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.... Show moreWe 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 compute Beilinson–Bloch heights in higher dimensions. Show less
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... Show moreWe 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, between the height of certain limit mixed Hodge structures and certain Beilinson–Bloch heights obtained from odd-dimensional hypersurfaces with a node. This congruence suggests a new method to compute Beilinson–Bloch heights. Here we explain how to compute the relevant limit mixed Hodge structures in practice, then apply our computational method to a nodal quartic curve and a nodal cubic threefold. In both cases we explain the nature of the primes occurring in the congruence. Show less
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... Show moreWe 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 tautological. Also we show that rings of tautological forms are always finite dimensional. Finally we characterize the Kawazumi–Zhang invariant as essentially the only smooth function on the moduli space of curves whose Levi form is a tautological form. Show less
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... Show moreEach 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 computable formula for the tropical moment of a tropical Jacobian in terms of potential theory on the underlying metric graph. We show that there exists a universal linear relation between the tropical moment, a certain capacity called the tau invariant, and the total length of a metric graph. To put our formula in a broader context, we relate our work to the computation of heights attached to principally polarized abelian varieties. Show less
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... Show moreWe 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 the results of [J. I. Burgos Gil et al., Lond. Math. Soc. Lect. Note Ser. 427, 45--77 (2016; Zbl 1378.14027)] to higher degrees. Show less
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... Show moreWe 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 involving the standard Riemann theta function, and another involving a theta function particular to genus three that was discovered by Frobenius. We review part of Frobenius’ work on his theta function and connect our results with a formula due to Bloch, Hain and Bost describing the archimedean height pairing of Ceresa cycles in genus three. Show less
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... Show moreWe 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 asymptotics are given a combinatorial interpretation in terms of S. Zhang’s theory of admissible Green’s functions on polarized metrized graphs. Show less
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... Show moreIn 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 the height jump divisors for families of pointed abelian varieties. The effectiveness of the height jump divisor was conjectured by Hain in the more general case of variations of polarized Hodge structures of weight −1 . Show less
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.... Show moreWe 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, we show that under these assumptions the so-called height jump divisors are always effective. This result is of particular interest in the context of biextension line bundles on Griffiths intermediate jacobian fibrations of polarized variations of Hodge structure of weight −1, pulled back along normal function sections. In the case of the normal function on M_g associated to the Ceresa cycle, our result proves a conjecture of Hain. As an application of our result we obtain that the Moriwaki divisor on M_g has non-negative degree on all complete curves in M_g not entirely contained in the locus of irreducible singular curves. Show less
ConclusionsThe high sensitivity of the U/DHU ratio at t=120min for detecting DPD deficiency, as defined by DPD activity in PBMCs, showed that the oral U loading dose can effectively identify... Show moreConclusionsThe high sensitivity of the U/DHU ratio at t=120min for detecting DPD deficiency, as defined by DPD activity in PBMCs, showed that the oral U loading dose can effectively identify patients with reduced DPD activity. Show less
