MAT 545Complex Geometry

Instructor:

Sorin Popescu (office: Math 3-109, tel. 632-8255, e-mail sorin at math.sunysb.edu)

Schedule:

TuTh 9:50am-11:10am Physics P123

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.

Textbook(s)

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 and part of Claire Voisin's textbook Hodge Theory and Complex Algebraic Geometry I. (These two textbooks will be in the library on reserve.)

 Other textbooks that you may find useful are the following: Complex analytic and algebraic geometry, Jean-Pierre Demailly (a PDF file of the current version). Download here a DjVu (searchable) version. Complex Differential Geometry, Sh. Kobayashi, H.H. Wu Basic Algebraic Geometry 1 & Basic Algebraic Geometry 2, I.R. Shafarevich, Springer 1995 Topologie Algebrique et Theorie des Faisceaux, R. Godement. Coherent Analytic Sheaves, H. Grauert and R. Remmert. Complex Algebraic Curves, F. Kirwan

Topics

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.
If time allows any of the following topics may perhaps be also discussed:
• 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.

Homework

I will assign problems in each lecture, ranging in difficulty from routine to more challenging. They are due the following week.