In this thesis we study metrics with special curvature properties on some homogeneous almost-Kähler manifolds. More precisely, given a symplectic manifold (M, ω) equipped with a compatible almost-complex structure J, we consider the homogeneous equation ρ = λω, where ρ is the Chern-Ricci form of J, that we call speciality condition. In particular, we focus on two classes of symplectic manifolds: symplectic T 2 -bundles over T 2 and adjoint orbits of semisimple Lie groups. Symplectic T 2 -bundles over T 2 are distributed in five classes. We prove that the ones belonging to four of these classes admit a special (Chern-Ricci flat) locally homogeneous compatible almost-complex structure, while the ones in the remaining class do not admit Chern-Ricci flat locally homogeneous compatible almost-complex structures. It is an open problem whether they admit non-locally homogeneous special compatible almost-complex structures. Adjoint orbits of semisimple Lie groups turn out to be naturally almost-Kähler mani- folds endowed with the Kirillov-Kostant-Souriau symplectic form and a canonically defined almost-complex structure. We give explicit formulae for the Chern-Ricci form, the Hermi- tian scalar curvature and the Nijenhuis tensor in terms of root data and we discuss the speciality condition, which may be translated in terms of root data as well. Moreover, we examine when compact quotients of these orbits are Kähler manifolds.

Special almost-Kähler geometry of some homogeneous manifolds

GATTI, ALICE
2019-12-13

Abstract

In this thesis we study metrics with special curvature properties on some homogeneous almost-Kähler manifolds. More precisely, given a symplectic manifold (M, ω) equipped with a compatible almost-complex structure J, we consider the homogeneous equation ρ = λω, where ρ is the Chern-Ricci form of J, that we call speciality condition. In particular, we focus on two classes of symplectic manifolds: symplectic T 2 -bundles over T 2 and adjoint orbits of semisimple Lie groups. Symplectic T 2 -bundles over T 2 are distributed in five classes. We prove that the ones belonging to four of these classes admit a special (Chern-Ricci flat) locally homogeneous compatible almost-complex structure, while the ones in the remaining class do not admit Chern-Ricci flat locally homogeneous compatible almost-complex structures. It is an open problem whether they admit non-locally homogeneous special compatible almost-complex structures. Adjoint orbits of semisimple Lie groups turn out to be naturally almost-Kähler mani- folds endowed with the Kirillov-Kostant-Souriau symplectic form and a canonically defined almost-complex structure. We give explicit formulae for the Chern-Ricci form, the Hermi- tian scalar curvature and the Nijenhuis tensor in terms of root data and we discuss the speciality condition, which may be translated in terms of root data as well. Moreover, we examine when compact quotients of these orbits are Kähler manifolds.
13-dic-2019
File in questo prodotto:
File Dimensione Formato  
tesi.pdf

accesso aperto

Descrizione: tesi di dottorato
Dimensione 742.27 kB
Formato Adobe PDF
742.27 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11571/1292132
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact