TOPICS IN ALGEBRAIC GEOMETRY

Tuesday -Thursday 2:40 -3:55 pm
Room 507 Math

In spite of the title, most of this course concerns homotopy theory and differential graded algebras of differential forms. The last few lectures will apply this theory to complex projective and quasi-projective varieties.

Here is a broad overview of the topics to be covered:

I. Generalities on Homotopy Theory
    Homotopy equivalence, weak homotopy equivalence, CW complexes, Whitehead's         Theorem, obsruction theory, Eilenberg-MacLane spaces, Postnikov towers

II. Rational Homotopy Theory
    Isomorphism on rational homotopy groups and on rational homology groups,
    formal definition inverting rational isomorphism, rational spaces, localization of             homotopy types, case of the fundamental group.

III. Homotopy theory of Differential Graded Algebras (DGA's)
        Homotopy of maps of DGA's, homotopy category of DGA's, minimal modelm in the         simply connected case, in the general case, Mal'cev completion of fundamental             group. Notionl of formality.

IV. Homotopy equivalence of DGA's and and rational homotopy theory of CW complexes
        simply connected case, rational nilpotent completion in the general case.

V. Kahler Identities for complex forms on a compact Kahler manifold 
        vanishing of Massey products via the principle of two types, formality theorem.
        consequences for the rational homotopy type of compact Kahler                                     manifolds/complex projective varieties.

VI. Mixed Hodge Structures
        basic definitions, existence on cohomology, extension to the entire rational                     homotopy type

            
 
Link to problems on CW complexes
Link to Terek's Kahler metrics paper.
Link to paper on Kahler identities.