K3 Surfaces

Alessandra Sarti
Laboratoire de Mathématiques et Applications
Université de Poitiers (France)
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 projective space.
The name K3 was given by André Weil in 1958 in honour of the three remarkable mathematicians: Kummer, Kähler and Kodaira and of the beautiful K2 mountain in Cachemire.
The topics of the lectures are the following (see, [1], [2], [4]):
  • K3 surfaces in the Enriques-Kodaira classification.
  • Examples: complete intersections, Kummer surfaces.
  • Basic properties of K3 surfaces.
  • Period domain: Torelli Theorem and the surjectivity of the period map.
  • Linear systems and embeddings in projective space.
  • Automorphisms of K3 surfaces
The preliminaries for this course are (see, the first chapters of [3]):
  • Basic notions of Algebraic Geometry such as: Zariski topology, algebraic variety, regular and rational functions, birational morphisms, blow-up.
  • Sheaves and vector bundles.
  • Divisors, intersection product.
  • Cech cohomology.
  • Some simplicial homology and cohomology.
References:
[1] W. P. Barth; K. Hulek; Ch. A. M. Peters; A. Van de Ven. Compact complex surfaces. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 4. Springer-Verlag, Berlin, 2004.
[2] A. Beauville. Complex algebraic surfaces. Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. Second edition. London Mathematical Society Student Texts, 34. Cambridge University Press, Cambridge, 1996.  
[3] R. Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
[4] D. Huybrechts. Lectures on K3 surfaces. http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf