 that there is a holomorphic map from S onto the ag manifold Z . From the proof of Lemma 6, we nd the following.

Lemma 7 The restriction of the map  on S is a locally trivial holomorphic bration from S onto Z with the product of elliptic curves E as ber. Lemma 8 The restriction of p to F is a locally trivial holomorphic bration with E as ber.

Proof: The holomorphic chart of the identity coset on Sp(1)= U(1) is g+ . The trivialization is given by the map from g+  E to F de ned by (v; [t; u; I]) 7! [t; u; exp(v)]: q. e. d.

3 Computation of Cohomology Groups In this section, we compute the cohomology groups which are relevant to the deformation theory.

Lemma 9 Let t = H 1 (E; OE ): Then for 0  j  n, H j (E; OE ) = ^j t. Proof: By construction E is the product of n elliptic curves. There exist complex coordinates f1 ; : : : ; n g on E such that the space of holomorphic (0; j )-forms is generated by the exterior products of d i , for 1  i  n, over the complex numbers. q. e. d.

Lemma 10 For all j , Rj pOF = OCP1 H j (E; OE ), Rj OS = OZ

H j (E; OE ), and Rj  OW = OZ H j (F; OF ). Proof: Since the maps pjF , jS and  are locally trivial brations, this lemma is a consequence of the Kunneth formula and the @ -Poincare lemma. q. e. d.

11

Lemma 11 For all j , H j (F; OF ) = H 0 (CP1; OCP1 ) H j (E; OE ) = H j (E; OE ); (12) H j (S; OS ) = H 0 (Z; OZ ) H j (E; OE ) = H j (E; OE ); (13) j 0 j j H (W; OW ) = H (Z; OZ ) H (F; OF ) = H (E; OE ): (14) Proof: Consider the Leray spectral sequence with

E2p;q = H p(CP1 ; Rq pOF ); and E1p;q ) H p+q (F; OF ): By the last lemma, E2p;q = H p (CP1 ; OCP1 ) H q (E; OE ). Since the space H p (CP1 ; OCP1 ) vanishes for all p  1, the Leray spectral sequence degenerates at the E2 -terms. The conclusion in (12) follows

from the convergence of the spectral sequence. By [2, Proposition 14.10], H p (Z; OZ ) vanishes for p  1. Therefore, the above argument is applied to the spectral sequence for Rp  OS and Rp  OW to verify (13) and the rst equality in (14). The last equality in (14) follows (12). q. e. d.

Lemma 12 For all j , H j (W;  Z ) = H 0 (Z; Z ) H j (E; OE ). Proof: The projection formula and Lemma 10 togehter imply that Rj (Z )  = Z H j (F; OF ). By [3, Theorem VII], H p(Z; Z ) vanishes for all p  1. Therefore, the spectral sequence of Rq  ( Z ) degenerates at the E2 -terms. The convergence of the spectral sequence yields H j (W;  Z ) = H 0 (Z; Z ) H j (F; OF ). Then the lemma follows (12). q. e. d.

Lemma 13 For all j , H j (W; p CP1 ) = sl(2; C) H j (E; OE ). Proof: The tangent sheaf on CP1 is isomorphic to O(2). When S1 and S2 are two distinct e ective divisors for the bundle p O(1), we have the exact sequence

0 ! OW ! p CP1 ! OS1 [S2 ! 0:

(15)

By Lemma 9 and Lemma 11, the coboundary map from the j -th cohomology level is identical to ^j t ^j t ! ^j +1t: Since the bundles involved are homogeneous with respect to the group T 2n?r  G ./ Sp(1), the cohomology spaces are representations of this group, and the coboundary maps are equivariant. Since ^j t and ^j +1t have di erent weights, by 12

Schur's Lemma the coboundary map is the zero map. Therefore, the induced long exact sequence of (15) splits. We have H j (W; p CP1 ) = H j (W; OW )  H j (S1 ; OS1 )  H j (S2 ; OS2 ) = C3 ^j t: As the Sp(1) factor in the semi-direct product acts on the sphere Sp(1)= U(1) by the left action, the spaces H j (W; OW ), H j (S1 ; OS1 ) and H j (S2 ; OS2 ) are not acted on invariantly. Therefore, the C3 -factor is an irreducible representation of Sp(1). It can only be the adjoint representation. q. e. d. Proposition 3 Let D be the kernel sheaf of the di erential of the map p from W onto CP1 . For all j , H j (W; W ) = nH j (W; OW )  H 0 (Z; Z ) H j (E; OE ) sl(2; C) H j (E; OE ): H j (W; D) = nH j (W; OW )  H 0 (Z; Z ) H j (E; OE ): Proof: The sheaf D is de ned by the exact sequence dp  0 ! D ! W ?! p CP1 ! 0: We claim that the induced map of dp on the j -th cohomology level, is surjective; and ker dpj = n H j (W; OW )  H j (W;  Z ). To verify these claims, we note that the group E = T 2n?r  U generates n holomorphic vector elds without zeroes on the twistor space W . The complex orbits of the holomorphic action of T 2n?r  U are the transversal intersection of the bers of the maps  and p, we have the exact sequence dp  (16) 0 ! nOW ! W d?!  Z  p CP1 ! 0: By Lemma 11, 12 and 13, the coboundary map from the j -th level is equivalent to H 0 (Z; Z ) H j (E; OE )  sl(2; C) H j (E; OE ) ! nH j+1 (E; OE ): Due to Lemma 9 and [3, Theorem VII], it is equivalent to (gC ^j t)  (sl(2; C) ^j t) ! n ^j +1 t = t ^j +1t: (17) As g is semi-simple, gC and sl(2; C) are irreducible representations of T 2n?r  G ./ Sp(1) while ^j+1t is completely reducible to a direct sum of one-dimensional representations, the coboundary map is the zero map. It follows that the induced long exact sequence of (16) splits, and the induced map of d  dp on each cohomology level is surjective. Now the lemma follows Lemma 12 and Lemma 13. q. e. d. 13

4 The Kuranishi Spaces Since the second cohomology of the twistor space W does not vanish, we apply Kuranishi theory to calculate deformations [13]. Let us review this theory brie y to set up notation. Let f 1 ; : : : ; n g be an orthonormal basis of the harmonic representatives of H 1 (W; W ). For any vector t = (t1 ; : : : ; tn ) in Cn, let 1 (t) = t1 1 + : : : + tn n . Let G be the Green's operator and @ be the adjoint operator of the @ -operator on W . For   2, de ne inductively  ?1 X 1  (t) = 2 @  G [ (t); ? (t)]; =1

(18)

where the operator [; ] takes exterior product on the form-components and Lie P bracket on the vector-components. Consider the formal sum (t) = 1  . Let f 1 ; : : : ; M g be an orthonormal basis for the space of harmonic (0; 2)-forms with values in W . De ne

fk (t) = ([(t); (t)]; k ):

(19)

According to the Kuranishi theory, there exists  such that ft 2 Cn : jtj < ; f1(t) = 0; : : : ; fM (t) = 0g forms a locally complete family of deformations of W [13, Theorem 1]. To describe the Kuranishi family of the complex structures on the twistor space W , we seek harmonic representatives for H j (W; W ). Since E is the product of n elliptic curves, there exists a coordinate i on the i-th elliptic curve so that !i = d i is a globally de ned (0; 1)form on E and that the real part of @=@ i , with 1  i  n, spans the real algebra a. By Proposition 3, H 1 (W; W ) = H 1 (E; OE ) (Cn  H 0 (Z; Z )  sl(2; C)): Let us identify each space in this formula. The summand Cn is generated by the holomorphic action of the group T 2n?r  B . It is naturally isomorphic to aC = H 0 (E; E ). On the product of elliptic curves E , the space H 1 (E; OE ) is isomorphic to the dual aC of H 0 (E; E ). By Bott [3], H 0 (Z; Z ) is isomorphic to gC. Therefore, Proposition 3 yields

H 1 (W; W )  = aC (a  g  sp(1))C ; and H 1 (W; D)  = aC (a  g)C ;

(20) In particular, the vector space H 1 (W; W ) is complex linearly spanned by !i (ai + vi + Ai ) where ai 2 a, vi 2 g and Ai 2 sp(1). As observed 14

in Proposition 3, the product of elliptic curves E is in the transversal intersection of the bers of the projection  and the holomorphic map p. In view of Lemma 6, the map   p is a locally trivial bration with E as bers. Therefore, the coordinates f1 ; : : : ; n g on E are ber coordinates on an open set V of the twistor space W . In particular, when d is the exterior di erential operator on W , the forms !i are holomorphic on the open set V . The computation in Lemma 11 demonstrates that these forms are global 1-forms on W . Their local representations reveal that they are holomorphic. Through the Dolbeault theory, !i (ai + vi + Ai ) are the harmonic representatives for H 1 (W; W ). Similarly, through Proposition 3, we see that !i ^ !j (aij + vij + Aij ), where aij 2 a, vij 2 g and Aij 2 sp(1) respectively, are harmonic representatives for H 2 (W; W ). If 1 (t) is a complex linear combination of !i (ai + vi + Ai ), [1 (t); 1 (t)] is a complex linear combination of !i ^ !j [ai + vi + Ai ; aj + vj + Aj ] with i < j . As [ai + vi + Ai ; aj + vj + Aj ] is a linear combinations of elements in the algebra a  g  sp(1), [1 (t); 1 (t)] is a harmonic (0; 2)-form with values in the algebra (a  g  sp(1))C . Therefore, it is in the kernel of the Green's operator. It follows from (18) that  (t) = 0 for all   2. From the de nitions of the functions fk (t) (19), we nd that the Kuranishi family for the twistor space W is de ned by the algebraic equations [ai + vi + Ai ; aj + vj + Aj ] = 0. This equation is equivalent to [vi ; vj ] = 0;

[Ai ; Aj ] = 0

and

[4(Ai ); vj ] = 0:

(21)

A deformation of the twistor space W is a twistor space only if the deformation carries a real structure [7, Section 2.5] [15, Remark 3.2]. If (t) = 1 (t) in H 1 (W; W ) is in the real linear span of elements of the form !i (ai + vi + Ai ), then the induced deformations is invariant of the real structure. Conversely, the in nitesimal deformation of any deformations of twistor spaces is real. Therefore, the Kuranishi family of deformations of the twistor space W with real structures is the real algebraic variety in the vector space a (a  g  sp(1))

(22)

de ned by the equations (21). Due to the correspondence between twistor spaces and quaternionic structures [14] [16], these deformations correspond to deformations of the underlying quaternionic structures on T 2n?r  G. We summarize our computations in the next theorem. 15

Theorem 1 The algebraic variety X Q = f !i (ai + vi + Ai) : ai 2 a; vi 2 g; Ai 2 sp(1); i [vi ; vj ] = 0; [Ai ; Aj ] = 0; [4(Ai ); vj ] = 0 for all i; j g is a locally complete family of deformations of the underlying quaternionic structures on T 2n?r  G. We call this family the Kuranishi family of the quaternionic structure on T 2n?r  G.

By Horikawa's deformation theory of maps [9] and the twistor correspondence [14], deformations of hypercomplex structures are contained in H 1 (W; D). By comparing this cohomology space with H 1 (W; W ) in Proposition 3, we conclude that the hypercomplex structure on T 2n?r  G deforms in a sub-family of the Kuranishi family. It is de ned by Ai = 0. Therefore, we have

Theorem 2 The algebraic variety X H = f !i (ai + vi) : ai 2 a; vi 2 g; [vi ; vj ] = 0 for i 6= j g i

is a locally complete family of deformations of the hypercomplex structure on T 2n?r  G. We call this family the Kuranishi family of the hypercomplex structure on T 2n?r  G.

The results in Theorem 2 can be re ned. Given (a; v) = (a1 ; : : : ; an ; v1 ; : : : ; vn ) in the Kuranishi space, the elements fv1 ; : : : ; vn g are contained in the Lie algebra v of a maximal torus V of the group G. By

Cathelineau [6], equivariant deformations are given by invariant elements of the cohomology spaces. The left-action of the group G on itself generates the adjoint action on gC  = H 0 (Z; Z ). Via the isomorphism (20), (a,v) is a V -invariant element in H 1 (W; W ). Therefore, the deformation generated by (a,v) is V -equivariant. Similarly, the deformation generated by (a; Adg v) is gV g?1 -invariant. As G is compact, the maximal tori are conjugate to each other and xed by the Weyl groups. Therefore, up to equivalence, the subspace of U -invariant elements in the Kuranishi family is complete. Since the U -invariant part of H 1 (W; D) is the linear space (a a  u a )C , the quotient space (a a  u a )=W , where W is the Weyl group of G with respect to U , is a complete family of hypercomplex structures on T 2n?r  G. 16

Note that deformations in this family have large symmetry group. Recall that a is the algebra of right translations generated by the group T 2n?r B . Since the induced action of T 2n?r  B on (a  u) a is trivial, every element in the complete family is (T 2n?r  B  U )-invariant. Since H 0 (W; D) = (a  g)C , the real subspace of (T 2n?r  B  U )invariant elements are precisely the algebra (2n ? r)u(1)  b  u = a  u. Therefore, the invariant part of H 0 (W; D) does not jump throughout an (T 2n?r  B  U )-equivariant deformation. It follows that a local moduli at a generic deformation is a smooth manifold whose dimension is equal to the dimension of the linear space (a  u) a . To summarize, we have

Theorem 3 Suppose G is a compact semi-simple Lie group of rank r, with a maximal torus U and Weyl group W . Let g = b nj=1 @j nj=1 fj

be the Joyce decomposition of G, and B be the subgroup of G whose Lie algebra is b. Let a be the algebra of right invariant vector elds generated by T 2n?r  B . Then (a a  u a)=W is a complete family of deformations of the left-invariant hypercomplex structures on T 2n?r  G. The identity component of the group of hypercomplex symmetries of a generic deformation in this family is the Abelian group T 2n . The local moduli at a generic deformation is a smooth manifold of dimension n(n + r).

The complete family of T 2n?r  G-invariant hypercomplex structures on the manifold T 2n?r  G is parametrized by a a . As observed in [10], these n2 parameters correspond to the choice of bases in the identi cation (2).

5 Examples The simplest compact semi-simple Lie groups are products of Sp(1), SO(4) and SO(3). For such manifolds, we construct the global moduli spaces based on the information in Theorem 3, and based on the techniques of constructing Abelian varieties.

5.1 Moduli of Hypercomplex Structures on (SU(2)U(1))n

Let (q1 ; : : : ; qn ; x1 ; : : : ; xn ) = (q; x) be coordinates for nS3  Rn . Here qj are unit quaternions. De ne a hypercomplex structure on (S3  R)n 17

by the multiplication (q; x), where  is a unit quaternion. For 1  j  n, de ne an action generated by

j (q; x) = (e2i1j q1 ; : : : ; e2inj qn; x + vj ): (23) The action of j is represented by the column vectors vj and j = (1j ; : : : ; nj )T , where ij are in R=Z. Assume that the vectors fv1 ; : : : ; vn g are linearly independent. Let ? = Zn be the group generated by f 1 ; : : : ; n g. We call (jV ) = (1 ; : : : ; n jv1 ; : : : vn ) (24) the period matrix of the manifold (S3  R)n =?. The groups ? are parameterized by the space of rank-n period matrices (25) P n = (R=Z)n2  GL(n; R): However, di erent period matrices may generate the same group. In fact, ? is generated by f 1 ; : : : ; n g and f ^1 ; : : : ; ^n g if and only if there is a matrix M = (mij ) in GL(n; Z) such that ^k = 1m1k : : : nmnk . It follows that the period matrices (jV ) and (^ jV^ ) generate the same group if and only if (^ jV^ ) = (M jV M ): (26) The quotient space (S3  R)n =? is a hypercomplex manifold because the actions of ? commute with the right multiplications of the quaternions on (q1 ; : : : ; qn). The quotient space is di eomorphic to (S3  S1 )n through the map from (S3  R)n to (S3  S1 )n : when y = V ?1 x,

P P (q; x) 7! (e?2i( j yj j1 ) q1 ; : : : ; e?2i( j yj jn ) qn; e2iy1 ; : : : ; e2iyn ): (27)

Lemma 14 The hypercomplex manifolds (S3  R)n=? and (S3  R)n =?0 are equivalent if and only if there exist period matrices (jV ) and (0 jV 0 ) for ? and ?0 respectively such that V = V 0 , and j = 0j . Proof: Let W be the twistor space of (S3  R)n . The actions of ? and ?0 are uniquely lifted to holomorphic actions on W . The twistor spaces for (S3  R)n =? and (S3  R)n =?0 are W=? and W=?0 respectively. Assume that there is a hyper-holomorphic isomorphism from (S3  R)n =? to (S3  R)n =?0 ; it lifts to a holomorphic isomorphism between the twistor spaces W=? and W=?0 . Such an isomorphism lifts to a holomorphic automorphism on W intertwining the groups ? and ?0 . 18

by

Through the map n S3  Rn ! n S3 n R+ ! Hn  HPn de ned

(28) (q; x) 7! (q1 ; : : : ; qn ; ex1 ; : : : ; exn ) 7! [1; ex1 q1 ; : : : ; exn qn ]; the hypercomplex manifold (S3  R)n is a quaternionic submanifold of Hn  HPn . It follows that its twistor space is a complex analytic subspace of the twistor space CP2n+1 of HPn . Since the co-dimension of W in CP2n+1 is equal to two, by the Hartog Theorem, the analytic automorphism on W is uniquely extended to an analytic automorphism of CP2n+1 . In particular, it is a linear map. It descends to a linear quaternionic map f from HPn to HPn . Since the map f sends (S3  R)n to (S3  R)n , it sends quaternionic coordinate hyperplanes to quaternionic coordinate hyperplanes. Therefore, f is in (GL(1; H))n+1 = GL(1; H)     GL(1; H). Since f is a hypercomplex transformations from Hn to Hn , it is contained in f1g  (GL(1; H))n . Therefore, there exists (ri ; qi ) in (R+ )n  (Sp(1))n  = (GL(1; H))n such that f (p1 ; : : : ; pn ; x1 ; : : : ; xn ) = (q1 p1 ; : : : ; qn pn ; ln r1 +x1 ; : : : ; ln rn +xn ). This map intertwines the groups ? and ?0 if and only if there exists generators f 1 ; : : : ; n g and f 10 ; : : : ; n0 g for these two groups respectively such that j  f = f  j0 . It follows that for each j , vj = vj0 and q` e2i`j = e2i`j0 q` for each `. It is possible if and only if 0q` is the unit quaternion j and e2i`j is the complex conjugate of e2i`j . Therefore, the lemma follows. q. e. d. Combining the last lemma with the identi cation given in (26), we conclude that the quotient space M = P n =(Zn2  GL(n; Z)) is a moduli space for hypercomplex structures on the product manifold (S3  S1 )n . In view of the dimension count in Theorem 3, we conclude that this moduli space is complete.

Theorem 4 The quotient space M = P n =(Zn2 GL(n; Z)) is a complete moduli space for hypercomplex structures on the product manifold (S3  S1 )n .

The set of singular points of the moduli space contains the quotients of the period matrices with non-trivial isotropy with respect to the action of Zn2 . The most singular part is given by the xed points of the action of Zn2 . This is the space of left-invariant hypercomplex structures. The strati cation of the singularities of the moduli space corresponds to the size of symmetries. For instance, a period matrix is a xed point of the j -th Z2 -action if and only if the corresponding 19

hypercomplex structure allows the left multiplication of the j -th factor in (SU(2)  U(1))n to be a group of hypercomplex transformations. The complete moduli space of hypercomplex structures on S3  S1 is the quotient fre2i : 0 < r < 1g=Z2 de ned by the equivalent relation re2i = re?2i . It is fre2i : 0 < r < 1; 0    0:5g. This result re nes our observations in (1.1).

5.2 Products of SO(4) and SO(3)

The results in the last section can be used to nd the complete moduli space of hypercomplex structures on (T 2  SO(4))m because SU(2)  SU(2) is a double covering of SO(4). Let a Zm2 action on 2m S3  R2m be generated by the maps (q1 ; : : : ; ?q2i?1 ; ?q2i ; : : : ; q2m ; x1 ; : : : ; x2m ) for 1  i  m. Then the quotient space (S3 R)2m =(?Zm2 ) is the manifold (T 2  SO(4))m with a hypercomplex structure. Let Zm2 act on the space of period matrices P 2m by (1 ; : : : ; 2i?1 + 1=2; 2i + 1=2; : : : ; 2m ; v1 ; : : : ; v2m ):

Then P 2m =(Z22m  GL(2m; R)  Zm2 ) is a complete moduli space of hypercomplex structures on (T 2  SO(4))m . The complete moduli spaces of hypercomplex structures on (SU(2)  S1 )n  (T 2  SO(4))m  (SO(3)  S1 )` can be similarly constructed.

5.3

SU(3)

as a Hypercomplex Quotient

In our previous paper [15], we studied the deformations of the hypercomplex structures on SU(3) from a di erent perspective. From Theorem 3, u(1)  (u(1)  u(1))=W is a complete parameter space of hypercomplex structures on SU(3). These parameters can be realized from the hypercomplex quotients as follows. We choose a hypercomplex structure on C3 C3 by I1 (; `) = (i; ?i`), I2 (; `) = (i`; i), I3 (; `) = (?`; ). De ne a function  = (1 ; 2 ; 3 ) on C6 nf0g by 1 (; `) = jj2 ? j`j2 and (2 + i3 )(; `) = 2h; `i, where h; `i is the Hermitian inner product on C3 . The function  is a hypercomplex moment map [10]. For each non-zero element  in the Lie algebra u(3), let ?  = R be the 1-parameter group acting on C6 t by (; `) 7! (e exp (t); et exp (t)`). It is a group of hypercomplex automorphisms, and it leaves the level set of  invariant. For any r in the open interval (0; 1), we consider the discrete group generated by (r; r`). 20

When  = 0, then the quotient space ?1 (0)=f?; hrig is SU(3). Therefore, one obtains the homogeneous hypercomplex structure on SU(3) as a hypercomplex quotient. This construction should be compared with the one in [15, Example 6.4] where we constructed deformations of the compact associated bundle of CP2 . For any  near 0 in u(3), the quotient is a deformation of the homogeneous hypercomplex structure. When exp (t) is a diagonal matrix, the hypercomplex manifolds has torus symmetry. Since the action of  induces a left-action on SU(3), the corresponding in nitesimal deformation coincides with the deformation determined by  when we use the identi cation given by Proposition 3. By Theorem 3, the algebra 3u(1) is a complete parameter space. Therefore, any small deformation of the hypercomplex structures on SU(3) can be obtained as a hypercomplex quotient. However, nding global moduli remains a dicult problem as seen in [5].

5.4 Remarks about T n  Sp(n)

The inclusion of n Sp(1) in Sp(n) as a diagonal subgroup gives an identi cation of the Kuranishi spaces of (T n  T n)-equivariant hypercomplex structures of T n n Sp(1) and T n  Sp(n). Therefore, the local moduli of hypercomplex structures on T n n Sp(1) and T n  Sp(n) are identical. This relation may be used to construct the global moduli space of hypercomplex structures on T n  Sp(n), much as in Example 5.1. Such comparison between deformations of hypercomplex structures on T 2n?r  G and deformations of hypercomplex structures on a subgroup occurs in a more general context. If the algebra b is trivial in the Joyce decomposition of G, let K be a subgroup of G such that its Lie algebra is @n  : : :  @1 . Then K is a semi-simple Lie group, and T 2n?r  K is a hypercomplex submanifold of T 2n?r  G with respect to the left-invariant hypercomplex structures. The inclusion map from T 2n?r  K to T 2n?r  G induces a one-to-one correspondence between the U -equivariant hypercomplex deformations of T 2n?r  K and of T 2n?r  G. Since the group K is a product of SO(4)'s, SO(3)'s and Sp(1)'s, one may again attempt to construct the global moduli space for the manifold T 2n?r  G using Example 5.2. This idea of using subgroups to nd global moduli for the ambient group has its limitation. For example, the subgroup K in SU(3) with algebra b  @1 in the Joyce decomposition is U(2). However, the moduli of hypercomplex structure on U(2) is two-dimensional while the local moduli for SU(3) is three-dimensional. 21

Acknowledgment We thank S. Salamon for proposing to us the topic

in this paper, D. Joyce for very useful comments which greatly improve our description of the Kuranishi families, C. Margerin for pointing out an error in the early stage of our work, and J.-P. Bourguignon for providing an excellent research environment at the I.H.E.S..

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