Programme:

The lectures will focus on two related classes of varieties: symplectic manifolds (K.O.) and Fano manifolds (J.W.).

Holomorphic symplectic manifolds form one of the three important building blocks of compact Kaehler manifolds with vanishing first Chern class. In the lectures, K. Oguiso will explain recent impressive progress on fiber space structures on a holomorphic symplectic manifold, which are mostly done by D. Matsushita and J.M. Hwang.

It is well known from Mori theory that in general varieties can be divided into some classes among which fibrations whose general fiber is Fano play a major role. The second part of the lectures will concern Fano manifolds and their study via rational curves.

Prerequisites: Basic knowledge of algebraic geometry.

Organizers: Adrian Langer and Jaroslaw Wisniewski. The school will take place in a Warsaw University pension in Lukecin (look here for more information), on Western part of Polish Baltic sea shore (see a map). The accommodation (full board, double room) will cost about 80 zloty (PLN) a day (1 Euro is approx. 3.5 PLN). The conference fee of 50 zloty will apply. Graduate students and postdocs with inadequate support from home institution may apply for accomodation cost waiver.

The school will be financially supported by Institute of Mathematics of Warsaw University and by Polish State Committee for Scientific Research.

All inquiries should be directed to Adrian Langer: alan at mimuw.edu.pl . Registration deadline: July 1st, 2008.


K.Oguiso's plan of the lectures on Lagrangian fibration of symplectic manifolds

1) general structure of fibered holomorphic symplectic manifold

2) a more explicit structure of smooth fibers and general singular fibers

3) the base space of fibered projective holomorphic symplectic manifold when it is smooth (main part of lectures)

4) structure theorem of Mordell-Weil group when the fibration admits a rational section (if time allows).

i) how Beauville-Fujiki's form on a holomorphic symplectic manifold can be effectively used in this study

ii) how one can use non-degenerate symplectic two form in concrete geometry

iii) beautiful harmony, found by J.M. Hwang, between geometry of rational curves on the base (Fano manifolds) and geometry of fibers of Lagrangian fibrations (abelian varieties and their degenerations) through non-degenerate symplectic two form.

1. Classical theory: projective manifolds with special linear sections, adjunction, canonical divisor, vanishings. Projective space, quadric and other hypersurfaces, del Pezzo manifolds and Mukai manifolds.

2. Rational curves on Fano manifolds: existence and consequences. Rationally connected varieties. Finitness of deformation types of Fano's.

3. Mori theory on Fano manifolds, cone of curves, cone of nef divisors. Contractions of Fano manifolds, some structural results. Rigidity of Mori cone under deformations.

4. Optional: rationality of Fano's, maps between Fano's.