If we want to study the constraints imposed on a complex manifold \(X\) by a large \(H_ 1 (M)\), we can use the Albanese map and so recover a variety (the Albanese image), which, in a sense, carries the information contained in that homology group. This suggests the following question. Does a formally similar set-up exist when the fundamental group replaces the first homology one? – The starting point is the following conjecture by Shafarevich: Let \(X\) be a smooth complex projective variety and \(X^*\) its universal cover. There exists a proper surjective morphism \(s^* : X^* \to \text{Sh} (X^*)\) onto a normal Stein space \(\text{Sh} (X^*)\). If the conjecture is true, the fundamental group \(\pi_ 1 (X)\) acts on \(\text{Sh} (X^*)\), this giving rise to a morphism onto the quotient \(s : X \to \text{Sh} (X) = \text{Sh} (X^*)/ \pi_ 1 (X)\). Even when the action above mentioned has fixed points, technical devices allow to still consider \(\text{Sh} (X)\). So, if the conjecture is true, we recover a morphism \(s\) and a variety \(\text{Sh} (X)\), taking into account how large \(\pi_ 1 (X)\) is and thus helping to show the constraints imposed on the algebro-geometric properties of \(X\) by a large fundamental group. – If \(s\) and \(\text{Sh} (X)\) exist, the fibres of \(s\) are characterized as those connected subvarieties \(Z\) – assuming that \(s\) has connected fibres – such that \[ \text{the image of } \pi_ 1 (Z) \text{ in } \pi_ 1 (X) \text{ is finite}. \tag{*} \] All that suggests the following definition. Let \(X\) be a normal and proper variety. A normal variety \(\text{Sh} (X)\) and a rational map \(s : X \to \text{Sh} (X)\) are called the Shafarevich variety and the Shafarevich map of \(X\) if \(s\) has connected fibres and if a condition similar to \((*)\) characterizes the irreducible components of the fibres out of the union \(U\) of countably many closed proper subvarieties. – As a first result the existence of the Shafarevich map is proved as well as that of variations of \(s\) defined by starting with the algebraic fundamental group or with normal subgroups. These maps are shown to be defined and proper on a large open set and their links with the Albanese morphism are pointed out according to the subgroups we start with. In view of condition \((*)\) the existence theorem decomposes the varieties into two classes according to whether \(\pi_ 1\) is finite or “generically large” \((s\) is birational, i.e. \(s\) contracts nothing out of \(U)\). In the former case this break-up applies to the fibres and so on. As to the smooth varieties \(X\) with generically large algebraic fundamental group, they should be built up by abelian varieties and varieties of general type. Actually such an \(X\) is conjectured to have a finite étale cover birational to a smooth family of abelian varieties over a projective variety of general type with generically large \(\pi_ 1\). The conjecture is proved if the Kodaira dimension of \(X\) is \(\geq \dim X - 2\).
Applications are given in many cases. Conditions on a normal analytic space are found ensuring that the surjection between the fundamental groups induced by a resolution of singularities is an isomorphism. A nonvanishing theorem is proved for varieties with generically large \(\pi_ 1\) and is applied to deduce information on the plurigenera of a smooth projective variety of general type. Further results on the plurigenera and on the pluricanonical map are given in the case of 3- folds of general type as well as numerical characterizations of varieties birational to Abelian ones.