Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes

Fédéric Campana (1)

1 IECN - Institut Élie Cartan de Nancy

This is the geometric-orbifold version of [Ca01/04]. We define the bimeromorphic {\it category} of geometric orbifolds. These interpolate between (compact K\" ahler) manifolds and such manifolds with logarithmic structure, and may be considered as ``virtual" ramified covers of the underlying manifold. These geometric orbifolds are here considered as fully geometric objects, and thus come naturally equipped with the usual invariants of varieties: morphisms and bimeromorphic maps, differential forms, fundamental groups and universal covers, fields of definition and rational points. The general expectation is that their geometry is qualitatively the same as that of manifolds with similar invariants. The most elementary of such geometric properties are established here. The arguments of [Ca01] can then be directly adapted to extend the main structure results established there to this geometric-orbifold category. We hope to come back to deeper aspects later. The motivation is that the natural frame for the theory of bimeromorphic classification of compact K\" ahler (and complex projective) manifolds without orbifold structure unavoidably seems to be the category of geometric orbifolds, as shown here (and in [Ca01] for manifolds) by the fonctorial decomposition of {\it special} orbifolds as tower of orbifolds with fibres having either $\kappa_+=-\infty$ or $\kappa=0$, and also, in a different context, by the minimal model program, in which most proofs naturally work only after the adjunction of a ``boundary". An orbifold is {\it special} if it does not map ``stably" onto a (positive-dimensional) orbifold of general type, while having $\kappa_+=-\infty$ means that it maps only onto orbifolds having $\kappa=-\infty$, and is expected to mean rationally connected in the orbifold category. Moreover, fibrations enjoy in the bimeromorphic category of geometric orbifolds extension (or ``additivity") properties {\it not satisfied} in the category of varieties without orbifold structure, permitting to express invariants of the total space as the extension (or ``sum") of those of the generic fibre and of the base. Indeed, the discrepancy between the invariants of the total space and the ``sum" of those of the base and general fibres is mainly due to the presence of multiple fibres, taken into account in the orbifold (but not in the manifold) category. For example, the natural sequence of fundamental groups always becomes exact in the orbifold category. Also the total space of a fibration is special if so are the generic fibre and the base. Both properties are very false without orbifold structures. This makes this category suitable to lift properties from orbifolds having either $\kappa_+=-\infty$ or $\kappa=0$ to those which are special. And even leads to expect that {\it specialness} is the exact geometric characterisation of some important properties (such as potential density or vanishing of the Kobayashi pseudometric). Let us notice that the notion of morphism used to treat fundamental groups, based on the classical {\it divisibility} differs, by necessity, from the one used to deal with other geometrical aspects.

Domaines: Géométrie algébrique [math.AG]

Fédéric Campana. Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes. 2009. ⟨hal-00356763⟩