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.
Author(s): Lucia Caporaso, James McKernan, Mircea Mustata, Mihnea Popa
Series: Mathematical Sciences Research Institute Publications
Edition: 1
Publisher: Cambridge University Press
Year: 2012
Language: English
Pages: 437
Contents
Preface
Fibers of projections and submodules of deformations
1. Introduction
2. Notation
3. Measuring the complexity of the bers
4. General projections whose bers contain given subschemes
5. A transversality theorem
Introduction to birational anabelian geometry
1. Abstract projective geometry
2. K-theory
3. Bloch-Kato conjecture
4. Commuting pairs and valuations
5. Pro-`-geometry
6. Pro-`-K-theory
7. Group theory
8. Stabilization
9. What about curves?
Periods and moduli
1. Attaching an abelian variety to an algebraic object
2. Periods and period maps
3. Is the period map injective?
4. Moduli spaces
The Hodge theory of character varieties
1. Introduction
2. The decomposition theorem
3. Ng�'s support theorem
4. The perverse ltration and the Lefschetz hyperplane theorem
5. Character varieties and the Hitchin bration: P D W0
6. Appendix: cup product and Leray ltration
Rigidity properties of Fano varieties
1. Introduction
2. General properties of Fano varieties
3. Mori-theoretic point of view
4. Deformations of the Cox rings
5. Deformations of the Mori structure
The Schottky problem
1. Introduction
2. Notation: the statement of the Schottky problem
3. Theta constants: the classical approach
4. Modular forms vanishing on the Jacobian locus:
5. Singularities of the theta divisor: the Andreotti-Mayer approach
6. Subvarieties of a ppav: minimal cohomology classes
7. Projective embeddings of a ppav: the geometry of the Kummer variety
8. The 000 conjecture
Interpolation
1. The interpolation problem
2. Reduced schemes
3. Fat points
4. Recasting the problem
Chow groups and derived categories of K3 surfaces
1. Introduction
2. Cohomology of K3 surfaces
3. Chow ring
4. Derived category
5. Chern classes of spherical objects
6. Automorphisms acting on the Chow ring
Geometry of varieties of minimal rational tangents
1. Introduction
2. Preliminaries on distributions
3. Equivalence of cone structures
4. Varieties of minimal rational tangents
5. Isotrivial VMRT
6. Distribution spanned by VMRT
7. Linear VMRT
8. Symmetries of cone structures
Quotients by nite equivalence relations
1. De nition of equivalence relations
2. First examples
3. Basic results
4. Inductive plan for constructing quotients
5. Quotients in positive characteristic
6. Gluing or pinching
Higher-dimensional analogues of K3 surfaces
0. Introduction
1. Examples
2. General theory
3. Complete families of HK varieties
4. Numerical Hilbert squares
Compacti cations of moduli of abelian varieties: an introduction
1. Introduction
2. Three perspectives on g
3. Degenerations
4. Compacti cations
5. Higher degree polarizations
The geography of irregular surfaces
1. Introduction
2. Irregular surfaces of general type
3. The Castelnuovo-de Franchis inequality
4. The slope inequality
5. The Severi inequality
Basic results on irregular varieties via Fourier-Mukai methods
1. Fourier-Mukai transform, cohomological support loci, and GV-sheaves
2. Generic vanishing theorems for the canonical sheaf and its higher
3. Some results of Ein and Lazarsfeld
4. The Chen-Hacon birational characterization of abelian varieties
5. On Hacon's characterization of theta divisors
Algebraic surfaces and hyperbolic geometry
1. Introduction
2. The main trichotomy
3. Ample line bundles and the cone theorem
4. Beyond Fano varieties
5. The cone conjecture
6. Outline of the proof of Sterk's theorem
7. Nonarithmetic automorphism groups
8. Klt pairs
9. The cone conjecture in dimension greater than 2
10. Outline of the proof of Theorem 8.7
11. Example
