Seminario de Geometría Algebraica

Daniele Angella

Universitá degli Studi di Firenze

"Cohomological decompositions and canonical metrics on complex non-Kähler manifolds"

 12:30. Sala 238

Recall that Kähler manifolds are manifolds endowed with a Hermitian metric that approximates the Euclidean metric at order two at any point. We can interpret Kähler geometry as a tentative to extend algebraic properties of projective manifolds to a wider class of non-algebraic manifolds by transcendental methods. But Kähler 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, . . . ; and many interesting examples of non-Kähler 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 Kähler to complex (possibly non-Kähler) 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 Kähler condition are provided by locally conformally Kähler 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.)