## Periodic Reporting for period 3 - TOSSIBERG (Theory of Stein Spaces in Berkovich Geometry)

Reporting period: 2018-07-01 to 2019-12-31

Complex Stein spaces may be thought of as analytic analogues of the affine schemes of algebraic geometry. They may be characterized in several manners: using convergence of holomorphic functions, topological properties or potential-theoretic properties, for instance. Especially useful for applications is the fact that their coherent cohomology vanishes. Despite the crucial importance of this theory in complex analytic geometry, its p-adic counterpart has hardly been sketched.

In the setting of Berkovich geometry (one among the several notions of p-adic geometry), recent developments have enabled to get a fine understanding of the topology of the spaces (work of Berkovich and Hrushovski-Loeser) and to define the basic tools of potential theory (work of Baker-Rumely, Thuillier, Boucksom-Favre-Jonsson and Chambert-Loir-Ducros). The conditions for a comprehensive study of p-adic Stein spaces are now met; this will be our first goal. The theory will then be used to investigate envelopes of holomorphy and meromorphy. As an application, we plan to derive rationality criteria for power series over function fields.

The second part of the project is devoted to the theory of Stein spaces for Berkovich spaces over rings of integers of number fields (where all the places appear on an equal footing). Those spaces have hardly been studied and only a very small part of the usual analytic machinery is available in this setting. Here, our main goal will consist in proving the basic and fundamental fact that relative polydisks are Stein spaces (in the cohomological sense). This will allow a deeper investigation of rings of convergent arithmetic power series (i.e. with integral coefficients) and will lead up to properties related to commutative algebra but also to the inverse Galois problem. Knowing that the coherent cohomology of polydisks vanishes also opens the road towards computing global cohomology groups for projective analytic spaces over ring of integers of number fields.

In the setting of Berkovich geometry (one among the several notions of p-adic geometry), recent developments have enabled to get a fine understanding of the topology of the spaces (work of Berkovich and Hrushovski-Loeser) and to define the basic tools of potential theory (work of Baker-Rumely, Thuillier, Boucksom-Favre-Jonsson and Chambert-Loir-Ducros). The conditions for a comprehensive study of p-adic Stein spaces are now met; this will be our first goal. The theory will then be used to investigate envelopes of holomorphy and meromorphy. As an application, we plan to derive rationality criteria for power series over function fields.

The second part of the project is devoted to the theory of Stein spaces for Berkovich spaces over rings of integers of number fields (where all the places appear on an equal footing). Those spaces have hardly been studied and only a very small part of the usual analytic machinery is available in this setting. Here, our main goal will consist in proving the basic and fundamental fact that relative polydisks are Stein spaces (in the cohomological sense). This will allow a deeper investigation of rings of convergent arithmetic power series (i.e. with integral coefficients) and will lead up to properties related to commutative algebra but also to the inverse Galois problem. Knowing that the coherent cohomology of polydisks vanishes also opens the road towards computing global cohomology groups for projective analytic spaces over ring of integers of number fields.

"Jérôme Poineau and Marco Maculan started investigating the notion of Stein space in the p-adic (and more generally non-archimedean) setting. Several definitions were available. In this context, the one mostly used is due to Kiehl (in ""Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie"", Inventiones mathematicae, 1966) and requires exhausting the space by affinoid spaces. In complex geometry, on the other hand, the definition is usually given in more topological terms through the notions of holomorphic separability and convexity. Finally, in the applications, one is especially interested in the vanishing of the cohomology of coherent sheaves, and this can also be taken as a definition. Jérôme Poineau and Marco Maculan managed to prove that those three definition actually give rise to the same notion in the case of spaces with boundary, provided the vanishing of cohomology is required to hold after an arbitrary scalar extension.

Jérôme Poineau and Velibor Bojkovic focused on the particular case of curves. In this setting, Michael Temkin recently proved deep results on the structure of morphisms. Jérôme Poineau and Velibor Bojkovic managed to give a new interpretation of the main object he introduced, the profile function, using fibers of scalar extensions. As a consequence, they are able to relate in a precise way, by means of an explicit formula, the ramification invariants of a morphism of p-adic curves to the radii of convergence of the associated differential equation. This was previously known only in the case of a tame morphism (where the result is trivial) and of the Frobenius morphism. They also derive a general formula computing the Laplacian of the height of the Newton polygon of a p-adic differential equation, a conjectured formula whose proof in the general case was still missing.

Ehud Hrushovski and François Loeser recently introduced a model-theoretic version of the analytification of a quasi-projective variety over a non-archimedean valued field. It gives rise to a strict pro-definable set in general and to a definable set in the case of curves. Jérôme Poineau and Pablo Cubides Kovacsics focused on the latter case and provide an alternative approach to endow the analytification of an algebraic curve with a definable structure. Along the way, they give definable versions of several usual notions of Berkovich analytic geometry: branch emanating from a point, residue curve at a point of type 2, etc. In addition, they derive a complete description of the definable subsets of analytic curves.

In the setting of global analytic geometry, i.e. for Berkovich spaces over Z, Jérôme Poineau managed to get a result of cohomological vanishing for higher-dimensional spaces. More precisely, he proved that closed Berkovich disks over Z have no higher coherent cohomology. This is a first step towards developing a full-fledged theory of Stein spaces in this context. Jérôme Poineau then derived results for rings of convergent arithmetic power series. The latter, roughly speaking series with coefficients in Z with positive radii of convergence, were introduced by David Harbater in a series of work related to the inverse Galois problem and he had proven several results about them in the case of one variable. Using cohomological techniques, Jérôme Poineau proved the noetherianity of rings of convergent arithmetic power series with an arbitrary number of variables.

In complex geometry, any Riemann surface of genus g admits a so-called Schottky uniformization, i.e. a uniformization by the complex projective line with 2g disks removed and whose group is the free group on g generators. David Mumford showed that this theory may partially be extended to a non-archimedean setting. Jérôme Poineau and Daniele Turchetti managed to give a uniform construction generalizing the previous ones. More precisely, they defined a universal Schottky space S as a Berkovich spaces over Z (an open subset of the affine"

Jérôme Poineau and Velibor Bojkovic focused on the particular case of curves. In this setting, Michael Temkin recently proved deep results on the structure of morphisms. Jérôme Poineau and Velibor Bojkovic managed to give a new interpretation of the main object he introduced, the profile function, using fibers of scalar extensions. As a consequence, they are able to relate in a precise way, by means of an explicit formula, the ramification invariants of a morphism of p-adic curves to the radii of convergence of the associated differential equation. This was previously known only in the case of a tame morphism (where the result is trivial) and of the Frobenius morphism. They also derive a general formula computing the Laplacian of the height of the Newton polygon of a p-adic differential equation, a conjectured formula whose proof in the general case was still missing.

Ehud Hrushovski and François Loeser recently introduced a model-theoretic version of the analytification of a quasi-projective variety over a non-archimedean valued field. It gives rise to a strict pro-definable set in general and to a definable set in the case of curves. Jérôme Poineau and Pablo Cubides Kovacsics focused on the latter case and provide an alternative approach to endow the analytification of an algebraic curve with a definable structure. Along the way, they give definable versions of several usual notions of Berkovich analytic geometry: branch emanating from a point, residue curve at a point of type 2, etc. In addition, they derive a complete description of the definable subsets of analytic curves.

In the setting of global analytic geometry, i.e. for Berkovich spaces over Z, Jérôme Poineau managed to get a result of cohomological vanishing for higher-dimensional spaces. More precisely, he proved that closed Berkovich disks over Z have no higher coherent cohomology. This is a first step towards developing a full-fledged theory of Stein spaces in this context. Jérôme Poineau then derived results for rings of convergent arithmetic power series. The latter, roughly speaking series with coefficients in Z with positive radii of convergence, were introduced by David Harbater in a series of work related to the inverse Galois problem and he had proven several results about them in the case of one variable. Using cohomological techniques, Jérôme Poineau proved the noetherianity of rings of convergent arithmetic power series with an arbitrary number of variables.

In complex geometry, any Riemann surface of genus g admits a so-called Schottky uniformization, i.e. a uniformization by the complex projective line with 2g disks removed and whose group is the free group on g generators. David Mumford showed that this theory may partially be extended to a non-archimedean setting. Jérôme Poineau and Daniele Turchetti managed to give a uniform construction generalizing the previous ones. More precisely, they defined a universal Schottky space S as a Berkovich spaces over Z (an open subset of the affine"

Jérôme Poineau and Marco Maculan plan to extend the equivalence result they proved to spaces with boundary. As shown by examples of Liu, this cannot be true as such and they are working on a suitable refinement of Kieh’ls definition of Stein space to overcome this difficulty.

Jérôme Poineau and Andrea Pulita are investigating specifically the case of p-adic curves. In this setting, they proved a very general analogue of Behnke-Stein approximation result, which enables to get a very precise understanding of Stein curves. They plan to apply those results to study the cohomology of p-adic differential equations on arbitrary quasi-smooth Stein curves and prove index formulas.

Jérôme Poineau and Velibor Bojkovic have the project to investigate higher-dimensional analogues of their results by systematically using base-change techniques. This would lead to a very well-behaved ramification theory even in the case of an imperfect residue field.

Jérôme Poineau and Pablo Cubides Kovacsics are working on an extension of their results to cover a wider class of curves (affinoid curves, which should be definable in the analytic language of Lipshitz), deal with fields with higher-rank valuations and obtain definability results in families.

Jérôme Poineau and Thibaud Lemanissier plan to extend the result of cohomological vanishing to open disks. They also have a longer-term project to study higher push-forwards of coherent sheaves by proper morphism.

Jérôme Poineau and Daniele Turchetti plan to pursue their study of the universal Schottky space by studying is properties: connectedness, fundamental group, cohomology, etc. They envision applications in two directions: for Teichmüller modular forms and for moduli spaces of curves.

Jérôme Poineau and Andrea Pulita are investigating specifically the case of p-adic curves. In this setting, they proved a very general analogue of Behnke-Stein approximation result, which enables to get a very precise understanding of Stein curves. They plan to apply those results to study the cohomology of p-adic differential equations on arbitrary quasi-smooth Stein curves and prove index formulas.

Jérôme Poineau and Velibor Bojkovic have the project to investigate higher-dimensional analogues of their results by systematically using base-change techniques. This would lead to a very well-behaved ramification theory even in the case of an imperfect residue field.

Jérôme Poineau and Pablo Cubides Kovacsics are working on an extension of their results to cover a wider class of curves (affinoid curves, which should be definable in the analytic language of Lipshitz), deal with fields with higher-rank valuations and obtain definability results in families.

Jérôme Poineau and Thibaud Lemanissier plan to extend the result of cohomological vanishing to open disks. They also have a longer-term project to study higher push-forwards of coherent sheaves by proper morphism.

Jérôme Poineau and Daniele Turchetti plan to pursue their study of the universal Schottky space by studying is properties: connectedness, fundamental group, cohomology, etc. They envision applications in two directions: for Teichmüller modular forms and for moduli spaces of curves.