Robin de Jong
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View article: Canonical local heights and Berkovich skeleta
Canonical local heights and Berkovich skeleta Open
We discuss canonical local heights on abelian varieties over non-archimedean fields from the point of view of Berkovich analytic spaces. Our main result is a refinement of Néron's classical result relating canonical local heights with inte…
View article: Heights on curves and limits of Hodge structures
Heights on curves and limits of Hodge structures Open
We exhibit a precise connection between Néron–Tate heights on smooth curves and biextension heights of limit mixed Hodge structures associated to smoothing deformations of singular quotient curves. Our approach suggests a new way to comput…
View article: Computing Heights via Limits of Hodge Structures
Computing Heights via Limits of Hodge Structures Open
We consider the problem of explicitly computing Beilinson–Bloch heights of homologically trivial cycles on varieties defined over number fields. Recent results have established a congruence, up to the rational span of logarithms of primes,…
View article: Rings of tautological forms on moduli spaces of curves
Rings of tautological forms on moduli spaces of curves Open
We define and study a natural system of tautological rings on the moduli spaces of marked curves at the level of differential forms. We show that certain 2-forms obtained from the natural normal functions on these moduli spaces are tautolo…
View article: Heights on curves and limits of Hodge structures
Heights on curves and limits of Hodge structures Open
We exhibit a precise connection between Néron–Tate heights on smooth curves and biextension heights of limit mixed Hodge structures associated to smoothing deformations of singular quotient curves. Our approach suggests a new way to comput…
View article: Metric graphs, cross ratios, and Rayleigh’s laws
Metric graphs, cross ratios, and Rayleigh’s laws Open
We systematically study the notion of cross ratios and energy pairings on metric graphs and electrical networks. We show that several foundational results on electrical networks and metric graphs immediately follow from the basic propertie…
View article: Computing heights via limits of Hodge structures
Computing heights via limits of Hodge structures Open
We consider the problem of explicitly computing Beilinson--Bloch heights of homologically trivial cycles on varieties defined over number fields. Recent results have established a congruence, up to the rational span of logarithms of primes…
View article: Rings of tautological forms on moduli spaces of curves
Rings of tautological forms on moduli spaces of curves Open
We define and study a natural system of tautological rings on the moduli spaces of marked curves at the level of differential forms. We show that certain 2-forms obtained from the natural normal functions on these moduli spaces are tautolo…
View article: Tropical moments of tropical Jacobians
Tropical moments of tropical Jacobians Open
Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus is its tropical moment. We give an explicit and efficiently…
View article: Chern-Weil and Hilbert-Samuel formulae for Singular Hermitian Line Bundles
Chern-Weil and Hilbert-Samuel formulae for Singular Hermitian Line Bundles Open
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 m…
View article: Rings of Siegel-Jacobi forms of bounded relative index are not finitely generated
Rings of Siegel-Jacobi forms of bounded relative index are not finitely generated Open
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 b…
View article: Faltings height and Néron–Tate height of a theta divisor
Faltings height and Néron–Tate height of a theta divisor Open
We prove a formula, which, given a principally polarized abelian variety $(A,\lambda)$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the N\'eron--Tate height of a symmetric theta divisor on $A$. Our fo…
View article: Chern–Weil and Hilbert–Samuel formulae for singular Hermitian line bundles
Chern–Weil and Hilbert–Samuel formulae for singular Hermitian line bundles Open
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 m…
View article: Faltings height and Néron–Tate height of a theta divisor
Faltings height and Néron–Tate height of a theta divisor Open
We prove a formula, which, given a principally polarized abelian variety $(A,\lambda )$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the Néron–Tate height of a symmetric theta divisor on $A$ . Our for…
View article: Jumps in the height of the Ceresa cycle
Jumps in the height of the Ceresa cycle Open
We study the jumps in the archimedean height of the Ceresa cycle, as introduced by R. Hain in his work on normal functions on moduli spaces of curves, and as further analyzed by P. Brosnan and G. Pearlstein in terms of asymptotic Hodge the…
View article: Frobenius’ theta function and Arakelov invariants in genus three
Frobenius’ theta function and Arakelov invariants in genus three Open
We give explicit formulas for the Kawazumi–Zhang invariant and Faltings delta-invariant of a compact and connected Riemann surface of genus three. The formulas are in terms of two integrals over the associated jacobian, one integral involv…
View article: Frobenius’ theta function and Arakelov invariants in genus three
Frobenius’ theta function and Arakelov invariants in genus three Open
We give explicit formulas for the Kawazumi-Zhang invariant and Faltings delta-invariant of a compact and connected Riemann surface of genus three. The formulas are in terms of two integrals over the associated jacobian, one integral involv…
View article: Chern–Weil Theory for Line Bundles with the Family Arakelov Metric
Chern–Weil Theory for Line Bundles with the Family Arakelov Metric Open
We prove a result of Chern-Weil type for canonically metrized line bundles on one-parameter families of smooth complex curves. Our result generalizes a result due to J.I. Burgos Gil, J. Kramer and U. Kühn that deals with a line bundle of J…
View article: Faltings delta-invariant and semistable degeneration
Faltings delta-invariant and semistable degeneration Open
We determine the asymptotic behavior of the Arakelov metric, the Arakelov–Green’s function, and the Faltings delta-invariant for arbitrary one-parameter families of complex curves with semistable degeneration. The leading terms in the asym…
View article: Metric graphs, cross ratios, and Rayleigh's laws
Metric graphs, cross ratios, and Rayleigh's laws Open
We study a notion of cross ratios on metric graphs and electrical networks. We show that several known results immediately follow from the basic properties of cross ratios. We show that the projection matrices of Kirchhoff have nice (and e…
View article: Tropical moments of tropical Jacobians
Tropical moments of tropical Jacobians Open
Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus is its tropical moment. We give an explicit and efficiently…
View article: Singularities of the biextension metric for families of abelian varieties
Singularities of the biextension metric for families of abelian varieties Open
In this paper we study the singularities of the invariant metric of the Poincaré bundle over a family of abelian varieties and their duals over a base of arbitrary dimension. As an application of this study we prove the effectiveness of th…
View article: SINGULARITIES OF THE BIEXTENSION METRIC FOR FAMILIES OF ABELIAN VARIETIES
SINGULARITIES OF THE BIEXTENSION METRIC FOR FAMILIES OF ABELIAN VARIETIES Open
In this paper we study the singularities of the invariant metric of the Poincaré bundle over a family of abelian varieties and their duals over a base of arbitrary dimension. As an application of this study we prove the effectiveness of th…
View article: On board monitoring of polluting emissions in sea shipping
On board monitoring of polluting emissions in sea shipping Open
The International Maritime Organization (IMO) has been working on the reduction of emissions in sea shipping. This has been done by stricter regulations over the past years in the form of emission limits. An important element which determi…
View article: Positivity of the Height Jump Divisor
Positivity of the Height Jump Divisor Open
We study the degeneration of semipositive smooth hermitian line bundles on open complex manifolds, assuming that the metric extends well away from a codimension two analytic subset of the boundary. Using terminology introduced by R. Hain, …
View article: Symmetric roots and admissible pairing
Symmetric roots and admissible pairing Open
Using the discriminant modular form and the Noether formula it is possible to write the admissible self-intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum, over a…
View article: Néron models and the height jump divisor
Néron models and the height jump divisor Open
We define an algebraic analogue, in the case of jacobians of curves, of the height jump divisor introduced recently by R. Hain. We give explicit combinatorial formulae for the height jump for families of semistable curves using labelled re…
View article: Local heights on Galois covers of the projective line
Local heights on Galois covers of the projective line Open
Let X be a smooth projective curve of positive genus defined over a number field K. Assume given a Galois covering map x from X to the projective line over K and a place v of K. We introduce a local canonical height on the set of K_v-value…
View article: Rousing reviews and instigative images: The impact of online reviews and visual design characteristics on app downloads
Rousing reviews and instigative images: The impact of online reviews and visual design characteristics on app downloads Open
Mobile apps are very popular. However, this is not true for every app, with some apps receiving millions of downloads, while other apps are mostly ignored. We investigate the popularity of apps in terms of downloads by focusing on two sali…