InMyCity Seminar on the conjectures of Campana, Lang and Vojta 

Where: Radboud University Nijmegen 

Meetings on Friday September 15th,   November 3rd,  December 1st

Organizers: Finn Bartsch and Ariyan Javanpeykar

In this seminar we will study some recent advances on the conjectures of Green-Griffiths, Lang, Vojta,  as well as the complementary conjectures of Campana.  We will focus mostly on the geometric picture (e.g., rational points over function fields) and  some of the  papers listed below. The goal is to familiarize the audience with enough techniques to be able to digest the work done in these recent works. We will most likely not get into the details of the papers below, but rather only discuss the tools needed to read parts of these papers. Notes will be uploaded in due course for each talk.  

Our main goal will be to read the following breakthrough papers 

•  Junyi Xie and Xinyi Yuan  -- Partial Heights, Entire Curves, and the Geometric Bombieri-Lang Conjecture  https://arxiv.org/abs/2305.14789 and https://arxiv.org/abs/2308.08117
  
  

 Day one.  (September 15th 2023) Classical finiteness theorems over function fields (De Franchis-Severi, Kobayashi-Ochiai, Manin-Grauert and Arakelov-Parshin)

ROOM: HG00.514

Talk 1.  (Ariyan, Nijmegen)   14:00h - 14:45h Present a proof of the following finiteness result due to De Franchis-Severi. Let Y be an algebraic variety. Then the   set of isomorphism classes of  smooth projective curves of genus at least two dominated by Y is finite (De Franchis) and, for every smooth projective curve C of genus at least two, the set of dominant rational maps from Y to C is finite (Severi). 

Talk 2. (notes)  (Finn, Nijmegen) 15:15h - 16h. Present a proof of Kobayashi-Ochiai's generalization of the theorem of De Franchis. Namely,  given X and Y, then the set of dominant rational maps from X to Y is finite, provided that Y is of general type (e.g.,  Y is a smooth projective curve of genus at least two).  We will briefly also discuss the following deep generalization of Severi's theorem: Given a variety X, the set of birational equivalence classes of smooth projective varieties Y of general type for which there exists a  dominant rational map X - - -> Y is finite. This is attributed to Maehara and Hacon-McKernan. We will explain some aspects of the (difficult) proof.

Talk 3.  (Ariyan, Nijmegen)  16:30 - 17:15h We will explain Shafarevich's conjecture over function fields, as proven by Arakelov-Parshin in the 70's. We explain how to deduce Manin-Grauert's theorem (i.e., the Mordell conjecture over function fields) from the Shafarevich conjecture.  The main statements we will discuss are as follows:  

Theorem 1. (Manin-Grauert) If B is a smooth quasi-projective curve and X->B is a smooth proper curve of genus at least two with quasi-finite moduli map (i.e., X->B is non-isotrivial), then X(B) is finite (i.e., the curve X_K has only finitely many K-points, where K =K(B) is the function field of B).

Theorem 2. (Arakelov-Parshin) If g>1 is an integer and B is a smooth quasi-projective curve, then the set of B-isomorphism classes of smooth proper curves X->B of genus g is finite.

Day two.  (November 3rd 2023) Campana's theory of orbifold pairs, Lang-Neron, the special locus of a variety, and finiteness results for rational points  

ROOM: HG00.068

Talk 1.  (Erwan Rousseau, Brest)  13:00h - 13:45h Campana's theory of orbifold pairs

Talk 2. (Ben, Nijmegen)  14:00h - 14:45h The Lang-Neron theorem and the trace of an abelian variety over a function field  

Talk 3.  (Zhelun, Leiden)  15:15h - 16h  The special locus of a subvariety of an abelian variety and Ueno's structure theorems for closed subvarietie.  A good reference here is Abramovich's paper  (which generalizes everything to semi-abelian varieties and to char p>0).

Talk 4.  (Ariyan, Nijmegen) 16:15 - 17h    An overview of finiteness results for closed subvarieties of abelian varieties and complements of divisors in abelian varieties over function fields (in the isotrivial and non-isotrivial case).

 Day three.  (December 1st 2023) An "old" proof of Parshin and the "new" work of Xie and Yuan on the geometric Bombieri-Lang conjecture

ROOM: HG00.307

Talk 1.  (Finn) 13:00h-14:00h Parshin's proof of Bombieri-Lang for subvarieties of abelian varieties

Talk 2.  (Ariyan) 14:30h - 1530h Overview of Xie-Yuan's work 

Talk 3. (Zhelun) 16h-17h  Partial heights in Xie-Yuan's work 

Some useful references for some of the talks:

•  Szpiro's Asterisque on Manin-Grauert and the Shafarevich conjecture (also in characteristic p>0) http://www.numdam.org/issues/AST_1981__86__R1_0/

• Lucio Guerra  and Gian Pietro Pirola   On the finiteness theorem for rational maps on a variety of general type  https://arxiv.org/abs/0807.2980v2 (see also Bibliography)

•  Shoshichi Kobayashi and Takushiro Ochiai  Meromorphic mappings onto compact complex spaces of general type    https://link.springer.com/article/10.1007/BF01389863

• Abramovich  Subvarieties of semi-abelian varieties http://www.numdam.org/item/CM_1994__90_1_37_0/

A list of possible papers to study in the future on fundamental groups, universal coverings, Nevanlinna theory, Shafarevich maps, ... Let me know if any of these interest you particularly.

•  Rodolfo Aguilar Aguilar and Frédéric Campana --  The nilpotent quotients of normal quasi-projective varieties with proper quasi-Albanese map  https://arxiv.org/abs/2301.11232

• Damian Brotbek and Yohan Brunebarbe -- Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications https://arxiv.org/abs/2007.12957

• Yohan Brunebarbe  -- The relative Green-Griffiths-Lang conjecture for families of varieties of maximal Albanese dimension https://arxiv.org/abs/2305.09613

• Yohan Brunebarbe -- Existence of the Shafarevich morphism for semisimple local systems on quasi-projective varieties https://arxiv.org/abs/2305.09741

• Yohan Brunebarbe -- Hyperbolicity in presence of a large local system  https://arxiv.org/abs/2207.03283

• Benoit Cadorel and Ya Deng and Katsutoshi Yamanoi  -- Hyperbolicity and fundamental groups of complex quasi-projective varieties https://arxiv.org/abs/2212.12225

• Ya Deng and Katsutoshi Yamanoi and Ludmil Katzarkov -- Reductive Shafarevich conjecture https://arxiv.org/abs/2306.03070

• Jürgen Jost, Shing-Tung Yau -- Harmonic Mappings and Algebraic Varieties Over Function Fields https://www.jstor.org/stable/2374964

• Katsutoshi Yamanoi -- Bloch's principle for holomorphic maps into subvarieties of semi-abelian varieties https://arxiv.org/abs/2304.05715