This course will be taught fully online. I will produce handwritten course notes, posted here (last update Nov 30). My intention is to update them weekly, as to keep up with the course. The course is also being taped and you can find the video recordings under the dates below.
Sept 14 Exterior algebra of a real inner product space, star operator, De Rham cohomology, Harmonic representation, complex versus real linear algebra.
Sept 16 Almost complex and complex manifolds, 𝜕 and related operators, Dolbeault lemma, holomorphic vector bundles, Dolbeault cohomology.
Sept 21 Hermitian forms, Levi form, strictly pseudoconvex function, Kähler metric (flat and Fubini-Study)
Sept 23 Exterior algebra of complex inner product space, review representation theory of sl(2), primitive decomposition.
Sept 28 Bilinear form on the primitive part, basic local identities on a Kähler manifold, proportionality of various Laplacians.
Sept 30 Hodge filtration and Hodge decomposition on the cohomology of kählerian manifold. Primitive decomposition of the cohomology of Kähler manifold, Hopf manifold, Hodge structure, polarization.
Oct 5 A compact Riemann surface is kählerian, polarizability of a Hodge structure as a restrictive condition, Albanese variety, Abel-Jacobi map and its basic properties.
Oct 7 Intermediate jacobians and associated Abel-Jacobi map.
Oct 12 Tate Hodge structure (and its motivation), Hodge structures are the objects of an abelian category, Deligne torus.
Oct 14 Sheaves, pre-image and direct image of a sheaf, ringed spaces and examples thereof, the modules over a ringed space make up an abelian category with enough injectives.
Oct 19 Right derived functors, cohomology of a sheaf, abstract De Rham theorem, a soft sheaf is acyclic for the direct image functor.
Oct 21 Singular, De Rham and Dolbeault cohomology as sheaf cohomology, the Hodge filtration as legacy of the holomorphic De Rham complex, the complex of relative differential forms, local systems, flat connection.
Oct 26 Gauss-Manin connection, Hodge filtration in a family, coherent modules and their higher direct images under proper mappings.
Oct 28 Griffiths transversality property, period domain and its horizontal distribution.
Nov 2 Siegel upper space as a period domain, moduli space of ppav’s, Gysin sequence, complex of logarithmic holomorphic forms, and the residue morphism.
Nov 4 The Hodge theory of the Gysin sequence, fitration by pole order and the Hodge filtration, cohomology of a smooth projective hypersurface. (Sorry for the mess I made here; I will redo some of this next time in a clearer, somewhat different manner.)
Nov 9 Continuation of the filtration by pole order and the associated graded pieces. Intermezzo on zero-dimensional graded complete intersections and their Cohen-Macaulay property.
Nov 11 Griffiths description of the Hodge structure on the primitive coomology of a hypersurface, the case of plane curve and examples of low degree. The meromorphic Picard group.
Nov 16 Picard group, canonical bundle of a hypersurface, Betti and De Rham descriptions of the Picard group.
Nov 20 Very ample and ample line bundles, Kodaira embedding theorem (no proof), theorem of Appell-Humbert.
Nov 23 Construction of Theta functions. Behavior of theta functions under translation. First step of the proof of the Lefschetz ampleness criterion.
Nov 30 Completion of the proof of Lefschetz ampleness criterion, connection of a holomorphic vectorbundle with Hermitian metric, Chern-Weil description of Chern classes. Application to holomorphic line bundles over a complex torus.
Dec 2 Mixed Hodge structure on a smooth complex variety.
Helpful references are (this list be will be expanded during the course).
R.O. Wells, Jr: Differential Analysis and Comples Manifolds, Spinger GTM 65
Ph. Griffiths, J. Harris: Principles of Algebraic Geometry, Wiley &Sons.
P. Deligne: Travaux de Griffiths Sém. Bourbaki 376 (June 1970) (in French).
Ph. Griffiths, On the periods of certain rational integrals I, II, Ann. of Math. 90 (1969), 460-495, 496-541.
H. Clemens, Ph. Griffiths: The intermediate Jacobian of the cubic threefold. Ann. of Math. 95 (1972), 281–356.
P. Deligne: Theorie de Hodge II (in French).