SEM. GEOMETRÍA ALGEBRAICA. Daniele Angella. "Cohomological decompositions and canonical metrics on complex nonKähler manifolds"
 jueves, 19 octubre
 Plaza de las Ciencias nº 3. Facultad de CC Matemáticas
Seminario de Geometría Algebraica
Daniele Angella
Universitá degli Studi di Firenze
"Cohomological decompositions and canonical metrics on complex nonKähler 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 nonalgebraic manifolds by transcendental methods. But Kähler manifolds do not exhaust the whole class of complex manifolds: the picture is wellstudied for compact complex surfaces thanks to the EnriquesKodairaSiu classification and to the works by, among others, Lamari, Buchdahl, Belgun, Apostolov, Dloussky, . . . ; and many interesting examples of nonKähler 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 nonKähler) manifolds. We focus in particular on three problems. First, the investigation of cohomological invariants: besides Dolbeault cohomology, we make use of the BottChern 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 coclosed 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.)
