Title of the course: Hyperbolicity of Possibly Non-Kähler Compact Complex Manifolds
Instructor: Assoc. Prof. Dan Popovici
Institution: Université Toulouse III – Paul Sabatier
Dates: 11-17 September 2023
Prerequisites: Basic notions in complex analysis (e.g. holomorphic functions, Cauchy-Riemann equations, (pluri)subharmonic functions), functional analysis (e.g. Hilbert, Banach and Fréchet spaces, weak topologies) and differential geometry (e.g. complex manifolds, vector bundles, Hermitian metrics).
Level: Graduate
Abstract: We will start by covering some basic material in complex analysis and geometry: holomorphic functions of several variables, the notions of complex structure and complex manifold (= a differentiable manifold with a holomorphic atlas), bidegrees for differential forms and currents (= a kind of forms with possibly singular coefficients), positivity notions for differential forms, Hermitian metrics (= smooth, positive definite forms of bidegree (1, 1)), complex vector bundles and the notions of connections and curvature thereon, etc. We will then introduce the Kähler metrics and will present their basic properties and point out several examples of complex manifolds carrying such metrics. Then if time permits we will first present the classical notions of Kobayashi and Brody hyperbolicity and we will then go on to present the very recent notions of balanced, divisorial and partial hyperbolicity that we introduced jointly with S. Marouani and H. Kasuya.
Language: EN