Cohomology and Characteristic Classes of (almost) complex manifolds

Abstract: - Joana Cirici (Universitat de Barcelona): “Dolbeault cohomology for almost complex manifolds”

Abstract. 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.

- Jean-Pierre Demailly (Institut Fourier, Université Grenoble Alpes) “On the approximate cohomology of quasi holomorphic line bundles”

Abstract. 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.

- Claude LeBrun (Stony Brook): "Einstein Metrics, Weyl Curvature, and Anti-Holomorphic Involutions"

Abstract. 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.

- Stefan Schreieder (Leibniz University Hannover): “Holomorphic one-forms without zeros on threefolds”

Abstract. 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.

complex variablesdifferential geometry

Audience: researchers in the topic

( chat | video )

Comments: The discussion is open at gitter.im/GTACOS-July2020/. The live discussion with the speakers for this series of talks will be held on July 21, see researchseminars.org/talk/tacos/1/

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Geometry and TACoS

Series comments: The idea of “Geometry & TACoS” is to have regular mathematical conferences on specific themes, related, in a broad sense, to Geometry and Topology of (Almost) Complex Structures. Each conference will consist of three blocks:

- A series of four lectures, by four speakers, which will be available on our YouTube channel at the beginning of each period.

- A discussion platform, available on Gitter, will be active for two weeks after the release of the videos. There, the participants will be able to interchange remarks, comments, and open questions concerning the topics of the four lectures, as well as interact with each other, the organizers and with the speakers (both in a public and a private way).

- The organizers will follow the development of the discussion and collect questions for the third part of the event, which is an informal coffee break (or TACoS break?) on Zoom, moderated by the organizers, during which the speakers may report on questions and comments that were asked in the chat rooms, and interact personally with the participants.

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