-- V. Muñoz, A. Gonzalez-Perito, J. A. Rojo, "Geometry and Topology of Manifolds: Surfaces and Beyond", AMS.
-- A. Moroianu, "Lectures on Kähler Geometry", LMS.
-- C. Voisin, "Hodge Theory and Complex Algebraic Geometry, I", Cambridge.
-- J.-P. Demailly, "Complex Analytic and Differential Geometry", https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
-- D. Huybrechts, "Complex Geometry", Springer.
-- W. Ballmann, "Lectures on Kähler Manifolds", EMS
-- W. M. Boothby, "An introduction to differentiable manifolds and riemannian geometry", Mathematics 120, Academic Press Inc. !986.
-- M. Audin, "Torus actions on symplectic manifolds", Progress in Mathematics 93, Birkhauser, 2nd ed., 2004.
-- A. Cannas da Silva, "Lectures on symplectic geometry", Lectures Notes in Mathematics 1764, Springer, 2008.
-- V. Guillemin, "Moment maps and combinatorial invariants of Hamiltonian T-spaces". Progress in Mathematics 122, Birkhauser 1994.
--L. Conlon. "Differentiable Manifolds" Second Edition, MBC Birkhauser, 2001.
--M. W. Hirsh, "Dfferential Topology" GTM Springer-Verlag 1976.
Learning Objectives
To acquire notions, tools, and methodologies aimed at investigation and research in Riemannian, complex, symplectic, and Kählerian geometry; to strengthen abilities in processing and presentation.
Prerequisites
Basics in differential geometry and Riemannian geometry.
Teaching Methods
Lessons, exercise sessions, and seminars.
Further information
For further information, contact the professors.
Type of Assessment
Oral exam.
Course program
-- Riemannian geometry: review of Riemannian manifolds, connections, curvatures, Einstein metrics.
-- Complex geometry: almost-complex manifolds and complex manifolds; integrability of almost-complex structures; examples; Riemann Uniformization Theorem.
-- Hermitian geometry: Hermitian metrics; connections and curvatures of Hermitian metrics.
-- Kählerian geometry: Kähler structures; projective algebraic manifolds and Kodaira embedding theorem; Hodge theory for Kähler manifolds; curvature of Kähler metrics; special Kähler metrics; Calabi-Yau manifolds.
-- Group actions on manifolds: introductory elements on Lie groups; group actions on manifolds, vector fields, statement and proof of the slice theorem (equivariant tubular neighborhood theorem).
-- Symplectic geometry: differential geometry of symplectic manifolds. Hamiltonian torus actions: convexity theorem and symplectic toric manifolds.