SCHOOL (AND WORKSHOP) ON
DIOPHANTINE GEOMETRY AND SPECIAL
VARIETIES
TRENTO, SEPTEMBER 16-21, 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 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.
- Special
manifolds: first definition by absence of Bogomolov sheaves of
differentials. Examples.
Conjectures.
Specialness vs Weak-specialness.
- 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 Cn,m.
Solution when the orbifold base is of general type.
The Core map c, its field of definition. Its conditional decomposition as c=(Jor)n.
Extension of Lang-Vojta's conjectures for
arbitrary smooth projective orbifolds
- 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.
- 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.
- Rational
and integral points. Different notions of integrality, examples.
- Lang-Vojta
conjectures, Siegel's and Faltings' Theorems. Algebraic groups, S-unit
equations.
- Heights,
Vojta's Main Conjecture. Campana's conjecture, abc
conjecture.
- Diophantine
approximation, the Subspace Theorem. Proof of the S-unit
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.