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Alma universitas studiorum parmensis A.D. 962 - Università di Parma
Università degli Studi di Parma

Real and Complex Manifolds: Geometry and Holomorphic Dynamics

Finanziato dall'Unione europea - NextGenerationEU
Ministero dell'Università e della Ricerca
Italia Domani - Piano Nazionale di Ripresa e Resilienza
Prin 2022
Dipartimento di Scienze Matematiche, Fisiche e Informatiche
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Details

ERC sector
PE1 - Mathematics
ERC subsector
PE1_6 - Geometry and global analysis
Project start date
CUP
D53D23005750006
Financial support received
€46.875,00

Description and purpose

The research activity of the Unit in Parma focuses on the following topics:

1) Harmonic forms for the Dolbeault Laplacian and Bott-Chern Laplacian on Hermitian manifolds. Special metrics on almost complex manifolds. Cohomological properties of complex non Kaehler manifolds. Kodaira dimension of almost complex manifolds.

2) Actions of real reductive groups G on a real submanifold X of a Kahler manifold Z. Riemannian and symplectic geometry of some non homogeneous domains in C^n.  Kaehler metrics with constant scalar curvaure.

3) Conformal Geometry of isotropic curves in the 3-dimensional complex quadric and their connections with surfaces theory. CR structures of homogeneous manifolds. Obstructions theory for k-nondegeneracy of real orbits in complex flag manifolds. 4) Holomorphic dynamics in C^2.


Website: https://sites.google.com/unifi.it/prin-2022-manifolds/home

Purpose

1) Study of Bott-Chern and Dolbeault harmonic forms on compact almost Hermitian manifolds. Existence of special metrics on complex Manifolds. Explicit computations of the Kodaira dimension of almost complex manifolds

2) Holomorphic reductive group actions on Kaehler manifolds admitting a gradient map.

3) Geometric and anlystic properties of microcanonical surfaces for the non linear Schrödinger equations. Special submanifolds of scalar flat ALE Kahler spaces.  In the framework of the CR geometry of transversal curves in the 3-sphere, we aim at investigating the total twist functional.  Provide existence conditions for an isometric and holomorphic immersion of a Kaehler manifold endowed with a canonical metric.

4) Dynamical properties of entire transcendental functions in C, especially the most exotic type of stable behaviour, namely wandering and Baker domains.

Achieved results

Study of the Bott-Chern cohomology of almost-complex manifolds. Computation of the Kodaira dimension for compact quotients of solvable groups with almost-complex structure. Construction of compact, non-Kaehler complex manifolds that satisfy the Del-Delbar Lemma. Formal, non-Kahler Aeppli-Bott-Chern complex manifolds. Compuation of Dolbeault numbers of almost Hermitian manifolds. Explicit constructions of balanced metrics and SKT metrics on solvmanifolds.

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