Math 549
Geometric Structures (Complex Manifolds)
John M. Lee
Spring 2004, Monday-Wednesday-Friday 1:30-2:20
Complex manifolds, like smooth (real) manifolds, are generalizations of
curves and surfaces to arbitrary dimensions, but with coordinate charts taking
their values in Cnand overlapping
holomorphically. Despite the formal similarity between the definitions, the
theory of complex manifolds is much deeper than just smooth real manifold theory
with "complex" substituted for "real" throughout. Just to get the flavor of how
the subject differs from smooth manifold theory, consider the following facts:
(1) All complex manifolds are orientable, and in fact carry a canonical
orientation. (2) The only global holomorphic functions on a compact complex
manifold are the constant functions. (3) There are no compact complex
submanifolds of Cn of dimension greater
than zero. (4) There is no such thing as an analytic bump function or partition
of unity, so most local analytic objects such as functions and vector fields
cannot be pieced together into global ones. (5) The space of holomorphic vector
fields on a compact complex manifold is finite-dimensional, and in many
interesting cases contains only the zero vector field.
Complex manifolds are ubiquitous in modern mathematics. They play key roles
in
- Riemannian geometry (Kähler metrics);
- Classical complex analysis (Riemann surfaces);
- Several complex variables (Stein manifolds);
- Algebraic geometry (nonsingular complex algebraic varieties);
- Low-dimensional topology (classification of 4-manifolds);
- Lie theory (complex Lie groups);
- Representation theory (complex flag varieties);
- String theory (Calabi-Yau manifolds); and even
- Number theory (elliptic curves, Jacobian varieties, modular
forms).
This course will introduce the basic concepts and machinery for studying
complex manifolds. The topics covered will definitely include the following:
- Basic definitions and examples of complex and almost complex manifolds
- Holomorphic vector bundles
- Complex projective space and its submanifolds
- Line bundles and divisors
- Hermitian and Kähler metrics
- The Dolbeault complex and Dolbeault cohomology
- Connections and curvature on Hermitian vector bundles
- Chern classes and Chern-Weil theory
Depending on time and the interests of the class, we might also cover some of
the following:
- Sheaves, sheaf cohomology, and the Dolbeault theorem
- The Hodge theorem for compact Kähler manifolds
- The Kodaira-Chow embedding theorem
- Riemann surfaces and the Riemann-Roch theorem
- Stein manifolds
Prerequisites: To succeed in this course, you need to have a
good understanding of smooth manifolds, Riemannian geometry, and basic complex
analysis. Successful completion of Math 547 and 534 should be sufficient.