arXiv subjects

Shou-Wu Zhang

Publications and source records attributed to Shou-Wu Zhang.

10 recordsLinked to original sources

arXiv2026

Rank of normal functions and Betti strata

In a recent work of the authors, we proved the generic positivity of the Beilinson-Bloch heights of the Gross-Schoen and Ceresa cycles. The geometric part of the proof was to prove the maximality of the rank of the associated normal function and the Zariski closedness of the Betti strata. In this paper, we generalize these geometric results to an arbitrary family of homologically trivial cycles. More generally, we prove a formula to compute the Betti rank and prove the Zariski closedness of the Betti strata, for any admissible normal function of a variation of Hodge structures of weight $-1$. We also define and prove results about degeneracy loci. In the end, we go back to the arithmetic setting and ask some questions about the rationality of the Betti strata and the torsion loci.

math.AG↗

arXiv2026

Heights of Ceresa and Gross-Schoen cycles

We study the Beilinson-Bloch heights of Ceresa and Gross-Schoen cycles in families. We construct that for any $g\ge 3$, a Zariski open dense subset $\mathcal{M}_g^{\mathrm{amp}}$ of $\mathcal{M}_g$, the coarse moduli of curves of genus $g$ over $\mathbb{Q}$, such that the heights of Ceresa cycles and Gross-Schoen cycles over $\mathcal{M}_g^{\mathrm{amp}}$ have a lower bound and satisfy the Northcott property.

math.NT↗

arXiv2025

Adelic line bundles on quasi-projective varieties

In this book, we establish a theory of adelic line bundles over quasi-projective varieties over finitely generated fields. Besides definitions of adelic line bundles, we consider their intersection theory, volume theory, and height theory, and apply these to study heights of algebraic points of quasi-projective varieties.

math.NT↗

arXiv2024

Height pairings for algebraic cycles on the product of a curve and a surface

For the product $X=C\times S$ of a curve and a surface over a number field, we construct unconditionally a Beilinson--Bloch type height pairing for homologically trivial algebraic cycles on $X$. Then for an embedding $f: C\to S$, we define an arithmetic diagonal cycle modified from the graph of $f$. This work extends previous work of Gross and Schoen when $S$ is the product of two curves, and is based on our recent work which relates the height pairings and the standard conjectures.

math.AG↗

arXiv2022

Standard Conjectures and Height Pairings

In this article, we extend Grothendieck's standard conjectures to cycles on degenerated fibers and use them to define some decompositions for the arithmetic Chow group of Gillet--Soulé. In the local setting, our decompositions provide non-archimedean analogues of "harmonic forms" on Kähler manifolds. In the global setting, our decompositions provide canonical arithmetic liftings called "L- liftings" of algebraic cycles on varieties over number fields, and thus provide a new height pairing called the L-height pairing as one extension of Beilinson--Bloch's pairing of homologically trivial cycles.

math.NT↗

arXiv2021

On the Averaged Colmez Conjecture

The Colmez conjecture, proposed by Colmez, is a conjecture expressing the Faltings height of a CM abelian variety in terms of some linear combination of logarithmic derivatives of Artin L-functions. The aim of this paper to prove an averaged version of the conjecture.

math.NT↗

arXiv2021

The arithmetic Hodge index theorem for adelic line bundles II

This is the second paper of a series. It extends the results of the first paper from number fields to finitely generated fields, based on the recent theory of adelic line bundles of the same authors. We prove an arithmetic Hodge index theorem for these adelic line bundles, and apply the theorem to obtain a rigidity property of the sets of preperiodic points of polarizable algebraic dynamical systems over any field.

math.NT↗

arXiv2013

The arithmetic Hodge index theorem for adelic line bundles I

This is the first paper of a series. We prove an arithmetic Hodge index theorem for adelic line bundles on projective varieties over number fields. It extends the arithmetic Hodge index theorem of Faltings, Hriljac and Moriwaki on arithmetic varieties. As consequences, we obtain the uniqueness part of the non-archimedean Calabi--Yau theorem, and a rigidity property of the sets of preperiodic points of polarizable algebraic dynamical systems over number fields.

math.NT↗

arXiv2010

Positivity of heights of codimension 2 cycles over function field of characteristic 0

In this note, we show how the classical Hodge index theorem implies the Hodge index conjecture of Beilinson for height pairing of homologically trivial codimension two cycles over function field of characteristic 0. Such an index conjecture has been used in our paper on Gross-Schoen cycles to deduce the Bogomolov conjecture and a lower bound for Hodge class (or Faltings height) from some conjectures about metrized graphs which have just been recently proved by Zubeyir Cinkir.

math.AG↗

arXiv2008

Gross--Schoen Cycles and Dualising Sheaves

The aim of this paper is to study the modified diagonal cycle in the triple product of a curve over a global field defined by Gross and Schoen. Our main result is an identity between the height of this cycle and the self-intersection of the relative dualising sheaf. We have some applications to the following problems in number theory and algebraic geometry.

math.NT↗