Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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5th Workshop "Complex Geometry and Lie Groups”

Isometric immersions of locally conformally K\"ahler manifolds

Daniele Angella (Dipartimento di Matematica e Informatica "Ulisse Dini", Università di Firenze)

created by daniele on 19 Aug 2018

11 jun 2018 -- 14:00

CDM-004, Firenze

Abstract.

The celebrated Kodaira Embedding Theorem gives geometric and cohomological conditions under which analytic geometry reduces to algebraic geometry. In general, such an embedding is not isometric. The problem of which real-analytic K\"ahler manifolds admit an isometric immersion into $\mathbb{C}{\mathrm P}^n$, or more in general into complex space forms has been studied by Eugenio Calabi.

{\em Locally conformally geometry} can be intepreted as a sort of "equivariant (homothetic) K\"ahler geometry" and a first specific non-K\"ahler setting. Despite the K\"ahler condition imposes strong topological obstructions, most of the known compact complex surfaces admit locally conformally K\"ahler structures. In the lcK context, the analogue of the projective space is played by {\em Hopf manifolds}, and an analogue of the Kodaira embedding has been proven by Liviu Ornea and Misha Verbitsky.

Inspired by Eugenio Calabi's work, we study isometric immersions of lcK manifolds into Hopf manifolds. In particular, we focus on non-K\"ahler compact complex surfaces. We also discuss problems concerning cohomological invariants of locally conformally K\"ahler or symplectic manifolds, and classifications of lcs and lcK structures on Lie groups.

The talk is based on a joint work with Michela Zedda, and on works with Adri\'an Andrada, Giovanni Bazzoni, Marcos Origlia, Alexandra Otiman, Maurizio Parton, Nicoletta Tardini, Luis Ugarte.

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