# Non-Archimedean and Tropical Geometry (2017)

Date & Time

Location

Schloss Elmau
In Elmau 2, 82493 Krün Germany

May 14-20, 2017

Note: Attendance is by invitation only.

Organizers:
Matt Baker, Georgia Institute of Technology
Sam Payne, Yale University

Previous years:

Organized by Matthew Baker (Georgia Tech) and Sam Payne (Yale University), the symposium will focus on setting a clear agenda for future developments in the related fields of tropical and non-Archimedean analytic geometry. One of the meeting goals is to produce high quality expository material presenting the methods, results and ambitions of these active research areas. Another is to identify problems in other fields of mathematics that could be amenable to tropical and non-Archimedean analytic methods and establish new rigorous links with those neighboring fields.

Discussion Topics Include:
• Combinatorial and tropical analogues of Hodge theory
• Refined curve counting, tropical geometry, and motivic invariants
• Wild coverings of curves and lifting problems for algebraic and tropical curves
• Rational points on curves, p-adic integration, and relations with tropical geometry

Talk Notes

• Agenda

Sunday
7:30 -9:30 PM Dinner at La Salle
Monday
8:00 – 9:30 AM Breakfast at La Salle
9:30 – 10:30 AM Ilia Itenberg
Tropical homology and Betti numbers of real algebraic varieties
10:30 – 11:00 AM Break
11:00 – 12:00 PM Kristin Shaw
Superforms and tropical cohomology
12:30 – 1:30 PM Lunch at La Salle
1:30 – 4:00 PM Discussion & Recreation*
4:00 – 4:30 PM Tea
4:30 – 5:30 PM Yifeng Liu
Tropical Dolbeault cohomology over non-Archimedean fields
5:30 – 5:45 PM Break
5:45 – 6:45 PM Melody Chan
Topology of the tropical moduli spaces of curves
7:30 – 9:30 PM Dinner at La Salle
Tuesday
8:00 – 9:30 AM Breakfast at La Salle
9:30 – 10:30 AM Mattias Jonsson
Tropical and non-Archimedean limits of degenerating families of volume forms
10:30 – 11:00 AM Break
11:00 – 12:00 PM José Ignacio Burgos Gil
A local approach to the non-Archimedean Monge-Ampère equation
12:30 – 1:30 PM Lunch at La Salle
1:30 – 4:00 PM Discussion & Recreation*
4:00 – 4:30 PM Tea
4:30 – 5:30 PM Tony Yue Yu
The Frobenius conjecture in dimension two
5:30 – 5:45 PM Break
5:45 – 6:45 PM Problem Session
7:30 – 9:30 PM Dinner at Wintergarden
Wednesday
8:00 – 9:30 AM Breakfast at La Salle
9:45 – 2:00 PM Guided Hike to Partnach Gorge
2:00 – 3:00 PM Lunch at Wintergarden
3:00 – 4:00 PM Recreation & Discussion
4:00 – 4:30 PM Tea
4:00 – 5:30 PM Joe Rabinoff
Potential theory on curves, p-adic integration, and rational points
5:30 – 5:45 PM Break
5:45 – 6:45 PM Eric Katz
Uniform bounds on torsion points on curves
7:30 – 9:30 PM Dinner at La Salle
Thursday
8:00 – 9:30 AM Breakfast at La Salle
9:30 – 10:30 AM Michael Temkin
On wild coverings of Berkovich curves
10:30 – 11:00 AM Break
11:00 – 12:00 PM Stefan Wewers
Wildly ramified covers I: Swan conductors and semistable reduction
12:30 – 1:30 PM Lunch at La Salle
1:30 – 4:00 PM Discussion & Recreation*
4:00 – 4:30 PM Tea
4:30 – 5:30 PM Andrew Obus
Wildly ramified covers, II: Swan conductors, the Berkovich different, and lifting
5:30 – 5:45 PM Break
5:45 – 6:45 PM Problem Session
7:30 – 9:30 PM Dinner at La Salle
Friday
8:00 – 9:30 AM Breakfast at La Salle
9:30 – 10:30 AM Jérôme Poineau
Radii of convergence of $$p$$-adic differential equations and ramification
10:30 – 11:00 AM Break
11:00 – 12:00 PM Sam Payne
Brill-Noether theory for curves of a fixed gonality, after Jensen and Ranganathan
12:30 – 1:30 PM Lunch at La Salle
1:30 – 4:00 PM Discussion & Recreation*
4:00 – 4:30 PM Tea
4:30 – 5:30 PM June Huh
Negative correlation and Hodge-Riemann relations
5:30 – 5:45 PM Break
5:45 – 6:45 PM Johannes Nicaise
A motivic Fubini theorem for the tropicalization map
7:30 – 9:30 PM Dinner at Kaminstüberl & Magic with Matt
Locations
SESSIONS Pavillion located at the Schloss Elmau Retreat
MEALS La Salle unless otherwise noted
TEA & DISCUSSION Pavillion located at the Schloss Elmau Retreat
EXCURSION Meet in Schloss Elmau Lobby
SATURDAY DEPARTURE Meet in Schloss Elmau Lobby

*Participants may explore the hotel property and its surrounding areas as well as engage in informal discussion with other participants.

### Start Time

The meeting will begin on Monday morning at 9:30 am in the Pavillon meeting room located at the Schloss Elmau Retreat. Breakfast will start at 8:00 am in La Salle restaurant.

### Meeting Location

All symposium activities will take place in the Pavillon meeting room located at the Schloss Elmau Retreat.

### AV

Participants will have access to a projector and screen for computer based talks as well as blackboards for those who prefer to give board-based talks. High-speed Internet access is available as well.

### Wednesday Excursion

On Wednesday symposia activities will be shortened so that anyone interested may attend a hiking excursion. Participants will embark on a fully guided hiking tour from Schloss-Elmau through the Partnach Gorge. The hike difficulty is moderate and will take approximately 3 hours.

### Dress Code

Business casual clothing should be worn during the symposia. The weather can change very quickly so we also advise bringing warm-weather clothing appropriate for spring in the mountains.If you plan on taking part in the hike to Partnach Gorge we advise you to wear hiking boots, or even better, light mountain boots, warm clothing (e.g. a sweater), sun protection (e.g. light cap) and take waterproofs (e.g. raincoat or umbrella) with you. Bringing along a small backpack or satchel in which to carry your water, camera and other items may also be useful to you.

• Participants

 Karim Adiprasito Hebrew University Matt Baker Georgia Institute of Technology Vladimir Berkovich Weizmann Institute of Science José Ignacio Burgos Gil Instituto de Ciencias Matemáticas Dustin Cartwright University of Tennessee Antoine Chambert-Loir Université Paris-Diderot Melody Chan Brown University Antoine Ducros Université Paris 6 June Huh Institute for Advanced Study Ilia Itenberg Université Pierre et Marie Curie Mattias Jonsson University of Michigan Eric Katz Ohio State University Yifeng Liu Northwestern Johannes Nicaise Imperial College London Andrew Obus University of Virginia Sam Payne Yale University Jérôme Poineau Université de Caen Joe Rabinoff Georgia Institute of Technology Kristin Shaw TU Berlin Michael Temkin Hebrew University Annette Werner Goethe University Stefan Wewers Ulm University Tony Yue Yu Université Paris-Diderot
• Abstracts

### José Ignacio Burgos Gil

#### A local approach to the non-Archimedean Monge-Ampère equation

Boucksom, Favre and Jonnson have proved the existence of solutions of the non-Archimedean Monge-Ampère equation in residue characteristic zero, in the case when the variety has a model over the function field of a curve. This algebraicity assumption allows them to use global methods.

In this talk we will explain how to remove the algebraicity hypothesis by using non-archimedean volumes, that are a local analogue of the arithmetic volumes. The non-Archimedean volumes are differentiable and allow us to prove an orthogonality property and the existence of solutions of the Monge-Ampère equation.

In the absence of a global hypothesis, to study volumes it is not enough to control the space of global sections but also certain $$H^{1}$$ groups. The main new input of this local approach is the use of a general version the Demailly-Angelini Holomorphic Morse inequalities (provided to us by R. Lazarsfeld) to control these first cohomology groups.

This is joint work with W. Gubler, P. Jell, K. Kunnemann and F. Martin.

### Melody Chan

#### Topology of the tropical moduli spaces of curves

The moduli space of n-marked, genus g tropical curves is a cell complex that was identified in work of Abramovich-Caporaso-Payne with the boundary complex of the complex moduli space $$M_{g,n}$$. I will give results on the topology of tropical $$M_{g,n}$$, obtaining as corollaries new calculations on the top-weight cohomology of the corresponding complex moduli spaces. Joint work with Soren Galatius and Sam Payne.

### June Huh

#### Negative correlation and Hodge-Riemann relations

Hodge-Riemann relations for finite graphs (and more generally matroids) are only partially understood. Huh will explain what is meant by this, and speculate on generalizations of known facts. This talk will be non-technical: Nothing will be assumed beyond basic linear algebra.

### Ilia Itenberg

#### Tropical homology and Betti numbers of real algebraic varieties

Under some assumptions, a tropical variety can be approximated by a one-parametric family of complex varieties, which provides an important link between complex and tropical geometries. The purpose of this talk is to discuss tropical homology together with its relations to Hodge decompositions (respectively, homology) in complex (respectively, real) world.

### Mattias Jonsson

#### Tropical and non-Archimedean limits of degenerating families of volume forms

Kontsevich and Soibelman have given a conjectural description of the Gromov-Hausdorff limit of a maximally degenerate family of polarized Calabi-Yau manifolds in terms of the Berkovich space attached to the degeneration. Motivated by this, Mustata, Nicaise and Xu recently studied the essential skeleton of this Berkovich space, which is a natural realization of the dual complex of a minimal model of the degeneration.

Jonsson will present joint work with Sebastien Boucksom, in which they show that the volume form induced by a holomorphic form of top degree on a fiber converges, in a suitable sense, to an explicit Lebesgue type measure on the essential skeleton.

### Eric Katz

#### Uniform bounds on torsion points on curves

Katz will discuss the recent proof with Joseph Rabinoff and David Zureick-Brown that there is a uniform bound for the number of rational points on genus g curves of Mordell-Weil rank at most g-3, extending a result of Stoll on hyperelliptic curves. Our work also gives unconditional bounds on the number of rational torsion points and bounds on the number of geometric torsion points on curves with very degenerate reduction type. Our bounds, like many in Diophantine geometry work by showing that the points to bounded are contained in the zero set of a collection of functions and then bounding the zeroes of those functions. Katz will focus on the challenges in making those zero bounds uniform, focusing on p-adic integration on bad reduction curves. He will also describe some work in progress towards unifying the Coleman-Chabauty method with Buium’s method of differential characters.

### Yifeng Liu

#### Tropical Dolbeault cohomology over non-Archimedean fields

In this talk, Liu will introduce the the notion of tropical Dolbeault cohomology for algebraic variety, or more generally analytic spaces, over non-Archimedean fields, using the construction of real forms by Chambert-Loir and Ducros. Liu will define the cycle class map into this cohomology and study some fundamental properties. If time permits, Liu will also discuss some analogue of Hodge theory for this cohomology.

### Johannes Nicaise

#### A motivic Fubini theorem for the tropicalization map

This talk is based on joint work with Sam Payne. Nicaise will present a Fubini theorem for the tropicalization map in the context of Hrushovski and Kazhdan’s theory of motivic integration. As an application, Nicaise will prove a conjectural description by Davison and Meinhardt of the motivic nearby fiber of a weighted homogeneous polynomial. This conjecture emerged in the theory of motivic Donaldson-Thomas invariants.

### Andrew Obus

#### Wildly ramified covers, II: Swan conductors, the Berkovich different, and lifting

Obus considers the question of whether a wildly ramified Galois cover of curves in characteristic p lifts to characteristic zero. This is equivalent to the Local Lifting Problem discussed in Stefan Wewers’s lecture. Obus will give a brief overview of some of the known results, but will then focus on a particular technique involved in the resolution of the “Oort conjecture” (that cyclic covers lift). Using this technique, one can show that under certain circumstances, a family of covers of a non-archimedean disk has at least one fiber that is itself a disk. This is a crucial step in the proof of the Oort conjecture, as well as in further work on lifting metacyclic covers. The major idea is to understand the location of certain “kinks” in the piecewise linear function given by Cohen-Temkin-Trushin’s Berkovich different.

### Sam Payne

#### Brill-Noether theory for curves of a fixed gonality, after Jensen and Ranganathan

Payne will present recent work of D. Jensen and and D. Ranganathan, proving a generalization of the Brill-Noether theorem for the variety of special divisors on a general curve of given genus and gonality. This work builds on earlier results of Pflueger, who analyzed the tropical geometry of special divisors on chains of loops to give an upper bound for the dimensions of these varieties. The main result is that this upper bound is sharp, and the proof blends techniques from logarithmic stable maps and nonarchimedean analytic geometry.

### Jérôme Poineau

Following F. Baldassarri, to any module with connection on a quasi-smooth $$p$$-adic Berkovich curve and any point of this curve, one may associate a family of radii of convergence. Given a finite étale morphism $$f : Y \to X$$ of quasi-smooth $$p$$-adic Berkovich curves and a module with connection $$(\mathcal{F},\nabla)$$ on $$Y$$, one can get one on $$X$$ by push-forward. We explain how to relate the radii of convergence of the two modules using invariants coming from the topological behavior of the morphism $$f$$. When $$(\mathcal{F},\nabla)$$ is the trivial module with connection, we use results of M. Temkin to prove that the radii of convergence of the push-forward at a point $$x$$ in $$X$$ of type 2 are the jumps of the upper ramification filtration of the various extensions $$\mathcal{H}(y)/\mathcal{H}(x)$$, where $$y$$ runs through the preimages of $$x$$. This is joint work with Velibor Bojković.

### Joe Rabinoff

#### Potential theory on curves, p-adic integration, and rational points

In this talk we show how potential theory on Berkovich curves leads to bounding the number of zeros of locally analytic functions on annuli. We then apply Chabauty’s method and work of Stoll to give a bound on the number of rational points of a hyperbolic curve which is uniform in the genus.

### Kristin Shaw

#### Superforms and tropical cohomology

This talk will present another approach to tropical cohomology via superforms. Superforms are bigraded real valued differential forms on polyhedral spaces introduced by Lagerberg. Shaw will begin by explaining our result that the Dolbeault cohomology of superforms is canonically isomorphic to tropical cohomology. She will also discuss our proof of a version of Poincaré duality for tropical manifolds. This is joint work with Philipp Jell and Jascha Smacka.

Shaw will then survey open questions about possible properties of tropical cohomology analogous to those of Weil cohomology theories and discuss how the theory of superforms may provide a fruitful approach to these questions. Finally, she will comment on recent joint work towards a tropical version of the Lefschetz (1, 1) Theorem with Philipp Jell and Johannes Rau.

### Michael Temkin

#### On wild coverings of Berkovich curves

The structure of residually tame morphisms f between Berkovich curves over an algebraically closed field k is adequately described by its skeleton enhanced to a morphism of metric graphs with reduction k-curves attached to the vertices. In particular, the corresponding morphism of enhanced graphs is the only discrete invariant of f, and a lifting theorem of Amini-Baker-Brugallé-Rabinoff shows that any suitable morphism of such graphs lifts to a morphism of Berkovich curves, see [ABBR15].

This lecture is devoted to the structure of residually wild morphisms f, which is far more complicated but was clarified to a large extent in recent works [CTT16] and [Tem14]. In particular, Temkin will describe the residually wild locus of f in terms of a new combinatorial invariant of the morphism, the profile function, and will explain how the profile function is related to the classical higher ramification theory. In the end of his talk Temkin will report on my work in progress with U. Brezner, where we study the lifting problem for “moderately wild” morphisms whose residual inseparability index does not exceed p. In this case, the only new invariant is the different function, and one can associate to reduction curves a new reduction invariant, an exact differential form. This invariant is finer than the different δf , in particular, it controls the slopes of δf and can be used to reprove the local Riemann-Hurwitz formulas in the moderately wild case. We expect that the lifting in this case is possible once one takes the reduction differential forms into account.

References

[ABBR15] Omid Amini, Matthew Baker, Erwan Brugallé, and Joseph Rabinoff, Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta, Res. Math. Sci. 2 (2015), Art. 7, 67. MR 3375652

[CTT16] Adina Cohen, Michael Temkin, and Dmitri Trushin, Morphisms of Berkovich curves and the different function, Adv. Math. 303 (2016), 800–858. MR 3552539

[Tem14] Michael Temkin, Metric uniformization of morphisms of Berkovich curves, ArXiv e-prints (2014), http://arxiv.org/abs/1410.6892.

### Stefan Wewers

#### Wildly ramified covers I: Swan conductors and semistable reduction

This talk and the next one (by Andrew Obus) are aimed at giving a survey of certain results on reduction and lifting of Galois covers of curves.

Wewers will start by recalling the definition (by Kato and Saito) of the refined Swan conductor, in the context of a wildly ramified Galois cover of a Berkovich curve. He will briefly compare this to the invariants defined in the previous talk by Michael Temkin. Wewers will then explain two applications. The first application is to the explicit computation of semistable reduction of curves. We focus on the case of covers of degree p, where we can give an essentially complete answer. The second application is to the Local Lifting Problem. Here we explain the so-called Hurwitz tree obstruction which can be used to show that certain Galois covers of germs of curves in characteristic p cannot be lifted to characteristic zero.

### Tony Yue Yu

#### The Frobenius conjecture in dimension two

We apply the counting of non-archimedean holomorphic curves to the construction of the mirror of log Calabi-Yau surfaces. In particular, we prove the Frobenius structure conjecture of Gross-Hacking-Keel in dimension two. This is joint work with Sean Keel.

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