New perspectives on
Holonomy and Submanifolds
Turin, 23-24 April 2004


Titles and abstracts 

of the talks


Bochner-flat weighted projective spaces as Kähler quotients
Paul Gauduchon, Ecole Polythecnique Paris

The existence of Bochner-flat smooth Kähler metrics on weighted projective spaces has been first established by Robert Bryant in his recent seminal paper Bochner-Kähler metrics. In this talk, which relies on a joint work with Liana David, we give a direct, simple construction of these metrics and we show that standard construction of the Fubini-Study metric as a Kähler reduction of the standard flat Kähler space can be non trivially deformed within the class of Bochner-flat Kähler structures.


Infinite dimensional symmetric spaces and Kac-Moody algebras
Ernst Heintze, University of Augsburg


Supersymmetric strings and special (SU(3)) holonomy geometries
Stefan Ivanov, University of Sofia

Based on joint work with Petar Ivanov, University of Sofia.
Electronic preprint available: arXiv:math.DG/0312094.

Riemannian manifolds admitting parallel spinor with respect to a metric connection with totally skew-symmetric torsion recently become a subject of interest in theoretical and mathematical physics. One of the main reasons is that the number of preserving supersymmetries in string theory depend essentially on the number of parallel spinors. We study the existence of parallel spinors with respect to a metric connection with skew-symmetric torsion mainly in dimensions 6 and related geometric structures.

We show that the existence of a parallel spinor with respect to a metric connection with skew-symmetric torsion in dimension 6 determines the connection in a unique way. A formula for the torsion 3-form of the SU(3)-connection and a formula for the Riemannian scalar curvature are found in terms of the given SU(3)-structure. Examples of SU(3)-instanton are obtained.


Singular "special fibrations" on Calabi-Yau manifolds
Diego Matessi,  Imperial College, London

Special submanifolds of a Calabi-Yau manifold are submanifolds on which the imaginary part of the holomorphic form vanishes and the real part is a non degenerate top dimensional form.
I will illustrate the construction of some singular "special fibrations" on (open) Calabi-Yau manifolds. The construction is motivated by the problem of compactifying open Calabi-Yau manifolds coming from singular affine structures on a 3 sphere.


Submanifolds and the Berger holonomy theorem
Carlos Olmos, Universidad Nacional de Cordoba (Argentina)
 
The Berger holonomy theorem is probably  the most important general (local) result of riemannian geometry, i. e. without making any curvature assumption (of course, combined with the basic facts as the de Rham decompositon theorem). We will relate in this talk submanifold and riemannian geometry.  We will refer to a relation between riemannian and normal holonomy  In this way we will give a conceptual proof of the Berger holonomy theorem (and also of Simons theorem on holonomy systems).


G2 metrics and M-theory
Simon M. Salamon, Politecnico di Torino

The cone C over a 7-manifold X with weak holonomy G2 has a metric with holonomy Spin(7), and this fact produced the first explicit example with exceptional holonomy. In the context of supergravity, it gives rise to solutions M x C and AdS x X on products with Minkowski 3-space and anti-de-Sitter 4-space respectively. More complicated 11-dimensional metrics are constructed from orbifolds with G2 and SU(3) structures. We discuss the resulting geometrical ideas that have been developed by Acharya, Atiyah, Witten and others, and we provide a new example based on Lagrangian foliations.


Parallel Kähler submanifolds of quaternionic Kähler symmetric spaces
Antonio J. Di Scala, Politecnico di Torino

The non totally geodesic parallel Kähler submanifolds (M2n,J1) of the quaternionic space HPn were classified by K. Tsukada. Here we give the complete classification of non totally geodesic immersions of parallel Kähler submanifolds (M2m,J1) in a quaternionic Kähler symmetric space (N4n,g' ,Q) of non zero scalar curvature, i.e. in a Wolf space W or in its non compact dual. They are exhausted by parallel Kähler submanifolds of a totally geodesic submanifold  which is either a hermitian symmetric space or a quaternionic projective space.


Hypersymplectic manifolds with a torus action
Andrew Swann, SDU Odense

Examples of hypersymplectic manifolds may be constructed using Marsden-Weinstein type quotients of flat space Cd,d by a compact Abelian group. I will discuss the geometry and topology of these quotients and how such information can be deduced from the geometry and combinatorics of configurations of cones. Joint work with Andrew Dancer.






back to the workshop webpage