Paolo Piccinni

Locally conformal Kähler structures in quaternionic geometry

We consider compact locally conformal quaternion Kähler manifolds M . This structure defines on M a canonical foliation, which we assume to have compact leaves. We prove that the local quaternion Kähler metrics are Ricci-flat and allow us to project M over a quaternion Kähler orbifold N with fibers conformally flat 4-dimensional real Hopf manifolds. This fibration was known for the subclass of locally conformal hyperkähler manifolds; in this case we make some observations on the fibers’ structure and obtain restrictions on the Betti numbers. In the homogeneous case N is shown to be a manifold and this allows a classification. Examples of locally conformal quaternion Kähler manifolds (some with a global complex structure, some locally conformal hyperkähler) are the Hopf manifolds quotients of Hn−{0} by the diagonal action of appropriately chosen discrete subgroups of CO+(4).

The canonical foliations of a locally conformal Kähler manifold

SummaryLet M be a locally conformal Kähler manifold. Then the Kähler form Ω of M satisfies dΩ=ωΩ for some closed 1 -form ω, called the Lee form of M. We show that M admits three canonical foliations (four if ω is parallel) and we prove several properties of them, improving previous results of I. Vaisman. In particular all of these foliations are totally geodesic and Riemannian, and one of them is also almost complex. If this latter foliation is regular on a compact M, then we prove that M is a locally trivial fiber bundle over a compact Kähler manifold M, and the fibers are totally geodesic flat 2-tori. Finally we study geometrical properties, the canonical class and the Godbillon-Vey class of the totally real foliation of a CR-submanifold N ⊂cM.

Reduction of Vaisman structures in complex and quaternionic geometry

We consider locally conformal Kahler geometry as an equivariant (homothetic) Kahler geometry: a locally conformal Kahler manifold is, up to equivalence, a pair (K,Γ), where K is a Kahler manifold and Γ is a discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kahler manifold (K,Γ) as the rank of a natural quotient of Γ, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kahler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover, we define locally conformal hyperKahler reduction as an equivariant version of hyperKahler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally, we show that locally conformal hyperKahler reduction induces hyperKahler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperKahler reduction.

Spin(9) and almost complex structures on 16-dimensional manifolds

For a Spin(9)-structure on a Riemannian manifold M16 we write explicitly the matrix ψ of its Kähler 2-forms and the canonical 8-form ΦSpin(9). We then prove that ΦSpin(9) coincides up to a constant with the fourth coefficient of the characteristic polynomial of ψ. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of ΦSpin(9) and

The even Clifford structure of the fourth Severi variety

{\Phi_{\rm Spin(9)}^2}

Toric Self-Dual Einstein Metrics as Quotients

in the special case of holonomy Spin(9).

3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients

We give a natural construction of an Einstein metric g on the products S 3 × S 2 and S 7 ×S 6, total spaces of some induced Hopf bundles. Since g is also a Sasakian metric, a locally conformai Kahler and conformally Ricci-flat metric h is induced by g on the products S 3 × S 2 × S 1 and S 7 ×S 6 ×S 1, that fiber also as twistor spaces over the hypercomplex and the Cayley Hopf manifolds S 3 × S 1 and S 7 ×S 1 . An extension of this construction is given to some Stiefel manifolds and induced Hopf bundles over Segre manifolds.

ON SOME MOMENT MAPS AND INDUCED HOPF BUNDLES IN THE QUATERNIONIC PROJECTIVE SPACE

Abstract TheHermitian symmetric spaceM = EIII appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure [19]. This means the existence of a real oriented Euclidean vector bundle E over it together with an algebra bundle morphism φ : Cl0(E) → End(TM) mapping Ʌ2E into skew-symmetric endomorphisms, and the existence of a metric connection on E compatible with φ. We give an explicit description of such a vector bundle E as a sub-bundle of End(TM). From this we construct a canonical differential 8-form on EIII, associated with its holonomy Spin(10) · U(1) ⊂ U(16), that represents a generator of its cohomology ring. We relate it with a Schubert cycle structure by looking at EIII as the smooth projective variety V(4) ⊂ CP26 known as the fourth Severi variety.