Special orbifolds and birational classification: a survey

Abstract

We shall show how to decompose, by functorial and canonical fibrations, arbitrary $n$-dimensional complex projective {Although the geometric results apply to compact K\" ahler manifolds without change, we consider here for simplicity this special case only.} varieties $X$ into varieties (or rather ` geometric orbifolds\rq $)$ of one of the three \pure geometries determined by the `sign' (negative, zero, or positive) of the canonical bundle. These decompositions being birationally invariant, birational versions of these \pure geometries, based on the \canonical (or ` Kodaira\rq $)$ dimension will be considered, rather. A crucial feature of these decompositions is indeed that, in order to deal with multiple fibres of fibrations, they need to take place in the larger category of `geometric orbifolds' $(X| \Delta)$. These are `virtual ramified covers' of varieties, which `virtually eliminate' multiple fibres of fibrations. Although formally the same as the `pairs' of the LMMP (see \cite{kmm}, \cite{KM}, \cite{BCHM} and the references there), they are here fully geometric objects equipped with the usual geometric invariants of varieties, such as sheaves of (symmetric) differential forms, fundamental group, Kobayashi pseudometric, integral points, morphisms and rational maps. It is intended to expose (with some addtional topics or developments), as briefly and simply as possible, and essentially skipping the proofs, the main content of arXiv:math/0110051 and arXiv:0705.0737 respectively published in ANN. Inst. Fourier (2004), and to appear in J.Inst. Math. Jussieu.Comment: Expanded version of a talk given in May 2009 at the Schiermonnikoog Conference on birational geometry (Holland, organised by C. Faber, E. Looijenga and G. Van der Gee