Conference on Kaehler and related geometries

Laboratoire Jean Leray, Nantes

Abstracts

APOSTOLOV V. (see GAUDUCHON P.)
Title : Extremal Kahler metrics on projective bundles over a curve II
Abstract : I will discuss the existence problem of extremal Kähler metrics (in the sense of Calabi) on the total space of a holomorphic projective bundle P(E) over a compact complex curve. The problem is not solved in full generality even in the case of a projective plane bundle over CP1. However, I will show that sufficiently "small" Kähler classes admit extremal Kahler metrics if and only if the underlying vector bundle E can be decomposed as a sum of stable factors. The talk will be based on a recent work with D. Calderbak, P. Gauduchon and C. Tonnesen-Friedman.

BOUCKSOM S.
Title : Canonically balanced metrics
Abstract : Using the varional characterization of solutions of Monge-Ampere equations explained in Zeriahi's talk, we will show the existence and convergence of (anti)canonically balanced metrics in the sense of Donaldson. A crucial ingredient will be the properness of the appropriate functionals, a result due to Tian and Phong-Song-Sturm-Weinkowe in the Fano case. This is joint work with Berman, Guedj and Zeriahi.

BOYER C.
Title : Maximal Tori in Contactomorphism Groups and Extremal Metrics
Abstract : I describe a general scheme for relating transverse almost complex structures on a contact manifold to conjugacy classes of maximal tori in the contactomorphism group. To a maximal torus of Reeb type there is an associated cone of Reeb vector fields, the Sasaki cone. I then consider the problem of the existence of extremal Sasakian or K-contact metrics related to a given conjugacy class of maximal tori. Examples are given and the moduli problem is discussed. In particular, I give an example where the moduli space of extremal Sasakian metrics is non-Hausdorff.

CHELTSOV I.
Title : Exceptional singularities
Abstract : The alpha-invariant itroduced by Tian plays an important role in Kähler Geometry and Birational Geometry. I will describe the role played by this invariant in Singularity Theory.

CHIOSSI S.G.
Title : Kaehler structures on Hermitian surfaces
Abstract : Hermitian 4-manifolds that admit an orthogonal Kaehler structure with "negative" orientation are studied. Holonomy issues are involved, and it is possible to give a local classification of Hermitian surfaces whose Chern holonomy is small. These structures are often obtained by a suitable deformation of a Kähler surface of Calabi type.

DEMAILLY J.-P.
Title : Regularity of plurisubharmonic envelopes, Morse inequalities and asymptotic cohomology (based in part on joint work with Robert Berman, arXiv: math.CV/0905.1246)
Abstract : Given a big line bundle, or more generally a big (1,1) cohomology class on a compact Kähler manifold, we show that potentials with minimal singularities have bounded second dd-bar derivative outside of the pole locus. Our regularity technique can be applied to study certain canonical metrics, or to investigate geodesics in the space of Kähler metrics, in the direction of a conjecture by Donaldson. It also yields a natural method for calculating Monge-Ampère masses, from which volumes and further asymptotic cohomology invariants introduced recently by algebraic geometers can be extended to arbitrary transcendental (1,1) classes.

EYSSIDIEUX P.
Title : Monge- Ampere equations in big cohomology classes

FINE J.
Title : Two conjectures, two geometric flows and a moment map
Abstract : (This is joint work with Dmitri Panov, although I will present more open problems than proven theorems!) Let M be a compact 4-manifold with three symplectic forms which at each point of M span a definite 3-plane in Lambda2. Donaldson has conjectured that M must be diffeomorphic to T4 or K3. I will describe a geometric flow designed to solve this conjecture. I will also discuss a related conjecture and flow which concerns a similar differential inequality over a 4-manifold, this time involving an SO(3)-connection. When the inequality is satisfied we conjecture that M is diffeomorphic to S4 or CP2. In each case the flow tries to deform the given solution of a differential inequality into a solution of a certain PDE. This in turn then defines an anti-self-dual Einstein metric on M. In the first conjecture the metric must have zero scalar curvature (so is hyperkahler) whilst in the second it must have positive scalar curvature. If solutions to the PDEs can be found then the known classification of such metrics would prove the two conjectures. This can be seen as analogous to the Ricci flow proof of the quater-pinched sphere theorem. Finally, if there is time, I will explain how the conjectures can be placed in the context of moment map geometry.

GAUDUCHON P. (see APOSTOLOV V.)
Title : Extremal Kähler metrics on projective bundles over a curve I

HASKINS M.
Title : Gluing methods in calibrated geometries.
Abstract : Gluing constructions are one of the few general techniques currently available for constructing calibrated geometries. In this talk we will give a brief overview of how gluing constructions have been used in several different calibrated geometries and then describe in more detail two recent such gluing constructions I have been involved with : (a) constructing new compact manifolds with holonomy group $G_2$ and rigid associative submanifolds of these $G_2$ holonomy manifolds, and (b) singular special Lagrangian submanifolds of $C^n$. In case (a) the building blocks arise from asymptotically cylindrical Calabi-Yau 3-folds constructed using Tian-Yau's work on noncompact versions of the Calabi conjecture. The most technical parts of the argument turn out to be complex algebraic in nature and not analytic. In case (b) the building blocks arise from considering degenerate limits of families of special Lagrangians with symmetry. In this case the main technical difficulties in the construction are analytic in nature. (a) is based on joint work with Alessio Corti and Tommaso Pacini. (b) is based on joint work with Nicos Kapouleas.

KOVALEV A.
Title : K3 surfaces with non-symplectic involution and compact irreducible G_2 manifolds
Abstract : I will describe a large new class of quasiprojective algebraic threefolds and their application to constructing many new examples of Ricci-flat compact 7-manifolds with holonomy G_2, via the connected-sum method. The G2 manifolds are constructed by gluing of two asymptotically cylindrical manifolds, each one being a product of a threefold and a circle. The threefolds are obtained using theory of K3 surfaces with non-symplectic involution due to Nikulin. The relation will also be explained between algebraic invariants of K3 surfaces and the "geography" of Betti numbers of the new and some previously known compact G_2-manifolds. Joint work with Nam-Hoon Lee.

LI H.
Title : Hamiltonian circle actions with minimal fixed sets

LOTAY J. D.
Title: Asymptotically conical associative 3-folds
Abstract : Associative 3-folds are examples of calibrated, hence minimal, submanifolds of 7-dimensional Riemannian manifolds which are connected with the geometry of the exceptional Lie group G_2. It is known that compact associative 3-folds have an obstructed deformation theory and that they are generically isolated. I will discuss a generalisation of this result for associative 3-folds in R7 which are asymptotic to a cone at infinity. In particular, I will show that, under certain conditions, such non-compact associative 3-folds can have a smooth moduli space of deformations of positive dimension.

NAGY P. A.
Title :The torsion and curvature of some classes of almost Kähler manifolds
Abstract : We study almost Kähler manifolds with Riemann curvature tensor Hermitian in the sense of Gray. This is a requirement on the first jet of the torsion of the first canonical Hermitian connection and shown to generically force the torsion of the latter to admit for a normal form. Examples, coming from classes of toric Kähler manifolds and generalisations of the Calabi construction, as well as classification results will be presented.

ORNEA L.
Title : Recent results in locally conformally Kähler geometry
Abstract : After a short introduction in the subject, I shall focus on explaining why locally conformally Kähler manifolds with automorphic potential on a covering are topologically Vaisman and derive some topological obstructions on their fundamental group. I shall also describe a criterion for the existence of such a potential in terms of an S1-action on the manifold. I shall end with some more information on the transformation groups of LCK manifolds. The talk is based on results obtained in collaboration with Misha Verbitsky and Andrei Moroianu.

PACARD F.
Title : End-to-end connected sum for Kähler Einstein metrics.

PANOV D.
Title : Polyhedral Kähler Manifolds
Abstract : Polyhedral Kähler manifolds are even dimensional polyhedral manifolds with unitary holonomy. In the 4-dimensional case we prove that such manifolds are smooth complex surfaces, and classify the singularities of the metric. The singularities form a divisor and the residues of the flat connection on the complement of the divisor give us a system of cohomological equations. Parabolic version of Kobayshi-Hitchin correspondence of T. Mochizuki permits us to characterize polyhedral Kähler metrics of non-negative curvature on CP2 with singularities at complex line arrangements.

SALAMON S.
Title : Non-integrable geometries and differential equations
Abstract : I shall discuss some Calabi-Yau type geometries and equations in the context of non-integrable almost complex structures. The focus will be on specific nilmanifolds and twistor spaces in real dimensions 4 and 6 which also carry integrable complex structures. Motivation comes from recent preprints of Tosatti-Weinkove arXiv:0906.0634 and Cecotti-Vafa ArXiv:0910.2615.

SANO Y.
Title : An observation on destabilizing test configurations and multiplier ideal subschemes

SANTORO B.
Title : Complete Kähler Ricci-flat metrics on resolutions of singularities
Abstract : We will discuss some existence results for complete Calabi-Yau metrics on crepant resoutions of singularities.

SINGER M.
Title : Scalar flat Kähler ALE metric

STOPPA J.
Title : A simple limit for slope instability
Abstract : I will present a special construction and discuss some open questions concerning the K- and slope-stability of general type varieties.

TONNESEN-FRIEDMAN C.
Title : Generalized quasi-Einstein metrics

VERBITSKY M.
Title : A global Torelli theorem for hyperkahler manifolds
Abstract : A Teichmuller space of $M$ is a space of complex structures on $M$ up to isotopies. We define a birational Teichmuller space by identifying certain points corresponding to bimeromorphically equivalent manifolds, and show that the period map gives an isomorphism of the birational Teichmuller space and the corresponding period space $SO(b_2-3, 3)/SO(2)\times SO(b_2 -3, 1)$. We use this result to state and prove a global Torelli theorem for hyperkaehler manifolds.

ZERIAHI A.
Title: A variational approach to complex Monge-Ampère equations
Abstract : We will show that degenerate complex Monge-Ampère equations in a big cohomology class of a compact Kähler manifold can be solved using a variationnel method independant of Yau's theorem.

Contacts: Yann Rollin or Gilles Carron