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Research Report 2001 - 2003 Faculty of Mathematics and Physics Department of Mathematics including Computer Sciences deutsch |
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Chair of Mathematics I
Complex Geometry, Algebraic Geometry Professor Dr. Thomas Peternell Tel: +921/55-3369 Fax: +921/55-2785 e-mail: thomas.peternell@uni-bayreuth.de www-Adresse: http://btm8x5.mat.uni-bayreuth.de/mathe1/ Scientists: Bauer, Thomas, Dr. (DFG-Schwerpunkt; wiss. Assistent) Eckl, Thomas, Dr. wiss.Assistent; bis 2003 Jahnke, Priska, Dr. (DFG-Schwerpunkt) Kebekus, Stefan, Dr. (wiss. Assistent, LST Mathe VIII; Heisenbergstipendiat seit 2002), bis 2003 Elsner, Detlev, Dipl.Math. (Graduiertenkolleg), Promotion 2001, bis 2001 Kühnel, Marko, Dipl.Math. (Graduiertenkolleg), Promotion 2001, bis 2001 Radloff, Ivo, Dr. (DFG-Schwerpunkt) Pukhlikov, Aleksandr, Dr., Humboldtstipendiat 2001-2003 Kronenthaler, Wolfgang, Dipl.Math., Doktorand seit 2002 Höring, Andreas, Doktorand seit 2003 Participation in Central Research Insitutes/Units: Global Methods in Complex Geometry Projects Manifolds of semipositive curvature We investigate compact Kähler manifolds admitting a metric of semipositive bisectional or Ricci curvature. In algebraic geometry these conditions are substituted by a more general version which is essentially of numerical nature and therefore easier to verify. We obtain manifolds with numerically effective tangent or anticanonical bundle. Our results include far reaching classification theorems. These projects were developed (and continue) in collaborations with J.P. Demailly and M.Schneider, with F. Serrano and with F.Campana and Q.Zhang. A joint project with P.Jahnke and I.Radloff is the classification of all threefolds with nef and "big" anticanonical class. It is the property "big" that guarentees that there up only finitely many daformation families, so that a complete description is indeed possible. Recently a joint project with T.Bauer studies the global structure of threefolds with nef anticanonical which is not "big". Principal Investigator (at the University of Bayreuth): Thomas Peternell Funding institution (contract number): Deutsche Forschungsgemeinschaft European Community Duration: seit 1991 Mori theory on KÄhler manifolds Since around 1980 Mori theory lead to deep results in the classification theory of projective manifolds, especially in dimension 3. The methods are algebraic and do not work in the Kähler case. Since 1995 I developped, partially in collaboration with F.Campana, methods for an analytic Mori theory in dimension 3. Recent work with J.P.Demailly (J.Diff.Geom). gives new results. At the moment W.Kronenthaler is working on this project (Ph.D. thesis). In algebraic Mori theory a joint paper with Boucksom, Demailly and Paun gives a numerical characterisation of pseudo-effective line bundles and applies that to the abundance problem. Principal Investigator (at the University of Bayreuth): Th. Peternell Funding institution (contract number): Deutsche Forschungsgemeinschaft European Community Duration: seit 1995 Calabi-Yau manifolds Calabi-Yau manifolds belong to the most important objects in algebraic geometry. In the last years I studied with K.Oguiso linear systems and the global structure of Calabi-Yau threefolds. Principal Investigator (at the University of Bayreuth): Th. Peternell Funding institution (contract number): Deutsche Forschungsgemeinschaft European Community Duration: seit 1991 Topology and complex structures One of the fundamental questions in complex geometry asks for the description of all complex structures on a given topologial or differential manifold. We studied the following specific topics: 1. Which algebraic complex structures exist on a Fano manifold? 2. Which complex structures exist on the 6-sphere? Question 1 was studied with P. Freitag and A. Summerer, Problem 2 with A.Huckleberry and S.Kebekus and with F.Campana and J.P.Demailly. Principal Investigator (at the University of Bayreuth): Th. Peternell Funding institution (contract number): Deutsche Forschungsgemeinschaft European Community Duration: seit 1992 Geometry of submanifolds In this project we study the influence of a submanifold on the global structure of the large manifold under suitable positivity assumptions on the normal bundle. A paper with M.Schneider and A.Sommese proves a relation between the Kodaira dimension of both manifolds. This is used to study submanifolds with non-positive Kodaira dimension and semipositive normal bundle. Principal Investigator (at the University of Bayreuth): Th. Peternell Funding institution (contract number): Deutsche Forschungsgemeinschaft European Community Duration: seit 1996 Contact manifolds and subsheaves in the tangent bundle Contact structures paly an imporant role in real differential geometry. In connection with quaternion Kähler manifolds and twistor spaces this notion also became important in complex geometry. In a project with S.Kebekus and A.Sommese I studied the structure of projective contact manifolds. Some aspects lead to interesting general problems on subsheaves in the tangent bundle. Recently I studied subsheaves in the tangent bundle in connection with the structure of the universal cover with F.Campana. Principal Investigator (at the University of Bayreuth): Th. Peternell Funding institution (contract number): Deutsche Forschungsgemeinschaft European Community Duration: seit 1997 The tangent bundle of algebraic manifolds A classical problem in the theory of (compact) complex manifolds is the question of global flatness resp. integrability of certain infinitesimal structures. For example, if the tangent bundle of a Kähler manifold is trivial, then it is already a torus, i.e., the triviality of the tangent bundle already implies global flatness. Joint work with F.Campana (Nancy) is the investigation of algebraic manifolds whose tangent bundle splits. The aim is to find a splitting of the universal cover, possibly under an integrability condition, at least in low dimensions. This project continues by the Ph.D. project of A. Höring. In twistor theory, uniformization theory and recently in Fano geometry, manifolds modeled after hermitian symmetric spaces play a central role. P.Jahnke and I.Radloff completely classified such manifolds in projective dimension 3. As in the case of dimension 2, treated by Kobayashi and Ochiai, it turns out that these structures are indeed always flat. As an example case, look at the quadric. In dimension 2, an infinitesimal quadric structure implies the splitting of the tangent bundle as a sum of line bundles (perhaps after some étale covering), and flatness means the universal covering space is indeed a product. In dimension 3, a Corollary to the result of P.Jahnke and I.Radloff is the complete classification of all projective threefolds whose tangent bundle is a symmetric square. Principal Investigator (at the University of Bayreuth): Th. Peternell Funding institution (contract number): DFG Duration: seit 1998 Geometry of coverings This is a joint project with A.Sommese (Notre Dame). We investigate the geometry of coverings of algebraic varieties which is to a large extent determined by a certain vector bundle on the base of the covering. We study the curvature and structure of this vector bundle. A new project with J.M.Hwang and S.Kebekus studies the space of all coverings between fixed manifolds. Principal Investigator (at the University of Bayreuth): Thomas Peternell Funding institution (contract number): DFG Duration: seit 1998 Publications 1. Towards a Mori theory on compact Kähler threefolds, III. Bull. Soc. Math. France 129, 339-356 (2001) 2. The Kodaira dimension of Kummer threefolds. Bull. Soc. Math. France 129, 357-359 (2001) 3.On the Albanese maps of compact Kähler manifolds with nef anticanonical bundles (mit F. Campana, Q.Zhang). Trans. Amer. Math. Soc. 2003 4. Ample vector bundles and branched coverings (mit A.J.Sommese). Comm. in Algebra 28, 5573-5599, 2001, special volume in honour of R.Hartshorne 5. Contact structures, rational curves and Mori theory. Proceedings 3rd European Congress of Mathematics, Barcelona 2000; Progress in Math. vol. 201, 509-518, Birkhäuser 2001 6. Subsheaves in the tangent bundle: integrability, stability and positivity. Lecture Notes Summerschool ``Vanishing Theorems and effective results in Algebraic Geometry, ICTP Lecture Notes, 2001 7. Pseudo-effective line bundles on compact Kähler manifolds (mit J.P. Demailly). Intl. J. Math. 12, 689-741 (2001) 8. Manifolds with nef subsheaves in the cotangent bundle (mit S.Kebekus, A.Sommese), in: Complex Geometry, Festschrift in honour of Hans Grauert, 157-163, Springer 2002 9. Manifolds with splitting tangent bundles,I (mit F.Campana). Math. Z. 241, 613-637 (2002) 10. Compact Kaehler threefolds with small Picard numbers. Applications of algebraic geometry to coding theory, physics and computation; 271-290, Kluwer 2001 11. A reduction map for nef line bundles (mit T.Bauer, F.Campana et al.). In Complex Geometry, Festschrift in honour of H.Grauert; 27-36, Springer 2002 12. The dual Kähler cone of a compact Kähler threefold (mit K.Oguiso). math.AG/0107224 13. Differential forms and terminal singularities. Preprint 2001 14. Generalized Tsen theorem and rationally connected fibrations (mit F.Campana, A.Pukhlikov). Doklady Mat. Nauk., 2003 15. 3. Line bundles on complex tori and a conjecture of Kodaira (mit J.P. Demailly, T.Eckl). math.AG/0212243 16. Ample vector bundles and branched coverings II (mit A.J.Sommese). math.AG/021120. To appear in the proceedings of the Fano conference, Torino 2002 17. A Kawamata-Viehweg vanishing theorem on compact Kähler manifolds (with J.P. Demailly). J. Differential Geometry 63, 231-277 (2003) 18. Nef reduction and the structure of threefolds with nef anticanonical bundles (mit T.Bauer). math.AG0310484, ersch. in Asian J. Math., volume in honour of Y.T.Siu 19. Holomorphic maps onto varieties of non-negative Kodaira dimension (mit J.M.Hwang und S.Kebekus). math.AG 0307220 20. The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension (mit S.Boucksom, J.P.Demailly und M.Paun). math.AG/0405285 T.Eckl: 1. Vector fields on smooth threefolds vanishing on complete intersections. manuscripta math. 107, 59-71 (2002) 2. Tsuji's numerical trivial fibrations. math.AG/0202270; to appear in Journal Alg. Geom. 3. Line bundles on complex tori and a conjecture of Kodaira (mit J.P. Demailly,T.Peternell. math.AG/0212243 Numerical trivial foliations. math.AG/0304312 P.Jahnke und I.Radloff: 1. Splitting jet sequences. math.AG/021054, ersch. in Math. Res. Letters 2. Threefolds with holomorphic normal projective connections. math.AG/0210117, ersch. in Math. Ann. 3. Fano 3-folds with sections in $\Omega_V^1(1)$. math.AG/0310390 4. Generatedness for Gorenstein Fano threefolds with canonical singularities. math.AG/0404156 5. Projective threefolds with holomorphic conformal structure. Preprint 2004 6. Submanifolds with splitting tangent bundles. math.AG/0304223 T.Bauer: 1. Stability and Futaki invariants of Fano hypersurface. Preprint 2001 Doctoral dissertations/PhD theses: Th.Bauer: Faserraumstrukturen auf 4-dimensionalen Calabi-Yau Mannigfaltigkeiten (2000) D.Elsner: Projektive Strukturen auf Fano 3-Faltigkeiten mit b_2 = b_3 = 2 (2000) M.Kühnel: Über gewisse Calabi-Yau-3-faltigkeiten mit Picardzahl 2 (2000) |
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