The Shafarevich Conjecture

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Place and time: Room 409, Snelius Building, Leiden. USUALLY 14:00-16:00 Tuesday.

Programme

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Number	Title	Speaker
0	Introduction	Ariyan Javanpeykar
1	Weak Boundedness 1	Ariyan Javanpeykar
2	Weak Boundedness 2	Ariyan Javanpeykar
3	Weak Boundedness 3	Ariyan Javanpeykar
4	Boundedness 1	David Holmes
5.1	Boundedness 2	David Holmes
5.2	Variants on Shafarevich Hyperbolicity 1	Ariyan Javanpeykar
6	Isom-schemes, isotrivality and the stack M_g	Lenny Taelman
7	Kodaira-Spencer and Rigidity	Bas Edixhoven
8	Variants of Shafarevich hyperbolicity 2	Ariyan Javanpeykar
9	Algebraic de Rham cohomology I	Bas Edixhoven	notes
10	Algebraic de Rham cohomology II	Bas Edixhoven	notes
11	The Lefschetz Hyperplane Theorem	David Holmes	notes
12	Rigidity and the iterated Kodaira-Spencer map	Ariyan Javanpeykar	24/06/2013, 13:00
13	TBA	TBA

Outline

In 1962, Shafarevich conjectured a finiteness theorem concerning families of curves. More precisely, he conjectured that for any curve C, there are only finitely many admissible families of curves over C. To prove this theorem, Parshin divided the problem into a number of steps:
-weak boundedness;
-boundedness;
-rigidity;
-hyperbolicity.
In this seminar, we aim at proving the Shafarevich conjecture following Arakelov and Parshin.

The proof of Arakelov-Parshin involves:
- vanishing theorems;
- explicit construction of parameter spaces;
- intersection theory;
- deformation theory;
- hom and Hilbert schemes;
- moduli stacks;
- ampleness of certain line bundles on moduli spaces.

Once we have completed the proof of Arakelov-Parshin, we will discuss numerous variants for higher dimension families (both higher dimensional bases and higher dimensional fibres). For instance, we will prove that simply connected projective varieties do not admit any admissible families of curves.

References:

Parshin, A. N. Algebraic curves over function fields. I. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 32 1968 1191-1219,

Arakelov, S. Ju. Families of algebraic curves with fixed degeneracies. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1269-1293.

Sur la theoreme de rigidite de Parshin et Arakelov, Journees de Geometrie Algebrique de Rennes (Rennes, 1978), Vol. II. Asterisque, 64, (1979), 169-202.

Seminaire sur les pinceaux de courbes de genre au moins deux, Asterisque, 86, (1981), 1-142.

Smooth families over rational and elliptic curves, Journal of Algebraic Geometry 1996;5(2):369-385. S. Kovacs

On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds, Eckart Viehweg and Kang Zuo, Duke Math. J. Volume 118, Number 1 (2003), 103-150.

Special families of curves, of Abelian varieties and of certain minimal manifolds over curves, Martin Moller, Eckart Viehweg and Kang Zuo

Organisers:

Robin de Jong Ariyan Javanpeykar Steffen Muller David Holmes

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If you are interested in participating, please contact one of the organisers.

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