Mathematical Institute of the University of Bonn

Real differential forms in non-archimedean geometry - Winter Term 2018/19

Lecturer:
Dr. Andreas Mihatsch
Endenicher Allee 60 · Zimmer 4.024
Tel.: 0228-73-62369
E-mail: mihatsch (add @math.uni-bonn.de)

Time and Place

Monday, 10-12h, N0.003
Friday, 16-18h, 0.011 (this has changed from the original time Thursdays, 10-12h)
First Lecture: Monday 08.10.2018.

Content of the course

In the Arakelov Intersection Theory of Gillet-Soulé, archimedean and non-archimedean places are treated differently: The local contribution to the intersection product from the finite places is given by an algebraic intersection number, while the contribution from the infinite places is the evaluation of the star-product of the Green currents of the Arakelov cycles.

This course is about an analytic approach to Arakelov Geometry which treats the non-archimedean places analogously to the archimedean ones. Following Chambert-Loir/Ducros and Gubler/Künnemann, we will introduce real differential forms on adic spaces by pulling back differential forms from R^n under tropicalization maps. This gives rise to (p,q)-forms on the adic space, with a calculus of differential operators similar to the one in complex geometry. It is also possible to define Green currents for cycles in this set-up and (in some cases) to define intersection products of such "non-archimedean Arakelov cycles".

We intend to also explain some results from Arakelov Theory using this new language. In particular we plan to explain an equidistribution result of Xinyi Yuan which generalizes a similar result of Shouw-Wu Zhang. An application of this (due to Shou-Wu Zhang and Ullmo) is the Bogomolov Conjecture.

Literature

• D. Abramovich, J.-F. Burnol, J. Kramer, C. Soulé: Lectures on Arakelov Geometry, Cambridge studies in advanced mathematics 33 (1992).
• A. Chambert-Loir, A. Ducros: Formes différentielles réelles et courants sur les espaces de Berkovich, arXiv:1204.6277, 2012.
• W. Gubler, K. Künnemann: A tropical approach to non-archimedean Arakelov theory, arXiv:1406.7637, 2014.
• E. Ullmo: Positivité et discrétion des points algébriques des courbes, Annales of Mathematics 147 (1998), no. 1. (see here)
• X. Yuan: Big Line Bundles over Arithmetic Varieties, Inventiones Mathematicae 173 (2008), no. 3. (see here)
• S.-W. Zhang: Admissible Pairing on a curve, Invent. math. 112 (1993).

Accreditation/Exams

This course is categorized as V5A4. There will be oral exams towards the end of the semester. Details will be announced during the lecture.