 Geometrie-Skripte

• Vorlesungsskript von Ursula Hamenstädt, Mathematisches Institut der Universität Bonn:

• Vorlesung von M.-A. Knus (Zürich WS 2001-02):

• Lecture Notes by Peter Michor (Wien, 1991/94/96):

• Lecture Notes by Sigmundur Gudmundsson, Lund University (2006)

• Lecures by Günter Harder, Mathematisches Institut der Universität Bonn:

• Lectures by James S. Milne: Contents:   Algorithms for polynomials   1.Algebraic sets   2.Affine algebraic varieties   3.Algebaic varieties   4.Local study: tangent spaces; tangent cones; singularities   5.Projective varieties; complete varieties   6.Finite maps   7.Dimension theory   8.Regular maps and their fibres   9.Algebraic geometry over an arbitrary field   10.Divisors and intersection theory   11.Coherent sheaves; invertible sheaves   12.Differentials   13.Algebraic varieties over the complex numbers   14.Further reading   Contents:   Introduction   Part I: Basic Theory of Abelian Varieties   1.Definitions; basic properties.   2.Abelian varieties over the complex numbers.   3.Rational maps into abelian varieties.   4.The theorem of the cube.   5.Abelian varieties are projective.   6.Isogenies   7.The dual abelian variety   8.The dual exact sequence.   9.Endomorphisms.   10.Polarizations and invertible sheaves.   11.The etale cohomology of an abelian variety.   12.Weil pairings.   13.The Rosati involution.   14.The zeta function of an abelian variety.   15.Families of abelian varieties.   16.Abelian varieties over finite fields.   17.Jacobian varieties.   18.Abel and Jacobi.   Part II: Finiteness Theorems.   19.Introduction.   20.Neron models; semistable reduction.   21.The Tate conjecture; semisimplicity.   22.Geometric finiteness theorems.   23.Finiteness I implies finiteness II.   24.Finiteness II implies the Shafarevich conjecture.   25.Shafarevich's conjecture implies Mordell's conjecture.   26.The Faltings height.   27.The modular height.    28.The completion of the proof of finiteness I. Appendix: Review of Faltings 1983 (MR 85g:11026) Contents: 1.Introduction   2.Etale Morphisms   3.The Etale Fundamental Group   4.The Local Ring for the Etale Topology   5.Sites   6.Sheaves for the Etale Topology   7.The Category of Sheaves on Xet.   8.Direct and Inverse Images of Sheaves.   9.Cohomology: Definition and the Basic Properties   10.Cech Cohomology   11.Principal Homogeneous Spaces and H1.   12.Higher Direct Images; the Leray Spectral Sequence   13.The Weil-Divisor Exact Sequence and the Cohomology of Gm   14.The Cohomology of Curves   15.Cohomological Dimension.   16.Purity; the Gysin Sequence.   17.The Proper Base Change Theorem.   18.Cohomology Groups with Compact Support.   19.Finiteness Theorems; Sheaves of Zl-modules   20.The Smooth Base Change Theorem.   21.The Comparison Theorem.   22.The Kunneth Formula.   23.The Cycle Map; Chern Classes   24.Poincare Duality   25.Lefschetz Fixed-Point Formula.   26.The Weil Conjectures.   27.Proof of the Weil Conjectures, except for the Riemann Hypothesis   28.Preliminary Reductions   29.The Lefschetz Fixed Point Formula for Nonconstant Sheaves   30.The MAIN Lemma   31.The Geometry of Lefschetz Pencils   32.The Cohomology of Lefschetz Pencils   33.Completion of the Proof of the Weil Conjectures.   34.The Geometry of Estimates

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