Algebraic geometry is one of the most diverse fields of research in mathematics. It has had an incredible evolution over the past century, with new subfields constantly branching off and spectacular progress in certain directions, and at the same time, with many fundamental unsolved problems still to be tackled. In the spring of 2009 the first main workshop of the MSRI algebraic geometry program served as an introductory panorama of current progress in the field, addressed to both beginners and experts. This volume reflects that spirit, offering expository overviews of the state of the art in many areas of algebraic geometry. Prerequisites are kept to a minimum, making the book accessible to a broad range of mathematicians. Many chapters present approaches to long-standing open problems by means of modern techniques currently under development and contain questions and conjectures to help spur future research.
Chapter 1. Fibers of projections and submodules of deformations
Chapter 2. Introduction to birational anabelian geometry
Chapter 3. Periods and moduli
Chapter 4. The Hodge theory of character varieties
Chapter 5. Rigidity properties of Fano varieties
Chapter 6. The Schottky problem
Chapter 7. Interpolation
Chapter 8. Chow groups and derived categories of K3 surfaces
Chapter 9. Geometry of varieties of minimal rational tangents
Chapter 10. Quotients by nite equivalence relations
Chapter 11. Higher-dimensional analogues of K3 surfaces
Chapter 12. Compacti cations of moduli of abelian varieties: an introduction
Chapter 13. The geography of irregular surfaces
Chapter 14. Basic results on irregular varieties via Fourier–Mukai methods
Chapter 15. Algebraic surfaces and hyperbolic geometry
