Летняя математическая школа «Алгебра и геометрия»

Birational classification and orbifold geometry, with a view toward arithmetic and hyperbolicity aspects

Frédéric Campana (Université de Lorraine)

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Резюме

We decompose any complex quasi-projective manifold $X$, by means of a single functorial fibration $c:$$X\to C$ (its `Core map') into parts of opposite geometry: `special' (the fibres) vs `general type' (the `orbifold base'). `Special' manifolds are the ones such that $\kappa(X,L)\neq p$, for any $p>0, L\subset \Omega^p_X$ of rank $1$; they generalise rational and elliptic curves, as opposed to hyperbolic curves.

The Core map appears to be relevant in topics others than birational classification: it permits to formulate for arbitrary $X's$ the conjectures of Lang-Vojta, about hyperbolicity and arithmetics for $X's$ of general type. It also allows to formulate (and prove) the higher dimensional version of Shafarevich's `hyperbolicity conjecture' for families of canonically polarised manifolds ( strengthening Viehweg's conjecture, which deals with bases of Log-general type).

The core map is (conditionally in a weak form of abundance) canonically decomposed as $c=(j\circ r)^n$, with $j$ (resp. $r$) orbifold versions of the Iitaka (resp. weak MRC) fibration. In particular: Special manifolds are just towers of fibrations with general fibres either with $\kappa=0$, or weakly rationally connected.

Материалы по курсу: Записки по курсу.