Lecturers.

Supporting lecturers.

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

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

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 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^or)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.