MAT 615: We will discuss some aspects of the geometry associated with (suitably) positive line bundles, e.g. The Kodaira Vanishing Theorem, the Weak Lefschetz Thoerem, The Hard Lefschetz Theorem, L^2-methods, Nadel Vanishing Theorem, non-vanishing Theorems.
Textbook(s): We shall use several sources, including R. Lazarsfeld's ``Positivity in Algebraic Geometry."
Lecture-by-lecture syllabus:
1. Review of terminology: Cartier and Weil divisors, Picard group, line bundles, locally free sheaves. Line bundles on P^n. Morphisms to projective space. Globally generated sheaf.
2. Evaluation map. Linear systems. Base Locus. Very ample. Very ample and separating points and vectors. Serre's Thm and resolution by special vector bundles on P^n. Ample. Cartan-Gorthendieck-Serre. Ample and finite maps. Ample and finite and surjective maps.
3. Kodaira Vanishing via cyclic covers.
4. Continued. Basic properties like: ample+free=ample, D= (very ample)-(very ample), etc.
5. Ampleness and very ampleness on smooth projective curves. Bertini. Kollar's example (for Matsusaka's).
6. Continued. Push-forwad for finite maps. Normal bundle of the diagonal.
7. Basic intersection theory: examples of quadrics (2 and 3 dim), normal surfaces, general definition and intepretation of intersection numbers. Numerical trivial, Neron-Severi and Thm of the base. Int. # well-defined on numerical classes.
8. Asymptotic RR. Producing singular sections. Maximal growth not on h^0. Nakai-Moishezon-Kleiman.
9. Continued. Consequences of Nakai Moshezon.
10. Nef line bundles. Kleiman's characterization. Consequences: nef as limit of ample, Seshadri etc.
11. No class.
12. Basic Hermitean algebra: h,g,omega. Fubini-Study on TP^n. Fubini-Study on O(1). Integrality of cohomology classes. Positive (1,1)-forms.
13. Connections on smooth complex bundles. Curvature. Cartan's. Cohomology class associated to line bundle via connection. Review of Check cohomology and the coboundary operator. Coincidence of the first Chern class and [1/(2pi i)\Theta].
14. Kodaira vanishing (Kodaira's proof).
15. Bochner technique, a priori inequality. Metrics on nef line bundles.
16. Positive (1,1)-forms and metrics via d-dbar lemma.
17. Kodaira Embedding Theorem.
18 Mention of THM B.... Positive (1,1)-currents. Psh functions.
19 Review of currents. Cauchy's integral formula in distribution from. Poincare'-Lelong.
20. Basics about Psh: Lelong number and multiplicity. Nadel Vanishing Theorem.
21. Effective global generation and very ampleness on curves via shm. Push forward of O_X and K_X for bimeromorphic maps of manifolds. Change of variables and multiplier ideal.
22. Analytic and algebraic singularities for psh fcns. Multiplier ideal for ncd. For analytic singularities via resolutions and change of variables.
23. Big divisors. Kodaira-type Vanishing for pull-back of ample via surjective map of surfaces. Statement of Kawamata-Vieheweg Vanishing for nef and big. Perturbed version. Implication K-V-VT --> Grauert-Riemannschneider.
24. Proof of perturbed K-V-VT. Coherent sheaves.
25. Coherent sheaves continued. Coherence of the multiplier ideal sheaf.
Special needs. If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. All information and documentation of disability is confidential. Arrangements should be made early in the semester (before the first exam) so that your needs can be accommodated.