UNITED NATIONS EDUCATIONAL SCIENTIFIC AND CULTURAL ORGANIZATION and INTERNATIONAL ATOMIC ENERGY AGENCY THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS Strada Costiera 11 Telephone: +39-040-2240111 I-34014 Trieste Telex: 460392 ICTP I Italy Telefax: +39-040-224163 SCHOOL ON VANISHING THEOREMS AND EFFECTIVE RESULTS IN ALGEBRAIC GEOMETRY Supported by the European Commission, Research DG, Human Potential Programme, High Level Scientific Conferences HPCF-CT-1999-00140 Co-sponsored by Max-Planck Gesellschaft and Humboldt Foundation 25 April - 12 May 2000 Miramare, Trieste, Italy Organizers J.-P. Demailly (France) & R. Lazarsfeld (U.S.A.) Advisory Committee J.-P. Demailly (France), Y. Kawamata (Japan), J. Kollar (U.S.A.), R. Lazarsfeld (U.S.A.), Th. Peternell (Germany), Y.T. Siu (U.S.A.) FINAL PROGRAMME =============== VENUE: MAIN LECTURE HALL, MAIN BUILDING Week 1 (25 - 29 April) Tuesday, 25 April ================= 8:30 - 12:00 Registration and administrative formalities 14:00 Opening 14:15 - 15:15 J.-P. Demailly (Institut Fourier, Grenoble, France) "Analytic methods and multiplier ideal sheaves"(I) 15:45 - 16:45 R. Lazarsfeld (University of Michigan, Ann Arbor, U.S.A.) "Algebraic methods and multiplier ideal sheaves" (I) Wednesday, 26 April =================== 9:00 - 10:00 J.M. Hwang (Korea Institute for Advanced Study, Seoul, Korea) "Geometry of minimal rational curves on Fano manifolds" (I) 10:30 - 11.30 Th. Peternell (Universitaet Bayreuth, Germany) "Applications of vanishing theorems and Mori theory to the classification theory of algebraic varieties" (I) 14:00 - 15:00 J.-P. Demailly (Institut Fourier, Grenoble, France) "Analytic methods and multiplier ideal sheaves" (II) 15:30 - 16:30 R. Lazarsfeld (University of Michigan, Ann Arbor, U.S.A.) "Algebraic methods and multiplier ideal sheaves" (II) Thursday, 27 April ================== 9:30 - 10:30 J.-P. Demailly (Institut Fourier, Grenoble, France) "Analytic methods and multiplier ideal sheaves" (III) 11:00 - 12:00 R. Lazarsfeld (University of Michigan, Ann Arbor, U.S.A.) "Algebraic methods and multiplier ideal sheaves" (III) 14:00 - 15:00 J.M. Hwang (Korea Institute for Advanced Study, Seoul, Korea) "Geometry of minimal rational curves on Fano manifolds" (II) 15:30 - 16:30 Th. Peternell (Universitaet Bayreuth, Germany) "Applications of vanishing theorems and Mori theory to the classification theory of algebraic varieties" (II) Friday, 28 April ================ 9:00 - 10:00 Th. Peternell (Universitaet Bayreuth, Germany) "Applications of vanishing theorems and Mori theory to the classification theory of algebraic varieties" (III) 10:30 - 11:30 J.M. Hwang (Korea Institute for Advanced Study, Seoul, Korea) "Geometry of minimal rational curves on Fano manifolds" (III) 14:00 - 15:00 J.-P. Demailly (Institut Fourier, Grenoble, France) "Analytic methods and multiplier ideal sheaves" (IV) 15:30 - 16:30 R. Lazarsfeld (University of Michigan, Ann Arbor, U.S.A.) "Algebraic methods and multiplier ideal sheaves" (IV) Saturday, 29 April ================== 9:00 - 10:00 J.M. Hwang (Korea Institute for Advanced Study, Seoul, Korea) "Geometry of minimal rational curves on Fano manifolds" (IV) 10:30 - 11:30 Th. Peternell (Universitaet Bayreuth, Germany) "Applications of vanishing theorems and Mori theory to the classification theory of algebraic varieties" (IV) Week 2 (1 - 6 May) Monday, 1 May ============= 9:00 - 10:00 S. Helmke (R.I.M.S., Kyoto, Japan) "Effective point separation and the Fujita Conjecture" (I) 10:30 - 11.30 K.E. Smith (University of Michigan, Ann Arbor, U.S.A.) "Tight Closure and Vanishing Theorems" (I) 14:00 - 15:00 E. Viehweg (Universitaet-GHS Essen, Germany) "Positivity of direct image sheaves and applications to families of higher dimensional manifolds"(I) 15:30 - 16:30 S. Helmke (R.I.M.S., Kyoto, Japan) "Effective point separation and the Fujita Conjecture" (II) Tuesday, 2 May ============== 9:00 - 10:00 E. Viehweg (Universitaet-GHS Essen, Germany) "Positivity of direct image sheaves and applications to families of higher dimensional manifolds" (II) 10:30 - 11:30 S. Helmke (R.I.M.S., Kyoto, Japan) "Effective point separation and the Fujita Conjecture" (III) 14:00 - 15:00 K.E. Smith (University of Michigan, Ann Arbor, U.S.A.) "Tight Closure and Vanishing Theorems" (II) 15:30 - 16:30 E. Viehweg (Universitaet-GHS Essen, Germany) "Positivity of direct image sheaves and applications to families of higher dimensional manifolds" (III) Wednesday, 3 May ================ 9:00 - 10:00 K.E. Smith (University of Michigan, Ann Arbor, U.S.A.) "Tight Closure and Vanishing Theorems" (III) 10:30 - 11:30 E. Viehweg (Universitaet-GHS Essen, Germany) "Positivity of direct image sheaves and applications to families of higher dimensional manifolds" (IV) 14:00 - 15:00 Y. Kawamata (University of Tokyo, Japan) "Vanishing theorems, extremal rays and the Fujita conjecture" (I) 15:30 - 16:30 S. Helmke (R.I.M.S., Kyoto, Japan) "Effective point separation and the Fujita Conjecture" (IV) Thursday, 4 May =============== 9:00 - 10:00 Y.T. Siu (Harvard University, Cambridge, U.S.A.) "Overview of hyperbolicity problems" 10:30 - 11:30 Y. Kawamata (University of Tokyo, Japan) "Vanishing theorems, extremal rays and the Fujita conjecture" (II) 14:00 - 15:00 K.E. Smith (University of Michigan, Ann Arbor, U.S.A.) "Tight Closure and Vanishing Theorems" (IV) 15:30 - 16:30 Y.T. Siu (Harvard University, Cambridge, U.S.A.) "Techniques of Nevanlinna theory" Friday, 5 May ============= 9:00 - 10:00 Y.T. Siu (Harvard University, Cambridge, U.S.A.) "Existence of one holomorphic jet differential" 10:30 - 11:30 Y. Kawamata (University of Tokyo, Japan) "Vanishing theorems, extremal rays and the Fujita conjecture" (III) 14:00 - 15:00 Y.T. Siu (Harvard University, Cambridge, U.S.A.) "Vector fields on the total space of a family of jet spaces" 15:30 - 16:30 Y. Kawamata (University of Tokyo, Japan) "Vanishing theorems, extremal rays and the Fujita conjecture" (IV) Saturday, 6 May =============== Morning: Final examination Week 3 (8 - 12 May) FINAL CONFERENCE Monday, 8 May ============= 9:00 - 10:00 Y.-T. Siu (Harvard University, Cambridge, U.S.A.) "Hyperbolicity of generic high-degree hypersurfaces and their complements" 10:30 - 11.30 K.E. Smith (University of Michigan, Ann Arbor, U.S.A.) "Differential operators in prime characteristic" 14:00 - 15:00 K.H. Paranjape (Ins. of Mathematical Sciences, Chennai, India) "Specialization of sections of Jacobian fibrations" 15:30 - 16:30 M. Andreatta (Universita' degli Studi di Trento, Italy) "Projective manifolds with a group action" Tuesday, 9 May ============== 9:00 - 10:00 H. Tsuji (Tokyo Institute of Technology, Japan) "Study of pluricanonical systems via numerically trivial fibrations" 10:30 - 11:30 D. Varolin (University of Michigan, Ann Arbor, U.S.A.) "Very large holomorphic diffeomorphism groups" 14:00 - 15:00 F. Campana (Universite' Nancy 1, France) "Kaehler threefolds with negative Kodaira dimension" 15:30 - 16:30 M. Verbitsky (Max-Planck-Ins.fuer Mathematik, Bonn, Germany) "Singularities of sheaves over hyperkaehler manifolds" Wednesday, 10 May ================= 9:00 - 10:00 M. Popa (University of Michigan, Ann Arbor, U.S.A.) "Generalized theta linear series on moduli spaces of vector bundles on curves" 10:30 - 11:30 Th. Peternell (Universitaet Bayreuth, Germany) "Manifolds with splitting tangent bundles" 14:00 - 15:00 J.M. Hwang (Korea Institute for Advanced Study, Seoul, Korea) "Seshadri numbers of compact quotients of bounded symmetric domains" 15:30 - 16:30 S. Helmke (R.I.M.S., Kyoto, Japan) "Effective point separation" Thursday, 11 May ================ 9:00 - 10:00 S. Takayama (Kyushu University, Fukuoka, Japan) "Very ample line bundles over quasi-abelian varieties" 10:30 - 11:30 A. Lopez (Universita' degli Studi di Roma 3, Italy) "Subvarieties of generic surfaces and hypersurfaces" 14:00 - 15:00 M. Mella (Universita' degli Studi di Ferrara, Italy) "On the birational geometry of quartic threefolds" 15:30 - 16:30 D. Kaledin (Independent University of Moscow, Russia) "Period map for non-compact holomorphically symplectic manifolds" Friday, 12 May ============== 9:00 - 10:00 Ch. Mourougane (Universite' Paris 6, France) "Nakano positivity and the L^2-metric on the direct image of an adjoint positive line bundle" 10:15 - 11:15 S. Kebekus (Univ. Bayreuth, Germany/Kyoto University, Japan) "Families of rational curves of small degree" 11:30 - 12:30 Y. Kawamata (University of Tokyo, Japan) "On a generalization of Fujita's freeness conjecture" ============================================================================== THEMES OF MAIN LECTURES (in alphabetical order of lecturer) 1. J.-P. Demailly (Institut Fourier, Grenoble, France) "Analytic methods and multiplier ideal sheaves" Subtitles/contents: I. Basic L^2 estimates and Kodaira vanishing II. Multiplier ideal sheaves, Nadel and Kawamata-Viehweg vanishing, effective results in algebraic geometry III. Ohsawa-Takegoshi L^2 extension theorem and adjunction IV. Applications: invariance of plurigenera, approximate Zariski decomposition. 2. S. Helmke (R.I.M.S., Kyoto, Japan) "Effective point separation and the Fujita Conjecture" 3. J.M. Hwang (Korea Institute for Advanced Study, Seoul, Korea) "Geometry of minimal rational curves on Fano manifolds" Abstract: This lecture is an introduction to my joint project with N. Mok where we develop a geometric theory of Fano manifolds of Picard number 1 by studying the collection of tangent directions of minimal rational curves through a generic point. After a sketch of some historical background, the fundamental object of this project, the variety of minimal rational tangents, is defined and various examples are discussed. Then some basic results about the geometric structure defined by the varieties of minimal rational tangents are explained, including an extension theorem for holomorphic maps preserving the geometric structure. A number of applications of this theory are given in the last part of the lecture. 4. Y. Kawamata (University of Tokyo, Japan) "Vanishing theorems, extremal rays and the Fujita conjecture" I. Singular hermitian metric on the Hodge bundle arising from an algebraic fiber space. II. Logarithmic generalization of the semipositivity theorem and applications. III. How to use the adjunction theorem IV. Length of extremal rays - another effective result related to Fujita conjecture. 5. R. Lazarsfeld (University of Michigan, Ann Arbor, U.S.A.) "Algebraic methods and multiplier ideal sheaves" I. Preliminaries; Definition of multiplier ideals; Examples. II. Vanishing theorems for multiplier ideals; geometric properties of multiplier ideals. III. First applications; asymptotic multiplier ideals. IV. Some algebraic applications of multiplier ideals. 6. Th. Peternell (Universitaet Bayreuth, Germany) "Applications of vanishing theorems and Mori theory to the classification theory of algebraic varieties" I. Integrability: foliations and generic nefness for non-uniruled varieties II. Maximal Nonintegrability: contact structures III. Stability: Fano manifolds IV. Positivity: Wahl's theorem, Mori's theorem, interpolations and adjunction theory. Abstract: Im my lectures I want to study how different properties of the tangent bundle influence the geometry of the underlying manifold. If E is a subbundle of the tangent bundle or, more generally, a coherent subsheaf, then one can ask whether E is integrable. In that case X carries a foliation. Special foliations are the subject of Lecture I. In Lecture II we study subbundles E which are maximal non-integrable, i.e. contact structures on X. In Lecture III we discuss Fano manifolds and the problem of stability of the tangent bundle. This means again the study of subsheaves E in the tangent bundle but now the focus is on the size of c_1(E) compared to c_1(X). Finally, in Lecture IV, ample subsheaves E of the tangent bundle are investigated. 7. Y.T. Siu (Harvard University, Cambridge, U.S.A.) "Hyperbolicity of algebraic varieties, jets and Nevanlinna theory" I. Overview of hyperbolicity problems, II. Techniques of Nevanlinna theory, III. Existence of one holomorphic jet differential, IV. Vector fields on the total space of a family of jet spaces. 8. K.E. Smith (University of Michigan, Ann Arbor, U.S.A.) "Tight Closure and vanishing theorems" 9. E. Viehweg (Universitaet-GH Essen, Germany) "Positivity of direct image sheaves and applications to families of higher dimensional manifolds" Subtitles/contents: I. Introduction: The Shafarevich conjectures for family of curves over a curve and differential forms on the moduli space of stable curves. II. Positivity and effective positivity results for families of higher dimensional manifolds over curves. III. Boundedness results and isotriviality of certain morphisms. IV. Some open questions about differential forms on moduli.