Special orbifolds and birational classification: a survey

Abstract

Among other things, we show the existence, for any $n$-dimensional connected complex projective manifold $X$, of a unique canonical and functorial fibration $c:X\to C(X)$, called ‘the core of $X$’, such that its general fibre is ‘special’, and its ‘orbifold base’ is of general type.

Assuming an orbifold version of Iitaka’s $C_{n,m}$-conjecture, we next show that $c=(jr{)}^n$, where $j$ (resp. $r$) is an orbifold version of the Moishezon-Iitaka fibration (resp. of the ‘rational quotient’ – sometimes also called the MRC-fibration). In particular, the ‘special’ manifolds are also simply the orbifold combination of orbifolds with either $\kappa =0$ or ${\kappa }_{+}=-\infty$ (a weak version of rational connectedness).

The class of ‘special’ manifolds may be seen of the class of manifolds which are ‘opposite’ to those of general type. It is for example conjectured that they are exactly the ones having an identically vanishing Kobayashi pseudometric, and also exactly the ones which are ‘potentially dense’, when defined over a number field.

Because it is necessary to work with orbifolds pairs in order to get meaningful fibrations, we start by developing the geometry of such orbifold pairs, and obtain the above decomposition results in this enlarged orbifold context.This text is intended to be an introduction to the papers [Ca04] and [Ca07].