ON HYPER KÄHLER MANIFOLDS ASSOCIATED TO LAGRANGIAN KÄHLER SUBMANIFOLDS OF T^{∗}ℂ^{n}

VICENTE CORTÉS

Abstract. For any Lagrangian Kähler submanifold M ⊂ T^{∗}ℂ^{n}, there exists a canonical hyper Kähler metric on T^{∗}M. A Kähler potential for this metric is given by the generalized Calabi Ansatz of the theoretical physicists Cecotti, Ferrara and Girardello. This correspondence provides a method for the construction of (pseudo) hyper Kähler manifolds with large automorphism group. Using it, an interesting class of pseudo hyper Kähler manifolds of complex signature (2,2n) is constructed. For any manifold N in this class a group of automorphisms with a codimension one orbit on N is specified. Finally, it is shown that the bundle of intermediate Jacobians over the moduli space of gauged Calabi Yau 3-folds admits a natural pseudo hyper Kähler metric of complex signature (2,2n).

Introduction

The generalized Calabi Ansatz of Cecotti, Ferrara and Girardello was discovered in the context of super string theory [C-F-G]. Nevertheless, it provides a simple method for the construction of such classical geometric structures as hyper Kähler metrics.

In the first part of this paper this construction is reviewed. We explain how to a Lagrangian (pseudo) Kähler submanifold M ⊂ T^{∗}ℂ^{n} one canonically associates a (pseudo) hyper Kähler metric on the complex symplectic manifold T^{∗}M, s. Thm. 1.5.

Then we study natural group actions on M and on its cotangent bundle T^{∗}M preserving the special geometric structures. This opens the way for a systematic construction of (pseudo) hyper Kähler manifolds of small cohomogeneity, s. Prop. 2.1 and Cor. 2.2. Using the classification of certain Lagrangian cones given in [dW-VP] and [C2], we construct examples of pseudo hyper Kähler manifolds of complex signature (2,2n) admitting a group of automorphisms of cohomogeneity one, s. Thm. 2.3. If M is a cone of appropriate signature, then M can be interpreted as formal moduli space of gauged Calabi Yau 3-folds, s. Prop. 3.2 and Thm. 3.4. In particular, the Lagrangian cones classified in [dW-VP] and [C2] are models for moduli spaces with large automorphism group, s. Prop. 3.3 and Rem. 3.

Finally, we prove that the pseudo hyper Kähler structure on the cotangent bundle T^{∗}M of a Lagrangian pseudo Kähler submanifold M ⊂ T^{∗}ℂ^{n} is always defined on a discrete fibre preserving quotient of T^{∗}M which is a torus bundle over M, s. Thm. 3.1. As a consequence, we obtain a natural pseudo hyper Kähler structure of complex signature (2,2n) on the bundle of intermediate Jacobians over the moduli space of gauged Calabi Yau 3-folds, where n = h^{2,1}, s. Thm. 3.5.

Supported by the Alexander von Humboldt Foundation and MSRI (Berkeley). Research at MSRI is supported in part by grant DMS-9022140.

Acknowledgements The author is very grateful to the Mathematical Sciences Research Institute, especially to S.-S. Chern and R. Osserman, for hospitality and support. He would also like to thank D.V. Alekseevsky, R.L. Bryant, A.B. Givental, P.A. Griffiths, S. Katz, M. Kontsevich and A. Van Proeyen for discussions related to the subject of this paper.

1. The canonical pseudo hyper Kähler metric on the cotangent bundle of a Lagrangian pseudo Kähler submanifold M ⊂ T^{∗}ℂ^{n}

Consider the following fundamental algebraic data:

1) A complex symplectic vector space (V, ω), dimℂ V = 2n.

2) A compatible real structure τ : V → V, i.e. a ℂ-antilinear involution such that the restriction of ω to its fixed point set V^{τ} is a real symplectic structure.

Up to isomorphism, we can assume that V = T^{∗}ℂ^{n}, V^{τ} = T^{∗}ℝ^{n} and that ω is the standard symplectic structure of T^{∗}ℂ^{n}, which is a real symplectic structure when restricted to T^{∗}ℝ^{n}.

Recall that a linear subspace of a (real or complex) symplectic vector space is called Lagrangian if it is maximally isotropic and that a submanifold M of a symplectic vector space is called Lagrangian submanifold if T_{m}M is a Lagrangian subspace for every point m ∈ M.

Given (V, ω, τ) we can define a Hermitian form γ of signature (n,n) on V by

γ(u, v) = \sqrt{−1}ω(u, τv) , u, v ∈ V .  (1)

(V, γ) is a pseudo Kähler manifold (of complex signature (n,n)) and hence the notion of pseudo Kähler submanifold is defined. In fact, a complex submanifold M ⊂ V is called a pseudo Kähler submanifold of (V, γ) if γ|M is nondegenerate.

Recall that a complex symplectic structure on a complex manifold is a holomorphic, closed and nondegenerate 2-form. A (pseudo) Kähler manifold is called (pseudo) hyper Kähler manifold if it admits a parallel complex symplectic structure. Remark 1: The complex symplectic structure ω defines on the (flat) pseudo Kähler manifold (V, γ) the structure of pseudo hyper Kähler manifold.

Proposition 1.1. Given (V, ω, τ) and γ as above, a Lagrangian submanifold M ⊂ (V, ω) is a pseudo Kähler submanifold M ⊂ (V, γ) if and only if T_{m}M ∩ τT_{m}M = 0 for all m ∈ M. In particular, V^{τ} ∩ T_{m}M = 0 is necessary.

Proof: Let L ⊂ (V, ω) be a Lagrangian subspace. Then γ|L is nondegenerate if and only if L ∩ τL = 0. □

A connected Lagrangian pseudo Kähler submanifold M ⊂ (V, ω, τ) has a well defined complex signature (k, l), k + l = n, namely the signature of the Hermitian form γ|T_{m}M, m ∈ M arbitrary. If l = 0, then γ|T_{m}M is positively defined and M is a Kähler submanifold of (V, γ).

Proposition 1.2. For any pseudo Kähler submanifold M ⊂ (V, γ) the function K^{M}(u) := γ(u, u), u ∈ M, is a pseudo Kähler potential. Any subgroup of the linear group Aut(V, γ) = {φ ∈ GL(V)|φ^{∗}γ = γ} ≅ U(n, n) which preserves M acts on M by holomorphic isometries.

The linear automorphism group Aut(V, ω, τ) ≅ Aut(V^{τ}, ω|V^{τ}) ≅ Sp(n, ℝ) of our fundamental algebraic data acts on the hyper Kähler manifold (V, γ, ω) by automorphisms, i.e. by holomorphic isometries preserving the complex symplectic structure. Here Sp(n, ℝ) denotes the real symplectic group in 2n variables.

The group

Aut_{d}(M) = {φ ∈ Aut(V, ω, τ)|φM = M}

is called the duality group of the Lagrangian pseudo Kähler submanifold M ⊂ (V, ω, γ). The next proposition follows from Prop. 1.2 since Aut_{d}(M) ⊂ Aut(V, γ) preserves M.

Proposition 1.3. The duality group Aut_{d}(M) acts on M holomorphically and isometrically.

To define the generalized Calabi Ansatz of [C-F-G], we have to choose a Lagrangian splitting for V^{τ}, i.e. a decomposition

V^{τ} = L_{0} ⊕ L′_{0}  (2)

into two Lagrangian subspaces. A Lagrangian splitting for V^{τ} induces a Lagrangian splitting for V:

V = L ⊕ L′ ,  (3)

where L and L′ are determined by

L = τL , L′ = τL′ and L_{0} = L^{τ} , L′_{0} = (L′)^{τ} .  (4)

Moreover, we have canonical isomorphisms associated to (2) and (3):

V^{τ} ≅ T^{∗}L_{0} , V ≅ T^{∗}L .

Definition 1.1. A Lagrangian submanifold M ⊂ (V, ω) is in general position with respect to the splitting V = L⊕L′ if the projection p : V → L induces an isomorphism of M onto its image.

Proposition 1.4. A Lagrangian Kähler submanifold M ⊂ (V, ω, γ) is in general position with respect to any splitting V = L ⊕ L′ induced by a Lagrangian splitting of V^{τ}, s. (2), (3) and (4).

Proof: It is sufficient to show that T_{m}M ∩ L′ = 0 for all m ∈ M. This follows from the fact that γ is positively defined on T_{m}M and zero on L′ = τL′. □

Given fundamental algebraic data (V, ω, τ), compatible Lagrangian splittings (2)– (4) and a Lagrangian pseudo Kähler submanifold M ⊂ (V, ω, γ) in general position with respect to the given splitting, there is a canonical real structure ρ′ on T M. In fact, the projection p : V → L induces an isomorphism T_{m}M →^{∼} L and hence we can define a real form (T_{m}M)^{ρ′} of T_{m}M by the equation dp (T_{m}M)^{ρ′} = L_{0}. Now we can define ρ′ (at the point m) as the ℂ-antilinear involution of T_{m}M with fixed point set (T_{m}M)^{ρ′}.

We denote by ρ the real structure on T^{∗}M which is dual to ρ′ and by g_{m}^{−1} the Hermitian metric on T_{m}^{∗}M which is inverse to g_{m} = γ|T_{m}M. Then the generalized Calabi Ansatz of [C-F-G] is given by the following potential on T^{∗}M:

K(σ) = K^{M}(π(σ)) + g_{π(σ)}^{−1}(σ + ρ(σ), σ + ρ(σ)) , σ ∈ T^{∗}M ,  (5)

where π : T^{∗}M → M is the natural projection and K^{M} is the pseudo Kähler potential of M, s. Prop. 1.2.

Theorem 1.5. [C-F-G] Let (V, ω, τ) be fundamental algebraic data, s. p. 1, γ the Hermitian form defined in (1), V = L ⊕ L′ a Lagrangian splitting as in (2)–(4) and M ⊂ (V, ω, γ) a Lagrangian Kähler submanifold (resp. pseudo Kähler submanifold of complex signature (k, l) in general position, s. Def. 1.1). Then the function K on T^{∗}M associated to these data, s. (5), is the Kähler potential of a hyper Kähler metric G on T^{∗}M (resp. the pseudo Kähler potential of a pseudo hyper Kähler metric G of complex signature (2k, 2l) on T^{∗}M). More precisely, the standard complex symplectic structure Ω on the cotangent bundle T^{∗}M is parallel with respect to the canonical connection of the Kähler (resp. pseudo Kähler) manifold N = (T^{∗}M, G).

Remark 2: In [C-F-G] the correspondence M ↦ N is called the c-map in rigid super symmetry. There is also a c-map in local super symmetry M ↦ N as was proven in [F-S]. In the latter case, N is a quaternionic Kähler manifold of negative Ricci curvature. Example 1: The simplest example of Lagrangian pseudo Kähler submanifold M ⊂ T^{∗}ℂ^{n} is a Lagrangian subspace L such that L ∩ τL = 0. If e.g.

L = spanℂ\{\frac{∂}{∂q^{1}} + i\frac{∂}{∂p_{1}} , …, \frac{∂}{∂q^{k}} + i\frac{∂}{∂p_{k}} , \frac{∂}{∂q^{k+1}} − i\frac{∂}{∂p_{k+1}} , …, \frac{∂}{∂q^{n}} − i\frac{∂}{∂p_{n}} \} ,

i = \sqrt{−1}, then γ|L has complex signature (k,n-k) and (T^{∗}L, G) is the flat model of pseudo hyper Kähler manifold of quaternionic signature (k,n-k), i.e. of complex signature (2k,2n-2k).

2. A class of pseudo hyper Kähler manifolds admitting a group of automorphisms of cohomogeneity one

Let (V, ω, τ) be our fundamental algebraic data, s. p. 1, γ the Hermitian form defined in (1), V = L ⊕ L′ a Lagrangian splitting as in (2)–(4) and M ⊂ (V, ω, γ) a Lagrangian pseudo Kähler submanifold in general position, s. Def. 1.1 and Prop. 1.4. By Thm. 1.5 to these data we canonically associate the pseudo hyper Kähler manifold N = (T^{∗}M, G) with standard complex symplectic structure Ω.

Now we are interested in natural group actions on M and N preserving the given geometric structures. Recall that the duality group Aut_{d}(M) of M consists of those linear automorphisms of (V, ω, τ) which preserve M. It acts holomorphically and isometrically on the pseudo Kähler manifold (M, g), s. Prop. 1.3. Let us consider the subgroup Aut_{sd}(M) of Aut_{d}(M) preserving also the Lagrangian splittings (2)–(4), i.e.

Aut_{sd}(M) = {φ ∈ Aut_{d}(M)|φL = L , φL′ = L′} .

Since Aut_{sd}(M) ⊂ Aut_{d}(M) ↪ Sp(V^{τ}, ω|V^{τ}) ⊂ GL(V^{τ}) ⊂ Aff(V^{τ}) = V^{τ}⋊GL(V^{τ}), we can also consider the corresponding affine group V^{τ}⋊Aut_{sd}(M) ↪ Aff(V^{τ}).

We will show that V^{τ}⋊Aut_{sd}(M) acts on N preserving the pseudo hyper Kähler structure. First we define a fibre preserving action of the vector group V^{τ} ≅ ℝ^{2n} on T^{∗}M. For this we consider the quotient map V → V/T_{m}M. Using the inclusion V^{τ} ⊂ V and the isomorphism V/T_{m}M ≅ T_{m}^{∗}M given by the symplectic structure ω, the quotient map induces an isomorphism of real vector spaces, cf. Prop. 1.1:

ψ_{m} : V^{τ} → T_{m}^{∗}M .  (6)

Now we define the action of an element v ∈ V^{τ} on T^{∗}M by

T_{m}^{∗}M ∋ σ ↦ σ + \sqrt{−1}ψ_{m}(v) ∈ T_{m}^{∗}M .  (7)

Choosing linear coordinates (v^{1}, …, v^{n}) for L_{0} ≅ ℝ^{n} induces coordinates (v_{1}, …, v_{n}) for L′_{0} ≅ L_{0}^{∗}. Since M is a Lagrangean submanifold in general position it is the image of a closed and hence locally exact section of T^{∗}ℂ^{n}. In other words, we can describe M locally by equations of the form

p_{i} = \frac{∂F (q^{1}, …, q^{n})}{∂q^{i}} , i = 1, …, n ,  (8)

where F (q^{1}, … q^{n}) is a locally defined holomorphic function of n variables. Note that for any locally defined holomorphic function F the equations (8) define a Lagrangean submanifold in general position, namely the image of the exact section dF of T^{∗}ℂ^{n}.

Denote by z^{i} := q^{i}|M, i = 1, …, n, the natural complex coordinates on M and by (z^{1}, …, z^{n}, w_{1}, …, w_{n}) the corresponding complex coordinate system for T^{∗}M. In these coordinates the action of v = (v^{1}, …, v^{n}, v_{1}, …, v_{n}) on T^{∗}M defined in (7) reads (cf. [C-F-G]): (z^{1}, …, z^{n}, w_{1}, …, w_{n}) ↦ (z̃^{1}, …, z̃^{n}, w̃_{1}, …, w̃_{n}),

z̃^{i} = z^{i} ,  w̃_{i} = w_{i} − \sqrt{−1}(v_{i} − ∑_{j} F_{ij}v^{j}) ,  (9)

where F_{ij} = \frac{∂^{2}F}{∂z^{i}∂z^{j}} are the second derivatives of the local holomorphic function F (z^{1}, …, z^{n}) defining the Lagrangian submanifold M, s. (8). The pseudo hyper Kähler potential K of N, s. (5), has the following expression:

K(z^{1}, …, z^{n}, w_{1}, …, w_{n}) = K^{M}(z^{1}, …, z^{n}) + ∑_{ij} g^{ij}(w_{i} + \overline{w}_{i})(w_{j} + \overline{w}_{j}) , (10)

where

K^{M}(z^{1}, …, z^{n}) = γ(dF |_{(z^{1},... ,z^{n})}, dF |_{(z^{1},... ,z^{n})}) = \sqrt{−1} ∑_{ij}(z^{i}\overline{F}_{i} − \overline{z}^{i}F_{i}) , (11)

g_{ij} = g_{ji} = γ(\frac{∂}{∂z^{i}} , \frac{∂}{∂z^{j}}) = \sqrt{−1}(\overline{F}_{ij} − F_{ij})  (12)

and the Hermitian fibre metric g^{−1} = (g^{ij}) on T^{∗}M is defined by the equation

∑_{j} g^{ij}g_{jk} = δ_{k}^{i} .  (13)

Proposition 2.1. The group V^{τ}⋊Aut_{sd}(M) ↪ Aff(V^{τ}) acts naturally on N by automorphisms of the pseudo hyper Kähler structure (s. Thm. 1.5), i.e. by holomorphic isometries preserving the complex symplectic structure Ω. The action of Aut_{sd}(M) is by point transformations of N = T^{∗}M and that of V^{τ}, given by (7), is fibre preserving and simply transitive on each fibre.

Proof: Aut_{d}(M) and thereby its subgroup Aut_{sd}(M) acts holomorphically on M and hence by complex symplectomorphisms on T^{∗}M, namely by holomorphic point transformations. Moreover, Aut_{sd}(M) preserves the pseudo Kähler potential (5) on T^{∗}M associated to the choice of Lagrangian splitting (2). This shows that Aut_{sd}(M) acts on N by automorphisms of the pseudo hyper Kähler structure.

Consider the map ψ : M × V^{τ} → T^{∗}M, ψ(m, v) := ψ_{m}(v). From the fact that ψ is Aut_{d}(M)-equivariant it follows that (7) extends the action of Aut_{d}(M) on N to a holomorphic action of V^{τ}⋊Aut_{sd}(M) on N. We check that the action of v ∈ V^{τ}, s. (9), preserves the complex symplectic structure Ω = ∑_{i} dz^{i} ∧ dw_{i}:

∑_{i}(dz̃^{i} ∧ dw̃_{i} − dz^{i} ∧ dw_{i}) = \sqrt{−1} ∑_{ij} v^{j}dz^{i} ∧ dF_{ij} = \sqrt{−1} ∑_{ijk} v^{j}F_{ijk}dz^{i} ∧ dz^{k} = 0 .

Next we check that under the action of v ∈ V^{τ} the pseudo Kähler potential K of N changes only by a pluriharmonic function. Using (9)-(13) we obtain:

K(z̃^{1}, …, w̃_{n}) − K(z^{1}, …, w_{n}) = −2 ∑_{j} v^{j}(w_{j} + \overline{w}_{j}) + ∑_{ij} g_{ij}v^{i}v^{j} .

The result is pluriharmonic, since the v^{i} are constants and g_{ij} is pluriharmonic as sum of a holomorphic and an antiholomorphic function, s. (12). Now it only remains to show that V^{τ} acts simply transitively on each fibre of T^{∗}M → M. This is clear since (6) is an isomorphism of real vector spaces. □

Corollary 2.2. If Aut_{sd}(M) has an orbit of codimension r on M then V^{τ}⋊Aut_{sd}(M) has an orbit of codimension r on N.

Our aim is to use Cor 2.2 for the construction of pseudo hyper Kähler manifolds N with the smallest possible cohomogeneity of the group V^{τ}⋊Aut_{sd}(M). (Recall that the cohomogeneity of a Lie group acting on a manifold is the minimal codimension of its orbits.)

The pseudo Kähler potential K^{M}(u) = γ(u, u), u ∈ M, defines an Aut_{d}(M)- invariant function on M and hence K^{M} ∘ π, π : T^{∗}M → M the projection, is an V^{τ}⋊Aut_{d}(M)-invariant function on N. This function cannot be constant on an open set, since K^{M} is the potential of a (nondegenerate) metric. Therefore, V^{τ}⋊Aut_{d}(M) ⊃ V^{τ}⋊Aut_{sd}(M) has no open orbit on N and hence is of cohomogeneity at least one.

Next we will present a class of pseudo hyper Kähler manifolds N for which the cohomogeneity of V^{τ}⋊Aut_{sd}(M) ⊂ V^{τ}⋊Aut_{d}(M) is in fact one. This class is associated, via the correspondence of Thm. 1.5, to an interesting class of Lagrangian pseudo Kähler submanifolds M ⊂ T^{∗}ℂ^{n+1} in general position (with respect to the standard Lagrangian splitting T^{∗}ℂ^{n+1} = ℂ^{n+1} ⊕ (ℂ^{n+1})^{∗} into positions and momenta). The latter class can be defined by the following additional conditions

(i) M is a cone, i.e. λM = M for all λ ∈ ℂ − {0}. (ii) The third fundamental form of M ⊂ T^{∗}ℂ^{n+1} is given by a homogeneous cubic

polynomial h(x^{1}, …, x^{n}) with real coefficients, s. remarks below. (iii) The real hypersurface {h = 1} ⊂ ℝ^{n} admits an open orbit H ⊂ {h = 1}

of a subgroup of GL(n, ℝ). Moreover, the second fundamental form of H is negatively defined.

The complete classification of such manifolds M was first obtained in [dW-VP]; an alternative, conceptual approach was developped in [C2].

First of all let us explain the meaning of (ii). The notion of 3rd fundamental form of a Lagrangian cone was introduced by Bryant and Griffiths [B-G] in the study of period maps for general Calabi Yau 3-folds, s. Section 3. The third fundamental form θ of a Lagrangian submanifold M ⊂ T^{∗}ℂ^{n+1} in general position satisfying (i) is by definition the 3rd fundamental form of the immersion

ϕ : (q^{1}, …, q^{n}) ↦ (π|M)^{−1}(1, q^{1}, …, q^{n}) ,

where π : T^{∗}ℂ^{n+1} → ℂ^{n+1} is the natural projection. More precisely, θ is the section of S^{3}T^{∗}ℂ^{n} whose value θ_{q} ∈ S^{3}T_{q}^{∗}ℂ^{n} ≅ S^{3}(ℂ^{n})^{∗} at q = (q^{1}, …, q^{n}), (1, q) ∈ π(M), is given by the cubic form

θ_{q}(x^{1}, …, x^{n}) = ∑_{ijk=1}^{n} x^{i}x^{j}x^{k}ω(\frac{∂^{3}ϕ(q)}{∂q^{i}∂q^{j}∂q^{k}} , ϕ(q)) .

So the condition (ii) is satisfied if and only if the homogeneous cubic polynomial θ_{q} is independent of q, i.e. θ_{q}(x^{1}, …, x^{n}) = h(x^{1}, …, x^{n}), and has (constant) real coefficients ω(\frac{∂^{3}ϕ(q)}{∂q^{i}∂q^{j}∂q^{k}} , ϕ(q)) ∈ ℝ.

In the following discussion we recall all needed basic facts about the class of Lagrangian submanifolds defined above; for complete details the reader is referred to [dW-VP] and [C2]. Under the condition (i), we can consider the complex manifold P (M) ⊂ P (T^{∗}ℂ^{n+1}) ≅ Pℂ^{2n+1}, which is a projectivized cone. If the potential K^{M} does not vanish on M, then P (M) admits a canonical Hermitian form g^{P (M)} invariant under the projective action of the duality group Aut_{d}(M), cf. (15). (The corresponding potential is log |K^{M}|.) Under the conditions (ii) and (iii), there exists an open subcone C ⊂ M ⊂ T^{∗}ℂ^{n+1} such that Aut_{sd}(C) acts transitively on P (C) ⊂ P (M), the Hermitian form γ has complex signature (1,n) on C and −g^{P (C)} is a (positively defined) Kähler metric on P (C). Without restriction of generality we assume M = C. The group Aut_{sd}(M) has an orbit of codimension one on M and by Prop. 2.1 and Cor. 2.2 V^{τ}⋊Aut_{sd}(M), V^{τ} = T^{∗}ℝ^{n+1} ≅ ℝ^{2n+2}, is a group of automorphisms of the hyper Kähler manifold N and has cohomogeneity one.

Now we describe the group Aut_{sd}(M) in more detail. Note that the subgroup of Sp(n+1, ℝ) ⊂ GL(2n+2, ℝ) preserving the standard Lagrangian splitting T^{∗}ℝ^{n+1} = ℝ^{n+1} ⊕ (ℝ^{n+1})^{∗} is the group of linear point transformations of T^{∗}ℝ^{n+1}, which is canonically isomorphic to GL(n + 1, ℝ). This gives rise to the embedding

ι : Aut_{sd}(M) ↪ GL(n + 1, ℝ) , φ ↦ φ|ℝ^{n+1} ,

where ℝ^{n+1} ⊂ T^{∗}ℝ^{n+1} is the zero section. GL(n + 1, ℝ) contains the affine group Aff(n, ℝ) = ℝ^{n}⋊GL(n, ℝ) as a subgroup:

Aff(n, ℝ) ∋ (v, B) ↦ ( \begin{matrix}1 & 0 \\ v & B \end{matrix} ) ∈ GL(n + 1, ℝ) .

So we can define the subgroup

A(M) := ι^{−1}(Aff(n, ℝ)) ⊂ Aut_{sd}(M) ,

which is embedded in the affine group Aff(n, ℝ) via the map ι.

By construction, s. [dW-VP] and [C2], the class of submanifolds M ⊂ T^{∗}ℂ^{n+1} defined above has the following property: The group A(M) has an open orbit on P (M) and (by restricting to an open subcone of M if necessary) we can assume that A(M) acts transitively on P (M). Moreover, it was proven in [C2] that (P (M), −g^{P (M)}) is isomorphic, as A(M)-Kähler manifold, to a Kählerian Siegel domain (U, g) of first type. The group ι(A(M)) ≅ A(M) acts naturally on (U, g) by affine transformations which are (holomorphic) isometries for the Kähler metric g.

The complete list of Kählerian Siegel domains (U, g) corresponding to the class of submanifolds M ⊂ T^{∗}ℂ^{n+1} defined above, s. (i)-(iii), was given in [C2] Thm. 2.8. For convenience of the reader, we recall that U has rank 2 or 3. The rank 2 domains in the list are numerated by the nonnegative integers and the rank 3 domains by special isometric maps or, equivalently, by ℤ_{2}-graded Clifford modules (up to equivalence defined in [C1] Def. I.10 and Def. II.5, cf. Def. II.4, Prop. II.25 and Prop. II.26). Recall that a ℤ_{2}-graded Clifford module of order k is a ℤ_{2}-graded module Ψ = Ψ_{0} ⊕Ψ_{1} of the real Clifford algebra Cℓ_{k}, k = 0, 1, 2, …, s. e.g. [L-M]. Summarising our discussion we obtain the following theorem.

Theorem 2.3. To any p ∈ {0, 1, 2, … } (resp. ℤ_{2}-graded Clifford module Ψ of order k ∈ {0, 1, 2, … }) we can canonically associate a Lagrangian pseudo Kähler submanifold in general position M(p) ⊂ T^{∗}ℂ^{n+1}, n = 2 + p, (resp. M(Ψ) ⊂ T^{∗}ℂ^{n+1}, n = k + 3 + dim Ψ) satisfying conditions (i)-(iii) on p. 6. Conversely, any Lagrangian pseudo Kähler submanifold of T^{∗}ℂ^{n+1} in general position satisfying (i)-(iii) contains one of the manifolds M above (i.e. M = M(p) or M = M(Ψ)) as an open subcone. For the manifolds M above, the affine group A(M) ↪^{ι} Aff(n, ℝ) acts transitively and isometrically on the Kähler manifold (P (M), −g^{P (M)}) by projective linear transformations. Moreover, (P (M), −g^{P (M)}) is isomorphic as A(M)- Kähler manifold to a Kählerian Siegel domain of type I with transitive affine action of ι(A(M)) ⊂ Aff(n, ℝ).

The pseudo hyper Kähler manifold N = (T^{∗}M, G) associated to any of these manifolds M by Thm. 1.5 has complex signature (2, 2n). Finally, the affine group ℝ^{2n+2}⋊Aut_{sd}(M) ⊂ Aff(2n+2, ℝ) acts on N by automorphisms of the pseudo hyper Kähler structure with an orbit of codimension one.

3. The pseudo hyper Kähler metric of the bundle of intermediate Jacobians over the moduli space of gauged Calabi Yau 3-folds

Let (V, ω, τ) be our fundamental algebraic data, s. p. 1, γ the Hermitian form defined in (1), V = L ⊕ L′ a Lagrangian splitting as in (2)-(4), M ⊂ (V, ω, γ) a Lagrangian pseudo Kähler submanifold in general position, s. Prop. 1.4, and, finally, Γ ⊂ V^{τ} a (cocompact) lattice. Using the isomorphism V/T_{m}M ≅ T_{m}^{∗}M induced by the symplectic form ω, we can identify T^{∗}M with the normal bundle N → M of M in V, N_{m} = V/T_{m}M. Since V^{τ} ⊃ Γ has zero intersection with T_{m}M, s. Prop. 1.1, Γ projects to a lattice [Γ] in N_{m} = V/T_{m}M and N_{m}/[Γ] is a complex torus. Let us denote by N /Γ the corresponding (holomorphic) torus bundle over M. Note that via the isomorphism N_{m} ≅ T_{m}^{∗}M the lattice [Γ] ⊂ N_{m} corresponds to the lattice ψ_{m}(Γ) ⊂ T_{m}^{∗}M, s. (6), and we can identify N /Γ with the quotient of T^{∗}M by the action of Γ ⊂ V^{τ} defined in equation (7).

Theorem 3.1. The pseudo hyper Kähler structure on T^{∗}M ≅ N constructed in Thm. 1.5 induces a pseudo hyper Kähler structure on the torus bundle T^{∗}M/Γ ≅ N /Γ for any lattice Γ ⊂ V^{τ}.

Proof: It was proven in Prop. 2.1 that the action of V^{τ} ⊃ Γ preserves the complex symplectic structure Ω and the pseudo hyper Kähler metric G on T^{∗}M. □

The purpose of this section is to use Thm. 3.1 for the construction of a pseudo hyper Kähler structure on the bundle of intermediate Jacobians over the moduli space of gauged Calabi Yau 3-folds. For this we have to review some known facts about the moduli space, thereby relating it to Thm. 3.1.

Let X be a (general) Calabi Yau 3-fold, i.e. a compact Kähler 3-fold with holonomy group SU(3). This implies that X has a holomorphic volume form vol_{X} ∈ H^{3,0}(X), unique up to scaling. Such a pair (X, vol_{X}) is called a gauged Calabi Yau 3-fold. The (Kuranishi) moduli space S of X is smooth and can be identified with a neighborhood of zero in H^{2,1}(X) ≅ H^{1}(X, T), s. [Bo], [Ti] and [To]. Denote by X = (X_{s})_{s∈S} → S, X_{0} = X, the corresponding deformation of complex structure. The “intersection” form

ω(ξ, η) := ∫_{X} ξ ∧ η , ξ, η ∈ H^{3}(X, ℤ) ,  (14)

defines an integral nondegenerate skew symmetric bilinear form on H^{3}(X, ℤ). The corresponding complex symplectic form on H^{3}(X, ℂ) will be denoted by the same letter. Consider the holomorphic line bundle H^{3,0}(X) → S with fibre H^{3,0}(X_{s}) at s ∈ S. Denote by H^{3,0}(X) − S the ℂ^{∗}-bundle over S which is obtained from the complex line bundle H^{3,0}(X) by removing the zero section S ∋ s ↦ 0 ∈ H^{3,0}(X_{s}). We think of it as moduli space of gauged Calabi Yau 3-folds (X_{s}, vol_{s}), vol_{s} ∈ H^{3,0}(X_{s}) − {0}, s ∈ S.

The holomorphic vector bundle H^{3}(X , ℂ) → S has a canonical flat connection defined by the lattice bundle H^{3}(X , ℤ) ⊂ H^{3}(X , ℂ), which is known as Gauß-Manin connection. Since the moduli space S is local, we can assume that S is simply connected and that the bundle H^{3}(X , ℂ) → S is trivial. In particular, we have canonical identifications H^{3}(X_{s}, ℤ) ≅ H^{3}(X, ℤ) and H^{3}(X_{s}, ℂ) ≅ H^{3}(X, ℂ). So we can define the period map

P er : S → P (H^{3}(X, ℂ)) , s ↦ H^{3,0}(X_{s}) .

It follows from Kodaira, Spencer and Griffiths’ deformation theory that

dP er(T_{s}S) = dπ(H^{3,0}(X_{s}) + H^{2,1}(X_{s})) ,

where π : H^{3}(X, ℂ) → P (H^{3}(X, ℂ)) is the canonical projection. This implies that the period map is an immersion and since S is local we can assume that P er : S → P er(S) ⊂ P (H^{3}(X, ℂ)) is an isomorphism. As a consequence, the cone M_{X} = ∪_{s∈S}P er(s) − {0} ⊂ H^{3}(X, ℂ) over P er(S) = P (M_{X}) is canonically identified with the moduli space H^{3,0}(X) − S of gauged Calabi Yau 3-folds. By the first Hodge-Riemann bilinear relations the tangent space

T_{u}M_{X} = H^{3,0}(X_{s}) + H^{2,1}(X_{s}) , u ∈ P er(s) − {0} ,

is a Lagrangian subspace of H^{3}(X, ℂ) with respect to the intersection form ω. Note that, using the standard real structure τ on H^{3}(X, ℂ) with fixed point set H^{3}(X, ℝ), we can define the Hermitian form γ of complex signature (n + 1, n + 1), n = h^{2,1}(X), on H^{3}(X, ℂ) by equation (1). The first and second Hodge-Riemann bilinear relations imply that M_{X} is a pseudo Kähler submanifold M_{X} ⊂ (H^{3}(X, ℂ), γ) of complex signature (1, n). More precisely, we have the relations: γ(u, u) > 0, γ(v, v) < 0 and γ(u, v) = 0 for all u ∈ H^{3,0}(X_{s}) − {0} and v ∈ H^{2,1}(X_{s}) − {0}. This motivates the following definition.

Definition 3.1. Given fundamental algebraic data (V, ω, τ) as on p. 1 and γ defined in (1), a Lagrangian pseudo Kähler submanifold M ⊂ (V, ω, γ) is called a formal moduli space (of gauged Calabi Yau 3-folds) if the following conditions are satisfied:

(i) M is a cone, i.e. λM = M for all λ ∈ ℂ − {0}. (ii) γ(u, u) > 0 for all u ∈ M. (iii) γ(v, v) < 0 for all 0 ≠ v ∈ T_{u}M such that γ(u, v) = 0.

To a Calabi Yau 3-fold X we have associated the following algebraic data: V = H^{3}(X, ℂ), ω the intersection form (14), τ the real structure with fixed point set V^{τ} = H^{3}(X, ℝ) and γ = \sqrt{−1}ω(·, τ·). With this understood the next proposition gives a summary of the preceding discussion.

Proposition 3.2. The cone M_{X} ⊂ (V, ω, γ) over the image P er(S) = P (M_{X}) of the period map is a formal moduli space of gauged Calabi Yau 3-folds in the sense of Def. 3.1.

Proposition 3.3. The class of Lagrangian pseudo Kähler cones defined in the previous section on p. 6 and classified in [dW-VP] and [C2] consists of formal moduli spaces of gauged Calabi Yau 3-folds.

Note that any statement which is true for formal moduli spaces M is true when M = M_{X}(≅ H^{3,0}(X) − S) is the (actual) moduli space of gauged Calabi Yau 3-folds associated to the Kuranishi moduli space S of a Calabi Yau 3-fold X. Remark 3: For any formal moduli space M ⊂ (V, ω, γ) there is a canonical Kähler metric −g^{P (M)} on P (M) known as special Kähler metric, cf. p. 7 and [C2], which can be defined by

g_{πu}^{P (M)}(dπv, dπv) = \frac{γ(v, v)}{γ(u, u)} − |\frac{γ(u, v)}{γ(u, u)}|^{2} ,  (15)

ON HYPER KÄHLER MANIFOLDS ASSOCIATED TO LAGRANGIAN SUBMANIFOLDS 11

for u ∈ M, v ∈ T_{u}M, where π : M → P (M) is the canonical projection. In the case of actual moduli spaces of gauged Calabi Yau 3-folds M = M_{X} the metric g^{P (M)} is known as Weil-Petersson metric. The formal moduli spaces M of Prop. 3.3 provide all the known examples of homogeneous special Kähler manifolds, i.e. with transitive isometry group. It is noteworthy that “most” of these examples are not Hermitian symmetric, s. [dW-VP] and [C2]. Homogeneous special Kähler manifolds of semisimple group have been completely classified, s. [A-C2]. We may ask the following natural question: Which of the homogeneous special Kähler manifolds can be realized as moduli spaces of Calabi Yau 3-folds (equipped with the Weil-Petersson metric)?

To round up our presentation, we place the concept of formal moduli space in the context of infinitesimal variations of Hodge structure, s. [G] and [B-G]. Given fundamental algebraic data (V, ω, τ) as on p. 1, a polarized Hodge structure of weight 3 on V, dimℂ V = 2n + 2, with Hodge numbers h^{3,0} = 1 and h^{2,1} = n is given by a decomposition into complex subspaces

V = H^{3,0} ⊕ H^{2,1} ⊕ H^{1,2} ⊕ H^{0,3} ,  (16)

such that H^{p,q} = τH^{q,p}, h^{p,q} = dimℂ H^{p,q} and satisfying the first and second Hodge-Riemann bilinear relations. Moreover, one assumes that a lattice Γ ⊂ V^{τ} is given and that ω restricts to an integral nondegenerate skew symmetric bilinear form on Γ. Let us denote by D the classifying space for such Hodge structures on V. Note that an element of D, i.e. a Hodge structure as above, is determined by the pair (H^{3,0}, H^{2,1}).

To any formal moduli space M ⊂ (V, ω, γ), s. Def. 3.1, we associate a map M → D:

u ↦ (H^{3,0}(u), H^{2,1}(u)) , H^{3,0}(u) := ℂu , H^{2,1}(u) := T_{u}M ∩ u^{⊥} ,

where the orthogonal complement ⊥ is to be taken with respect to the Hermitian form γ. Obviously this map factorizes to a map φ_{M} : P (M) → D. If we denote by p : D → P (V) the projection (H^{3,0}, H^{2,1}) ↦ H^{3,0} then p ∘ φ_{M} : P (M) → P (V) is the trivial inclusion. For any Hodge decomposition (16) one has also the corresponding Hodge filtration: F^{3} ⊂ F^{2} ⊂ F^{1} ⊂ F^{0} = V, F^{p} = ⊕_{p≤k≤3}H^{k,3−k}. Consider now a variation of Hodge structure u ↦ (H^{p,q}(u))_{p,q}, u ∈ U, and denote by (F^{p}(u))_{p} the corresponding Hodge filtrations. If the variation of Hodge structure arises from a local deformation of complex structure (X_{u})_{u∈U}, i.e. H^{p,q}(u) = H^{p,q}(X_{u}), then it must satisfy Griffiths’ infinitesimal period relations [G]

∂F^{p}(u) ⊂ F^{p−1}(u) .  (17)

This means that holomorphic partial derivatives of a (local) holomorphic section of the vector bundle (F^{p}(u))_{u∈U} are sections of (F^{p−1}(u))_{u∈U}. The next theorem follows from the work of Bryant and Griffiths [B-G].

Theorem 3.4. Let M ⊂ (V, ω, γ) be a formal moduli space, dimℂ V = 2n+2 and D the classifying space for Hodge structures as above. Then the map φ_{M} : P (M) → D is a solution to the differential system on D defined by the infinitesimal period relations (17). Conversely, any solution φ : U → D, U a complex n-fold, to this differential system for which p ∘ φ : U → P (V) is an immersion is locally of the form φ_{M}.

Let X be a Calabi Yau 3-fold, S its Kuranishi moduli space and M_{X} = ∪_{s∈S}P er(s)− {0} ⊂ H^{3}(X, ℂ) the cone over the image of the period map P er : S → P (H^{3}(X, ℂ)). Recall that M_{X} is the moduli space of gauged Calabi Yau 3-folds associated to X. The intermediate Jacobian of X_{s}, s ∈ S, is the complex torus

J (X_{s}) = \frac{H^{3}(X, ℂ)}{H^{3,0}(X_{s}) + H^{2,1}(X_{s}) + H^{3}(X, ℤ)} .

The bundle of intermediate Jacobians J → M_{X} over M_{X} is the holomorphic torus bundle whose fibre at u ∈ P er(s) − {0} ⊂ M_{X} is J_{u} = J (X_{s}).

Theorem 3.5. For any Lagrangian splitting H^{3}(X, ℝ) = L_{0} ⊕ L′_{0} such that M_{X} ⊂ H^{3}(X, ℂ) is in general position, there is a pseudo hyper Kähler structure of complex signature (2, 2n), n = h^{2,1}(X), on the bundle of intermediate Jacobians J → M_{X}.

Proof: By Prop. 3.2 the moduli space of gauged Calabi Yau 3-folds M_{X} is a formal moduli space. In particular, s. Def. 3.1, it is a Lagrangian pseudo Kähler submanifold of (H^{3}(X, ℂ), ω, γ), where ω is the intersection form (14) and γ = \sqrt{−1}ω(·, τ·) is defined with the help of the standard real structure τ, i.e. H^{3}(X, ℂ)^{τ} = H^{3}(X, ℝ). By Thm. 1.5, we can associate to these data together with the Lagrangian splitting of H^{3}(X, ℝ) a pseudo hyper Kähler structure on T^{∗}M_{X}. Using the intersection form ω we can identify the cotangent bundle T^{∗}M_{X} → M_{X} with the normal bundle N → M_{X} of M_{X} ⊂ H^{3}(X, ℂ). The normal bundle has fibre

N_{u} = \frac{H^{3}(X, ℂ)}{T_{u}M_{X}} = \frac{H^{3}(X, ℂ)}{H^{3,0}(X_{s}) + H^{2,1}(X_{s})}

at u ∈ P er(s) − {0} ⊂ M_{X} and J → M_{X} is precisely the torus bundle N /iΓ ≅ N /Γ → M considered in Thm. 3.1 with M = M_{X}, V = H^{3}(X, ℂ) and Γ = H^{3}(X, ℤ). Now the theorem is an immediate consequence of Thm. 3.1. □ Remark 4: As checked in the proof of Prop. 2.1, the standard complex symplectic structure Ω on T^{∗}M_{X} ≅ N is invariant under the action of the lattice Γ = H^{3}(X, ℤ) ⊂ H^{3}(X, ℝ) = V^{τ}. So it factorizes to a complex symplectic structure on J = N /Γ ≅ T^{∗}M/Γ independent of any choice of Lagrangian splitting H^{3}(X, ℝ) = L_{0} ⊕ L′_{0}. This is the parallel complex symplectic structure associated to the pseudo hyper Kähler structure of Thm. 3.5 and it coincides with the complex symplectic structure which was recently constructed by Donagi and Markman [D-M].

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Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany

  E-mail address: vicente@math.uni-bonn.de