Frederic Campana: Orbifolds, Special Varieties and Classification Theory

Abstract: For complex projective manifolds X, 3 "pure geometries" can be
defined, according to the sign (negative, zero, or positive) of the
canonical bundle. We show how to intrinsically decompose any such X into
"pieces" of pure geometry, by means of canonical fibrations. The single
new ingredient is the orbifold structure on the base of any fibration
between two such X's. Conjecturally, extending Lang's picture, this
decomposition holds at the arithmetic and hyperbolicity levels.