December 12, 2001

Special Kähler manifolds: a survey

Vicente Cortés

Mathematisches Institut der Universität Bonn Beringstr. 1, 53115 Bonn, Germany vicente@math.uni-bonn.de

Abstract

This is a survey of recent contributions to the area of special Kähler geometry. It is based on lectures given at the 21st Winter School on Geometry and Physics held in Srni in January 2001.

Keywords: special Kähler manifolds MSC 2000: 53C26, 53C25, 53A15

1 Remarkable features of special Kähler manifolds

A (pseudo-) Kähler manifold (M, J, g) is a differentiable manifold endowed with a complex structure J and a (pseudo-) Riemannian metric g such that

(i) J is orthogonal with respect to the metric g, i.e. J^{∗}g = g and (ii) J is parallel with respect to the Levi Civita connection D, i.e. DJ = 0. In the following we will always allow pseudo-Riemannian, i.e. possibly indefinite metrics. The prefix “pseudo” will be generally omitted. The following definition is by now standard, see [F].

Definition 1 A special Kähler manifold (M, J, g, ∇) is a Kähler manifold (M, J, g) together with a flat torsionfree connection ∇ such that

(i) ∇ω = 0, where ω = g(·, J·) is the Kähler form and (ii) ∇J is symmetric, i.e. (∇_{X}J)Y = (∇_{Y} J)X for all vector fields X and Y.

More precisely, one should speak of affine special Kähler manifolds since there is also a projective variant of special Kähler manifolds. In fact, there is a class of (affine) special Kähler manifolds M, which are called conic special Kähler manifolds and which admit a certain ℂ^{∗}-action. The quotient of M by that action can be considered as projectivisation of M and is called a projective special Kähler manifold, see [ACD]. Originally [dWVP1], in the supergravity literature, by a special Kähler manifold one understood a projective special Kähler manifold. This terminology was mantained in the first mathematical papers on that subject [C1, C2, AC] and abandoned with the publication of [F].

Example 1: Let (M, J, g) be a flat Kähler manifold, i.e. the Levi Civita connection D is flat. Then (M, J, g, ∇ = D) is a special Kähler manifold and ∇J = 0. Conversely, any special Kähler manifold (M, J, g, ∇) such that ∇J = 0 satisfies ∇ = D = Levi Civita connection of the flat Kähler metric g. This is the trivial example of a special Kähler manifold.

Before giving a general construction of special Kähler manifolds, which yields plenty of non-flat examples, I would like to offer some motivation for that concept.

• The notion of special Kähler manifold was introduced by the physicists de Wit and Van Proeyen [dWVP1] and has its origin in certain supersymmetric field theories. More precisely, affine special Kähler manifolds are exactly the allowed targets for the scalars of the vector multiplets of field theories with N = 2 rigid supersymmetry on four-dimensional Minkowski spacetime. Projective special Kähler manifolds correspond to such theories with local supersymmetry, which describe N = 2 supergravity coupled to vector multiplets. N = 2 supergravity theories occur as low energy limits of type II superstrings and play a prominent role in the study of moduli spaces of certain two-dimensional superconformal field theories [CFG]. The structure of these moduli spaces is desribed as the product of a projective special Kähler manifold and a quaternionic Kähler manifold. Besides these strong physical motivations there is also a number of rather mathematical reasons to study special Kähler manifolds.

• Interesting moduli spaces carry the structure of a special Kähler manifold, for example:

– The (Kuranishi) moduli space M_{X} of gauged complex structures associated to a Calabi-Yau 3-fold X is a special Kähler manifold of complex signature (1, n), n = h^{2,1}(X). This fact can be found in the physical literature, see e.g. [S] and references therein. M_{X} parametrises pairs (J, vol ), where J is a complex structure and vol a J-holomorphic volume form on a given compact Calabi-Yau manifold of complex dimension 3. Let me recall that (from the Riemannian point of view) a Calabi-Yau n-fold is a Riemannian manifold with holonomy group SU(n). More generally, the affine cone over any abstract variation of polarized Hodge structure of weight 3 and with h^{3,0} = 1 is a (conic) special Kähler manifold, see [C2]. Such cones can be considered as formal moduli spaces, i.e. the underlying variation of Hodge structure is not necessarily induced by the deformation of complex structure of some Kähler manifold.

– The moduli space of deformations of a compact complex Lagrangian submanifold Y in a hyper-Kähler manifold X is a special Kähler manifold with positive definite metric [H1]. A hyper-Kähler manifold is a Riemannian manifold with holonomy group in Sp(n). Such a manifold X is automatically Kähler of complex dimension 2n and carries a holomorphic symplectic structure Ω. A complex submanifold Y ⊂ X of complex dimension n is called Lagrangian if ι^{∗}Ω = 0, where ι : Y → X is the inclusion map.

• The cotangent bundle of any special Kähler manifold carries the structure of a hyper-Kähler manifold. This corresponds to the dimensional reduction of N = 2 supersymmetric theories from four to three spacetime dimensions [CFG]. This construction, which is called the c-map in rigid supersymmetry, is discussed, applied and generalised in the mathematical literature [C2, F, H1, ACD]. For example, it is used in [C2] to obtain a hyper-Kähler structure (of complex signature (2, 2n)) on the bundle J → M_{X} of intermediate Jacobians over the above moduli space M_{X} associated to a Calabi-Yau 3-fold X. The fibre of the holomorphic bundle J over (J, vol ) ∈ M_{X} is the intermediate Jacobian

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

of (X, J).

• There is also a c-map in local supersymmetry, i.e. in supergravity, which to any projective special Kähler manifold of (real) dimension 2n associates a quaternionic Kähler manifold of dimension 4n+4 [FS]. It corresponds to the dimensional reduction of N = 2 supergravity coupled to vector multiplets from dimension four to three. For mathematical discussions of this deep construction, see [H3, K].

• The base of any algebraic completely integrable system is a special Kähler manifold, see [DW, F]. An algebraic completely integrable system is a holomorphic submersion π : X → M from a complex symplectic manifold X to a complex manifold M with compact Lagrangian fibres and a smooth choice of polarisation on the fibres. This is essentially the inverse construction of the rigid c-map. There should also exist an inverse construction for the local c-map.

• It was shown in [ACD] that the notion of special Kähler manifold has natural generalisations in the absence of a metric: “special complex” and “special symplectic” manifolds. The cotangent bundle of such manifolds carries interesting geometric structures which generalise the hyper-Kähler structure on the cotangent bundle of a special Kähler manifold. Special complex geometry (in the absence of a metric) may provide insight in physical theories for which no Lagrangian formulation (and for that reason no target metric) is available.

• There is a close relation between special Kähler manifolds and affine differential geometry discovered in [BC1]. In fact, any simply connected special Kähler manifold has a canonical realisation as a parabolic affine hypersphere. This will be explained in detail in section 3.

• Any projective special Kähler manifold has a canonical (pseudo-) Sasakian circle bundle which is realised as a proper affine hypersphere [BC3].

• It was discovered in [BC2] that special Kähler manifolds with a flat indefinite metric have a nontrivial moduli space, which is closely related to the moduli space of Abelian simply transitive affine groups of symplectic type.

• Homogeneous projective special Kähler manifolds were classified, under various assumptions in [dWVP2, C1, AC]. Under the c-map they give rise to homogeneous quaternionic Kähler manifolds. If one restricts attention to the homogeneous projective special Kähler manifolds of semi-simple group, then one finds a list of Hermitian symmetric spaces of non-compact type which shows a remarkable coincidence with the list of irreducible special holonomy groups of torsionfree symplectic connections [MS], as was noticed in [AC]. Finally, the classification [AC] may lead to the generalisation of recent ideas of Hitchin about special features of geometry in six dimensions to other dimensions [H2].

2 The construction of special Kähler manifolds

In this section we will see that the equations defining special Kähler manifolds are completely integrable, in the sense that the general local solution can be obtained from a free holomorphic potential. The discussion follows [ACD] and is based on the extrinsic approach to special Kähler manifolds developped in [C2]. For a similar discussion from the bi-Lagrangian point of view see [H1].

The ambient data in the extrinsic approach are the following: The complex symplectic vector space V = T^{∗}ℂ^{n} = ℂ^{2n} with canonical coordinates (z^{1}, …, z^{n}, w_{1}, …, w_{n}). In these coordinates the symplectic form is

Ω = ∑_{i=1}^{n} dz^{i} ∧ dw_{i} .

We denote by τ : V → V the complex conjugation with respect to V^{τ} = T^{∗}ℝ^{n} = ℝ^{2n}. The algebraic data (V, Ω, τ) induce on V the Hermitian form

γ := \sqrt{−1}Ω(·, τ·)

of complex signature (n, n).

Let M be a connected complex manifold of complex dimension n. We denote its complex structure by J.

Definition 2 A holomorphic immersion ϕ : M → V is called Kählerian if ϕ^{∗}γ is nondegenerate and it is called Lagrangian if ϕ^{∗}Ω = 0.

A Kählerian immersion ϕ : M → V induces on M the pseudo-Riemannian metric g = Re ϕ^{∗}γ such that (M, J, g) is a Kähler manifold.

Lemma 1 Let ϕ : M → V be a Kählerian Lagrangian immersion. Then the Kähler form ω = g(·, J·) of the Kähler manifold (M, J, g) is given by

ω = 2 ∑_{i=1}^{n} dx̃^{i} ∧ dỹ_{i} ,

where x̃^{i} := Re ϕ^{∗}z^{i} and ỹ_{i} := Re ϕ^{∗}w_{i}.

Proof: The metric g_{V} := Re γ is a flat Kähler metric of (real) signature (2n, 2n) on the complex vector space (V, J). Its Kähler form is

ω_{V} := ∑(dx^{i} ∧ dy_{i} + du^{i} ∧ dv_{i}) ,

where x^{i} := Re z^{i}, y_{i} := Re w_{i}, u^{i} := Im z^{i} and v_{i} := Im w_{i}. On the other hand, the two-form

Re Ω = ∑(dx^{i} ∧ dy_{i} − du^{i} ∧ dv_{i})

vanishes on M. This shows that

ω = ϕ^{∗}ω_{V} = 2 ∑ ϕ^{∗}(dx^{i} ∧ dy_{i}) = 2 ∑ dx̃^{i} ∧ dỹ_{i} .

The lemma implies that the functions x̃^{1}, …, x̃^{n}, ỹ_{1}, …, ỹ_{n} define local coordinates near each point of M. Therefore we can define a flat torsionfree connection ∇ on M by the condition ∇dx̃^{i} = ∇dỹ_{i} = 0, i = 1, …, n. Now we can formulate the following fundamental theorem.

Theorem 1 Let ϕ : M → V be a Kählerian Lagrangian immersion with induced geometric data (g, ∇). Then (M, J, g, ∇) is a special Kähler manifold. Conversely, any simply connected special Kähler manifold (M, J, g, ∇) admits a Kählerian Lagrangian immersion ϕ : M → V inducing the data (g, ∇) on M. The Kählerian Lagrangian immersion ϕ is unique up to an affine transformation of V = ℂ^{2n} with linear part in Sp(ℝ^{2n}).

For the proof of that result and its projective version see [ACD], where analogous extrinsic characterisations are obtained also for special complex and special symplectic manifolds.

The above theorem may be considered as an extrinsic reformulation of the intrinsic Definition 1. The important advantage of the extrinsic characterisation in terms of Kählerian Lagrangian immersions lies in the well known fact that Lagrangian immersions are locally defined by a generating function. More precisely, any holomorphic Lagrangian immersion into (V, Ω) is locally defined by a holomorphic function F (z^{1}, …, z^{n}), at least after suitable choice of canonical coordinates (z^{1}, …, z^{n}, w_{1}, …, w_{n}). In fact, such a function defines a Lagrangian local section ϕ = dF of T^{∗}ℂ^{n} = V. It is a Kählerian Lagrangian immersion if it satisfies the nondegeneracy condition det Im ∂^{2}F ≠ 0. Similarly, projective special Kähler manifolds are locally defined by a holomorphic function F satisfying a nondegeneracy condition and which in addition is homogeneous of degree 2.

3 Special Kähler manifolds as affine hyperspheres

The main object of affine differential geometry are hypersurfaces in affine space ℝ^{m+1} with its standard connection denoted by ˜∇ and parallel volume form vol. A hypersurface is given by an immersion φ : M → ℝ^{m+1} of an m-dimensional connected manifold. We assume that M admits a transversal vector field ξ and that m > 1. This induces on M the volume form ν = vol(ξ, …), a torsionfree connection ∇, a quadratic covariant tensor field g, an endomorphism field S (shape tensor) and a one-form θ such that

˜∇_{X}Y = ∇_{X}Y + g(X, Y)ξ ,
˜∇_{X}ξ = SX + θ(X)ξ .

We will assume that g is nondegenerate and, hence, is a pseudo-Riemannian metric on M. This condition does not depend on the choice of ξ. According to Blaschke [B], once the orientation of M is fixed, there is a unique choice of transversal vector field ξ such that ν coincides with the metric volume form vol^{g} and ∇ν = 0. This particular choice of transversal vector field is called the affine normal and the corresponding geometric data (g, ∇) are called the Blaschke data. Notice that, for the affine normal, θ = 0 and S is computable from (g, ∇) (Gauß equations). Henceforth we use always the affine normal as transversal vector field.

Definition 3 The hypersurface φ : M → ℝ^{m+1} is called a parabolic (or improper) hypersphere if the affine normal is parallel, ˜∇ξ = 0. It is called a proper hypersphere if the lines generated by the affine normals intersect in a point p ∈ ℝ^{m+1}, which is called the centre. For parabolic hyperspheres the centre is at ∞.

Notice that ˜∇ξ = 0 ⇔ S = 0 ⇔ ∇ is flat. For proper hyperspheres S = λId, λ ∈ ℝ − {0}.

The main result of [BC1] is the following:

Theorem 2 Let (M, J, g, ∇) be a simply connected special Kähler manifold. Then there exists a parabolic hypersphere φ : M → ℝ^{m+1}, m = dim_{ℝ} M = 2n, with Blaschke data (g, ∇). The immersion φ is unique up to a unimodular affine transformation of ℝ^{m+1}.

The proof of Theorem 2 makes use of the Fundamental Theorem of affine differential geometry [DNV], which is the generalisation of Radon’s theorem [R] to higher dimensions:

Theorem 3 Let (M, g, ∇) be a simply connected oriented pseudo-Riemannian manifold with a torsionfree connection ∇ such that the Riemannian volume form vol^{g} is ∇-parallel. Then there exists an immersion φ : M → ℝ^{m+1} with Blaschke data (g, ∇) if and only if the g-conjugate connection \overline{∇} is torsionfree and projectively flat. The immersion is unique up to unimodular affine transformations of ℝ^{m+1}.

Recall that the g-conjugate connection \overline{∇} on M is defined by the equation:

Xg(Y, Z) = g(∇_{X}Y, Z) + g(Y, \overline{∇}_{X}Z) for all vector fields X, Y, Z .

Proof: (of Theorem 2) Let (M, J, g, ∇) be a simply connected special Kähler manifold. For any Kähler manifold we have vol^{g} = \frac{ω^{n}}{n!} , n = dim_{ℂ} M. This implies the first integrability condition ∇vol^{g} = 0, since ∇ω = 0. The conjugate connection is torsionfree and flat. This will follow from:

Lemma 2 [BC1] Let (M, J, g, ∇) be a special Kähler manifold. Then \overline{∇} = ∇^{J}, where the connection ∇^{J} is defined by

∇_{X}^{J}Y := J∇_{X}(J^{−1}Y) = −J∇_{X}(JY) = ∇_{X}Y − J(∇_{X}J)Y

for all vector fields X and Y.

From the formula defining ∇^{J} we see that a vector field X is ∇-parallel if and only if JX is ∇^{J}-parallel. Therefore, if X_{i}, i = 1, …, 2n = dim_{ℝ} M, is a parallel local frame for the flat connection ∇ then JX_{i}, i = 1, …, 2n, is a ∇^{J}-parallel local frame. This shows that \overline{∇} = ∇^{J} is flat. Similarly the torsionfreedom of \overline{∇} = ∇^{J} = ∇ − J∇J follows from that of ∇ and the symmetry of the tensor ∇J. So the assumptions of Theorem 3 are satisfied and we conclude the existence of a hypersurface φ : M → ℝ^{m+1}, m = 2n. inducing on M the Blaschke data (g, ∇). Now the flatness of ∇ implies that φ is a parabolic hypersphere.

As applications we obtain:

Corollary 1 Any holomorphic function F on a simply connected open set U ⊂ ℂ^{n} with det Im ∂^{2}F ≠ 0 defines a parabolic hypersphere of dimension m = 2n.

This is a generalisation of a classical theorem of Blaschke about 2-dimensional parabolic spheres. An explicit representation formula for the parabolic hypersphere in terms of the holomorphic function F was given in [C3].

Corollary 2 Let (M, ∇, g) be a special Kähler manifold with (positive) definite metric g. If g is complete then ∇ is the Levi Civita connection and g is flat.

Proof: This follows by combining Theorem 2 with the following classical theorem of Calabi and Pogorelov [Ca, P].

Theorem 4 If the Blaschke metric g of a parabolic affine hypersphere (M, g, ∇) is definite and complete, then M is affinely congruent to the paraboloid x^{m+1} = ∑_{i=1}^{m}(x^{i})^{2} in ℝ^{m+1}. In particular, ∇ is the Levi Civita connection and g is flat.

Lu [L] proved that any special Kähler manifold (M, g, J, ∇) with a definite and complete metric g is flat without making use of Calabi and Pogorelov’s Theorem. Special Kähler manifolds (M, J, g, ∇) with a flat indefinite and geodesically complete metric g for which ∇ is complete and is not the Levi Civita connection were constructed in [BC2].

For projective special Kähler manifolds \overline{M} it has been established in [BC3] that a natural circle bundle S → \overline{M} can be canonically realised as a proper hypersphere. Moreover, the metric cone over S is a conic special Kähler manifold M, which is in turn realised as a parabolic hypersphere in a compatible way. There are also projective analogues of Corollaries 1 and 2.

References

[ACD] D. V. Alekseevsky, V. Cortés, C. Devchand, Special complex manifolds, to appear in J. Geom. Phys., available as math.DG/9910091

[AC] D. V. Alekseevsky, V. Cortés, Classification of stationary compact homogeneous special pseudo Kähler manifolds of semisimple group, Proc. London Math. Soc.

(3) 81 (2000), 211-230

[BC1] O. Baues and V. Cortés, Realisation of special Kähler manifolds as parabolic spheres, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2403-2407

[BC2] O. Baues and V. Cortés, Abelian simply transitive affine groups of symplectic type, preprint 2001-39, Max-Planck-Institut für Mathematik, Bonn, math.DG/0105025

[BC3] O. Baues and V. Cortés, Proper affine hyperspheres which fiber over projective special Kähler manifolds (in preparation)

[B] W. Blaschke, Vorlesungen über Differentialgeometrie II. Affine Differentialgeometrie, Grundlehren der Mathematischen Wissenschaften VII, Springer Verlag, Berlin 1923

[Ca] E. Calabi, Improper affine hypersurfaces of convex type and a generalization of a theorem of Jörgens, Michigan Math. J. 5 (1958), 105-126

[CFG] S. Cecotti, S. Ferrara and L. Girardello, Geometry of type II superstrings and the moduli space of superconformal field theories, Int. J. Mod. Phys. A4 (1989), 2475-2529

[C1] V. Cortés, Homogeneous special geometry, Transform. Groups 1 (1996), no. 4, 337-373.

[C2] V. Cortés, On hyper-Kähler manifolds associated to Lagrangian Kähler submanifolds of T^{∗}C^{n}, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3193-3205

[C3] V. Cortés, A holomorphic representation formula for parabolic hyperspheres, to appear in the Proceedings of the international conference ”PDEs, Submanifolds and Affine Differential Geometry”, Warsaw, September 2000, as publication of the Stefan Banach International Mathematical Center, math.DG/0107037

[dWVP1] B. de Wit and A. Van Proeyen, Potentials and symmetries of general gauged N=2 supergravity–Yang-Mills models, Nucl. Phys. B245 (1984), 89-117

[dWVP2] B. de Wit and A. Van Proeyen, Special geometry, cubic polynomials and homogeneous quaternionic spaces, Commun. Math. Phys. 149 (1992), 307-333 [DNV] F. Dillen, K. Nomizu and L. Vrancken, Conjugate connections and Radon’s

theorem in affine differential geometry, Monatsh. Math. 109 (1990), 221-235

[DW] R. Y. Donagi and E. Witten, Supersymmetric Yang–Mills theory and integrable systems, Nucl. Phys. B460 (1996), 299-334

[FS] S. Ferrara, S. Sabharwal, Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces, Nucl. Phys. B332 (1990), 317-332

[F] D. S. Freed, Special Kähler manifolds, Commun. Math. Phys. 203 (1999), no. 1, 31-52

[H1] N. J. Hitchin, The moduli space of complex Lagrangian submanifolds, Asian J. Math. 3 (1999), no. 1, 77-91

[H2] N. J. Hitchin, Generalized complex structures in 6 dimensions, in Arbeitstagung 2001, preprint 2001 (50), Max-Planck-Institut für Mathematik, Bonn

[H3] N. J. Hitchin, The quaternionic Kähler manifold of Ferrara and Sabharwal, manuscript

[K] M. Krahe, Diplomarbeit, Universität Bonn, 2001

[L] Z. Lu, A note on special Kähler manifolds, Math. Ann. 313 (1999), no. 4, 711-713

[MS] S. Merkulov, L. Schwachhöfer, Classification of irreducible holonomies of torsion-free affine connections, Ann. of Math. 150 (1999), no. 3, 77-149

[P] A. V. Pogorelov, On the improper affine hypersurfaces, Geom. Dedicata 1 (1972), 33-46

[R] J. Radon, Die Grundgleichungen der affinen Flächentheorie, Leipziger Berichte 70 (1918), 91-107

[S] A. Strominger, Special geometry, Comm. Math. Phys. 133 (1990), no. 1, 163-180