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Xinyi Yuan

Publications and source records attributed to Xinyi Yuan.

35 records · Page 2Linked to original sources
arXiv2023

The Geometric Bombieri-Lang Conjecture for Ramified Covers of Abelian Varieties

In this paper, we prove the geometric Bombieri-Lang conjecture for projective varieties which have finite morphisms to abelian varieties of trivial traces over function fields of characteristic 0. The proof is based on the idea of constructing entire curves in the pre-sequel "Partial heights, entire curves, and the geometric Bombieri-Lang conjecture." A new ingredient is an explicit description of the entire curves in terms of Lie algebras of abelian varieties.

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arXiv2023

Partial Heights, Entire Curves, and the Geometric Bombieri-Lang Conjecture

We introduce a new approach to the geometric Bombieri--Lang conjecture for hyperbolic varieties in characteristic 0. The main idea is to construct an entire curve on a special fiber of a variety over a complex function field from an infinite sequence of rational points of the variety. The construction relies on the classical Brody lemma in complex geometry and is conditional on a non-degeneracy conjecture of partial heights. In particular, we prove the geometric Bombieri-Lang conjecture for hyperbolic projective varieties which have finite morphisms to abelian varieties over function fields of characteristic 0.

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arXiv2022

Information-driven Path Planning for Hybrid Aerial Underwater Vehicles

This paper presents a novel Rapidly-exploring Adaptive Sampling Tree (RAST) algorithm for the adaptive sampling mission of a hybrid aerial underwater vehicle (HAUV) in an air-sea 3D environment. This algorithm innovatively combines the tournament-based point selection sampling strategy, the information heuristic search process and the framework of Rapidly-exploring Random Tree (RRT) algorithm. Hence can guide the vehicle to the region of interest to scientists for sampling and generate a collision-free path for maximizing information collection by the HAUV under the constraints of environmental effects of currents or wind and limited budget. The simulation results show that the fast search adaptive sampling tree algorithm has higher optimization performance, faster solution speed and better stability than the Rapidly-exploring Information Gathering Tree (RIGT) algorithm and the particle swarm optimization (PSO) algorithm.

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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.

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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.

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arXiv2021

Geometric Bogomolov conjecture in arbitrary characteristics

The goal of this paper is to prove the full geometric Bogomolov conjecture. We first reduce it to the case that the extension of the base fields has transcendence degree 1, and then we prove the later case by intersection theory in algebraic geometry. The proof uses Yamaki's reduction theorem on the geometric Bogomolov conjecture and the Manin--Mumford conjecture proved by Raynaud and Hrushovski.

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arXiv2020

Comparison of arithmetic Brauer groups with geometric Brauer groups

Let $X$ be a projective and smooth variety over a field $k$. The goal of this paper is to prove that the cokernel of the canonical map $Br(X)\to Br(X_{k^s})^{G_k}$ has a finite exponent. Both groups are natural invariants arising from consideration of the Tate conjecture of divisors over $X$.

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arXiv2014

Genus Periods, Genus Points and Congruent Number Problem

In this paper, based on an ideal of Tian, we establish a new sufficient condition for a positive integer to be a congruent number in terms of Legendre symbols of prime factors of the positive integer. Our criterion generalizes previous criterions of Heegner, and Birch--Stephens, Monsky, and Tian, and conjecturally provides a list of positive density of congruent numbers. Our method of proving our criterion is to give formulae for the analytic Tate--Shafarevich number in terms of the so-called genus periods and genus points. These formulae are derived from the Waldspurger formula and the generalized Gross--Zagier formula of Yuan-Zhang-Zhang.

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arXiv2013

Relative Noether inequality on fibered surfaces

We prove effective upper bounds on the global sections of nef line bundles of small generic degree over a fibered surface over a field of any characteristic. It can be viewed as a relative version of the classical Noether inequality for surfaces. As a consequence, we give a new proof of the slope inequality for fibered surface without using any stability method. The treatment is essentially different from these of Xiao, Cornalba--Harris and Moriwaki. We also study the geography problem of surfaces in positive characteristics and show that the Severi inequality is true for surfaces of general type in positive characteristic whose Albanese map is generically finite. Moreover, the geography of surfaces with Albanese fibrations is studied.

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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.

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arXiv2012

Effective bound of linear series on arithmetic surfaces

We prove an effective upper bound on the number of effective sections of a hermitian line bundle over an arithmetic surface. It is an effective version of the arithmetic Hilbert--Samuel formula in the nef case. As a consequence, we obtain effective lower bounds on the Faltings height and on the self-intersection of the canonical bundle in terms of the number of singular points on fibers of the arithmetic surface.

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arXiv2012

Volumes of Arithmetic Okounkov Bodies

This paper proves the volume of the arithmetic Okounkov body, constructed from a hermitian line bundle on an arithmetic variety by the author in a previous paper, is equal to the the volume of the hermitian line bundle up to a simple constant multiple. It is an improvement and simplification of the previous work.

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arXiv2012

Big Line Bundles over Arithmetic Varieties

We prove a Hilbert-Samuel type result of arithmetic big line bundles in Arakelov geometry, which is an analogue of a classical theorem of Siu. An application of this result gives equidistribution of small points over algebraic dynamical systems, following the work of Szpiro-Ullmo-Zhang. We also generalize Chambert-Loir's non-archimedean equidistribution.

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arXiv2009

On Volumes of Arithmetic Line Bundles II

For a hermitian line bundle over an arithmetic variety, we construct a convex continuous function on the Okounkov body associated to the generic fibre of the line bundle. The integration of the continuous function gives the growth of the Euler characteristic of the hermitian line bundle. It is the global version of the recent work of Nystrom.

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arXiv2008

On Volumes of Arithmetic Line Bundles

We show an arithmetic generalization of the recent work of Lazarsfeld-Mustata which uses Okounkov bodies to study linear series of line bundles. As applications, we derive a log-concavity inequality on volumes of arithmetic line bundles and an arithmetic Fujita approximation theorem for big line bundles.

math.AG↗