MATH 666 lectures:

8/27: gave overview of the class, defined complex and almost complex manifolds (with a few definitions to
be supplied later), reviewed definition of Riem. manifold, stated Cauchy formula
8/29: gave differential-geometric backround for Cauchy formula (e.g. Stoke's theorem) and proved it.
8/31: proved Cauchy formula in several variables, Riemann and Hartogs extension theorems (sect. 1.2)
9/3 : existence of solutions to d-bar (1.3), generalities on vector bundles, decomposition of complexified
tangent bundle
9/5: went over hw 1,  stated Frobenius, Newlander-Nirenberg and d-bar Poincare theorems
9/7:  explained why \overline\partial_E is well defined even though d_E is not, gave examples
of complex manifolds including Grassmannians
9/10: began discussion of Riemannian, symplectic and complex geometry and how they
come together in Kahler geometry - stated Fundamental Lemma of Riem. geometry.
also described tautological quotient bundle on the Grassmannian
9/12: went over hw, discussed connections on Riem, symplectic and hermitian manifolds
9/14: equivalence of 3 definitions of Kahler, flatness of Kahler to order 2 at a point in coordinates
9/17 examples of Kahler manifolds (3.3)
10/1 examples cont'd , blowups
10/3 blowups cont'd, began sheaves (chap 4)
10/5 presheaves, their morphisms
10/8 sheaf maps, complexes
10/10 examples of sheaves, resolutions, poor man's sheaf cohomology
10/12 abelian categories
10/15 injective resolutions
10/17 higher direct images, sheaf cohomology
10/18 flasque, the ever-useful double complex trick
10/22 homework covered, interpretations of H^1
10/24 interpretations cont'd, intro to chap 5
10/26 Laplacians on Riem. manifolds
10/29 Laplacians cont'd
10/31  fundamental theorem on elliptic operators
11/2 covered homework, Kahler identities
11/5 more Kahler identitites, \del\bar\del lemma
11/7 Hodge index theorem
11/9 Hodge index cont'd, c_1(L)
11/12 c_1(L) and Kodaira vanishing