Etiquette: Phone should be turned off and inside pockets, backpacks, etc. No texting, web surfing etc. No food. Please use email only for emergencies (you can always talk to me in class, office hours, ...).

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 2:20pm-3:40am, 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. The following book of mine contains some of this material and exercises: The Hodge Theory of Projective Manifolds. The follwoing are notes for a summer school course on algebraic maps and contain plenty of examples and exercises on sheaves: The Hodge Theory of maps.

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 (made in person in class; please use email for emergencies only) on TU 4-6pm and TH 4-5pm. 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: Announced in class. Students are responsible for keeping track of these announcements and should consult with each other if they miss classes.

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