## Shafarevich maps and plurigenera of algebraic varieties.(English)Zbl 0819.14006

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.

### MSC:

 14E05 Rational and birational maps 14E20 Coverings in algebraic geometry 14F35 Homotopy theory and fundamental groups in algebraic geometry 14J40 $$n$$-folds ($$n>4$$)
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