Fano Manifolds

Jaroslaw Wisniewski, Warsaw University

ruled surface Ciechanow water tower source: Wikipedia

Contents of lectures. Most topics are covered in standard references [CKM], [D*], [M], [KM]. I add only references which are specific for the particular subject. Introduction: Fano manifolds, Kobayashi-Ochiai theorem. Existence of rational curves on Fano manifolds, Frankel-Hartshorne conjecture [M1]. Cone of curves on Fano manifolds, Mori theorem on the cone of curves [M2]. Mukai's conjecture about Fano's of large (pseudo)index [W3*]. Contractions of faces of Mori's cone; Kawamata-Shokurov base-point freeness. Castelnuovo theorem. Types of Mori's contractions. Estimates on the locus [W4]. Homogeneous varieties and their contractions. Remmert - Van de Ven problem and related problems [L], [OW]. Uniruled and rationally connected varieties.

Problem sheets:

1. First problem set, intersection of curves and divisors.
2. Second problem set, ruled surfaces.
3. Third problem set, various problems.
4. Fourth problem set, contractions of curves on surfaces.
5. Fifth problem set, cones of Del Pezzo's.
6. Sixth problem set, Del Pezzo's polytopes, part (1).
7. Seventh problem set, Del Pezzo's polytopes, part (2).
8. Eigth problem set, Root systems associated to Del Pezzo's.
9. Nineth problem set, Combinatorics of S4.

Readings:

• [D1] Debarre, Introduction to Mori theory, notes at the author's webpage.
• [CKM] Clemens. Koll\'ar, Mori, Higher-dimensional complex geometry. Asterisque 166 (1988)
• [D2] Debarre, Higher-dimensional algebraic geometry. Universitext. Springer-Verlag, New York, 2001.
• [IP] Iskovskih, Prokhorov, Fano Varieties, Enclycopeadia of Mathematical Sciences, Algebraic Geometry V, Springer 1999.
• [KM] Koll\'ar, Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge 1998.
• [M] Matsuki, Introduction to the Mori program. Universitext. Springer-Verlag, New York, 2002
• [K] Koll\'ar, Rational curves on algebraic varieties. Springer-Verlag, Berlin, 1996.

• Hartshorne, Algebraic geometry, Springer.
• Vakil, Foundations of algebraic geometry
• Ueno, Algebraic geometry, AMS.

• [M1] Mori, Projective manifolds with ample tangent bundles. Ann. of Math. 110
• [M2] Mori, Threefolds whose canonical bundles are not numerically effective. Ann. of Math. 116
• [MP] Miyaoka, Peternell, Geometry of higher dimensional algebraic varieties, DMV Seminar 26, Birkhauser 1997.
• [R] Reid, Decomposition of toric morphisms. Arithmetic and geometry, Vol. II, 395, Birkhauser 1983.
• [HK] Hu, Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331-348.
• [L] Lazarsfeld, Some applications of the theory of positive vector bundles, Complete Intersections (Acireale, 1983), Lecture Notes in Math. 1092.
• [OW] Occhetta, Wisniewski, On Euler-Jaczewski sequence and Remmert-van de Ven problem for toric varieties. Math. Z. 241.
• [W1] Wisniewski, On deformation of nef values. Duke Math. J. 64 (1996), 325-332;
• [W1a] Wisniewski, Cohomological invariants of complex manifolds coming from extremal rays, Asian J. Math. 2 (1998) 289-301.
• [W2] Wisniewski, Toric Mori theory and Fano manifolds. Geometry of toric varieties, 249-272, Semin. Cong 6, SMF 2002.
• [W3] Wisniewski, On a conjecture of Mukai. Manuscripta Math. 68.
• [W3a] Wisniewski, On Fano manifolds of large index. Manuscripta Math. 70.
• [W4] Wisniewski, On contractions of extremal rays of Fano manifolds. J. Reine Angew. Math. 417.