Complex surfaces | Algebraic surfaces | Differential geometry
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface in complex projective 3-space. Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy to classify) and the negatively curved surfaces of general type (which are essentially unclassifiable). K3 surfaces can be considered the simplest algebraic varieties whose structure does not reduce to curves or abelian varieties, and yet where a substantial understanding is possible. A complex K3 surface has real dimension 4, and it plays an important role in the study of smooth 4-manifolds. K3 surfaces have been applied to Kac–Moody algebras, mirror symmetry and string theory. It can be useful to think of complex algebraic K3 surfaces as part of the broader family of complex analytic K3 surfaces. Many other types of algebraic varieties do not have such non-algebraic deformations. (Wikipedia).
Alessandra Sarti: Topics on K3 surfaces - Lecture 2: Kummer surfaces
Abstract: Aim of the lecture is to give an introduction to K3 surfaces, that are special algebraic surfaces with an extremely rich geometry. The most easy example of such a surface is the Fermat quartic in complex three-dimensional space. The name K3 was given by André Weil in 1958 in hono
From playlist Algebraic and Complex Geometry
Alessandra Sarti: Topics on K3 surfaces - Lecture 3: Basic properties of K3 surfaces
Abstract: Aim of the lecture is to give an introduction to K3 surfaces, that are special algebraic surfaces with an extremely rich geometry. The most easy example of such a surface is the Fermat quartic in complex three-dimensional space. The name K3 was given by André Weil in 1958 in hono
From playlist Algebraic and Complex Geometry
Alessandra Sarti: Topics on K3 surfaces - Lecture 5: Finite automorphism groups
Abstract: Aim of the lecture is to give an introduction to K3 surfaces, that are special algebraic surfaces with an extremely rich geometry. The most easy example of such a surface is the Fermat quartic in complex three-dimensional space. The name K3 was given by André Weil in 1958 in hono
From playlist Algebraic and Complex Geometry
Alessandra Sarti: Topics on K3 surfaces - Lecture 4: Nèron-Severi group and automorphisms
Abstract: Aim of the lecture is to give an introduction to K3 surfaces, that are special algebraic surfaces with an extremely rich geometry. The most easy example of such a surface is the Fermat quartic in complex three-dimensional space. The name K3 was given by André Weil in 1958 in hono
From playlist Algebraic and Complex Geometry
Christian Liedtke: Crystalline cohomology, period maps, and applications to K3 surfaces
Abstract: I will first introduce K3 surfaces and determine their algebraic deRham cohomology. Next, we will see that crystalline cohomology (no prior knowledge assumed) is the "right" replacement for singular cohomology in positive characteristic. Then, we will look at one particular class
From playlist Algebraic and Complex Geometry
Alessandra Sarti: Topics on K3 surfaces - Lecture 6: Classification
Abstract: Aim of the lecture is to give an introduction to K3 surfaces, that are special algebraic surfaces with an extremely rich geometry. The most easy example of such a surface is the Fermat quartic in complex three-dimensional space. The name K3 was given by André Weil in 1958 in hono
From playlist Algebraic and Complex Geometry
Alessandra Sarti: Topics on K3 surfaces - Lecture 1: K3 surfaces in the Enriques Kodaira...
Lecture 1: K3 surfaces in the Enriques Kodaira classification and examples Abstract: Aim of the lecture is to give an introduction to K3 surfaces, that are special algebraic surfaces with an extremely rich geometry. The most easy example of such a surface is the Fermat quartic in complex
From playlist Algebraic and Complex Geometry
S. Filip - K3 surfaces and Dynamics (Part 1)
K3 surfaces provide a meeting ground for geometry (algebraic, differential), arithmetic, and dynamics. I hope to discuss: - Basic definitions and examples - Geometry (algebraic, differential, etc.) of complex surfaces - Torelli theorems for K3 surfaces - Dynamics on K3s (Cantat, McMullen)
From playlist Ecole d'été 2018 - Teichmüller dynamics, mapping class groups and applications
Complex surfaces 1: Introduction
This talk is part of a series giving an informal survey of complex algebraic surfaces. We give an overview of the Enriques-Kodaira classification, with examples of most of the different types of surfaces. We conclude by giving an example of a non-algebraic surface: the Hopf surface. Furth
From playlist Algebraic geometry: extra topics
Alessandra Sarti, Old and new on the symmetry groups of K3 surfaces
VaNTAGe Seminar, Feb 9, 2021
From playlist Arithmetic of K3 Surfaces
Edgar Costa, From counting points to rational curves on K3 surfaces
VaNTAGe Seminar, Jan 26, 2021
From playlist Arithmetic of K3 Surfaces
Hodge theory and derived categories of cubic fourfolds - Richard Thomas
Richard Thomas Imperial College London September 16, 2014 Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics with K3s associated to them at the le
From playlist Mathematics
Bianca Viray, The Brauer group and the Brauer-Manin obstruction on K3 surfaces
VaNTAGe seminar, February 23, 2021
From playlist Arithmetic of K3 Surfaces
Tony Varilly Alvarado, Descent on K3 surfaces: Brauer group computations and challenges
VaNTAGe seminar March 23, 2021 License: CC-BY-NC-SA
From playlist Arithmetic of K3 Surfaces
Francesca Balestrieri, The arithmetic of zero-cycles on products of K3 surfaces and Kummer varieties
VaNTAGe seminar, March 9, 2021
From playlist Arithmetic of K3 Surfaces
Frank Gounelas : Rational curves on K3 surfaces
Bogomolov and Mumford proved that every complex projective K3 surface contains a rational curve. Since then, a lot of progress has been made by Bogomolov, Chen, Hassett, Li, Liedtke, Tschinkel and others, towards the stronger statement that any such surface in fact contains infinitely many
From playlist Algebraic and Complex Geometry
Complex surfaces 5: Kodaira dimension 0
This talk is an informal survey of the complex projective surfaces of Kodaira number 0. We first explain why there are 4 types of such surfaces (Enriques, K3, hyperelliptic, and abelian) and then give a few examples of each type.
From playlist Algebraic geometry: extra topics
Richard Thomas - The Katz-Klemm-Vafa formula
Richard THOMAS (Imperial College London, UK)
From playlist Algèbre, Géométrie et Physique : une conférence en l'honneur