New Perspectives on the classification
of Fano Manifolds

Preliminary Programs

The School consists of two courses of 10 hours each. Lessons in the morning start at 9.00am. There will be a coffee break at 11.00am. After lunch break there will be supporting lectures on the courses' topics and exercise classes.

Mirror Symmetry and Fano manifolds (A. Corti)

Prerequisites:

Basic algebraic geometry (for example R. Hartshorne, Algebraic Geometry, 1977, Springer, or I. Shafarevich, Basic Algebraic Geometry 1 and 2, 2007, Springer).

Program:

The object of this course is to introduce a new perspective on the classification of Fano manifolds coming from work in mirror symmetry. The quantum period of a Fano manifold X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. Indeed, we say that a Fano manifold X is mirror to a Laurent polynomial f if the quantum period of X equals the classical period integral of f. [3] shows by explicit computation that the 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by a combinatorially defined collection of Laurent polynomials called Minkowski polynomials. This strongly suggests a new approach to the classification of Fano manifolds: by proving an appropriate mirror theorem and then classifying Fano mirrors. The distinct advantage of this approach is that the mirrors are objects of combinatorics: given enough computing power, their classification is a trivial matter and involves none of the subtle projective and algebraic geometry that one needs in the classical approach.
The only known theoretical framework for mirror symmetry which is general enough and constructive enough for this new approach to Fano classification is the Gross-Siebert program. In the Gross-Siebert program, pairs (X; D) of a Fano manifold and anticanonical divisor D arise, together with their mirrors, as smoothings of degenerate pairs made of toric components glued torically (the Laurent polynomial f arises from a tropical disk count). We expect that, at the time of this school, we will have made some progress interpreting the work of [3] in the framework of the Gross-Siebert program.
The lectures will cover the following topics:

1. a discussion of the works [1], [2], [3] and some of the necessary background on quantum cohomology, periods of integrals, and combinatorics of lattice polytopes and mutations;
2. an introduction to the Gross-Siebert program and the construction of pairs (X; D) by smoothing degenerate pairs made of toric components glued torically, by scattering diagrams;
3. examples of Fano 4-folds (depending on time and progress).

[1] M. Akhtar, T. Coates, S. Galkin and A. Kasprzyk, Minkowski polynomials and mutations, SIGMA Symmetry Integrability Geom. Methods Appl. 8 (2012), Paper 094, 17 pp.
[2] T. Coates, A. Corti, S. Galkin, V. Golyshev and A. Kasprzyk, Mirror Symmetry and Fano Manifolds, arXiv:1212.1722, to appear in the Proceedings of the 6th European Congress of Mathematics.
[3] T. Coates, A. Corti, S. Galkin and A. Kasprzyk, Quantum Periods for 3-Dimensional Fano Manifolds, arXiv:1303.3288.

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Explicit birational geometry and Fano varieties (Y. Prokhorov)

Prerequisites:

1. Basic algebraic geometry (for example R. Hartshorne, Algebraic Geometry, 1977, Springer, or I. Shafarevich, Basic Algebraic Geometry 1 and 2, 2007, Springer).
2. Basic Mori theory (e.g. K. Matsuki, Introduction to the Mori Program, 2002, Springer).

It would be also desirable that participants knew something about Fano varieties (e.g. V. Iskovskikh and Y. Prokhorov, Algebraic Geometry V: Fano varieties, 1999, Springer).

Program:

The object of this course is to reflect the present state of the classification theory of Fano varieties. This classification is closely related to the one of the finite subgroups of the group Bir(X) of birational automorphisms of a Fano variety X and these subgroups can be studied by a method coming from the Mori theory. In the course we will present many recent applications of this method obtained by many mathematicians (Beauville, Blanc, Dolgachev, de Fernex, Iskovskikh, Lamy, Cantat, Cheltsov, Shramov and others).
The following topics will be covered:

1. basic properties, lower-dimensional Fano varieties;
2. equivariant Mori theory: arithmetic and geometric cases;
3. singular Fano varieties and "good" degenerations;
4. applications: rationality problems, Cremona groups, etc.