# Non-Archimedean and tropical geometry

This is the webpage of a workshop on non-Archimedean and tropical geometry.
The event, originally scheduled to take place at the Institut für Mathematik of the Goethe-Universität Frankfurt am Main in 2020, has been replaced by a series of online talks between October and December 2021. More specifically, the seminars take place online on Zoom every other Friday afternoon (Central European Time) starting at the beginning of October, that is on the following dates: October 1st, 15th, and 29th, November 12th and 26th, and December 10th.

The session of November 26th will take place in a hybrid format at the Centre de Mathématiques Laurent Schwartz of the École Polytechnique.

Organizing committee: Lorenzo Fantini, Martin Ulirsch, Annette Werner.

Registration: Zoom links will be sent out by email to all registered participants. If you wish to be added to the mailing list please send an email to Lorenzo or to Martin.

Videos and notes: Videos of most talks are posted to the NonArchTrop YouTube channel a few days after each session. Some notes and slides are available here.

### Speakers

• Erwan Brugallé
• Melody Chan
• Antoine Ducros
• Claudia Fevola
• Elise Goujard
• Walter Gubler
• David Holmes
• Klaus Künnemann
• Vlerë Mehmeti
• Jan Steffen Müller
• Navid Nabijou
• Jérôme Poineau
• Joseph Rabinoff
• Kris Shaw
• Daniele Turchetti
• Stefan Wewers
• Dimitri Wyss
• Tony Yue Yu

## Schedule and abstracts

### Friday, October 1st

This session took place online on Zoom.

14:00-15:00 (CET)
Antoine Ducros (Sorbonne University)
Non-standard analysis and non-archimedean geometry
video and notes
In a joint work with E. Hrushovski and F. Loeser, we show that certain one-parameter families of complex integrals have a limit that can be expressed as an integral on a Berkovich space over the field ℂ((t)) (in the sense of the theory of integration of real differential forms on Berkovich spaces, developed by Chambert-Loir and myself). In this talk I will present this result, but rather focus on the general method we introduced to prove it, which we hope will be useful for a lot of other situations involving a "t-adic limit of one-parameter families of complex objects". It consists in introducing a huge non-standard model of ℂ also equipped with a non-archimedean absolute value; working on such a model enables by design to deal at the same time with limits of usual complex objects (through non-standard analysis) as well as with non-archimedean objects, allowing for a direct comparison between these two worlds.

15:15-16:15 (CET)
Joe Rabinoff (Duke University)
Weakly smooth forms and Dolbeault cohomology of curves
video and slides
Gubler and I work out a theory of weakly smooth forms on non-Archimedean analytic spaces closely following the construction of Chambert-Loir and Ducros, but in which harmonic functions are forced to be smooth. We call such forms "weakly smooth". We compute the Dolbeault cohomology groups of rig-smooth, compact non-Archimedean curves with respect to this theory, and show that they have the expected dimensions and satisfy Poincaré duality. We carry out this computation by giving an alternative characterization of weakly smooth forms on curves as pullbacks of certain "smooth forms" on a skeleton of the curve. This yields an isomorphism between the Dolbeault cohomology of the skeleton, which can be computed using standard combinatorial methods, and the Dolbeault cohomology of the curve.
This work is joint with Walter Gubler.

16:30-17:30 (CET)
Tony Yue Yu (California Institute of Technology)
Generalizing GKZ secondary fan using Berkovich geometry
video and notes
Gelfand-Kapranov-Zelevinski introduced the notion of secondary fan in the study of the Newton polytopes of discriminants and resultants. It also controls the geometric invariant theory for toric varieties. We propose a generalization of the GKZ secondary fan to general Fano varieties using ideas from Berkovich geometry and Mori theory. Furthermore, inspired by mirror symmetry, we propose a synthetic construction of a universal family of Kollár-Shepherd-Barron-Alexeev stable pairs over the toric variety associated to the generalized secondary fan. This generalizes the families of Kapranov-Sturmfels-Zelevinski and Alexeev in the toric case. We gave a detailed construction and proved the stability in the case of del Pezzo surfaces. This is joint work with Hacking and Keel.

### Friday, October 15th

This session took place online on Zoom.

14:00-15:00 (CET)
Kris Shaw (University of Oslo)
A tropical approach to the enriched count of bitangents to quartic curves
Using A1 enumerative geometry Larson and Vogt have provided an enriched count of the 28 bitangents to a quartic curve. In this talk, I will explain how these enriched counts can be computed combinatorially using tropical geometry. I will also introduce an arithmetic analogue of Viro’s combinatorial patchworking for real algebraic curves which, in some cases, retains enough data to recover the enriched counts. Finally, I will comment on a possible tropical approach to the enriched count of the 27 lines on a cubic surface of Kass and Wickelgren. This talk is based on joint work with Hannah Markwig and Sam Payne.

15:15-16:15 (CET)
Melody Chan (Brown University)
Homology representations of compactified configurations on graphs applied to M2,n
video and notes
The homology of a compactified configuration space of a graph is equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. We construct an efficient free resolution for these homology representations. Using the Peter-Weyl Theorem for symmetric groups, we consider irreducible representations individually, vastly simplifying the calculation of these homology representations from the free resolution. As our main application, we obtain computer calculations of the top weight rational cohomology of the moduli spaces M_{2,n}, equivalently the rational homology of the tropical moduli spaces Δ_{2,n}, as a representation of S_n acting by permuting point labels for all n≤10. We further give new multiplicity calculations for specific irreducible representations of S_n appearing in cohomology for n≤17. Our approach produces information about these homology groups in a range well beyond what was feasible with previous techniques. Joint work with Christin Bibby, Nir Gadish, and Claudia He Yun.

16:30-17:30 (CET)
Daniele Turchetti (University of Warwick)
Schottky spaces and moduli of curves over Z
video and slides
Schottky uniformization is the description of an analytic curve as the quotient of an open dense subset of the projective line by the action of a Schottky group. All Riemann surfaces can be uniformized in this way, as well as some non-archimedean curves, called Mumford curves. In this talk, I will present a construction of universal Mumford curves: analytic spaces that parametrize both archimedean and non-archimedean uniformizable curves of a fixed genus. This result relies on the existence of suitable moduli spaces for marked Schottky groups, that can be built using the theory of Berkovich spaces over rings of integers of number fields developed by Poineau.
After introducing Berkovich analytic geometry from the beginning, I will describe universal Mumford curves and explain how these can be related to combinatorial structures arising from the theories of tropical moduli and geometric group theory. This is based on joint work with Jérôme Poineau.

### Friday, October 29th

This session took place online on Zoom.

14:00-15:00 (CET)
Elise Goujard (Bordeaux University)
Counting (tropical) covers: quasimodularity of generating functions
notes
Counting (weighted) ramified coverings of the sphere or the torus lead to several applications, one of them is the (weighted) count of integer points in some moduli spaces of flat surfaces, leading to the evaluation of the Masur-Veech volumes or the Siegel-Veech constants of these moduli spaces. Both these quantities are relevant to the study of dynamics in polygonal billiards for instance, but also to other dynamical problems on flat surfaces (such as the count of closed geodesics). With that motivation in mind, I will explain a joint work with Martin Möller on the generating series of these counts : they are quasimodular and this property holds "graph by graph". Some of our results can be stated in terms of tropical covers and I will detail the relation of our results with a previous work of Böhm-Bringmann-Buchholz-Markwig. I will also give several questions and conjectures about the "completed cycles" that appear naturally in this setting.

15:15-16:15 (CET)
Walter Gubler (University of Regensburg)
Forms on Berkovich spaces based on harmonic tropicalizations
video and slides
Chambert-Loir and Ducros introduced smooth forms and currents on Berkovich spaces using tropicalization maps induced by morphisms to tori. In joint work with Philipp Jell und Joe Rabinoff, we allow more generally harmonic tropicalization maps to define a larger class of weakly smooth forms which has essentially the same properties as the smooth forms, but have a better cohomological behavior.

16:30-17:30 (CET)
Navid Nabijou (University of Cambridge)
Tropical expansions and rubber torus actions
video and notes
Given a normal crossings pair (X,D), its tropicalization can be defined as the cone over the dual intersection complex of D. A polyhedral subdivision of the tropicalization induces a degeneration of X, called a tropical expansion. To first approximation, this is obtained by attaching additional "bubble" irreducible components to X, along strata in D. Tropical expansions form a natural class of degenerations which admit nice combinatorial descriptions. They have been studied and exploited by many authors, in many contexts.
We investigate automorphisms of tropical expansions covering the identity on X. We discover a purely tropical description: the so-called rubber torus is the torus associated to the moduli space of tropical edge lengths in the polyhedral subdivision, and its action on each component of the expansion is encoded in a linear "tropical position map." Our main application is to logarithmic enumerative geometry, which I will motivate, but the talk will not focus on this. This is joint work with Francesca Carocci.

### Friday, November 12th

This session took place online on Zoom.

14:00-15:00 (CET)
Jan Steffen Müller (University of Groningen)
video
I will discuss a new construction of p-adic height functions on abelian varieties over number fields using Besser's p-adic Arakelov theory. In analogy with Zhang's construction of Néron-Tate heights via adelic metric, these heights are given in terms of canonical p-adic adelic metrics on line bundles. As an application, I will describe a new and simplified approach to the quadratic Chabauty method for the computation of rational points on certain curves. This is joint work in progress with Amnon Besser and Padmavathi Srinivasan.

15:15-16:15 (CET)
Klaus Künnemann (University of Regensburg)
Pluripotential theory for tropical toric varieties and non-archimedean Monge-Ampère equations
video and slides
Tropical toric varieties are partial compactifications of finite dimensional real vector spaces associated with rational polyhedral fans. We introduce pluripotential theory on tropical toric varieties. This theory provides a canonical correspondence between complex and non-archimedean pluripotential theories of invariant plurisubharmonic functions on toric varieties. We apply this correspondence to solve invariant non-archimedean Monge-Ampère equations on toric and abelian varieties over arbitrary non-trivially valued non-archimedean fields. This is joint work with José Ignacio Burgos Gil, Walter Gubler and Philipp Jell.

16:30-17:30 (CET)
Stefan Wewers (Ulm University)
Explicit models of curves via nonarchimedian geometry
video and slides
I will report on my long term efford to make the computation of the semistable reduction of curves over p-adic fields effective and practical. I will focus on some particular cases, and on the use of methods from nonarchimedian analytic geometry to achieve this goal.

### Friday, November 26th

This session will take place in the Seminar Room of the Centre de Mathématiques Laurent Schwartz of the École Polytechnique (Palaiseau, France). Information on how to reach the department with public transport can be found here.
The talks will also be livestreamed on Zoom. Contact an organizer to receive a link.

14:00-15:00 (CET)
Vlerë Mehmeti (University Paris-Saclay)
A Hasse Principle on Berkovich Analytic Curves
Patching techniques, under various forms and inspired from results in complex analysis, have in the past been used as an approach to the inverse Galois problem. Recently, these techniques have become a very important tool in the study of local-global principles. I will explain how patching can be adapted to Berkovich analytic curves. Working in this setting, one can then obtain several local-global principles, all of which are applicable to quadratic forms.

15:15-16:15 (CET)
Erwan Brugallé (Nantes University)
Euler characteristic and signature of real semi-stable degenerations
It is interesting to compare the Euler characteristic of the real part of a real algebraic variety to the signature of its complex part. For example, a theorem by Itenberg and Bertrand states that both quantities are equal for "primitive T-hypersurfaces". After defining these latter, I will give a motivic proof of this theorem via the motivic nearby fiber of a real semi-stable degeneration. This proof extends in particular the original statement by Itenberg and Bertrand to non-singular tropical varieties.

16:30-17:30 (CET)
Jérôme Poineau (University of Caen Normandy)
Analytic dynamics over Z and torsion points of elliptic curves
Let Y be a Berkovich space over Z. Recall that such a space naturally contains non-Archimedean parts (such as usual p-adic Berkovich spaces) and Archimedean parts (such as complex analytic spaces). Denote by X the relative projective line over Y. For each point y in Y, let µ_y be a measure defined on the fiber X_y (which is an analytic projective line over the complete residue field associated to y). Inspired by the work of Favre on endomorphisms on hybrid Berkovich spaces, we prove general continuity results for families of measures of the form $(µ_y)_{y∈Y}$ coming from dynamical systems on X. Following a strategy by DeMarco-Krieger-Ye, we then deduce new cases of a conjecture of Bogomolov-Fu-Tschinkel on uniform bounds on the number of common images on ℙ^1 of torsion points of two elliptic curves.

### Friday, December 10th

This session will take place online on Zoom. Contact an organizer to receive a link.

14:00-15:00 (CET)
Claudia Fevola (Max Planck Institute Leipzig)
Kp Solitons from Tropical Limits
In this talk, we study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann’s theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which parametrizes all tau functions arising from such a sum. After introducing solitons solutions, we compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces.
This is joint work with Daniele Agostini, Yelena Mandelshtam and Bernd Sturmfels.

15:15-16:15 (CET)
David Holmes (Leiden University)
Piecewise polynomials and intersection theory
Rather than studying the Chow ring of a variety X, it is becoming increasing popular to study some kind of limit of Chow rings over blowups of X; for example this arises in enumerative geometry and in the study of singular hermitian metrics. Philosophically this limit might be thought of as the Chow ring of the Riemann-Zariski space of X, or perhaps of the valuativisation of X; but we will not dwell on such questions. Rather, we will explain how piecewise-polynomial functions on the tropicalisation of X give an efficient way to write down certain 'tautological' elements of this 'limit Chow ring', and describe applications to enumerative problems, and a Sage implementation for the moduli space of stable curves.

16:30-17:30 (CET)
Dimitri Wyss (École Polytechnique Fédérale de Lausanne)
DT-invariants from non-archimedean integrals
Let M(ß,χ) be the moduli space of one-dimensional semi-stable sheaves on a del Pezzo surface S, supported on an ample curve class ß and with Euler-characteristic χ.
Working over a non-archimedean local field F, we define a natural measure on the F-points of M(ß,χ). We prove that the integral of a certain gerbe on M(ß,χ) with respect to this measure is independent of χ if S is toric. A recent result of Maulik-Shen then implies that these integrals compute the Donaldson-Thomas invariants of M(ß,χ). A similar result holds for suitably twisted Higgs bundles. This is joint work with Francesca Carocci and Giulio Orecchia.

## Practical information

If you have any questions you can get in touch with Lorenzo Fantini and/or Martin Ulirsch.

The workshop is funded by the LOEWE Research Unit Uniformized structures in arithmetic and geometry and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) projects 456557832 and 444845124, the SFB 326 Geometry and Arithmetic of Uniformized Structures.

The image in the header, a view of Frankfurt's skyline at night, was taken by Mathias Konrath. It is covered by a Creative Commons CC0 1.0 Universal (CC0 1.0) Public Domain Dedication license and can be found here.

This website has been designed by Lorenzo Fantini picking many bits and pieces of code from various places online; in particular it makes use of Rokas Lengvenis's jQuery plugin scroller.