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Annette Werner

Publications and source records attributed to Annette Werner.

34 records · Page 2Linked to original sources
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↗
arXiv2014

Symmetric Alcoved Polytopes

Generalized alcoved polytopes are polytopes whose facet normals are roots in a given root system. We call a set of points in an alcoved polytope a generating set if there does not exist a strictly smaller alcoved polytope containing it. The type $A$ alcoved polytopes are precisely the tropical polytopes that are also convex in the usual sense. In this case the tropical generators form a generating set. We show that for any root system other than $F_4$, every alcoved polytope invariant under the natural Weyl group action has a generating set of cardinality equal to the Coxeter number of the root system.

math.CO↗
arXiv2012

Automorphisms of Drinfeld half-spaces over a finite field

We show that the automorphism group of Drinfeld's half-space over a finite field is the projective linear group of the underlying vector space. The proof of this result uses analytic geometry in the sense of Berkovich over the finite field equipped with the trivial valuation. We also take into account extensions of the base field.

math.AG↗
arXiv2011

Mustafin Varieties

A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special fiber is reduced and Cohen-Macaulay, and its irreducible components form interesting combinatorial patterns. For configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision. This connects our study to tropical and toric geometry. For general configurations, the irreducible components of the special fiber are rational varieties, and any blow-up of projective space along a linear subspace arrangement can arise. A detailed study of Mustafin varieties is undertaken for configurations in the Bruhat-Tits tree of PGL(2) and in the two-dimensional building of PGL(3). The latter yields the classification of Mustafin triangles into 38 combinatorial types.

math.AG↗
arXiv2010

A tropical view on Bruhat-Tits buildings and their compactifications

We relate some features of Bruhat-Tits buildings and their compactifications to tropical geometry. If G is a semisimple group over a suitable non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of G and in some of its compactifications are described by tropical linear algebra. The compactifications we consider arise from algebraic representations of G. We show that the fan which is used to compactify an apartment in this theory is given by the weight polytope of the representation and that it is related to the tropicalization of the hypersurface given by the character of the representation.

math.GR↗
arXiv2009

Bruhat-Tits theory from Berkovich's point of view. II. Satake compactifications

In our previous paper "Bruhat-Tits theory from Berkovich's point of view. I ? Realizations and compactifications of buildings", we investigated realizations of the Bruhat-Tits building B(G,k) of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry, and we studied in detail the compactifications of the building which arise from this point of view. In this paper, we give a representation theoretic flavor to these compactifications, following Satake's original constructions for Riemannian symmetric spaces. We first prove that Berkovich compactifications of a building coincide with the compactifications previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding B(G,k) in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.

math.GR↗
arXiv2009

A tropical view on the Bruhat-Tits building of SL and its compactifications

We describe the stabilizers of points in the Bruhat-Tits building of the group SL with tropical geometry. There are several compactifications of this building associated to algebraic representations of SL. We show that the fans used to compactify apartments in this theory are given by tropical Schur polynomials.

math.CO↗
arXiv2009

Bruhat-Tits Theory from Berkovich's Point of View. I - Realizations and Compactifications of Buildings

We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group G over a suitable non-Archimedean field k we define a map from the Bruhat-Tits building B(G,k) to the Berkovich analytic space Gan asscociated with G. Composing this map with the projection of G^an to its flag varieties, we define a family of compactifications of B(G,k). This generalizes results by Berkovich in the case of split groups. Moreover, we show that the boundary strata of the compactified buildings are precisely the Bruhat-Tits buildings associated with a certain class of parabolics. We also investigate the stabilizers of boundary points and prove a mixed Bruhat decomposition theorem for them.

math.AG↗
arXiv2006

Compactifications of Bruhat-Tits buildings associated to linear representations

Let G be a connected semisimple group over a non-Archimedean local field. For every faithful, geometrically irreducible linear representation of G we define a compactification of the associated Bruhat-Tits building X(G). This yields a finite family of compactifications of X(G) which contains the polyhedral compactification. Moreover, this family can be regarded as a non-Archimedean analogue of Satake compactifications for symmetric spaces.

math.AG↗
arXiv2005

Vector bundles on p-adic curves and parallel transport

We define functorial isomorphisms of parallel transport along etale paths for a class of vector bundles on a p-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise to representations of the algebraic fundamental group of the curve. This may be viewed as a partial analogue of the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces.

math.AG↗
arXiv2004

Line bundles and p-adic characters

For a certain class of vector bundles E on abelian varieties A over local fields containing all line bundles algebraically equivalent to zero we define a canonical representation of the Tate module of A on the fibre of E in the zero section. This extends an old construction of Tate for line bundles to vector bundles of higher rank. We also compare this construction to the theory of parallel transport for vector bundles on p-adic curves developed in mathAG/0403516. Relations with the Hodge-Tate decomposition are also explained.

math.AG↗
arXiv2002

Compactification of the Bruhat-Tits building of PGL by seminorms

We construct a compactification of the Bruhat-Tits building associated to the group PGL(V) which can be identified with the space of homothety classes of seminorms on V endowed with the topology of pointwise convergence. Then we define a continuous map from the projective space to this compactification which extends the reduction map from Drinfeld's p-adic symmetric domain to the building.

math.AG↗
arXiv2000

Arakelov intersection indices of linear cycles and the geometry of buildings and symmetric spaces

This paper generalizes Manin's approach towards a geometrical interpretation of Arakelov theory at infinity to linear cycles on projective spaces. We show how to interpret certain non-Archimedean Arakelov intersection numbers of linear cycles on $P^{n-1}$ with the combinatorial geometry of the Bruhat-Tits building associated to PGL(n). This geometric setting has an Archimedean analogue, namely the Riemannian symmetric space associated to SL(n,C), which we use to interpret analogous Archimedean intersection numbers of linear cycles in a similar way.

math.AG↗