This is the website for the seminar Abelian varieties held at the Johannes-Gutenberg University of Mainz during the Wintersemester of 2014-2015
Practical information
Organizers: Ariyan Javanpeykar, Ronan Terpereau
Schedule: Tuesday 16h00-17h30 in Room 04-422
Literature
- GENERALITIES
- David Mumford. Abelian varieties, pages viii+242. Tata Institute of Fundamental. Research Studies in Mathematics, No. 5. Published for the Tata. Institute of Fundamental Research, Bombay, 1970.
- Christina Birkenhake and Herbert Lange. Complex abelian varieties. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 302. Springer-Verlag, Berlin, 2004.
- Alexander Polishchuk. Abelian varieties, theta functions and the Fourier transform. Cambridge Tracts in Mathematics, 153. Cambridge University Press, Cambridge, 2003.
- Gerard van der Geer and Ben Moonen. Abelian varieties. Book available on-line.
- Arnaud Beauville. Classical theta functions and their generalization. Notes on-line.
- Greg H. Anderson. Logarithmic derivatives of Dirichlet L-functions and the periods of abelian varieties. Compositio Mathematica 1982, tome 45, no.3, p. 315-332.
- Oleksandr Iena. Vector bundles on elliptic curves and factors of automorphy. Rend. Istit. Mat. Univ. Trieste 43 (2011), 61–94.
- Frans Oort. http://www.math.nyu.edu/~tschinke/books/finite-fields/final/05_oort.pdf
- Janos Kollar. https://web.math.princeton.edu/~kollar/book/chap1.pdf
- ABOUT THE MODULI SPACE OF ABELIAN VARIETIES:
- Gary Cornell, Joseph Silverman. Arithmetic geometry. Springer-Verlag New-York, 1986.
- Michel Brion. Compactification de l'espace des modules des variétés abéliennes principalement polarisées. Séminaire Bourbaki, 48 (p1-32), 2005-2006.
- Yung-sheng Tai. On the Kodaira dimension of the moduli space of abelian varieties. Inventiones mathematicae 1982, Volume 68, Issue 3, pp 425-439.
- Carel Faber, Gerard van der Geer, Frans Oort. Moduli of Abelian Varieties. Progress in Mathematics Volume 195, 2001, pp 345-416.
- Iku Nakamura. Compactification by GIT-stability of the moduli space of abelian varieties. arXiv:1406.0174
- Samuel Grushevsky. Geometry of A_g and its compactifications. arXiv:0711.0094
- Martin Olsson. https://math.berkeley.edu/~molsson/Overview3.pdf
- Martin Olsson. https://math.berkeley.edu/~molsson/mono020807.pdf
Schedule
- Holomorphic line bundles on complex tori (Laura Biroth), 28th of October, notes
content: Appel-Humbert theorem, dual complex torus, Poincaré line bundle.
references: [Polishchuk,I.1], [Mumford, I.1 and I.2], [Birkenhake-Lange, 2.1-2.5] - Theta functions I (Xin Lu), 4th of November, notes
content: isogenies, Riemann forms, theta functions, algebraizability of complex tori.
references: [Mumford, I.3], [Cornell-Silverman, IV.1-IV.3] - Theta functions II (Helge Ruddat), 11th of November, notes
content: isogenies, Riemann forms, theta functions, algebraizability of complex tori.
references: [Mumford, I.3], [Cornell-Silverman, IV.1-IV.3] - Cohomology of line bundles (Markus Pauly), 18th of November, notes
content: vanishing theorem of Mumford and Kempf, dimension of the cohomology group of a line bundle, Riemann-Roch theorem.
references: [Polishchuk,I.7], [Mumford III.16], [Birkenhake-Lange, 3.1-3.6] - Abelian varieties and theorem of the cube (Carolin Peternell), 25th of November, notes
content: algebraic group point of view, rigidity lemma, theorem of the cube, ampleness criterion.
references: [Polishchuk,II.8], [Mumford, II.4-II.6] - 2nd of December: NO TALK!
- The dual abelian variety (Levente Nagy), 9th of December, notes
content: construction of the dual abelian variety, functorial point of view, (principal) polarization, construction of the quotient of abelian varieties.
references: [Polishchuk,II.9], [Mumford, II.7-II.9], [Birkenhake-Lange, 4.1] - The Jacobian (Adam Czaplinski), 16th of December, notes
content: symmetric power of a curve, construction of the jacobian of a curve, (statement of) the Torelli theorem.
references: [Polishchuk,III.16 and III.21], [Birkenhake-Lange, 11.1], [Cornell-Silverman, VII.1-VII.4 and VII.12] - Singular cohomology of complex tori vs étale cohomology of abelian varieties (Thomas Weissschuh), 6th of January, notes
content: étale cohomology groups of abelian varieties, singular cohomology groups of complex tori.
references: [Birkenhake-Lange, 1.3 and 1.4], [Cornell-Silverman, V.15] - Abelian surfaces (Maksim Zhykhovich), 13th of January, notes
content: Kummer surfaces and other examples, Reider's theorem, product of elliptic curves.
references: [Birkenhake-Lange, 10.1-10.6] - Period distribution of Abelian varieties (Maximilian Preisinger), 20th of January, notes
content: periods of Abelian varieties, endomorphism ring of Abelian varieties, CM fields
references: [Anderson] - 27th of January: NO TALK!
- Moduli theory of abelian varieties (Ariyan Javanpeykar), 2nd of February (at 16.00 in room 04-422), notes
content: representable functors, examples of moduli functors, moduli of abelian varieties and its basic properties (smoothness, separatedness, non-properness, etc) - Abelian varieties over finite fields and Weil numbers (Axel Stäbler), 3rd of February, notes
content: abelian varieties over finite fields, Weil numbers, Honda-Tate theory
references: [Oort] - Compactification of the moduli space of abelian varieties (Ronan Terpereau), 10th of February, notes
content : case of elliptic curves , minimal compactification, modular compactification of Alexeev, semi-abelian varieties
references: [Brion], [Olsson]