Maurizio Parton

Examples of non-trivial rank in locally conformal Kähler geometry

We consider locally conformal Kähler geometry as an equivariant, homothetic Kähler geometry (K, Γ). We show that the de Rham class of the Lee form can be naturally identified with the homomorphism projecting Γ to its dilation factors, thus completing the description of locally conformal Kähler geometry in this equivariant setting. The rank rM of a locally conformal Kähler manifold is the rank of the image of this homomorphism. Using algebraic number theory, we show that rM is non-trivial, providing explicit examples of locally conformal Kähler manifolds with

Reduction of Vaisman structures in complex and quaternionic geometry

{1\nless{\text{\upshape \rmfamily r}_{M}}\nless b_1}

Locally conformal Kähler reduction

. As far as we know, these are the first examples of this kind. Moreover, we prove that locally conformal Kähler Oeljeklaus-Toma manifolds have either rM = b1 or rM = b1/2.

A tool which mines partial execution traces to improve static analysis

We propose a new technique combining dynamic and static analysis of programs to find linear invariants. We use a statistical tool, called simple component analysis, to analyze partial execution traces of a given program. We get a new coordinate system in the vector space of program variables, which is used to specialize numerical abstract domains. As an application, we instantiate our technique to interval analysis of simple imperative programs and show some experimental evaluations.

The even Clifford structure of the fourth Severi variety

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.

Hopf surfaces: locally conformal Kähler metrics and foliations

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

Families of strong KT structures in six dimensions

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