Construction of projective special K\"ahler manifolds

In this paper we present an intrinsic characterisation of projective special K\"ahler manifolds in terms of a symmetric tensor satisfying certain differential and algebraic conditions. We show that this tensor vanishes precisely when the structure is locally isomorphic to a standard projective special K\"ahler structure on $\SU(n,1)/S(\U(n)\U(1))$. We use this characterisation to classify 4-dimensional projective special K\"ahler Lie groups.


Introduction
Projective special Kähler manifolds are a special class of Kähler quotients of conic special Kähler manifolds which is a class of pseudo-Kähler manifolds endowed with a symplectic, flat, torsion-free connection and an infinitesimal homothety.
Explicit examples can be found in [1], where homogeneous projective special Kähler manifolds of semisimple Lie groups are classified. A notable case appearing in this list is the complex hyperbolic n-space. Many projective special Kähler manifolds can be constructed via the so called r-map [8], which is a construction arising from supergravity and string theory allowing to build a projective special Kähler manifold starting from a homogeneous cubic polynomial. See [6] for a classification of 6-dimensional manifolds that can be constructed via the r-map. Another example is obtained by taking the Weil-Petersson metric on the space of complex structure deformations on a Calabi-Yau 3-dimensional manifold [5].
Projective special Kähler manifolds appear in the study of supergravity and mirror symmetry with the name local special Kähler manifolds (see [10] and [11] for more details on their story and applications to physics, and in particular [4] for their importance in mirror symmetry). The name projective special Kähler was given by Freed in [11] where he also shows how such manifolds are quotients

Definitions
In this section we are introducing the basic objects that we are going to discuss in this work.
The coming definition involves a flat connection ∇ and its exterior covariant derivative operator d ∇ .
Definition 2.4. A projective special Kähler manifold is a Kähler manifold M endowed with a C * -bundle π : M → M with ( M , g, I, ω, ∇, ξ) conic special Kähler such that ξ and Iξ are the fundamental vector fields associated to 1, i ∈ C respectively and M is the Kähler quotient with respect to the induced U(1)action. In this case we say that M has a projective special Kähler structure.
For brevity, we will often denote a projective special Kähler manifold by (π : M → M, ∇).
Remark 2.5. We shall see later that by construction, the action is always Hamiltonian with moment map − g(ξ, ξ), and the choice of the level set affects the quotient only up to scaling.
Concerning the notation for projective special Kähler manifolds as in Definition 2.4, when a tensor or a connection is possessed by both M and M , we will write them and everything concerning them (torsion, curvature forms, covariant exterior differentials) on M with (·) above, whereas the corresponding objects on M will be denoted without it.

Difference tensor
This section is devoted to the tensor obtained as difference between the flat and Levi-Civita connection on a conic special Kähler manifold. We present the known symmetry of this tensor and write the flatness condition in terms of it [11, p.9-11].
Let ( M , g, I, ω, ∇, ξ) be a conic special Kähler manifold of dimension n + 1. We define η as the (1,2)-tensor such that for all vector fields X, Y on M we have η X Y = ∇ X Y − ∇ LC X Y , where the employed notation η X Y means η(X, Y ). Consider frames adapted to the pseudo-Kähler structure, hence such that the linear model is (R 2n+2 , g 0 , I 0 , ω 0 ), where g 0 = 2k k=1 (e k ) 2 − (e 2n+1 ) 2 − (e 2n+2 ) 2 , Ie 2k−1 = e 2k for k = 1, . . . , n + 1 and ω 0 = g 0 (I 0 ·, ·). Let ω ∇ and ω LC be the connection forms corresponding respectively to the flat and the Levi-Civita connections represented with respect to an adapted frame. Thus we have Since both connections are symplectic, the corresponding forms, and thus η, will have values in sp(2n + 2, R) which can be described as where A t is the transposed of A with respect to g 0 , that is such that g 0 (AX, Y ) = g 0 (X, A t Y ). It follows that η corresponds to a section of T * ⊗sp(2n+2, R) where T is the standard real representation of U(n, 1). Throughout this section we will use the following notation found in [18]: if V is a complex representation with a real structure σ, we define where V is the conjugate representation of V .
In particular, the the following complex Lie algebra isomorphisms hold: The Lie algebra sp(2n + 2, R) is closed with respect to transposition, and thus it is also closed with respect to symmetrisation and antisymmetrisation. As a consequence, we have the following splitting: sp(2n + 2, R) = (sp(2n + 2, R) ∩ sym(2n, 2)) ⊕ (sp(2n + 2, R) ∩ so(2n, 2)) The first summand consists of symmetric matrices A ∈ gl(2n + 2, R) such that 0 = A t I 0 + I 0 A = AI 0 + I 0 A and thus, as complex endomorphisms, its elements are all the real, C-antilinear and symmetric ones and therefore it is [[S 2,0 ]]. By contrast, the second summand is u(n, 1), which is isomorphic to [Λ 1,1 ]. As a u(n, 1)-representation, the subspace containing η is then isomorphic to  Both the Levi-Civita and the flat connection are torsion-free, therefore η must be symmetric in the two covariant indices. We already know that A( η S ) vanishes thanks to the previous condition and moreover, (1) is injective (actually an isomorphism) when restricted to T * ⊗so(2n+2, R), so A( η A ) = 0 if and only if η A = 0. The torsion-free condition is then equivalent to η = η S , so in conclusion, η is in the irreducible component isomorphic to [[S 3,0 ]]. The isomorphism is constructed with the musical isomorphisms ♭ and ♯ corresponding to the metric; explicitly, it is a restriction of We have then proven Lemma 3.1. On a conic special Kähler manifold ( M , g, I, ω), the tensor η is a Notice that in the process we have also proven Using the flatness of ∇, we observe: where Ω LC and d LC are respectively the curvature and exterior covariant derivative of the Levi-Civita connection on M .
and with a connection ∇ with connection form ω ∇ = ω LC + η, then Proof. The Levi-Civita connection form takes values in u(n, 1), so Ω LC is of type S 2 (u(n, 1)) and therefore, if ♭ is the map lowering the contravariant index, we get that ♭ Ω LC belongs to Ω 2 ( M ,

Conic and projective special Kähler metrics
In this section we will consider the case of a projective special Kähler manifold (π : M → M, ∇) and we will give the explicit relation between the metric on M and the one on M (see e.g. [7, Section 1.1]).
The mapping π : M → M is a C * -principal bundle with infinitesimal principal action generated by ξ and Iξ. We can always build the function r = − g(ξ, ξ) : M → R + and define S = r −1 (1) ⊆ M with inclusion map ι S : S ֒→ M . Now r has no critical points, since and g is non-degenerate. It follows that S is a submanifold of dimension 2n + 1 whose tangent bundle corresponds to ker(dr) ⊂ T M. Notice that dr( Iξ) = − g( Iξ,ξ) r = − ω(ξ,ξ) r = 0, so Iξ is a vector field tangent to S and it induces a principal U(1)-action. The induced metric on S is g S = ι * S g and thus L Iξ g S = ι * S L Iξ g = 0.
The principal action of C * on M induces by inclusion an R + -action, and in addition we have Lemma 4.1. The map r : M → R + is degree 1 homogeneous with respect to the action of R + ⊆ C * on M , i.e. for all s ∈ R + and p ∈ M r(ps) = r(p)s As a consequence of this lemma, we can now define a retraction It is well defined, since r(p(u)) = r(u 1 r(u) ) = r(u) r(u) = 1. Moreover, pι S = id S implies the surjectivity of p, which allows us to see p : M → S as a principal R + -bundle and π S := πι S : S → M as a principal S 1 -bundle; the composition of the two gives π. Lemma 4.2. If (π : M → M, ∇) is projective special Kähler, then M is diffeomorphic to S × R + , and moreover g = r 2 p * g S − dr 2 Proof. Let a : S × R + → M be the restriction of the principal right action M × R + → M to S × R + and consider also (p, r) : M → S × R + . These maps are smooth and each an inverse to the other, in fact if u ∈ M , a(p, r)(u) = a(p(u), r(u)) = u 1 r(u) r(u) = u and for all (q, s) ∈ S × R + , (π S , r)a(q, s) = (p(qs), r(qs)) = (q s r(qs) , r(q)s) = (q, s). For the second statement consider the symmetric tensor We want to prove it is basic, that is horizontal and invariant with respect to the principal R + -action.
Since there is only one vertical direction, and since g ′ is symmetric, it is enough to check whether g ′ vanishes when evaluated on the fundamental vector field ξ in one component. Using (3) we obtain And now for the R + -invariance: Therefore g ′ is basic, which in turn implies it is of the form p * g ′′ for some tensor g ′′ ∈ T 2 S, so that g = r 2 p * g ′′ − dr 2 The proof is ended by the following observation: The C * -bundle π : M → M has a unique principal connection orthogonal to the fibres with respect to g; the connection form can be written as Explicitly, we can describe ϕ using the metric: If we restrict it to S, we obtain a connection form ϕ = ι * S ϕ = −ι * S (ι ξ ω) corresponding to the S 1 -action on S.
Notice that p * ϕ = ϕ, because the connection form (4) is right-invariant, so ϕ = p * ϕ ′ for some ϕ ′ , and thus ϕ = ι * S ϕ = ι * S p * ϕ ′ = (pι S ) * ϕ ′ = ϕ ′ . The moment map for the action generated by Iξ is µ : 2 , so up to an additive constant, we can assume is a level set of the moment map and M is the Kähler quotient, π S : S → M is a pseudo-Riemannian submersion and thus we can write g S = π * S g − ϕ 2 .
Proof. From the previous arguments For the Kähler form it is enough to notice that π is holomorphic, M being a Kähler quotient, and that For future reference we give the following S ω M in fact, the restriction to S of ω maps fixes r = 1 and thus kills dr.
It will be useful to compute also

Lifting the coframe
The purpose of this section is to lift a generic unitary coframe on a projective special Kähler manifold to one on the corresponding conic special Kähler. This will enable us to give a more explicit formulation of the Levi-Civita connection and associated curvature tensor on the conic special Kähler manifold. In our convention, on a Kähler manifold (M, g, I, ω), the hermitian form is h = g + iω. Given a projective special Kähler manifold (π : M → M, ∇), and an open subset U ⊆ M , consider a unitary coframe θ = (θ 1 , . . . , θ n ) ∈ Ω 1 (U, C n ) on M , then we can build a coframe θ ∈ Ω 1 (π −1 (U ), C n+1 ) on M as follows: This coframe is compatible with the U(n, 1)-structure because it takes complex values and We will denote the dual frame to a given coframe by the same symbol, but with lower indices.
Remark 5.1. Given a connection on a Kähler manifold, it can be represented by a connection form ω with values in u(n, 1) whose complexification is gl(n + 1, C) ∼ = T 1,0 ⊗ T 1,0 ⊕ T 0,1 ⊗ T 1,0 , so we obtain projections in each component, respectively ω 1,0 1,0 and ω 0,1 0,1 such that ω = ω 1,0 1,0 + ω 0,1 0,1 . Notice that ω 0,1 0,1 = ω 1,0 because ω comes from a real representation and to give the first component is equivalent to give the whole form. Notice also that ([[T ]], I), as complex representation, is isomorphic to T 1,0 and the component A 1,0 1,0 of an endomorphism A gives the corresponding endomorphism of T 1,0 . We will often present connection forms by giving only the T 1,0 1,0 component. We will call R the projection from the complex tensor algebra to the real representation, defined so that R(α) = α + α where the conjugate is the real structure.
and its curvature form is Proof. The connection form (6) is metric if and only if the matrix is antihermitian with respect to g and since ω LC is antihermitian with respect to g, we get The torsion form of this connection is Θ LC = d θ + ω LC ∧ θ, so for 1 ≤ k ≤ n In the last component ω LC is metric and torsion-free, therefore by uniqueness it must be the Levi-Civita connection.
Let us now compute its curvature form Ω LC = d ω LC + ω LC ∧ ω LC . For 1 ≤ k, h ≤ n we have Since the curvature form must also be antihermitian, we also get Finally, is a curvature tensor of the complex projective space of dimension n; in fact, Ω P n C is the curvature with respect to the Fubini-Study metric (see for example [16,II,p.277]). In order to verify that Ω P n C is exactly the curvature of the Fubini-Study rather than a multiple, we compute the Ricci tensor: Then, scal P n C = 2(n + 1) Thus Ω P n C corresponds exactly to the curvature of P n C with the Fubini-Study metric. Now, whenever we have a smooth map f : M → N between Riemannian manifolds, we can extend the pull-back f * : T • N → T • M on the covariant tensor algebra to the whole tensor algebra, using the musical isomorphisms in each contravariant component. Explicitly, for X vector field on N , we define f * X := ♯f * ♭X = (f * X ♭ ) ♯ . Notice that this extension of the pull-back is still functorial, since if f : M → N , g : N → L are smooth maps, then f * g * X = ♯f * ♭♯g * ♭X = ♯f * g * ♭X = ♯(gf ) * ♭X = (gf ) * X.
Since M and M are Riemannian manifolds, we have π * : T • • M → T • • M , and in particular, for 1 ≤ k ≤ n we have Remark 5.4. In this notation,

Deviance
In this section we will continue the analysis of the tensor η started in section 3.
The aim is to reduce it to a locally defined tensor on M that we call deviance. We will then use it to give an explicit local description of the Ricci tensor and the scalar curvature.
Lowering the contravariant index of the curvature form, for Z ∈ X M , thanks to the symmetries of the Riemannian tensor we obtain proving that Ω LC (ξ, X)Y = 0, which implies the statement.

As before
Proceeding as in the previous point This quantity is zero as shown in the previous point, so it follows that L Iξ η = ∇ I, so (2) ends the proof.
We can now use a coframe θ as in section 5 in order to progress in the study of η. We then write η = R( η j k,h θ k ⊗ θ j ⊗ θ h ). Since every operator we use is C-linear, we can study only the component in Because of Lemma 6.1, the coefficients η j k,h vanish if any one of the indices is n + 1; moreover, η j k,h is completely symmetric in its indices. The last statement follows from the fact that ♭ 2 η is a tensor in π * S 3,0 M , and such tensors are expressed using only π * θ k for 1 ≤ k ≤ n, where the metric is positive definite, and thus ♭ 2 does not change the signs of the coefficients of η.
We are now ready to reduce η to an object defined locally on the base space. Proposition 6.3. Given a projective special Kähler (π : M → M, ∇) and a section s : Proof. For every point p ∈ M we can find a local unitary coframe θ defined on an open set containing p, and the corresponding coframe θ on M as in (5).
For the coming arguments we first compute the following Lie derivatives Independent components must vanish, so we obtain a family of differential equa- We define η, as the component in Notice that since πs = id M , the pullbacks satisfy s * π * = id T • • M , so And this time, Let now z : π −1 (U ) → C * be as in the statement, then in particular for all Definition 6.4. Given a section s : U → S with U open subset of M , we will call the corresponding tensor η found in Proposition 6.3 the deviance tensor with respect to s.
We can give a more global formulation of Proposition 6.3 in the following terms Proposition 6.5. Given a projective special Kähler manifold (π : Proof. Let u ∈ M , then there exists an open neighbourhood U ⊆ M of u and local trivialisation (π| π −1 (U) , z) : π −1 (U ) → U × C * induced by a section s : U → S so, for all w ∈ π −1 (U ) we have w = s(π(w))z(w). Let now η : U → S 3,0 M be the deviance corresponding to s; we define γ(u) := z(u) 2 η(p) where p = π(u). This definition is independent on the choice of s. In order to prove it take another s ′ : U ′ → S with p ∈ U ′ and the corresponding z ′ and η ′ , then, on U ∩ U ′ , there is a map c := z • s ′ : U ∩ U ′ → C whose image is in S 1 , as both s and s ′ are sections of S. By definition, We can define the homomorphism L ⊗ L → ♯ 2 S 3,0 M locally: given a section s : . This map does not depend on the choice of the section as one can see from the relations above, and it is also independent on the representatives chosen of these classes; for the first class for example z(ua)w = z(u)aw.
This map commutes with the projections on M and it is C-linear on the fibres, so it is a complex vector bundle map. Definition 6.6. We call γ : S → ♯ 2 S 3,0 M of Proposition 6.5 the intrinsic deviance of the projective special Kähler manifold.
Remark 6.7. Given a section s : U → S and the corresponding function z ∈ C ∞ π −1 (U ), C * such that sz = id π −1 (U) , we can compute dz = z( 1 r dr + idϑ), since locally z = re iϑ . Notice that ϑ is not globally defined on π −1 (U ), but dϑ and e iϑ are. Moreover, is a principal connection form, in fact it is equivariant for the action of C * as z(ua) = az(u) for all a ∈ C and, given a complex number a and its corresponding fundamental vector field a * ∈ X M , and thus on π −1 S (U ): ϕ = dϑ| S + π * S τ If we consider in fact the form ϕ−dϑ, we notice that it is basic, as it can also be seen as the difference of two connection forms on π −1 (U ) (namely (4) and (9)) up to a multiplication by i. Therefore, ϕ − dϑ = π * τ for some τ ∈ Ω 1 (U ). The second equation is simply obtained from the first by restriction to S ⊆ M .

Characterisation theorem
In this section we prove our main theorem, characterising projective special Kähler manifolds in terms of the deviance. We start by deriving necessary conditions on the deviance, reflecting the curvature conditions of Proposition 3.2.
Proposition 7.1. For a projective special Kähler manifold (π : M → M, ∇) with ( M , g, I, ω, ∇, ξ), and a local section s : U → S, then the corresponding deviance η satisfies Proof. Thanks to Proposition 6.3, we know that there exists z = r 2 e 2iϑ and The next step is to compute d LC π * η, but since we are using the Levi-Civita connection, it is equivalent to compute ♯ 2 ( d LC π * σ), where σ = ♭ 2 η ∈ S 3,0 U . Let us consider a local coframe θ in M and the corresponding lifting θ as in (5), so that we can denote explicitly σ = σ k,j,h θ k ⊗ θ j ⊗ θ h . We have We can now compute the following for X ∈ X π −1 (U ) : In general then, if σ = θ k ⊗ σ k , where σ k = σ k,j,h θ j , θ h ∈ S 2,0 U , we have by symmetry Notice in particular that the last two rows are symmetric in the first two indices. In order to compute d LC π * σ we need to antisymmetrise ∇ LC π * σ in the first two indices and multiply by two, so only the first row survives and we get Substituting this value in (10), we obtain As observed in Remark 6.8, ϕ − dϑ = π * τ , so we have From Proposition 3.2, we know that d LC η = 0, and since η ∈ Ω 1 (U, T 0,1 ⊗ T 1,0 ), η and η are linearly independent, so this quantity vanishes if and only if z 2 π * d LC η − 2iτ ∧ η does and therefore Let us now look at the final ingredient of the curvature tensor, that is 1 2 [ η∧ η]. In the setting of Proposition 6.3, given a section s : U → S, and the induced deviance η, then We can compute this tensor for a local coframe θ on M . Since we have Remark 7.2. Note that [η ∧ η] is independent on the local coframe, and if we consider another section such that s ′ = sa on the intersection of their domains, with a taking values in S 1 , if η ′ is the deviance corresponding to s, For a projective special Kähler manifold (π : M → M, ∇) of real dimension 2n, Proposition 3.2 says that 0 = r 2 π * (Ω LC + Ω P n C + [η ∧ η]), thus we have the following equation: This is a curvature tensor, so we can compute its Ricci and scalar component.
Proof. The first summand in (12) gives the Ricci tensor of M , the second gives the Ricci tensor of the projective space (7). In order to compute the last term, consider a unitary frame θ; from previous computations, Thus we obtain (13).
From this tensor we can now compute the scalar component by taking the trace, raising the indices with g and then dividing it by the dimension of M . Thus the first summand gives scal M , the second gives 2(n + 1) and the third proving (14).
In particular, since the norm of η is non negative, we obtain a lower bound for the scalar curvature: D1 In this case, 3 is satisfied by every such family of sections.
Proof. Given a projective special Kähler manifold, we define S := r −1 (1) ⊂ M and ϕ := −ι ξ ω| S . The principal action on S is generated by Iξ which is tangent to S since T u S = ker(dr) and dr(Iξ) = − 1 r ξ ♭ (Iξ) = − g(ξ,Iξ) r . The curvature is then dϕ = −2π * S ω as shown in Remark 4.4, so the first point is satisfied. The second condition holds thanks to Proposition 6.5. For the third point, we get D1 from the arguments leading to equation (12) and D2 from Proposition 7.1.
Let now θ be a unitary coframe on an open subset U ⊆ M , then we can lift it to a complex coframe θ on π −1 (U ) defined as in (5). It is straightforward to check that θ is adapted to the pseudo-Kähler structure of M . Notice that the proof of Proposition 5.2 is still valid in this situation even though we do not know whether M → M has a structure of projective special Kähler manifold; this gives us a description of the Levi-Civita connection form on M with respect to θ. Notice that θ k (ξ) = 0 for k ≤ n and θ n+1 (ξ) = dt(t∂ t ) + i ϕ(t∂ t ) = t so ξ = R(t θ n+1 ). We can thus compute Each section s α corresponds to the trivialisation (π| π −1 (U) , z α ) : π −1 U → U × C * in the sense that s(π(u)) · z α (u) = u for all u ∈ π −1 (U α ). For all α on π −1 (U α ), define the tensor η α := R(z 2 α π * η α ). The family { η α } α∈A is compatible on intersections U 1 ∩ U 2 , in fact if s 1 = cs 2 for c ∈ U(1), then z 2 = cz 1 and Therefore, this family glues to form a tensor η ∈ ♯ 2 S 3 M . We can build another connection ∇ := ∇ LC + η. Notice that ∇ξ = ∇ LC ξ + η(ξ) = id + R(z 2 α π * η α )(ξ) = id because locally η α is horizontal for all α. In order to prove that ∇ is symplectic, since the Levi-Civita connection is symplectic, it is enough to prove that ω( η, ·) + ω(·, η) = 0. Locally, ω = 1 2i n+1 k=1 θ k ∧ θ k and in fact, for all X = R(X k θ k ), Y = R(Y k θ k ), Z = R(Z k θ k ) vector fields on M : By the symmetry of η, this quantity vanishes. Proving that d ∇ I = 0, is equivalent to proving that ∇ I is symmetric in the two covariant indices, and thus ∇I = which is symmetric, proving d ∇ I = 0. For the flatness of ∇, we compute the curvature locally By Proposition 5.2, Ω LC = r 2 π * (Ω LC + Ω P n C ). For the same reasoning exposed in the proof of Proposition 7.1, d LC η = 0 if and only if d LC η − 2is * ϕ ∧ η = 0, which is granted by D2.
We are only left to prove that M is the Kähler quotient or M with respect to the U(1)-action and in order to do so, notice that ω(Iξ, ·) = − g(ξ, ·) = rdr = d r 2 2 , so µ := r 2 2 is a moment map for Iξ. Notice that µ −1 ( 1 2 ) = S × {1} and S is a principal bundle so, by definition of g and ω, S/U(1) is isometric to M and this ends the proof.
Remark 7.8. Instead of requiring a section η as in Corollary 7.7, we could use a section σ of S 3,0 M such that ♯ 2 σ = η.
Theorem 7.5 allows to find a whole class of projective special Kähler structures from a given one, as shown in the following Proposition 7.9. Let (π : M → M, ∇) be a projective special Kähler manifold with intrinsic deviance γ : S → ♯ 2 S 3,0 M , then for all β ∈ C ∞ (M ) there is a new projective special Kähler manifold (π : M → M, ∇ β ) with intrinsic deviance γ β = e 2iβ γ : S → ♯ 2 S 3,0 M .
Proof. We want to use Theorem 7.5, so consider the same bundle π S : S → M , but with a new connection form, that is ϕ β := dπ * S β + ϕ. Notice that dϕ β = dϕ = −2π * ω, so this is an acceptable principal connection form. The bundle map γ β is still homogeneous of degree 2. We are only left to prove the two conditions of point 3, so consider a family of sections {(U α , s α )} α∈A corresponding to a trivialisation of S and let η β α = γ β •s α , so that η β α = e 2iβ γ•η α . We thus have As for the curvature condition D1, it still holds because Finally, we give a notion of isomorphism that, by using our characterisation, will identify all the cases provided by Proposition 7.9 This definition is natural for classification purposes, and also behaves well with respect to the c-map. The idea is to have a notion of isomorphism which is not so rigid as to ask the full preservation of the structures involved in Definition 2.4, but still rigid enough so that applying the c-map to isomorphic projective special Kähler manifolds, we will get isomorphic quaternion Kähler manifolds. The details will be presented in a future work.

Complex hyperbolic n-space
In this section we are going to describe a special family of projective special Kähler manifolds, which can be thought of as the simplest possible model in a given dimension.
Let C n,1 be the Hermitian space C n+1 endowed with the hermitian form z, w = z 1 w 1 + · · · + z n w n − z n+1 w n+1 It is a complex vector space, so it makes sense to consider the projective space associated to it, that is P(C n,1 ) = (C n,1 \{0})/C * with the quotient topology and the canonical differentiable structure, where C * acts by scalar multiplication. We will denote the quotient class corresponding to an element z ∈ C n,1 by [z]. We can define the following open subset: ∈ H n C , then |v 1 | 2 + · · · + |v n | 2 − |v n+1 | 2 < 0 so |v n+1 | 2 > |v 1 | 2 + · · · + |v n | 2 ≥ 0 which implies v n+1 = 0. We thus have a global differentiable chart H n C → C n by restricting the projective chart Remark 8.1. The inverse of this chart C n → P(C n,1 ) maps z = (z 1 , . . . , z n ) ∈ C n to [(z 1 , . . . , z n , 1)], which is in H n C if and only if z 2 < 1. We have proven that H n C is diffeomorphic to the complex unitary ball and thus in particular it is contractible.
Consider now the Lie group SU(n, 1) of the matrices with determinant 1 that are unitary with respect to the Hermitian metric on C n,1 . We define a left action of SU(n, 1)

on H C such that A[v] = [Av]; it is well defined by linearity and invertibility and it is smooth.
This action is also transitive, in fact given [v], [w] ∈ H n C , without loss of generality, we can assume that v, v = −1 = w, w . Because of this, we can always complete v and w to an orthonormal basis with respect to the hermitian product, obtaining {v 1 , . . . , v n , v} and {w 1 , . . . , w n , w}. Consider the following block matrices V = (v 1 | . . . |v n |v) and W = (w 1 | . . . |w n |w) which, up to permuting two of the first n-columns, belong to SU(n, 1). The matrix A = W V −1 ∈ SU(n, 1) maps v in w and thus [v] in [w].
We shall now compute the stabiliser of the last element of the canonical basis e n+1 for this action, that is, the set of matrices A ∈ SU(n, 1) such that Ae n+1 = λe n+1 for λ ∈ C. Observe that λ ∈ U(1) since −1 = e n+1 , e n+1 = Ae n+1 , Ae n+1 = λe n+1 , λe n+1 = −|λ| 2 Moreover, the last column of A is A n+1 = Ae n+1 = λe n+1 . This forces A to assume the form B 0 0 λ Since A belongs to SU(n, 1), we must infer that B belongs to U(n) and λ = det(B) −1 . The stabiliser of e n+1 is thus S(U(n)U(1)), which is isomorphic to U(n). We deduce that H n C is a symmetric space SU(n, 1)/S(U(n)U(1)). We will adopt the nomenclature of [13] for the following Definition 8.2. We call the Kähler manifold H n C of complex dimension n the complex hyperbolic n-space.
There is a natural Kähler structure on H n C coming from its representation as a symmetric space G/H. B(X, Y ) = 2(n + 1)tr(XY ), ∀X, Y ∈ u(n, 1) We restrict the Killing form to m in order to define an Ad(H)-invariant bilinear form, that is, given x, y ∈ C n , if X, Y are the corresponding tangent vectors, We define g [en+1] := θ ⋆ θ, which is Ad(U(n))-invariant, so it extends to a global Riemannian metric g. By using the same idea, we can also define an almost complex structure I on m as the map corresponding to the scalar multiplication by i on C n . This structure is compatible with the metric and it is Ad(U(n))invariant, so it defines a Kähler structure (see [16, II, Proposition 9.3, p.260]). The Kähler form ω is then: Proposition 8.3. The manifold H n C has curvature tensor −Ω P n C and is projective special Kähler for all n ≥ 1 with constant zero deviance.
Proof. The computation of the curvature tensor is standard. By Remark 8.1, we know that H n C is contractible, so in particular its Kähler form is exact, allowing us to apply Corollary 7.7. If we choose as tensor η of type ♯ 2 S 3,0 M the 0-section, then the differential condition (16) is trivially satisfied, while the condition (15) follows from the computation of the curvature tensor.
Notice that the deviance measures the difference of a projective special Kähler manifold of dimension 2n from being the complex hyperbolic n-space. More precisely, we have In particular, for any section of S defined on an open neighbourhood of p, the corresponding local deviance vanishes at p whenever the two curvatures coincide.
Proof. One direction follows from condition D1. For the opposite one, if Ω M = Ω H n C = −Ω P n C , then scal M = −2(n + 1) and the intrinsic deviance vanishes as the norm of any local deviance vanishes by (14).
We can also prove Proposition 8.5. The only complete connected and simply connected projective special Kähler manifold of dimension 2n with zero deviance is H n C .
Proof. Let (π : M → M, ∇) be such a projective special Kähler manifold. Consider a point p ∈ M , then (T p M, g, I) can be seen as a complex vector space compatible with the metric and can thus be identified with the tangent space at a point of H n C via an isomorphism F as they are both isomorphic to C n with the standard metric. Being complex manifolds, H n C and M are analytic, and since the curvature of M is forced to be −Ω P n C , which corresponds to a u(n)-invariant map from the bundle of unitary frames to S 2 (u(n)), it is also parallel with respect to the Levi-Civita connection. It follows that the linear isomorphism F preserves the curvature tensors and their covariant derivatives. It follows that F can be extended to a diffeomorphism f : M → H n C (See [16, I, Corollary 7.3, p.261]) such that F is its differential at p.
Since F preserves I and ω which are parallel, f is an isomorphism of Kähler manifolds, as the latter maps parallel tensors to parallel tensors. Since the deviance of both manifolds is zero, we also have an isomorphism of projective special Kähler manifolds.

Classification of 4-dimensional projective special Kähler Lie groups
If M is a Lie group, the conditions of Theorem 7.5 are simpler, because a Lie group is always parallelisable. As a consequence, the bundle ♯ 2 S 3,0 (M ) is trivial, and in particular we have a global coordinate system to write the local deviances.
Definition 9.1. A projective special Kähler Lie group is a Lie group with projective special Kähler structure such that the Kähler structure is left-invariant.
Notice that we do not require the deviance to be left-invariant. An example is H n C , since the Iwasawa decomposition SU(n, 1) = KAN (see [14, Theorem 1.3, p.403]) gives a left-invariant Kähler structure on the solvable Lie group AN . We denote by H λ the hyperbolic plane with curvature −λ 2 , which is actually just a rescaling of H 1 C . With Definition 9.1, we are able to classify 4-dimensional projective special Kähler Lie groups; we obtain exactly two, which coincide with the two 4-dimensional cases appearing in the classification of projective special Kähler manifolds homogeneous under the action of a semisimple Lie groups ( [1]).
Theorem 9.2. Up to isomorphisms of projective special Kähler manifolds, there are only two connected simply connected projective special Kähler Lie groups of dimension 4: H √ 2 × H 2 and the complex hyperbolic plane. Up to isomorphisms that also preserve the Lie group structure, there are four projective special Kähler Lie groups of dimension 4, listed in Table 4.
Proof. We will start from the classification of pseudo-Kähler Lie groups provided by [12]. Table 1   Among these families, only the ones in Table 2 have a positive definite metric: It is now straightforward to find a unitary frame u for each case, that is such that g =   Table 3 we notice that the curvature tensors are of two types:

From the computations in
(i) a 2 H 1 + b 2 H 2 for a, b ≥ 0; (ii) −a 2 (Ω P 2 C + 6bH 2 ) for a > 0 and b ∈ {0, 1}; Consider the globally defined complex coframe θ 1 = u 1 + iu 2 , θ 2 = u 3 + iu 4 . If M has a projective special Kähler structure, thanks to Theorem 7.5, there is an S 1 -bundle π S : S → M and a suitable family of sections. Choose in this family a section s : U → S with U containing the identity element of M . Let η = γ • s which is a section of ♯ 2 S 3,0 U , then applying ♭ 2 we obtain a section σ of S 3,0 U which better displays the symmetry.
We write σ in its generic form with respect to θ: for some functions c 1 , c 2 , c 3 , c 4 ∈ C ∞ (U, C). By raising the second index, we obtain η = ♯ 2 σ which is then we have In other words, the coefficients of [η ∧ η] are the pairwise hermitian products of v 1 , v 2 , v 3 .
Returning to the classification, if we write H 1 , H 2 , Ω P 2 C with respect to the complex coframe, we notice that the positions corresponding to the mixed Hermitian products are always zero.
As a consequence, for all cases, if (12) holds, then v 1 , v 2 , v 3 must be orthogonal. Now we will treat each case of possible curvature tensor separately.
(i) Let a, b ≥ 0 and Ω LC = a 2 H 1 + b 2 H 2 , then These equalities translate to a linear system in the squared norms of x, y, z, w introduced in (17), namely Notice that because of (18), y and z cannot vanish simultaneously, so we have (at each point) three different cases: • Suppose at first that z = 0, then s = 0 and y = 1, so y = 0 and • the remaining case has z = 0 and y = 0. In order to solve it, let us call t := yz = 0, then (18) and (19) give in contradiction with t = 0.
In conclusion, for this class of curvature tensors, the only solutions are for We deduce that in Table 3 there are no solutions for the cases I, II, IV, V, and the only solutions in case III are the ones mentioned before. Moreover, these solutions are isomorphic to one another and the isomorphism is obtained by swapping u 1 with u 3 and u 2 with u 4 . The simply connected Lie group corresponding to this case is H √ 2 × H 2 . (ii) Let now a > 0, b ∈ {0, 1} and Ω LC = −a 2 (Ω P 2 C + 6bH 2 ), then Therefore we obtain the equations Giving the conditions We now impose the vanishing of We have four different cases: • Suppose at first that y = z = 0, then s = 0 and a = 1, so (19) is always satisfied, while (20) becomes that has solutions (x, y, z, w) = (0, 0, 0, √ 3be iα ) for α ∈ C ∞ (U ) and then (c 1 , c 2 , c 3 , c 4 ) = (0, 0, 0, • The remaining case has z = 0 and y = 0. In order to solve it, let us call t := yz = 0, then (20) and (19) give The latter implies a = 1, and from (20), we deduce a contradiction: 0 < |y| 2 = −s < 0.
In conclusion, the only solutions for this type of curvature tensors are obtained for In Table 3, these results correspond to: VI for a = 1 and σ = √ 3 2 e iα (θ 2 ) 3 for α ∈ C ∞ (U ); VII for a = 1 and σ = 0; VIII and IX for a = 1 √ δ , δ > 0 and σ = 0. Table 4 summarises (up to isomorphisms) the cases satisfying the curvature condition, showing the non vanishing differentials of the coframe and the Levi-Civita connection.
We can immediately say that cases VII, VIII, IX are all projective special Kähler, because since σ = 0, the differential condition is trivially satisfied.
Concerning case III, we can compute d LC σ by understanding how the Levi-Civita connection behaves on the unitary complex coframe θ.
Suppose that VI is projective special Kähler, than by Theorem 7.5, locally we must have the differential condition D2. Consider the unitary global complex coframe θ.
Notice that this is never of the form required by condition D2 for any available choice of σ, since evaluating the last component at θ 1 , we obtain i √ 3 2 θ 2 ∧ θ 2 ⊗ θ 2 whereas the same operation on a form of type iτ ∧ σ would evaluate to zero. We deduce that VI does not admit a projective special Kähler structure.
We are now left with cases III, VII, VIII, IX. At the level of Lie groups, case III corresponds to the connected simply connected Lie group H √ 2 × H 2 with σ = 3 2 (θ 1 ) 2 θ 2 up to isomorphism. The other deviances are in fact obtained by taking e iα σ and thus we are in the situation of Proposition 7.9. The Lie groups corresponding to the cases VII, VIII and IX, are in particular homogeneous, and they all have zero deviance, so by Proposition 8.5 we deduce that they are all isomorphic to H 2 C as projective special Kähler manifolds.
Remark 9.3. It is striking that in case III, which is obtained via the r-map from the polynomial x 2 y, the deviance is a global tensor which is a multiple of this polynomial with respect to a Kähler holomorphic coframe.
It turns out that all 4-dimensional projective special Kähler Lie groups are simply connected, so this theorem already presents all possible cases.
Proposition 9.4. Let (π : M → M, ∇) be a projective special Kähler manifold, then the universal cover p : U → M admits a projective special Kähler structure. In particular, if γ : S → ♯ 2 S 3,0 M is the intrinsic deviance for M , then p * S → U is an S 1 -bundle and if we call p ′ the canonical map p * S → S, then U has deviance p * • γ • p ′ : p * S → ♯ 2 S 3,0 U on U .
If M is a projective special Kähler Lie group, then so is U .
Proof. Since p : U → M is a cover, we can lift the whole Kähler structure of M to U by pullback (U, p * g, p * I, p * ω) (the pullback of I makes sense, since p is a local diffeomorphism). We will now use Theorem 7.5. The S 1 -bundle S lifts to an S 1 -bundle π p * S : p * S → U , where the right action can be defined locally, since p is a local diffeomorphism. The principal connection ϕ on S lifts to ϕ ′ = p ′ * ϕ and its curvature is, as expected, dϕ ′ = p ′ * dϕ = −2p ′ π * S ω = −2π * p * S p * ω. Let γ ′ = p * • γ • p ′ : p * S → ♯ 2 S 3,0 U , then γ ′ (ua) = a 2 γ ′ (u) holds, as the action is defined on the fibres, which are preserved by the pullback. The remaining properties also follow from the fact p is a local diffeomorphism.
Finally, if M is a Lie group with left invariant Kähler structure, then U is a Lie group and its Kähler structure is also left invariant.
Given a universal cover p : U → M of a projective special Kähler Lie group, ker(p) is a discrete subgroup and when M is connected, ker(p) is forced to be in the centre Z(U ) of U .
From this observation we obtain the following corollary Corollary 9.5. A connected 4-dimensional projective special Kähler Lie group is isomorphic to one of the following: • H √ 2 × H 2 with deviance ♭ 2 ( 3 2 (θ 1 ) 2 θ 2 ) in the standard complex unitary coframe θ; • complex hyperbolic n-space with zero deviance.
Proof. The proof follows from Theorem 9.2 with Proposition 9.4, as a connected group M with universal cover p : U → M is isomorphic to U/ ker(p) and, if M is a projective special Kähler Lie group, so is U by Proposition 9.4. Since U is also simply connected, Theorem 9.2 provides all the possibilities up to isomorphisms preserving the Lie structure. The statement follows from the fact that these possibilities for U have trivial centre.