Proposed Course:
Complex Manifolds
John M. Lee

Complex manifolds, like smooth (real) manifolds, are generalizations of curves and surfaces to arbitrary dimensions, but with coordinate charts taking their values in Cⁿ and 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 decomposition 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.