# 2019 Torino/Trento School (and Workshop)

created by daniele on 21 Mar 2019

16 sep 2019 - 21 sep 2019

Trento

SCHOOL (AND WORKSHOP) ON DIOPHANTINE GEOMETRY AND SPECIAL VARIETIES
TRENTO, SEPTEMBER 16-21, 2019

Second announcement

Lecturers.
M. Campana (Université de Lorraine – Fr)
P. Corvaja (Università degli studi di Udine – It)

Supporting lecturers.
A. Turchet (University of Washington – USA)

Organizers.
The School/Workshop is organized by C. Bertolin, G. Casnati, F. Galluzzi, R. Notari, F. Vaccarino.
For contacting the organizers send a mail to

geometryschool2019nospamgmail.com

The School/Workshop is supported by CIRM-Fondazione Bruno Kessler (formerly CIRM-ITC), Dipartimento di Scienze Matematiche – Politecnico di Torino, Foundation Compositio Mathematica, Journal de Théorie des Nombres de Bordeaux, Dipartimento di Matematica – Università degli Studi di Torino. The School and the Workshop will take place at

Fondazione Bruno Kessler-IRST
via Sommarive, 18
38050 Povo (Trento) - Italy

Aim of the School.
The School is mainly aimed to PhD students and young researchers in Algebraic Geometry, introducing the participants to research, beginning from a basic level with a view towards the applications and to the most recent results. A tentative program is as follows.

F. Campana.
1) Special manifolds: first definition by absence of Bogomolov sheaves of differentials. Examples. Conjectures. Specialness vs Weak-specialness.
2) Orbifold pairs and their invariants. Multiple fibres. Orbifold base of a fibration. Bijection between Bogomolov sheaves and fibrations with orbifold base of general type. Special manifolds: second definition by absence of fibrations of general type.
3) The orbifold version of Iitaka's Conjecture C{n,m}. Solution when the orbifold base is of general type. The Core map c, its field of definition. Its conditional decomposition as c=(J\circ r)n. Extension of Lang-Vojta's conjectures for arbitrary smooth projective orbifolds
4) Mordell conjecture: orbifold version. Hyperbolic analogue via Nevanlinna theory. Solution of Lang's conjectures for some simply-connected surfaces. Examples of Weakly-special, but non special threefolds. Description of their Kobayashi pseudometric.
5) The fundamental group. Abelianity conjecture. Solution for linear representations. Solution under existence of a Zariski dense entire curve (after K. Yamanoi). Potential Hilbert Property and specialness. Hyperbolic analogue (after Corvaja-Zannier).

P. Corvaja.
1) Rational and integral points. Different notions of integrality, examples.
2) Lang-Vojta conjectures, Siegel's and Faltings' Theorems. Algebraic groups, S-unit equations.
3) Heights, Vojta's Main Conjecture. Campana's conjecture, abc conjecture.
4) Diophantine approximation, the Subspace Theorem. Proof of the S-unit equation theorem.
5) Integral points on curves; a proof of Siegel's theorem. Some applications to algebraic surfaces.

Aim of Workshop.
The Workshop is intended to discuss the state of the art. Up to now the following speakers have confirmed their participation: E. Amerik, A. Cadoret, E. Rousseau, J. Winkelmann. People interested in delivering a short communication are kindly requested to submit the title and an abstract within July 9th to

geometryschool2019nospamgmail.com

Schedule.
The School will start on Monday 16th at 13.00 and it will end on Friday 20th at 13.00. Workshop will start on Friday 20th at 14.30 and it will end on Saturday 21st at 14.00. On Friday evening a social dinner open to the participants of the School/Workshop will be organized.

Financial supports for Participants.
There are some grants covering lodging expenses in double rooms for young participants. Applicants must fill in the online application form at the web-site

before July, 1st. The organizing committee will examine the applications and will send out notifications of acceptance by July 14th.

Registration.
Participants who do not require financial support are expected to fill in the on line registration form at the web-site

before July 29th.

Further announcements.
A third announcement will follow, probably in June.

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