arXiv subjects

Ana María Botero

Publications and source records attributed to Ana María Botero.

12 recordsLinked to original sources

arXiv2026

Pure extension of the theta divisor over the moduli space of abelian varieties

A theta divisor on the universal principally polarised abelian variety can be extended to a compactification either by taking the Zariski closure, or by taking the unique extension which is pure of weight 2. For the latter, following ideas of Yuan and Zhang, we need to pass to the category of adelic- or b-divisors. We show that the two choices of extension differ by a tropicalisation of the Riemann theta function. We prove an extension of Moret-Bailly's ''key formula'' that features the pure weight 2 extension of the theta divisor, and discuss various arithmetic applications, including a ''universal'' formula for the Néron--Tate height of a point. A key technical input is the systematic use of the theory of logarithmic abelian varieties due to Kajiwara, Kato, and Nakayama.

math.AG↗

arXiv2026

Monge-Ampère measures on balanced polyhedral spaces

We study classes of convex functions on balanced polyhedral spaces and establish various structural properties, including a compactness theorem for polyhedrally plurisubharmonic functions. Using tropical intersection theory, we construct Monge--Ampère measures, first associated with piecewise affine functions, and then we extend it to polyhedrally plurisubharmonic functions. We investigate polyhedral Monge--Ampère equations on balanced polyhedral spaces via a variational approach, providing sufficient conditions for the existence of solutions as well as explicit counterexamples. Finally, we relate our framework to non-archimedean pluripotential theory and explore its connection with the non-archimedean Monge--Ampère equation.

math.AG↗

arXiv2025

Equivariant non-archimedean Arakelov theory of toric varieties

We develop an equivariant version of the non-archimedean Arakelov theory of [BGS95] in the case of toric varieties. We define the equivariant analogues of the non-archimedean differential forms and currents appearing in \emph{loc.~cit.} and relate them to piecewise polynomial functions on the polyhedral complexes defining the toric models. In particular, we give combinatorial characterizations of the Green currents associated to equivariant cycles and combinatorial descriptions of the arithmetic Chow groups.

math.AG↗

arXiv2024

Rings of Siegel-Jacobi forms of bounded relative index are not finitely generated

We show that the ring of Siegel-Jacobi forms of fixed degree and of fixed or bounded ratio between weight and index is not finitely generated. Our main tool is the theory of toroidal b-divisors and their relation to convex geometry. As a byproduct of our methods, we prove a conjecture of Kramer about the representation of all Siegel-Jacobi forms as sections of certain line bundles and we recover a formula due to Tai for the asymptotic dimension of the space of Siegel-Jacobi forms of given ratio between weight and index.

math.AG↗

arXiv2024

Generalized Minkowski weights and Chow rings of T-varieties

We give a combinatorial characterization of Fulton's operational Chow cohomology groups of a complete , $\Q$-factorial, rational T-variety of complexity one in terms of so called generalized Minkowsky weights in the contraction-free case. We also describe the intersection product with Cartier invariant divisors in terms of the combinatorial data. In particular this provides a new way of computing top intersection numbers of invariant Cartier divisors combinatorially.

math.AG↗

arXiv2024

On Chow groups of toric schemes over a DVR

We give an explicit combinatorial presentation of the Chow groups of a toric scheme over a DVR. As an application, we compute the Chow groups of several toric schemes over a DVR and of their special fibers.

math.AG↗

arXiv2024

Equivariant Chern Classes of Toric Vector Bundles over a DVR and Bruhat--Tits Buildings

We define equivariant Chern classes of a toric vector bundle over a proper toric scheme over a DVR. We provide a combinatorial description of them in terms of piecewise polynomial functions on the polyhedral complex associated to the toric scheme, which factorize through to an extended Bruhat--Tits building. We further motivate this definition from an arithmetic perspective, connecting to the non-Archimedean Arakelov theory of toric varieties.

math.AG↗

arXiv2021

Chern-Weil and Hilbert-Samuel formulae for singular hermitian line bundles

We show a Chern-Weil type statement and a Hilbert-Samuel formula for a large class of singular plurisubharmonic metrics on a line bundle over a smooth projective complex variety. For this we use the theory of b-divisors and the so-called multiplier ideal volume function. We apply our results to the line bundle of Siegel-Jacobi forms over the universal abelian variety endowed with its canonical invariant metric. This generalizes a result of the second author for the universal family of elliptic curves to higher degrees.

math.AG↗

arXiv2021

Convex analysis on polyhedral spaces

We introduce notions of concavity for functions on balanced polyhedral spaces, and we show that concave functions on such spaces satisfy several strong continuity properties.

math.CO↗

arXiv2021

Toroidal b-divisors and Monge-Ampère measures

We generalize the intersection theory of nef toric (Weil) b-divisors on smooth and complete toric varieties to the case of smooth and complete toroidal embeddings. As a key ingredient we show the existence of a limit measure, supported on the weakly embedded rational conical polyhedral space attached to the toroidal embedding, which arises as a limit of discrete measures defined via tropical intersection theory on the polyhedral space. We prove that the intersection theory of nef toroidal Cartier b-divisors can be extended continuously to nef toroidal Weil b-divisors and that their degree can be computed as an integral with respect to this limit measure. As an application, we show that a Hilbert--Samuel type formula holds for big and nef toroidal Weil b-divisors.

math.AG↗

arXiv2018

Intersection theory of toric $b$-divisors in toric varieties

We introduce toric $b$-divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions toric $b$-divisors are integrable and that their degree is given as the volume of a convex set. Moreover, we show that the dimension of the space of global sections of a nef toric $b$-divisor is equal to the number of lattice points in this convex set and we give a Hilbert--Samuel type formula for its asymptotic growth. This generalizes classical results for classical toric divisors on toric varieties. Finally, we relate convex bodies associated to $b$-divisors with Newton--Okounkov bodies. The main motivation for studying toric $b$-divisors is that they locally encode the singularities of the invariant metric on an automorphic line bundle over a toroidal compactification of a mixed Shimura variety of non-compact type.

math.AG↗

arXiv2017

Canonical decomposition of a difference of convex sets

Let $N$ be a lattice of rank $n$ and let $M = N^{\vee}$ be its dual lattice. In this note we show that given two compact, bounded, full-dimensional convex sets $K_1 \subseteq K_2 \subseteq M_{\R} \coloneqq M \otimes_{\Z} \R$, there is a canonical convex decomposition of the difference $K_2 \setminus K_1$ and we interpret the volume of the pieces geometrically in terms of intersection numbers of toric $b$-divisors.

math.AG↗