Math 582D
Calabi-Yau Manifolds
Charles Doran
Winter 2006, Monday/Wednesday/Friday, 11:30
Ever since the proof by S.-T. Yau of the Calabi conjecture in the
mid-1970s, manifolds admitting a Ricci-flat Kähler-Einstein metric
(so-called Calabi-Yau manifolds) have fascinated the mathematics
community. Yau's proof—of the existence of a unique such metric within
each Kähler class on a manifold with c1=0—was
entirely nonconstructive. Nevertheless, for many applications in both
differential and algebraic geometry, the existence of such a metric
satisfying this simple topological criterion has sufficed to yield a host
of important consequences. Over the past 20 year Calabi-Yau manifolds have
taken on new importance through their role as compactification spaces for
the extra dimensions in string theory. String theory has provided more than
just further motivation to study Calabi-Yau manifolds: ideas from string
theory have steered the development of the subject in exciting and
mathematically unexpected directions.
The goal of this course is to provide an introduction to Calabi-Yau
manifolds and their associated structures: e.g., Picard-Fuchs differential
equations, Bryant-Griffiths contact structure on the period domain,
intermediate Jacobians and "special geometry", special Lagrangian
submanifolds, etc. This will include a balance between
differential/algebro-geometric proofs and applications, as well as an
emphasis on the concrete realization (e.g., as hypersurfaces or complete
intersections in toric varieties) and manipulation of Calabi-Yau
manifolds. Physical applications, including certain forms of Mirror
Symmetry will also be discussed.
Prerequisites: The course should be accessible to those who have
completed either Math 547 (the first quarter of Geometric Structures) or
Math 507 (the first quarter Algebraic Geometry),
or whose background lies in string theory.