Walter Gubler

Tropical varieties for non-archimedean analytic spaces

Generalizing the construction from tropical algebraic geometry, we associate to every (irreducible d-dimensional) closed analytic subvariety of

The Bogomolov conjecture for totally degenerate abelian varieties

\mathbb{G}_{m}^{n}

Forms and currents on the analytification of an algebraic variety (after Chambert-Loir and Ducros)

a tropical variety in ℝn with respect to a complete non-archimedean place. By methods of analytic and formal geometry, we prove that the tropical variety is a totally concave locally finite union of d-dimensional polytopes. For an algebraic morphism f:X’→A to a totally degenerate abelian variety A, we give a bound for the dimension of f(X’) in terms of the singularities of a strictly semistable model of X’. A closed d-dimensional subvariety X of A induces a periodic tropical variety. A generalization of Mumford’s construction yields models of X and A which can be handled with the theory of toric varieties. For a canonically metrized line bundle L̄ on A, the measures c1(L̄|X)∧d are piecewise Haar measures on X. Using methods of convex geometry, we give an explicit description of these measures in terms of tropical geometry. In a subsequent paper, this is a key step in the proof of Bogomolov’s conjecture for totally degenerate abelian varieties over function fields.

Non-Archimedean Geometry and Applications

We prove the Bogomolov conjecture for a totally degenerate abelian variety A over a function field. We adapt Zhang’s proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place v. As a corollary, we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v.

Non-Archimedean Analytic Geometry

Chambert-Loir and Ducros have recently introduced real differential forms and currents on Berkovich spaces. In these notes, we survey this new theory and we will compare it with tropical algebraic geometry.

Heights in Diophantine Geometry

The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to other fields, in particular to number theory and algebraic geometry. These applications included Mirror Symmetry, the Langlands program, p-adic Hodge theory, tropical geometry, resolution of singularities and the geometry of moduli spaces. Much emphasis was put on making the list of talks to reflect this diversity, thereby fostering the mutual inspiration which comes from such interactions. Mathematics Subject Classification (2010): 03 C 98, 11 G 20, 11 G 25, 14 F 20, 14 G 20, 14 G

Skeletons and tropicalizations

The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to arithmetic and algebraic geometry. These applications include questions in Arakelov theory, p-adic differential equations, p-adic Hodge theory and the geometry of moduli spaces. Various methods were used in combination with analytic geometry, in particular perfectoid spaces, model theory, skeleta, formal geometry and tropical geometry. Mathematics Subject Classification (2010): 03C98, 11G25, 12H25, 14F20, 14G20, 14G22, 32P05. Introduction by the Organisers The half-size workshop Non-Archimedean Analytic Geometry, organized by Vladimir Berkovich (Rehovot), Walter Gubler (Regensburg) and Annette Werner (Frankfurt) had 26 participants. Non-Archimedean analytic geometry is a central area of arithmetic geometry. The first analytic spaces over fields with a nonArchimedean absolute value were introduced by John Tate and explored by many other mathematicians. They have found numerous applications to problems in number theory and algebraic geometry. In the 1990s, Vladimir Berkovich initiated a different approach to non-Archimedean analytic geometry, providing spaces with good topological properties which behave similarly as complex analytic spaces. Independently, Roland Huber developed a similar theory of adic spaces. Recent years have seen a growing interest in such spaces since they have been used to solve several deep questions in arithmetic geometry. We had 19 talks in this workshop reporting on recent progress in non-Archimedean analytic geometry and its applications. All talks were followed by lively 3212 Oberwolfach Report 53/2012 discussions. Several participants explained work in progress. The workshop provided a useful platform to discuss these new developments with other experts. During the workshop, we saw applications to complex singularity theory and to Brill–Noether theory in algebraic geometry. Progress was made in the study of Berkovich spaces over Z, and they were used for an arithmetic Hodge index theorem with applications to the non-archimedean Calabi-Yau problem. An analog of complex differential geometry was developed on Berkovich spaces which allows us to describe non-archimedean Monge-Ampere measures as a top-dimensional wedge product of first Chern forms or currents. Two talks focused on p-adic differential equations where Berkovich spaces help to understand the behaviour of radii of convergence. Scholze’s perfectoid spaces, which have led to spectacular progress regarding the monodromy weight conjecture, and their relations to padic Hodge theory were the topic of two other lectures. Methods from Model Theory become increasingly important in arithmetics, and we have seen two talks adressing this in connection with analytic spaces. Skeleta and tropical varieties are combinatorial pictures of Berkovich spaces, and these tools were used in several talks. In the one-dimensional case these methods lead to a better understanding of well-studied objects of algebraic geometry such as moduli spaces of curves or component groups. Non-Archimedean Analytic Geometry 3213 Workshop: Non-Archimedean Analytic Geometry