Math 539 Topics in Algebraic Geometry

Time: Mon, Wed, Fri 13:00–13:50 Crow/205
Office Hours: Mon 14:00–14:50 and Fri 14:00–14:50 in Cupples I/205
Exam: My preferred approach is for students to have individual projects with talk at end of course.

Syllabus

The central theme of the course is mirror symmetry for Fano varieties, also known as the Fano/Landau–Ginzburg correspondence. The following is an approximate list of possible topics: finer decisions will be made along the way. I will try to make contact with topics of interest to the audience.

Mirror symmetry for nonsingular algebraic surfaces as a correspondence between algebraic families of Fano surfaces and mutation equivalence classes of plane lattice polygons.

Elements of a general theory of cluster varieties, volume-preserving birational maps, mutations. Factorization of volume-preserving birational maps in terms of mutations. The holomorphic symplectic case and cluster algebras.

The mirror of a Fano as a cluster variety. Toric degenerations of a (nonsingular) Fano surface are in one-to-one correspondence with the torus charts on the mirror cluster variety. Generalization to orbifold surfaces. Conjectural generalization to Fano varieties in higher dimensions. Higher dimensional examples eg from Grassmannians (audience please help!).

Formulation of mirror symmetry in terms of quantum cohomology and quantum orbifold cohomology. Introduction to quantum orbifold cohomology.

Mirror symmetry for Fano 3-folds: several approaches (Dolgachev, Golyshev, KKP, Corti et al). Sufficient condition for smoothing Gorenstein toric Fano 3-folds from admissible decomposition data. Construction of the mirror from admissible decomposition data. 0-mutable polynomials. The case of Q-Fano 3-folds.