Giuseppe Barbaro
Pluriclosed flow on complex surfaces
We recall the fundamental aspects of the pluriclosed flow introduced by Streets and Tian to generalize the Ricci flow in the Hermitian non-Kähler context. Building on this, a conjectural description of the long time behavior of the flow on complex surfaces is formulated. For class VII surfaces, this results in a parabolic approach to the GSS conjecture.
We thus describe some of the recent advances in the theory of pluriclosed flow, focusing on existence and convergence results.
Anna Fino
Balanced Hermitian geometry from K3 surfaces
A Hermitian metric on a complex manifold is called balanced if its fundamental form is co-closed. Balanced metrics provide a natural class of Hermitian metrics extending Kähler geometry to the non-Kähler setting. In this talk, I will discuss several constructions of balanced non-Kähler manifolds arising from K3 surfaces and K3 orbifolds. I will also present some results related to the Hull--Strominger system.
Vlad Marchidanu
Holomorphic tensors on products of Sasaki manifolds
The product of two Sasaki manifolds always supports a family, indexed by a complex nonreal parameter, of complex structures, induced as quotients of the product of their Kähler cones by a $(\mathbb C, +)$ action. They admit no locally conformally Kähler metrics. Nevertheless, motivated by the Vaisman case we show that holomorphic tensors on these products are invariant under the Reeb flows of both factors. More generally, on products of algebraic cones invariance under a contraction implies invariance under its Zariski closure. Besides this fact, for compact Sasaki manifolds we fundamentally exploit an explicit embedding of the product of their cones into affine space, equivariant for the complexified Reeb torus, which has a normal Zariski closure. The embedding also gives a holomorphic embedding of the product into a weighted Calabi–Eckmann manifold.
Michael McQuillan
Foliations on non-Kahler surfaces.
A priori the classification of foliations on non-Kahler surfaces appears to hit the same
block as usual Enriques-Kodaira classification, i.e. h^1,0 isn't h^0,1. However, I've
only just started thinking about it, so this could change. Irrespectively, the talk will
consist in whatever I can prove about non-Kahler surfaces admitting a foliation in curves.
Liviu Ornea
Two generalisations in higher dimensions of the Inoue surface $S_M$.
I shall briefly recall the construction and some properties of the Inoues surfaces $S_M$, then
I shall discuss Oeljeklaus-Toma and Endo-Pajitnov manifolds, reviewing old and new results on their (analytic and metric) geometry, and topology.
Lorenzo Sillari
Towards an Enriques–Kodaira classification of almost complex 4-manifolds
Abstract: The Enriques–Kodaira classification organises compact complex surfaces according to their Kodaira dimension κ ∈ {−∞, 0, 1, 2}. Chen and Zhang recently extended plurigenera and Kodaira dimension to arbitrary almost complex manifolds, using pseudoholomorphic sections of the pluricanonical bundles. It is natural to ask how much of the classification survives once integrability is dropped. In real dimension four, the almost complex Kodaira dimension still shows classification-like behaviour: maximal Kodaira dimension forces integrability, and κ = 1 yields elliptic-fibration structures. It also shows new phenomena, such as the failure of deformation invariance. I will focus on almost complex 4-manifolds with torsion first Chern class. Under a tameness assumption, the Kodaira dimension is either 0 or −∞, and it vanishes exactly when the canonical bundle is pseudoholomorphically torsion. On K3 and Enriques surfaces, a Bogomolov–Tian–Todorov-type unobstructedness theorem gives a local description of the space of almost complex structures with vanishing Kodaira dimension. I will conclude with work in progress and open problems. Partly based on joint work with A. Tomassini.
Andrei Teleman
On the classification of Inoue surfaces
Matei Toma
Dynamical properties of non-kählerian surfaces
We report on joint work with Ionuț Chiose studying exact positive currents on compact non-kählerian surfaces with a view towards their classification from a dynamical point of view.
We will describe the place of the "known" non-kählerian surfaces in this classification and propose a plan of attack of the Global Spherical Shell Conjecture along these lines.