19 oct 2017 -- 12:30
Universidad Complutense de Madrid
Abstract.
Recall that K\"ahler manifolds are manifolds endowed with a Hermitian metric that approximates the Euclidean metric at order two at any point. We can interpret K\"ahler geometry as a tentative to extend algebraic properties of projective manifolds to a wider class of non-algebraic manifolds by transcendental methods. But K\"ahler manifolds do not exhaust the whole class of complex manifolds: the picture is well-studied for compact complex surfaces thanks to the Enriques-Kodaira-Siu classification and to the works by, among others, Lamari, Buchdahl, Belgun, Apostolov, Dloussky, \dots; and many interesting examples of non-K\"ahler complex manifolds occur in higher dimension, --- also with applications in theoretical physics.
The aim of the talk is to try to extend methods and results from projective and K\"ahler to complex (possibly non-K\"ahler) manifolds. We focus in particular on three problems. First, the investigation of cohomological invariants: besides Dolbeault cohomology, we make use of the Bott-Chern cohomology to measure the failure to complex cohomological decompositions. Second, the existence of special, or canonical, Hermitian metrics: two natural ways to weaken the K\"ahler condition are provided by locally conformally K\"ahler metrics and co-closed Hermitian metrics; on the other hand, we can ask for curvature properties with respect to the Chern connection. Third, we need to construct examples: in particular, techniques like deformations or modifications yield new structures from the existing ones.
(The talk is based on joint works with Simone Calamai, Cristiano Spotti, Tatsuo Suwa, Nicoletta Tardini, Adriano Tomassini.)