**Time：**March 15-27, 2024

**Venue：**E4-233, Yungu Campus, Westlake Univeristy, Hangzhou, China

**ZOOM ID: **932 4062 7883

**Passcode: **863416

Friday, March 15

**9:00-10:30**

**Speaker: **Jose Ignacio Burgos Gil

**Title: **Introduction to the theory of heights I

**Abstract:** In this course we will introduce the concept of height and show how Arakelov theory provides a flexible framework to define heights. As an application we will study heights on Elliptic curves and recall the proof of Mordell's theorem. If time permits, we will also discuss equidistribution properties and the proof of Bogomolov's conjecture.

**14:00-15:30**

**Speaker:** Dennis Eriksson (Chalmers)

**Title: **Relative intersection theory and Arakelov geometry I

Monday, March 18

**9:00-10:30**

**Speaker: **Jose Ignacio Burgos Gil

**Title: **Introduction to the theory of heights II

**11:00-12:00**

**Speaker: **Francesco Tropeano (Università della Calabria)

**Title: **Finite translation orbits on double families of abelian varieties

**Abstract:** Consider two families of g-dimensional abelian varieties induced by two distinct rational maps on the same variety and having big common domain of definition. Two non-torsion sections of these families induce two birational fiberwise translations on the total space, respectively. We study the action of the subgroup of rational automorphisms generated by both translations and show that points with finite orbit lie in a proper Zariski closed subset (that can be described to a certain extent) when our families satisfy a condition of Manin-Mumford type. Our work is a higher dimensional generalization of a result of Corvaja, Tsimermann and Zannier. This is a joint work with Paolo Dolce (Westlake University, Hangzhou).

**14:00-15:30**

**Speaker: **Jose Ignacio Burgos Gil

**Title: **Introduction to the theory of heights III

Wednesday, March 20

**9:00-10:30**

**Speaker:** Francesco Zucconi (Udine)

**Title: **Recent results on the slope of fibrations I

**14:00-15:30**

**Speaker:** Dennis Eriksson (Chalmers)

**Title: **Relative intersection theory and Arakelov geometry II

Friday, March 22

**9:00-10:30**

**Speaker:** Francesco Zucconi (Udine)

**Title: **Recent results on the slope of fibrations II

**10:45-11:45**

**Speaker:** Tong Zhang (East China Normal University)

**Title: **Slope inequality for threefolds fibred over curves

**Abstract: **In the 1980s, Cornalba-Harris and Xiao established the optimal slope inequality for surfaces fibred over curves. The Arakelovian version for arithmetic surfaces was later obtained by Bost. Based on Xiao's method, various slope inequalities for threefolds fibred over curves were obtained, but it is not clear whether they are optimal. In this talk, I will introduce a recent joint work in progress with Y. Hu, proving that the optimal lower bound of the slope for threefolds fibred over curves is 4/3.

**14:00-15:30**

**Speaker:** Dennis Eriksson (Chalmers)

**Title: **Relative intersection theory and Arakelov geometry III

Monday, March 25

**9:00-10:30**

**Speaker:** Francesco Zucconi (Udine)

**Title: **Recent results on the slope of fibrations III

**11:00-12:00**

**Speaker: **Nuno Hultberg (University of Copenhagen)

**Title: **Arakelov geometry of semiabelian varieties and toric bundles

**Abstract: **Contrary to canonical heights on abelian varieties or toric varieties, heights on semiabelian varieties and their compactifications do not arise from a polarized dynamical system. Small points on semiabelian varieties are therefore more mysterious than in the previous cases. For instance, points on the boundary of the compactification may have negative height. Equidistribution of small points on semiabelian varieties was in turn first proven by Lars Kuehne in 2018. I study Zhang minima, arithmetic intersection numbers, and Okounkov bodies on semiabelian varieties placed in the context of toric bundles. For this I prove an arithmetic version of a theorem of Hofscheier, Khovanskii and Monin.

**15:30-17:00**

**Speaker: **Jose Ignacio Burgos Gil

**Title: **Introduction to the theory of heights IV

Wednesday, March 27

**9:30-10:30**

**Speaker:** Chunhui Liu, Harbin Institute of Technology

**Title:** Arithmetic Fujita approximation over adelic curves

**Abstract: **Fujita approximation is an approximative version of Zariski decomposition of pseudo-effective divisors. More precisely, it says that a power of a big line bundle can be decomposed as the sum of an ample and an effective line bundle under a birational morphism, where the volume of this big line bundle can be approximated by that of the ample one in some sense. An arithmetic analogue over number fields was proved by H. Chen and X. Yuan. In this talk, I will introduce a generalization under the framework of Arakelov geometry over adelic curves.