DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 1–11

SOME GEOMETRIC ASPECTS OF COMPATIBLE COMPLEX

STRUCTURES ON A QUATERNION KÄHLER MANIFOLD

D.V. Alekseevsky, S. Marchiafava, M. Pontecorvo

Abstract. This is a report on joint work concerning almost complex structures on a quaternion Kähler manifold (M^{4n}, g, Q) which are compatible with the quaternionic structure Q, see [AM1,2], [P1,2] and [AMP1,2,3]. We point out the interest for conformally balanced complex structures, which are related to solutions of a “twistor equation” and on a compact positive quaternion Kähler manifold may be seen as Hamiltonian forms of Killing vector fields on M^{4n} or as Hamiltonian functions on the twistor space Z and also as eigenforms of the Laplacian acting on self-dual 2-forms.

1. Introduction

A main subject of our research during the past years was the study of complex and hypercomplex structures on a quaternionic manifold and, in particular, on a quaternion Kähler manifold. Here we focus on quaternion Kähler manifolds by reporting on several results that were obtained recently, see [AM1,2], [P1,2] and [AMP1,2,3].

A quaternion Kähler manifold^{1} is a Riemannian manifold (M^{4n}, g), n ≥ 2, whose holonomy group Hol(g) is isomorphic to a subgroup of Sp(n) · Sp(1). We recall that the enlarged quaternionic unitary group Sp(n)·Sp(1) may be presented as the group of ℝ−linear transformations T : ℍ^{n} ⟶ ℍ^{n} of the numerical ndimensional (right) quaternionic space ℍ^{n} which are of the form

T : ξ ↦ ξ′ = Aξq  ξ, ξ′ ∈ ℍ^{n}

1991 Mathematics Subject Classification. 53C10, 53C15, 32C10, 58G25. Key words and phrases. Quaternion Kähler - Compatible complex structure - Conformally

balanced complex structure - Hamiltonian form - Self-dual form.

Work done under the program of G.N.S.A.G.A. of C.N.R.; partially supported by M.U.R.S.T (Italy) and E.S.I. (Vienna).

This paper is in final form and no version of it will be submitted for publication elsewhere. 1For an updated introduction on this subject, see the lecture of S. Salamon in these proceed-

ings, [S4].

where A ∈ Sp(n) is a quaternionic unitary transformation (with respect to the canonical Hermitian product η ·ξ = ∑_{α} \overline{η^{α}}ξ^{α}) and q is a unitary quaternion which multiplies on the right. Note that for any quaternion p ∈ ℍ one has T (ξp) = (T ξ)p′ with p′ = q̄pq; moreover the Euclidean scalar product is preserved, T η × T ξ = ℝe(T η · T ξ) = ℝe(q̄(η · ξ)q) = η × ξ, see [Ma]. There is an identification

Sp(n) · Sp(1) ≡ Sp(n) × Sp(1)/ℤ_{2} , ℤ_{2} = {(Id, 1), (−Id, −1)} .

Classical examples of compact quaternion Kähler manifolds are the Wolf spaces: homogeneous symmetric Riemannian manifolds of the form G/N(K) where G is a compact simple Lie group and N(K) is the normalizer in G of the regular 3dimensional subgroup K of G associated with the maximal root. The Wolf spaces of the classical groups are the following:

ℍP_{n} = \frac{Sp(n + 1)}{Sp(n) × Sp(1)},

G_{2}(ℂ^{n+2}) = \frac{SU(n + 2)}{S(U(n) × U(2))} , G_{4}^{+}(ℝ^{n+4}) = \frac{SO(n + 4)}{S(O(n) × O(4))} ,

see for ex. [Be, p. 408].

A quaternion Kähler manifold is always an Einstein manifold , whose scalar curvature we denote by K. It carries a quaternionic structure Q which can be identified with a rank-3 subbundle Q →^{π} M^{4n} of skew-symmetric endomorphisms of the tangent bundle T M, with (local) admissible frames H = (J_{1}, J_{2}, J_{3} = J_{1}J_{2}) where J_{1}, J_{2}, J_{3} are anti-commuting almost complex structures. Notice that because of the right factor Sp(1) of the holonomy group, a global admissible frame does not exist in general.

The bundle Q is parallel with respect to the Levi-Civita connection of g, denoted by ∇ = ∇^{g}, that is for any admissible frame H = (J_{α}) one has

(1.1.1)  ∇J_{α} = ω_{γ} ⊗ J_{β} − ω_{β} ⊗ J_{γ}

for any circular permutation (α, β, γ) of (1, 2, 3), where the ω_{α}, α = 1, 2, 3, are 1-forms.

We shall denote by (M^{4n}, g, Q) a quaternion Kähler manifold.

Duality with respect to the metric g defines a rank-3 subbundle Λ_{+} → M of the bundle of 2-forms Λ_{2} which we call the bundle of self-dual forms.

\begin{matrix}Q & ⟶ & End^{sk}(T M) \\ π & ↓ \\ M \end{matrix}  ⟹^{g}  \begin{matrix}Λ_{+} & ⟶ & Λ_{2}(T M) \\ π′ & ↓ \\ M \end{matrix}

Λ_{+} is equipped with (local) admissible frames (F_{α} = g ∘ J_{α}), α = 1, 2, 3, where H = (J_{α}) is an admissible frame of Q and, of course,

(1.1.2)  ∇F_{α} = ω_{γ} ⊗ F_{β} − ω_{β} ⊗ F_{γ}

SOME GEOMETRIC ASPECTS OF COMPATIBLE COMPLEX … 3

The following integrability conditions hold:

(1.2)  dω_{α} + ω_{β} ∧ ω_{γ} = −νF_{α}  (α = 1, 2, 3)

where ν = K/4n(n + 2) is the reduced scalar curvature

We recall that ν = 0 if and only if the bundle Q → M^{4n} (resp., Λ_{+} → M^{4n}) is locally flat and there exist local admissible frames H = (J_{α}) (resp., (F_{α} = g ∘ J_{α})) which are parallel. In turn, this conditions are equivalent to the restricted holonomy group Hol_{0}(g) of g being isomorphic to a subgroup of Sp(n), that is (M^{4n}, g) is locally hyperKähler .

2. Compatible almost complex structures on (M^{4n}, g, Q)

Let (M^{4n}, g, Q) be a quaternion Kähler manifold. A compatible almost complex structure , shortly a compatible a.c.s. , is a section J ∈ Γ(Q) of Q → M^{4n} such that J^{2} = −Id, i.e. for any H = (J_{α}),

J = ∑_{α=1}^{3} c_{α}J_{α}

for some smooth functions c_{α}, α = 1, 2, 3, ∑_{α=1}^{3} c_{α}^{2} = 1.

When J is integrable one says that it is a compatible complex structure.

From the integrability conditions (1.2) it follows immediately that if there exists a compatible complex structure J which is parallel then the manifold must be locally hyperKähler . In fact, for any admissible frame H = (J_{1} = J, J_{2}, J_{3}) we get ∇J = ω_{3} ⊗ J_{2} − ω_{2} ⊗ J_{3} = 0. Hence ω_{2} = ω_{3} = 0 and

0 = dω_{2} + ω_{3} ∧ ω_{1} = −νF_{2}

which implies ν = 0.

Global hypothesis on M^{4n} imply the following stronger result.

Proposition ([AMP2]). Let J be a parallel complex structure (not necessarily compatible). Assume M^{4n} simply connected, complete, irreducible. Then

(1) J is compatible if and only if ν = 0 and there exists a (global) parallel admissible frame H = (J_{α}) such that

J = ∑_{α=1}^{3} c_{α}J_{α}

where the c_{α} are constants.

(2) Otherwise, (M^{4n}, g) is homothetic to the complex Grassmannian G_{2}(ℂ^{n+2}) or its non compact dual and J is the unique parallel complex structure.

From now on J will denote a compatible a.c.s. and F = g ∘ J its Kähler form. The following result shows that local conditions on J apparently weaker are

still equivalent to the condition that J is parallel.

Proposition ([AMP2]).

(1) J is locally conformally symplectic, i.e. dF = α∧F, if and only if ∇J = 0.

(2) J is cosymplectic, i.e. δF = 0, only if ν ≤ 0 and ν = 0 only when ∇J = 0.

(3) A self-dual form ω ∈ Γ(Λ_{+}) is closed, dω = 0, only if it is parallel, ∇ω = 0.

Let us also mention another useful local result.

Proposition ([AMP2]). If J is integrable then the differential dθ of its Lee form θ := −δF ∘ J is a Q-Hermitian harmonic 2-form.

On M^{4n} the fundamental 4-form Ω,

Ω = ∑_{α=}^{3} F_{α} ∧ F_{α}

is globally defined, non degenerate and parallel. Let assume that the scalar curvature is non-zero, that is ν ≠ 0, then

Ω = \frac{1}{ν^{2}} d( ∑_{α} ω_{α} ∧ dω_{α} + 2ω_{1} ∧ ω_{2} ∧ ω_{3})

with respect to any admissible frame H = (J_{α}); this was shown in [AM1] and used to prove that on a compact quaternion Kähler manifold (M^{4n}, g, Q) with ν ≠ 0 no global admissible frame H = (J_{α}) exists.

On the other hand, if ν < 0 a complete M^{4n} may admit a global admissible frame, and even a hypercomplex frame H = (J_{α}), that is the J_{α} are integrable, α = 1, 2, 3: e.g. the hyperbolic quaternionic space ℍ̃P_{n}.

Proposition ([AMP2]). Let (M^{4n}, g, Q) be compact and ν > 0. Then no compatible a. c. s. J exists on M^{4n}.

S. Salamon proved that locally there exist many compatible almost complex structures which are integrable [S2]. On the other hand M. Pontecorvo proved that on a compact M^{4n} a (global) compatible complex structure may exist only if it is parallel and hence the manifold is locally hyperKähler, see [P1].

There are general interesting questions about existence of compatible almost complex structures from the local point of view which as far as we know are still open.

Problem 1. [AM2] Beside ℍP^{n} and its non compact dual ℍ̃P_{n}, which other Wolf space admits local hypercomplex frames H = (J_{α})?

The solution would be a first step to solve the following more general problem.

Problem 2. Which quaternion Kähler manifolds are locally hypercomplex?

Other natural questions concerning global or local compatible complex structures will arise from the following discussion.

From now on we will assume ν ≠ 0.

3. Killing vector fields and conformally balanced compatible complex structures J

Associated to any self-dual form η ∈ Γ(Λ_{+}) there is a compatible almost complex structure J = \frac{g^{−1}∘η}{|η|} defined on the open set U = \{x ∈ M, η_{x} ≠ 0\} and in what follows we will point out a special class of compatible complex structures J which are associated to Hamiltonian 2-forms defined by Killing fields.

Let X be a Killing vector field on M and denote by the same symbol the 1-form g ∘ X associated to it by the metric g. Then the 2-form ∇X = \frac{1}{2}dX decomposes as

(3.1)  ∇X = ∇X^{𝔰𝔭^{1}} + ∇X^{𝔰𝔭^{n}}

where

∇X^{𝔰𝔭^{1}} = −\frac{1}{4n} ∑_{α=1}^{3} T race(J_{α}∇X)F_{α}

is the self-dual part of ∇X and ∇X^{𝔰𝔭^{n}} = ∇X −∇X^{𝔰𝔭^{1}} is a field of endomorphisms which commute with Q, [∇X^{𝔰𝔭^{n}}, Q] = 0.

The Hamiltonian 2-form of X is

η_{X} = \frac{4}{ν} ∇X^{𝔰𝔭^{1}}

and satisfies

(3.2)  dη_{X} = i_{X}Ω

As a self-dual 2-form the Hamiltonian η_{X} is uniquely defined by (3.2) and the correspondence

𝔦𝔰𝔬𝔪(g) ∋ X ↦ η_{X} ∈ Γ(Λ_{+})

is injective.

The following result was first proved in [S1], [GL]; see also [AMP2].

Proposition. On a compact quaternion Kähler manifold (M^{4n}, g, Q) with ν ≠ 0, the Hamiltonian 2-forms η_{X} of Killing vector fields X ∈ 𝔦𝔰𝔬𝔪(g) are exactly the solutions of the following twistor equation

D̄η := ∇η + \frac{1}{3} ∑_{α} δη ∘ J_{α} ⊗ J_{α} = 0

We proved the same result by looking at the solutions of the twistor equation from the following other point of view.

We recall that an (integrable) complex structure J on M is called conformally balanced if its Lee form θ = −δ(g ∘ J) ∘ J is exact: θ = dh.

Proposition ([AMP2]). Let (M^{4n}, g, Q) be a quaternion Kähler manifold with ν ≠ 0. If η = η_{X} is the Hamiltonian 2-form of a Killing vector field, then the associated almost complex structure J is a conformally balanced complex structure with Lee form θ = d log |η|^{2}.

Conversely, if the almost complex structure J associated with a self-dual 2form η is a conformally balanced complex structure then η is the Hamiltonian of a Killing vector field X given by X = (df ∘ J)^{#} where θ = dlogf^{2} and f = |η|.

Our next result shows that compatible conformally balanced complex structures can also be seen as eigenforms of the Laplacian.

Proposition ([AMP3]). On a quaternion Kähler manifold (M^{4n}, g, Q) with ν ≠ 0 the Hamiltonian 2-form η_{X} of a Killing vector field X is an eigenform of the Laplacian with eigenvalue λ = 2(n + 2)ν, i.e. λ = \frac{K}{2n}.

The statement can be proved by recalling a classical result which states that for a Killing vector field X on an m-dimensional Einstein manifold the 2-form dX is an eigenform which is associated to the eigenvalue λ = 2K/m and, moreover, by observing that the decomposition (3.1) is holonomy invariant and hence the orthogonal projection onto the space of self-dual forms commutes with the Laplacian.

A more direct proof that the conformally balanced compatible complex structures correspond to eigenforms of the Laplacian is presented in the last section of this paper, see also [M].

In the next section we will discuss the following natural question: is it true that any eigenform η ∈ Λ_{+}^{2} of the Laplacian with eigenvalue λ = \frac{K}{2n} is the Hamiltonian form of a Killing vector field X = (g)^{−1} ∘ (df ∘ J) = (δη)^{#} and hence satisfies also the twistor equation? .

4. Spectral properties of the twistor fibration

Let (M^{4n}, g, Q) be a positive quaternion Kähler manifold, that is ν > 0.

The unit bundle Z := S_{1}(Q) = {J ∈ Q|J^{2} = −Id} has a natural complex structure J′ and also a Kähler-Einstein metric g′ such that the twistor fibration t : Z → M is a Riemannian submersion with totally geodesic fiber isometric to ℂP_{1}:

\begin{matrix}ℂP_{1} & ⟶ & Z \\ t & ↓ \\ & & M \end{matrix}  ( prototype: \begin{matrix}ℂP_{1} & ⟶ & ℂP_{2n+1} \\ t & ↓ \\ & & ℍP_{n} \end{matrix} )

The scalar curvatures of g and g′ are related by K′ = [\frac{(n+1)(2n+1)}{2n(n+2)}]K, which can also be written as

\frac{K′}{2n + 1} = 2(n + 1)ν

where m = 2n + 1 is the complex dimension of Z.

The value \frac{K′}{m} is related to spectral properties of Z. Indeed, let us recall the Lichnerowicz-Matsushima theorem, see [Be p. 330].

Theorem. Let (Z, g′, J′) be a compact Kähler-Einstein manifold with K′ > 0 , dim_{ℂ}Z = m. Then:

(Lichnerowicz) the first non-zero eigenvalue of △ on functions is greater or equal to \frac{K′}{m};

(Matsushima) there is an isomorphism

Ψ : 𝔦𝔰𝔬𝔪(g′) → E_{\frac{K′}{m}} (C^{∞}(Z′))

where E_{\frac{K′}{m}} (C^{∞}(Z′)) is the eigenspace for the Laplacian with eigenvalue λ = \frac{K′}{m}. Precisely:

Ψ : 𝔦𝔰𝔬𝔪(g′) ∋ X ↦ f = −\frac{m}{K′} trace(J′∇X) ∈ E_{\frac{K′}{m}} (C^{∞}(Z′))

where f = H_{X} is a Hamiltonian of X. Viceversa:

Ψ^{−1} : E_{\frac{K′}{m}} (C^{∞}(Z′)) ∋ f ↦ X = (J′df)^{#} ∈ 𝔦𝔰𝔬𝔪(g′)

Since by a classical result of B. Watson, see [B-B], for any function f : M → ℝ one has (△^{M}f) ∘ t = △^{Z}(f ∘ t), we are able to deduce immediately that 2(n + 1)ν is the minimal non-zero eigenvalue of the Laplacian acting on functions on M^{4n}, because otherwise \frac{K′}{2n+1} would not be minimal for functions on Z; this result was already proved in [Marg] and [AM3] using different methods.

A quaternionic version of the Lichnerowicz-Matsushima Theorem is obtained by the following correspondence between Hamiltonians of Killing fields on the positive quaternion Kähler manifold (M^{4n}, g, Q) and on its twistor space. In fact, for any Killing vector field X ∈ 𝔦𝔰𝔬𝔪(g) there is a natural extension X^{Z} on Z, which is also Killing:

(4.1)  \begin{matrix}𝔦𝔰𝔬𝔪(g) ∋ X & ⟶ & X^{Z} \\ ↓ & & ↓ \\ η_{X} & ⟶ & H_{X^{Z}} \end{matrix}

The lower arrow in the diagram above is induced by a natural injective correspondence

Φ : Γ(Λ_{+}) ∋ η ⟶ f_{η} ∈ C^{∞}(Z)

given by

f_{η}(z) = c⟨(g^{−1} ∘ η), J_{z}⟩  ∀z ∈ Z

where ⟨·, ·⟩ is the natural scalar product in Q_{π(z)} ⊂ End(T_{π(z)}M), J_{z} is the complex structure of the tangent space T_{π(z)}M^{4n} represented by z and c = −1/8(n + 1) is an appropriate constant. Indeed, by restricting Φ to Hamiltonian forms of Killing vector fields X and using the notation

f_{X} := Φ(η_{X})

the following proposition holds.

Proposition ([AMP3]). For any Killing vector field X ∈ 𝔦𝔰𝔬𝔪(g) one has

f_{X} = H_{X^{Z}}

The following theorem says that Φ relates the spectra of the Laplacian acting on self-dual 2-forms of M with the spectra of functions on Z . (We denote by E_{λ}(A) the eigenspace of a Laplacian acting on the space A).

Theorem ([AMP3]). For all real numbers λ the restriction of Φ to E_{λ}(Λ_{+}(M)) is a linear injection

Φ : E_{λ}(Λ_{+}(M)) ⟶ E_{λ−2ν}(C^{∞}(Z))

In particular we have the following result which can be thought of as a quaternionic version of Lichnerowicz-Matsushima Theorem.

Proposition ([AMP3]). Let (M^{4n}, g, Q) be a compact, positive quaternion Kähler manifold. Then the first non-zero eigenvalue λ̄ of the Laplacian on self-dual forms is bounded below by 2(n + 2)ν,

λ̄ ≥ 2(n + 2)ν = \frac{K}{2n}

Equality holds if and only if there are non trivial Killing vector fields on M and on Z.

Remark. The equality always holds if n ≤ 4, see [S1]. By the above results, the following theorem relates compatible conformally balanced complex structures on M^{4n} to eigenfunctions of the Laplacian on the twistor space Z.

Theorem ([AMP3]). Let (M^{4n}, g, Q) be a compact, positive quaternion Kähler manifold. The map

Φ : E_{\frac{K}{2n}} (Λ_{+}(M)) ⟶ E_{\frac{K′}{2n+1}} (C^{∞}(Z))

is a linear isomorphism unless M^{4n} is the quaternionic projective space.

We conclude the paper by giving another approach to the relationship between conformally balanced complex structures and eigenforms of the Laplacian.

5. Self-dual 2-forms and eigenforms of the Laplacian

Let σ ∈ Γ(Λ_{+}) be a self-dual form and J the associated compatible complex structure which is defined on U = {x ∈ M^{4n}, σ_{x} ≠ 0} by J = g^{−1} ∘ F where F = |σ|^{−1}σ. In the formulas below we will refer to an admissible frame H = (J_{1} = J, J_{2}, J_{3}).

We recall the following formula

(5.1)  \begin{aligned}△(fF) & = [f(4ν + ∥ω_{2}∥^{2} + ∥ω_{3}∥^{2}) + △f]F \\ & + [f(δω_{3} − ⟨ω_{1}, ω_{2}⟩) − 2⟨df, ω_{3}⟩]F_{2} \\ & + [−f(δω_{2} + ⟨ω_{1}, ω_{3}⟩) + 2⟨df, ω_{2}⟩]F_{3} \end{aligned}

see [AMP2]. We recall also the following result which holds for a complex J.

Lemma ([AMP2]). Let J = J_{1} be a compatible complex structure on M^{4n}. Then

(5.2)  δω_{2} + ⟨ω_{1}, ω_{3}⟩ = 0  ,  δω_{3} − ⟨ω_{1}, ω_{2}⟩ = 0

Proof. By expressing that for α = 3 the 2-form at left member of integrability condition (1.2) is Q-Hermitian one has the identities

\begin{matrix}(∇_{X}ω_{3})(J_{3}Y) − (∇_{J_{3}Y} ω_{3})(X) + ω_{1}(X)ω_{2}(J_{3}Y) − ω_{1}(J_{3}Y)ω_{2}(X) \\ = (∇_{J_{ρ}X}ω_{3})(J_{3}J_{ρ}Y) − (∇_{J_{3}J_{ρ}Y} ω_{3})(J_{ρ}X) \\ + ω_{1}(J_{ρ}X)ω_{2}(J_{3}J_{ρ}Y) − ω_{1}(J_{3}J_{ρ}Y)ω_{2}(J_{ρ}X) \end{matrix} (ρ = 1, 2, 3)

We take also into account the identity

(5.3)  ∇_{X}ω_{3} = −(∇_{X}ω_{2}) ∘ J_{1} + ω_{2}(X)(ω_{2} ∘ J_{3}) + ω_{2}(J_{1}X)(ω_{2} ∘ J_{2})

which is obtained by differentiating the identity ω_{2} ∘J_{2} = ω_{3} ∘J_{3} (equivalent to the integrability of J). Then for ρ = 1, by substituting J_{2}Y to Y and then contracting with respect to g, one obtains δω_{2} + ⟨ω_{1}, ω_{3}⟩ = 0. By interchanging J_{3} and J_{2} one gets also δω_{3} − ⟨ω_{1}, ω_{2}⟩ = 0 □

Proposition. On a quaternion Kähler manifold (M^{4n}, g, Q) with ν ≠ 0, a selfdual 2-form σ whose associated a. c. s. J is a conformally balanced complex structure with θ = d log |σ|^{2} is an eigenform of the Laplacian with eigenvalue λ = 2(n + 2)ν, i.e. λ = \frac{K}{2n} .

Proof. We prove that a self-dual 2-form σ = fF where F is the Kähler form of a conformally balanced complex structure J with Lee form θ = dlogf^{2}, is an eigenform for the Laplacian with eigenvalue λ = 2(n + 2)ν.

Let H = (J_{1} = J, J_{2}, J_{3}) be an admissible basis of Q. We have df = \frac{1}{2}fθ and ω_{2} = −\frac{1}{2}θ ∘ J_{2} , ω_{3} = −\frac{1}{2}θ ∘ J_{3} since θ = ω_{2} ∘ J_{2} + ω_{3} ∘ J_{3} and ω_{2} ∘ J_{2} = ω_{3} ∘ J_{3} because of the integrability, see [AMP1]. Then, by taking account also of (5.2), formula (5.1) reduces to

(5.4)  △(fF) = [f(4ν + \frac{1}{2}∥θ∥^{2}) + △f]F

Moreover △f = δdf = \frac{1}{2}δ(fθ) = \frac{1}{2}(−⟨df, θ⟩ + fδθ) = \frac{1}{2}(−\frac{1}{2}f∥θ∥^{2} + fδθ) = \frac{1}{4}f(2δθ − ∥θ∥^{2}) and by using the identity^{2}

(5.5)  ∥θ∥^{2} + 2δθ = 8nν

see [AMP2], it follows

△f = f(2nν − \frac{1}{2}∥θ∥^{2})

Hence by substitution in previous formula (5.4) we get △σ = 2(n + 2)νσ as wanted. □

2This identity holds for any compatible complex structure and is also useful to prove the above mentioned result of Pontecorvo, see [AMP2].

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Gen. Antonova 2 - 99, 117279 Moskow, RUSSIA E-mail: daleksee@esi.ac.at ; fort@rsuh.ru

Dipartimento di Matematica, Università di Roma “La Sapienza” P.le A. Moro 2, 00185 Roma, ITALY E-mail: marchiafava@axrma.uniroma1.it

Dipartimento di Matematica, Università di Roma Tre Largo S.L.Murialdo 1, 00146 Roma, ITALY E-mail: max@mat.uniroma3.it