Antoine Ducros

Antoine Ducros (Paris 6) посетил Лабораторию алгебраической геометрии в мае 2013 года.

30 мая в 15.40 в ауд. 310 Независимого Московского Университета (Б. Власьевский, 11) Antoine Ducros выступил на семинаре "Глобус" с докладом "REAL DIFFERENTIAL FORMS AND CURRENTS ON p-ADIC ANALYTIC SPACES".

Аннотация: I will present a joint work with Antoine Chambert-Loir, in which we develop kind of a 'harmonic analysis' formalism on Berkovich spaces. More precisely, we define:

- real differential forms of bidegree (p,q) on a Berkovich space X of dimension n;

- the integral of a (n,n) form (with compact support) on X;

- the boundary integral of a (n,n-1) form.

We have Stokes and Green formulas in this context. We define currents by duality, and the Poincaré-Lelong formula holds.

We are also able to associate to a metrized line bundle (L,||.||) a curvature form c_1(L,||.||) (if ||.|| is not smooth, this is not a form in general, but a current). If (L,||.||) comes from a formal model, c_1(L,||.||)^n is shown to be a measure, which coincides with a measure previously defined by Chambert-Loir in terms of intersection theory on the special fiber (in his work on p-adic equidistribution of points of small height).

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В период с 21 по 29 мая 2013 г. Antoine Ducros и Jerome Poineau (Strasbourg) прочитали миникурс «Introduction to Berkovich analytic spaces».

Лекции состоялись

21.05 (вт., 17.00, ауд. 317), 22.05 (ср., 17.00, ауд. 311), 23.05 (чт., 15.30, ауд. 311), 27.05 (пн., 17.00, ауд. 1001), 28.05 (вт., 17.00, ауд. 317) и 29.05 (ср., 17.00, ауд. 311)
на факультете математики НИУ ВШЭ (ул. Вавилова, д.7).

Abstract: At the end of the eighties, Vladimir Berkovich introduced a new way to define p-adic analytic spaces. A surprising feature is that, although p-adic fields are totally discontinuous, the resulting spaces enjoy many nice topological properties: local compactness, local path-connectedness, etc. On the whole, those spaces are very similar to complex analytic spaces. They already have found numerous applications in several domains: arithmetic geometry, dynamics, motivic integration, etc.

In this course, we will introduce Berkovich spaces and study their basic properties. The program will cover the following topics: - non-Archimedean fields, absolute values - Tate algebras, affinoid algebras and their properties - affinoid spaces - Berkovich spaces - analytification of algebraic varieties - analytic curves (local structure, homotopy type).

The course will be understandable to those who know the definition of the field of p-adic numbers Q_p.


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