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