Comology and Characteristic Classes of (Almost) Complex Manifolds

I will introduce a Frölicher-type spectral sequence that is valid for all almost complex manifolds, yielding a natural Dolbeault cohomology theory for non-integrable structures. I will revise the harmonic theory surrounding Dolbeault cohomology and explain some applications to nilmanifolds and nearly Kähler manifolds. This is joint work with Scott Wilson.

Given a non rational Bott-Chern cohomology class of type (1,1) on a complex manifolds, there exists a sequence of “quasi holomorphic” line bundles whose Chern classes approximate very closely certain multiples of the given cohomology class. We will report on spectral estimates provided by L. Laeng in his PhD thesis (2002), in relation with a number of newer ideas emerging e.g. from our recent study of Bergman vector bundles. We hope that these techniques could possibly be helpful to approach the conjectures on transcendental holomorphic Morse inequalities and Kähler invariance of plurigenera.

A Riemannian metric is said to be Einstein if it has constant Ricci curvature. Dimension four is in many respects a privileged realm for Einstein metrics. In particular, there are certain 4-manifolds, such as K3 and complex ball-quotients, where every Einstein metric comes from Kaehler geometry, and where the moduli space of Einstein metrics can therefore be shown to be connected. In this lecture, I will discuss analogous but weaker results that characterize the known Einstein metrics on the ten smooth compact 4-manifolds that arise as del Pezzo surfaces, as well as on a family of five closely-related 4-manifolds that do not even admit almost-complex structures.

We show that a smooth complex projective threefold admits a holomorphic one-form without zeros if and only if the underlying real 6-manifold is a smooth fibre bundle over the circle, and we give a complete classification of all threefolds with that property. Our results prove a conjecture of Kotschick in dimension three. Joint work with Feng Hao.