SCHOOL (AND WORKSHOP) ON
DIOPHANTINE GEOMETRY AND SPECIAL
VARIETIES
TRENTO, SEPTEMBER 1621, 2019
First 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
gianfranco.casnati [nospam] polito.it
The School/Workshop is supported by CIRMFondazione Bruno Kessler
(formerly CIRMITC), 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 KesslerIRST
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.
 Special
manifolds: first definition by absence of Bogomolov sheaves of
differentials. Examples.
Conjectures.
Specialness vs Weakspecialness.
 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.
 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^{o}r)^{n}.
Extension of LangVojta's conjectures for
arbitrary smooth projective orbifolds
 Mordell
conjecture: orbifold version. Hyperbolic analogue via Nevanlinna
theory. Solution of Lang's conjectures for some simplyconnected
surfaces.
Examples of Weaklyspecial, but non special threefolds. Description
of their Kobayashi pseudometric.
 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 CorvajaZannier).
P. Corvaja.
 Rational
and integral points. Different notions of integrality, examples.
 LangVojta
conjectures, Siegel's and Faltings' Theorems. Algebraic groups, Sunit
equations.
 Heights,
Vojta's Main Conjecture. Campana's conjecture, abc
conjecture.
 Diophantine
approximation, the Subspace Theorem. Proof of the Sunit
equation theorem.
 Integral
points on curves; a proof of Siegel's theorem. Some applications
to algebraic surfaces.
Further
announcements.
A more detailed second announcement (containing information on
accomodation, registration and financial supports) will follow probably
January in 2019.