8 jan 2019 -- 16:00
LMU München
Abstract.
In the tentative to move from the K\"ahler to the non-K\"ahler setting, we consider several problems concerning holomorphic cohomological invariants and existence of ``special'' Hermitian metrics on complex manifolds.
We focus on the role of the {\em Bott-Chern and Aeppli cohomologies}. On the one side, they allow to numerically characterize the strong Hodge decomposition in cohomology given by the {\em $\partial\overline\partial$-Lemma}. On the other side, the first Bott-Chern class in Bott-Chern cohomology encodes information on the curvature of the Chern connection.
Several problems naturally arise. The behaviour of the strong Hodge decomposition under natural operations on the complex structure like {\em deformations} or {\em modifications} has been studied by several authors, including recent work by Jonas Stelzig. In the same spirit, the understanding of a notion of (geometric) {\em Bott-Chern formality} related to the algebra structure of the Bott-Chern cohomology is stimulating. The problem of existence of special metrics ({\itshape e.g.} balanced metrics in the sense of Michelsohn) under cohomological assumptions ({\itshape e.g.} $\partial\overline\partial$-Lemma) exhibits difficulties when attacked with analytic techniques and pde's.
We focus on the role of the {\em Chern connection} when looking for Hermitian metrics with special curvature properties. The {\em Chern-Yamabe problem} acts as an analogue of the Yamabe problem for Hermitian manifolds, and concerns Hermitian metrics having constant scalar curvature with respect to the Chern connection in a conformal class. When the expected curvature is non-positive, some results can be shown; and difficulties arise in the positive curvature case. This problem relates also to several notions of {\em Chern-Einstein metrics}. Note the plural ``notions'', due to the lack of {\em symmetries of the curvature tensor} of the Chern connection.
The talk is based on and inspired by joint works with: Adriano Tomassini; Nicoletta Tardini; Simone Calamai, Cristiano Spotti; Antonio Otal, Raquel Villacampa, Luis Ugarte; Michela Zedda.