arXiv subjects

Annette Werner

Publications and source records attributed to Annette Werner.

At least 19 recordsLinked to original sources

arXiv2026

Benchmarks in Leipzig

Between April 1 and May 15, 2026, a group of 49 mathematicians compiled a dataset of research-level mathematics questions with known answers. Most of the work was done during the 3-day workshop *Benchmarks in Leipzig* with 35 participants at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. We present the resulting collection of 100 questions. We evaluated these questions in three stages: a single attempt by five state-of-the-art LLMs, followed by a 20-runs-per-model evaluation with three of these models, and finally a 3-run attempt with two heavy-thinking models. After Stage 1, 41 questions remained completely unsolved; after Stage 2, this count dropped to 16; and we concluded Stage 3 with only 2 unsolved questions. This demonstrates that the mathematical reasoning capabilities of LLMs are becoming impressive.

math.HO↗

arXiv2024

Combinatorial Cremona automorphisms and Coxeter arrangement matroids

We explore birational geometry of matroids by investigating automorphisms of their coarse Bergman fans. Combinatorial Cremona maps provide such automorphisms of Bergman fans which are not induced by matroid automorphisms. We investigate the structure of matroids allowing combinatorial Cremona maps and prove a realizability criterion in the presence of two different Cremonas. We also prove that for all matroids associated to Coxeter arrangements the group of coarse automorphisms of the Bergman fan is generated by the matroid automorphisms and at most one combinatorial Cremona map.

math.AG↗

arXiv2023

Semi-homogeneous vector bundles on abelian varieties: moduli spaces and their tropicalization

Let $A$ be an abelian variety with totally degenerate reduction over a non-Archimedean field. We describe the moduli space of semihomogeneous vector bundles on $A$ from the perspective of non-Archimedean uniformization and show that the essential skeleton may be identified with a tropical analogue of this moduli space. For $H=0$ our moduli space may be identified with the moduli space $M_{0,r}(A)$ of semistable vector bundles with vanishing Chern classes on $A$. In this case we construct a surjective analytic morphism from the character variety of the analytic fundamental group of $A$ onto $M_{0,r}(A)$, which naturally tropicalizes. One may view this construction as a non-Archimedean uniformization of $M_{0,r}(A)$.

math.AG↗

arXiv2023

$p$-adic Simpson correspondences for principal bundles in abelian settings

We explore generalizations of the $p$-adic Simpson correspondence on smooth proper rigid spaces to principal bundles under rigid group varieties $G$. For commutative $G$, we prove that such a correspondence exists if and only if the Lie group logarithm is surjective. Second, we treat the case of general $G$ when $X$ is itself an ordinary abelian variety, in which case we prove a generalisation of Faltings' ``small'' correspondence to general rigid groups. On abeloid varieties, we also prove an analog of the classical Corlette-Simpson correspondence for principal bundles under linear algebraic groups.

math.AG↗

arXiv2022

On the birational geometry of matroids

This paper investigates isomorphisms of Bergman fans of matroids respecting different fan structures, which we regard as matroid analogs of birational maps. We show that isomorphisms respecting the fine fan structure are induced by matroid isomorphisms. We introduce Cremona automorphisms of the coarse structure of Bergman fans, which are not induced by matroid automorphisms. We show that the automorphism group of the coarse fan structure is generated by matroid automorphisms and Cremona maps in the case of rank 3 matroids which are not parallel connections and for modularly complemented matroids.

math.AG↗

arXiv2021

An intrinsic characterization of Bruhat-Tits buildings inside analytic groups

Given a semisimple group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat-Tits builidng into the analytic space associated to the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding.

math.AG↗

arXiv2021

The $p$-adic Corlette-Simpson correspondence for abeloids

For an abeloid variety $A$ over a complete algebraically closed field extension $K$ of $\mathbb Q_p$, we construct a $p$-adic Corlette-Simpson correspondence, namely an equivalence between finite-dimensional continuous $K$-linear representations of the Tate module and a certain subcategory of the Higgs bundles on $A$. To do so, our central object of study is the category of vector bundles for the $v$-topology on the diamond associated to $A$. We prove that any pro-finite-\'etale $v$-vector bundle can be built from pro-finite-\'etale $v$-line bundles and unipotent $v$-bundles. To describe the latter, we extend the theory of universal vector extensions to the $v$-topology and use this to generalise a result of Brion by relating unipotent $v$-bundles on abeloids to representations of vector groups.

math.AG↗

arXiv2020

Local systems on diamonds and $p$-adic vector bundles

We use Scholze's framework of diamonds to gain new insights in correspondences between $p$-adic vector bundles and local systems. Such correspondences arise in the context of $p$-adic Simpson theory in the case of vanishing Higgs fields. In the present paper we provide a detailed analysis of local systems on diamonds for the \'etale, pro-\'etale, and the $v$-topology, and study the structure sheaves for all three topologies in question. Applied to proper adic spaces of finite type over $\mathbb{C}_p$ this enables us to prove a category equivalence between $\mathbb{C}_p$-local systems with integral models, and modules under the $v$-structure sheaf which modulo each $p^n$ can be trivialized on a proper cover. The flexibility of the $v$-topology together with a descent result on integral models of local systems allows us to prove that the trivializability condition in the module category may be checked on any normal proper cover. This result leads to an extension of the parallel transport theory by Deninger and the second author to vector bundles with numerically flat reduction on a proper normal cover.

math.AG↗

arXiv2019

Strongly semistable reduction of syzygy bundles on plane curves

We investigate degenerations of syzygy bundles on plane curves over $p$-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fiber consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fiber is a syzygy bundle and which become trivial when restricted to each projective line in the special fiber. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamental open problem in $p$-adic Simpson theory to determine the category of Higgs bundles corresponding to continuous representations of the \'etale fundamental group of a curve. Faltings' $p$-adic Simpson correspondence and work of Deninger and the second author shows that bundles with Higgs field zero and potentially strongly semistable reduction fall into this category. Hence the results in the present paper determine a class of syzygy bundles on plane curves giving rise to a $p$-adic local system. We apply our methods to a concrete example on the Fermat curve suggested by Brenner and prove that this bundle has potentially strongly semistable reduction.

math.AG↗

arXiv2017

Realizability of tropical canonical divisors

We use recent results by Bainbridge-Chen-Gendron-Grushevsky-Moeller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in equicharacteristic zero. Given a pair $(Γ, D)$ consisting of a stable tropical curve $Γ$ and a divisor $D$ in the canonical linear system on $Γ$, we give a purely combinatorial condition to decide whether there is a smooth curve $X$ over a non-Archimedean field whose stable reduction has $Γ$ as its dual tropical curve together with a effective canonical divisor $K_X$ that specializes to $D$. Along the way, we develop a moduli-theoretic framework to understand Baker's specialization of divisors from algebraic to tropical curves as a natural toroidal tropicalization map in the sense of Abramovich-Caporaso-Payne.

math.AG↗

arXiv2017

On Endomorphisms of Arrangement Complements

Let $\Omega$ be the complement of a connected, essential hyperplane arrangement. We prove that every dominant endomorphism of $\Omega$ extends to an endomorphism of the tropical compactification $X$ of $\Omega$ associated to the Bergman fan structure on the tropicalization of $\Omega$. This generalizes a previous result by R\'emy, Thuillier and the second author which states that every automorphism of Drinfeld's half-space over a finite field $\mathbb{F}_q$ extends to an automorphism of the successive blow-up of projective space at all $\mathbb{F}_q$-rational linear subspaces. This successive blow-up is in fact the minimal wonderful compactification by de Concini and Procesi, which coincides with $X$ by results of Feichtner and Sturmfels. Whereas the previous proof is based on Berkovich analytic geometry over the trivially valued finite ground field, the generalization discussed in the present paper relies on matroids and tropical geometry.

math.AG↗

arXiv2017

Parallel transport for vector bundles on p-adic varieties

We develop a theory of étale parallel transport for vector bundles with numerically flat reduction on a $p$-adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous $p$-adic representation of the étale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a $p$-adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings' $p$-adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.

math.AG↗

arXiv2017

Wonderful compactifications of Bruhat-Tits buildings

Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, is also investigated.

math.GR↗

arXiv2017

Tropical Skeletons

In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let $K$ be a complete non-Archimedean field, and let $X$ be a closed subscheme of a toric variety over $K$. We define the tropical skeleton of $X$ as the subset of the associated Berkovich space $X^{\rm an}$ which collects all Shilov boundary points in the fibers of the Kajiwara--Payne tropicalization map. We develop polyhedral criteria for limit points to belong to the tropical skeleton, and for the tropical skeleton to be closed. We apply the limit point criteria to the question of continuity of the canonical section of the tropicalization map on the multiplicity-one locus. This map is known to be continuous on all torus orbits; we prove criteria for continuity when crossing torus orbits. When $X$ is schön and defined over a discretely valued field, we show that the tropical skeleton coincides with a skeleton of a strictly semistable pair, and is naturally isomorphic to the parameterizing complex of Helm--Katz.

math.AG↗

arXiv2016

Skeletons and tropicalizations

Let $K$ be a complete, algebraically closed non-archimedean field with ring of integers $K^\circ$ and let $X$ be a $K$-variety. We associate to the data of a strictly semistable $K^\circ$-model $\mathscr X$ of $X$ plus a suitable horizontal divisor $H$ a skeleton $S(\mathscr X,H)$ in the analytification of $X$. This generalizes Berkovich's original construction by admitting unbounded faces in the directions of the components of H. It also generalizes constructions by Tyomkin and Baker--Payne--Rabinoff from curves to higher dimensions. Every such skeleton has an integral polyhedral structure. We show that the valuation of a non-zero rational function is piecewise linear on $S(\mathscr X, H)$. For such functions we define slopes along codimension one faces and prove a slope formula expressing a balancing condition on the skeleton. Moreover, we obtain a multiplicity formula for skeletons and tropicalizations in the spirit of a well-known result by Sturmfels--Tevelev. We show a faithful tropicalization result saying roughly that every skeleton can be seen in a suitable tropicalization. We also prove a general result about existence and uniqueness of a continuous section to the tropicalization map on the locus of tropical multiplicity one.

math.AG↗

arXiv2015

Analytification and Tropicalization over non-archimedean fields

This paper provides an overview of recent progress on the interplay between tropical geometry and non-archimedean analytic geometry in the sense of Berkovich. After briefly discussing results by Baker, Payne and Rabinoff in the case of curves, we explain a result by Cueto, Häbich and the author comparing the tropical Grassmannian of planes to the analytic Grassmannian. We also give an overview of the general higher-dimensional theory developed by Gubler, Rabinoff and the author. In particular, we explain the construction of generalized skeleta in which are polyhedral substructures of Berkovich spaces lending themselves to comparison with tropicalizations. We discuss the slope formula for the valuation of rational functions and explain two results on the comparison between polyhedral substructures of Berkovich spaces and tropicalizations.

math.AG↗

arXiv2015

Bruhat-Tits buildings and analytic geometry

This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry canbe used to compactify buildings. We discuss in detail the example of the special linear group. Moreover, we give an intrinsic description of Bruhat-Tits buildings in the framework of non-Archimedean analytic geometry.

math.GR↗

arXiv2014

Faithful tropicalization of the Grassmannian of planes

We show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich's sense by constructing a continuous section to the tropicalization map. Our main tool is an explicit description of the algebraic coordinate rings of the toric strata of the Grassmannian. We determine the fibers of the tropicalization map and compute the initial degenerations of all the toric strata. As a consequence, we prove that the tropical multiplicities of all points in the tropical projective Grassmannian are equal to one. Finally, we determine a piecewise linear structure on the image of our section that corresponds to the polyhedral structure on the tropical projective Grassmannian.

math.AG↗