Page 116 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 116
TOPICS IN COMPLEX AND QUATERNIONIC GEOMETRY (MS-74)
Geometry of 3-(α, δ)-Sasaki manifolds and submersions over quaternionic
Kähler spaces
Ilka Agricola, agricola@mathematik.uni-marburg.de
Philipps-Universität Marburg, Germany
Coauthors: Leander Stecker, Giulia Dileo
We give a gentle introduction to the new class of 3-(α, δ)-Sasaki manifolds, which are a nat-
ural generalisation of 3-Sasaki manifolds. We prove that any such manifold admits a locally
defined Riemannian submersion over a quaternionic Kähler manifold. In the non-degenerate
case (δ = 0) we describe all homogeneous 3-(α, δ)-Sasaki manifolds fibering over symmetric
Wolf spaces and their noncompact dual symmetric spaces. In the compact base case, this yields
a complete classification of homogeneous 3-(α, δ)-Sasaki manifolds, while for non-compact
bases, we provide a general construction of homogeneous 3-(α, δ)-Sasaki manifolds fibering
over nonsymmetric Alekseevsky spaces, the lowest possible dimension of such a manifold be-
ing 19.
References
[1] Ilka Agricola, Giulia Dileo, Generalizations of 3-Sasakian manifolds and skew torsion,
Adv. Geom. 20 (2020), 331-374.
[2] Ilka Agricola, Giulia Dileo, Leander Stecker, Homogeneous non-degenerate 3-(α, δ)-
Sasaki manifolds and submersions over quaternionic Kähler spaces, to appear in Ann.
Global Anal. Geom.
Flags and Twistors
Amedeo Altavilla, amedeoaltavilla@gmail.com
Università di Bari, Italy
Coauthors: Edoardo Ballico, Maria Chiara Brambilla, Simon Salamon
In this talk, I will present some first results on the geometry of the flag manifold F as twistor
space of the complex projective plane. Firstly, I will present some general facts on low-degree
curves and surfaces in the flag manifold. Afterward, I will introduce the twistor fibration as-
sociated with the standard Hermitian metric in CP2 and describe the set of twistor fibers. In
the second part, I will give a description of the family of automorphisms of F that come from
unitary automorphisms of CP2 and I will show a classification result for a family of algebraic
surfaces in F, up to such transformations. For a special sub-family of these surfaces, namely
those which are j-invariant, I will give a deeper geometric description.
114
Geometry of 3-(α, δ)-Sasaki manifolds and submersions over quaternionic
Kähler spaces
Ilka Agricola, agricola@mathematik.uni-marburg.de
Philipps-Universität Marburg, Germany
Coauthors: Leander Stecker, Giulia Dileo
We give a gentle introduction to the new class of 3-(α, δ)-Sasaki manifolds, which are a nat-
ural generalisation of 3-Sasaki manifolds. We prove that any such manifold admits a locally
defined Riemannian submersion over a quaternionic Kähler manifold. In the non-degenerate
case (δ = 0) we describe all homogeneous 3-(α, δ)-Sasaki manifolds fibering over symmetric
Wolf spaces and their noncompact dual symmetric spaces. In the compact base case, this yields
a complete classification of homogeneous 3-(α, δ)-Sasaki manifolds, while for non-compact
bases, we provide a general construction of homogeneous 3-(α, δ)-Sasaki manifolds fibering
over nonsymmetric Alekseevsky spaces, the lowest possible dimension of such a manifold be-
ing 19.
References
[1] Ilka Agricola, Giulia Dileo, Generalizations of 3-Sasakian manifolds and skew torsion,
Adv. Geom. 20 (2020), 331-374.
[2] Ilka Agricola, Giulia Dileo, Leander Stecker, Homogeneous non-degenerate 3-(α, δ)-
Sasaki manifolds and submersions over quaternionic Kähler spaces, to appear in Ann.
Global Anal. Geom.
Flags and Twistors
Amedeo Altavilla, amedeoaltavilla@gmail.com
Università di Bari, Italy
Coauthors: Edoardo Ballico, Maria Chiara Brambilla, Simon Salamon
In this talk, I will present some first results on the geometry of the flag manifold F as twistor
space of the complex projective plane. Firstly, I will present some general facts on low-degree
curves and surfaces in the flag manifold. Afterward, I will introduce the twistor fibration as-
sociated with the standard Hermitian metric in CP2 and describe the set of twistor fibers. In
the second part, I will give a description of the family of automorphisms of F that come from
unitary automorphisms of CP2 and I will show a classification result for a family of algebraic
surfaces in F, up to such transformations. For a special sub-family of these surfaces, namely
those which are j-invariant, I will give a deeper geometric description.
114
