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).

All are of very geometric flavour in both statement and proof. In the course of proof, we would like to emphasize on: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.