28 mar 2018 -- 15:00
Aula Magna, DM, Pisa
Abstract.
K\"ahler geometry provides analytic techniques to extend results from the algebraic to the transcendental setting. Non-K\"ahler geometry is then the attempt to perform a separate analysis of the complex and symplectic contributions. The very ultimate aim is a tentative classification of compact complex manifolds, by means of cohomological invariants and canonical Hermitian metrics.
In this talk, we summarize some results in these directions. From the cohomological point of view, we investigate the {\em Bott-Chern cohomology} of complex manifolds. Beside the de Rham and the Dolbeault cohomologies, it provides further information on the holomorphic structure. The "difference" between Bott-Chern and de Rham cohomologies is measured: we characterize its vanishing, that is, the property of cohomological decomposition known as $\partial\overline\partial$-Lemma. Recall that the {\em $\partial\overline\partial$-Lemma} is a fundamental tool in K\"ahler geometry, for example in translating the statement of the Calabi conjecture to a Monge-Ampère equation. We study the behaviour of the $\partial\overline\partial$-Lemma under natural operations such as deformations of the complex structure and modifications. In particular, it turns out to be a bimeromorphic invariant of compact complex threefolds.
From the metric point of view, the first issue is the identification of a notion of special or even canonical metric. For example, we study the existence of Hermitian metrics with {\em constant scalar curvature with respect to the Chern connection} in conformal classes. This provides an analogue of the Yamabe problem in complex geometry. On the other side, the Strominger system for spacetime supersymmetry in heterotic string theory suggests a role for {\em balanced metrics}, namely, Hermitian metrics with coclosed associated form. This notion too is a bimeromorphic invariant of compact complex manifolds.
It is conjectured that metrics of balanced type always exist on compact complex manifolds satisfying the $\partial\overline\partial$-Lemma. This and similar problems can be rephrased in solving equations of Monge-Ampère and $k$-Hessian type.
(The talk in based on joint works with Simone Calamai, Hisashi Kasuya, Cristiano Spotti, Tatsuo Suwa, Nicoletta Tardini, Adriano Tomassini, and others.)