Course Description. The course aims to give brief introduction to basic notions and techniques in complex differential geometry and complex algebraic geometry. The language will be mainly geometric and/or analytic and we will assume material covered in MAT 530/531, MAT 534/535 as well as MAT 539 (Algebraic Topology), and MAT 542/543 (Complex Analysis). It should however be possible to fill in some of the gaps during the semester.

MEETING PLACE AND TIME. TuTh 9:50am-11:10am, Physics P122.

TEXTBOOK. No particular text will be used exclusively for the course, but the plan is to cover parts of chapters 0-1-2 in Ph. Griffiths' and J. Harris' textbook Principles of Algebraic Geometry.

MAT 543: The following is a tentative list of what we will try to cover in class:

• Notions of several complex variables/analytic geometry: Cauchy's Integral Formula, Hartog's theorem, Weierstrass preparation theorems, analytic Nullstellensatz, Riemann's extension theorem, etc.
• Basic Sheaf theory: presheaves, sheaves, cohomology of sheaves, long exact sequence of cohomology, basic computations and applications (among them the de Rham and Dolbeault theorems).
• Complex differential geometry: complex vector bundles, hermitian metrics, connections, hermitian vector bundles, the metric connection, sub-bundles and quotient bundles, tensor bundles, curvature, Chern classes (differential geometric approach), positivity.
• Kaehler differential geometry: The Kaehler condition, projective manifolds, basic topological consequences of the Kaehler condition, the Kaehler identities.
• Hodge theory: The Hodge Theorem: statement (no proof), and consequences (harmonic forms=Dolbeault cohomology, finite dimensionality, Poincare' duality, Serre duality), Hodge decomposition for compact Kaehler manifolds. The Kodaira (Akizuki-Nakano) vanishing theorem, the Hard Lefschetz Theorem.
• Divisors and line bundles: Divisors and line bundles, linear systems and maps to projective space Bertini's Theorem, exponential sequence and Chern classes, adjunction formula, Cartan-Serre's Theorems A and B, Lefschetz theorem on (1,1) classes, blowing up, Kodaira embedding theorem, projective bundles, Chern classes revisited.

• Topology of projective varieties: Weak Lefschetz Theorem, Lefschetz pencils, monodromy, Picard-Lefschetz, relation with the Hard Lefschetz Theorem.
• Curves: Riemann-Roch, compact Riemann surfaces are projective, Riemann-Hurwitz, genus formula, Abel's Theorem, the Jacobian, Jacobi inversion, tori, Albanese variety.

There will be no homework and no exams. Students are expected to give in-class presentations on selected related topics/exercises. I will assist students in preparing these presentations.

Exams: There will be no exams.

Office Hours: By appointment: By appointment: on TU and TH ??? and on TU ????. MAT Tower 5-108 (on TU am I could be in P-143).

Grade: Based on in-class presentations and in-class participation.

Schedule of in-class presentations: To be announced.

Quod non est in web non est in mundo