From: "Dmitry V. Egorov" Title: A new equation on the low-dimensional Calabi-Yau metrics . In this talk I will introduce a new equation on the compact Kahler manifolds. Solutions of this equation correspond to the Calabi--Yau metric. New equation describes deformation of complex structure, while Monge-Ampere equation describes deformation of symplectic structure. http://arxiv.org/abs/1104.5575 Ruxandra Moraru Here are the title an abstract for my talk. Title: Compact moduli spaces of stable bundles on Kodaira surfaces Abstract: In this talk, I will examine the geometry of moduli spaces of stable bundles on Kodaira surfaces, which are non-Kaehler compact surfaces that can be realised as torus fibrations over elliptic curves. These moduli spaces are interesting examples of holomorphic symplectic manifolds whose geometry is similar to the geometry of Mukai's moduli spaces on K3 and abelian surfaces. In particular, for certain choices of rank and Chern classes, the moduli spaces are themselves Kodaira surfaces. This is joint work with Marian Aprodu and Matei Toma. From alesker.semyon75@gmail.com TITLE: Quaternionic Monge-Ampere equations and HKT-geometry. ABSTRACT: A notion of quaternionic Monge-Ampere equation will be introduced. These are non-linear second order elliptic equations which make sense on so called hypercomplex manifolds, in particular on the flat quaternionic space. They admit an interpretation in the framework of Hyper Kahler with Torsion (HKT) geometry (to be explained in the talk). We formulate a quaternionic version of the Calabi conjecture, and state a number of partial results towards its proof. Part of the results are joint with M. Verbitsky. Subject: Paul Andi Nagy, nagyp@uni-greifswald.de Title: Symplectic forms on Kaehler surfaces Abstract: Necessary and sufficient conditions for the existence of orthogonal almost-Kaehler structures on Kaehler surfaces will be given. We will explain how these conditions work on several classes of examples. The relation to the problem of finding a symplectic form on a Kaehler surface will be outlined. From: Anna Fino Title: Speciali Hermitiane structures and symplectic geometry Abstract. Symplectic forms taming complex structures on compact manifolds are strictly related to a special type of Hermitian metrics, known in the literature as "strong Kaehler with torsion" metrics. I will present general results on "strong K\"ahler with torsion" metrics, their link with symplectic geometry and more in general with generalized complex gometry. Moreover, I will show for certain $4$-dimensional non-Kaehler symplectic $4$-manifolds some recent results about the Calabi-Yau equation in the context of symplectic geometry. From: "Vicente Cortes" "From cubic polynomials to complete quaternionic K??hler manifolds" Abstract I will explain two supergravity constructions, which allow to construct certain special Riemannian manifolds starting with other special Riemannian manifolds. We show that the resulting manifolds are complete if the original manifolds are. By composition of the two constructions we obtain complete quaternionic K??hler manifolds out of certain cubic hypersurfaces. At the end I will formulate two open problems concerning such hypersurfaces. The talk is based on arXiv:1101.5103 (hepth, mathdg). From: Vladlen Timorin Title: Maps that take lines to conics Abstract: we will discuss generalizations of the classical theorem of Moebius (1827): a one-to-one self-map of a real projective space that takes all lines to lines is a projec tive transformation. E.g. we study sufficiently smooth local maps taking line segments to parts of conics. A description of local maps taking line segments to circle arcs depends n on-trivially on the dimension (the descriptioninvolves classical geometries, quaternionic Hopf fibrations, representations of Clifford algebras) . For most dimensions, it is still missing. From: Sonke Rollenske Lagrangian fibrations on hyperkaehler manifolds Hyperkaehler (also called irreducible holomorphic symplectic) manifolds form an important class of manifolds with trivial canonical bundle. One fundamental aspect of their structure theory is the question whether a given hyperkaehler manifold admits a Lagrangian fibration. I will report on a joint project with Daniel Greb and Christian Lehn investigating the following question of Beauville: if a hyperkaehler manifold contains a complex torus T as a Lagrangian submanifold, does it admit a (meromorphic) Lagrangian fibration with fibre T? I will describe a complete positive answer to Beauville's Question for non-algebraic hyperkaehler manifolds, and give explicit necessary and sufficient conditions for a positive solution in the general case using the deformation theory of the pair (X,T). From: Gueo Grantcharov Calibrations in hyperkaehler geometry Abstract: We describe a family of calibrations arising naturally on a hyperkaehler manifold M. These calibrationscalibrate the holomorphic Lagrangian, holomorphic isotropic and holomorphic coisotropic subvarieties. When Mis an HKT (hyperk\"ahler with torsion) manifold with holonomy SL(n, H), we construct another family ofcalibrations, which calibrates holomorphic Lagrangian and holomorphic coisotropic subvarieties. They are (generally speaking) not parallel with respect to any torsionless connection on M. We note also that there are examples of complex isotropic submanifolds in SL(n, H) manifolds with HKT structure, which can not be calibrated by any form, unlike the Kaehler case. Isabel Dotti idotti@famaf.unc.edu.ar 'Some restrictions on existence of abelian complex structures' We describe the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. More precisely, we consider a distinguished class of Lie algebras admitting abelian complex structures given by abelian double products. The structure of these Lie algebras can be described in terms of a pair of commutative associative algebras satisfying a compatibility condition. We will show that when g is a Lie algebra with an abelian complex structure J, and g decomposes as g = u + Ju, with u an abelian subalgebra, then g is an abelian double product. Joint work with A. Andrada and M. L. Barberis From: Semyon Alesker TITLE: Quaternionic Monge-Ampere equations ABSTRACT: A notion of quaternionic Monge-Ampere equation will be introduced. These are non-linear second order elliptic equations which make sense on so called hypercomplex manifolds, in particular on the flat quaternionic space. They admit an interpretation in the framework of Hyper Kahler with Torsion (HKT) geometry (to be explained in the talk). We formulate a quaternionic version of the Calabi conjecture, and state a number of partial results towards its proof. Part of the results are joint with M. Verbitsky. Paul Gauduchon pg@math.polytechnique.fr Almost complex structures on quaternion-Kaehler manifolds The aim of this lecture is to show that, apart from the complex Grassmannianss Gr2(Cn+2), the compact quaternionic quaternion-Kaehler manifolds of positive type admit no almost complex structure, even in the weak sense (joint work with Andrei Moroianu and Uwe Semmelmann). From: "piccinni" "Some curiosities on Spin(9) and the sphere S^{15}" Although holonomy Spin(9) is only possible for the two 16-dimensional symmetric spaces $\mathbb OP^2$ and $\mathbb O H^2$, weakened holonomy Spin(9) conditions have been proposed and studied, in particular by Th. Friedrich. A basic problem is to have a simple algebraic formula for the canonical $8$-form $\Phi_{Spin}(9)}$, similar to the usual definition of the quaternionic 4-form $\Phi_{\mathrm{Sp}(n)\cdot \mathrm{Sp}(1)}= \omega_I^2+\omega_J^2+\omega_K^2$, witten in terms of local compatible almost hypercomplex structures (I,J,K). In the talk, a simple formula for $\Phi_{\mathrm{Spin}(9)}$ is presented, discussing a family of local almost hypercomplex structures associated with a Spin(9)-manifold $M^{16}$. Some of these complex structures, now on model spaces $\R^{16^q}$, are then used to give an approach through Spin(9) to the very classical problem of writing down a maximal system of tangent vector fields on spheres $S^{N-1} \subset \R^N$. If time permits, some properties of manifolds equipped with a locally conformal parallel Spin(9) metric will be also discussed. From: "R Bielawski" "Hypercomplex and pluricomplex geometry" ABSTRACT: I'll will describe a new type of geometric structure on complex manifolds. It can be viewed as a deformation of hypercomplex structure, but it also leads to a special type of hypercomplex geometry. These structures have both algebro-geometric and differential-geometric descriptions, and there are interesting examples arising from physics. From: "Keizo_Hasegawa" "Locally conformally Kahler structures on homogeneous spaces". Keizo Hasegawa (Niigata University, JAPAN) Locally conformally Kaehler structures on homogeneous spaces A homogeneous Hermitian manifold M with its homogeneous Hermitian structure h, defining a locally conformally Kaehler structure w is called a homogeneous locally conformally Kaehler or shortly a homogeneous l.c.K. manifold. If a simply connected homogeneous l.c.K. manifold M=G/H, where G is a connected Lie group and H a closed subgroup of G, admits a free action of a discrete subgroup D of G from the left, then a double coset space D\G/H is called a locally homogeneous l.c.K. manifold. We discuss explicitly homogeneous and locally homogeneous l.c.K. structures on Hopf surfaces and Inoue surfaces, and their deformations. We also classify all complex surfaces admitting locally homogeneous l.c.K. structures. We show as a main result a structure theorem of compact homogeneous l.c.K. manifolds, asserting that it has a structure of a holomorphic principal fiber bundle over a flag manifold with fiber a 1-dimensional complex torus. As an application of the theorem, we see that only compact homogeneous l.c.K. manifolds of complex dimension 2 are Hopf surfaces of homogeneous type. We also see that there exist no compact complex homogeneous l.c.K. manifolds; in particular neither complex Lie groups nor complex paralellizable manifolds admit their compatible l.c.K. structures. We show as a main result a structure theorem of compact homogeneous l.c.K. manifolds, asserting that it has a structure of a holomorphic principal fiber bundle over a flag manifold with fiber a 1-dimensional complex torus. As an application of the theorem, we see that only compact homogeneous l.c.K. manifolds of complex dimension 2 are Hopf surfaces of homogeneous type. We also see that there exist no compact complex homogeneous l.c.K. manifolds; in particular neither complex Lie groups nor complex paralellizable manifolds admit their compatible l.c.K. structures. This talk is based on a joint work with Y. Kamishima "Locally conformally Kaehler structures on homogeneous spaces" (arXiv:1101.3693). To: "Stefan Ivanov" "Extremals for the Sobolev-Folland-Stein inequality, the quaternionic contact Yamabe problem and related geometric structures" We describe explicitly non-negative extremals for the Sobolev inequality on the quaternionic Heisenberg groups and determine the best constant in the $L^2$ Folland-Stein embedding theorem involving quaternionic contact (qc) geometry and the qc Yamabe equation. Translating the problem to the 3-sasakian sphere, we determine the qc Yamabe invariant on the spheres. We describe explicitly all solutions to the qc Yamabe equation on the seven dimensional quaternionic Heisenberg group. The main tool is the notion of qc structure and the Biquard connection. We define a curvature-type tensor invariant called qc conformal curvature in terms of the curvature and torsion of the Biquard connection and show that a qc manifold is locally qc conformal (gauge equivalent) to the standard flat qc structure on the Heisenberg group, or equivalently, to the 3-sasakian sphere if and only if the qc conformal curvature vanishes. Possibly, this will help to reduce the qc Yamabe problem to that of the spherical qc manifolds. From: stefan@mi.ras.ru To: From: "Dmitri V Alekseevsky" Universal coverings of strictly pseudoconvex domains Abtract: The universal covering of a strictly pseudoconvex domain in a Stein manifold is completely determined by the local CR-geometry of its boundary. I will discuss various results and problems related to this general principle. This is joint work with Rasul Shafikov. "Lorentzian manifolds with large isometry group" I give a survey of results about isometry group of Lorentzian manifolds and will describe some classes of homogeneous Lorentzian manifolds including homogeneous manifolds of a semisimple Lie group and manifolds with weak??y irreducible isotropy group.