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**Shafarevich maps and automorphic forms.**
*(English)*
Zbl 0871.14015

M. B. Porter Lectures. Princeton, NJ: Princeton University Press (ISBN 978-0-691-04381-4/hbk). ix, 199 p. (1995).

The aim of this book is to study the geometric properties of a smooth complex projective variety \(X\) (usually expressed in terms of cohomological invariants obtained starting with the canonical class \(K_X)\) under the assumption that \(X\) has a large fundamental group.

The author makes a systematic use of the Shafarevich maps, a non-abelian generalization of the classical Albanese morphisms. Let \(X\) be a normal proper variety. Then a normal proper variety \(\text{Sh} (X)\) and a morphism \(\text{sh}_X: X\to \text{Sh} (X)\) are called the Shafarevich variety of \(X\) and, respectively, the Shafarevich morphism of \(X\) if \(\text{sh}_X\) has connected fibers and for any closed subvariety \(Z\subset X\) either \(\text{sh}_X (Z)\) is a point, or \(\text{image} (\pi_1(Z) \to\pi_1 (X))\) is infinite. – The existence of Shafarevich morphisms in general depends on an open conjecture by Shafarevich going back to 1972. What the author shows is that the rational version of a Shafarevich morphism (called a Shafarevich map) always exists.

The following notion describes roughly the varieties \(X\) for which the Shafarevich map is birational. One says that a proper variety \(X\) has a generically large fundamental group if \(\text{image} (\widehat \pi_1 (Z)\to \widehat \pi_1(X))\) is infinite for “most” subvarieties \(Z\subset X\), where \(\widehat \pi_1\) denotes the algebraic fundamental group. – Here are an example of a main result and an example of an open question from this book:

Theorem. Let \(X\) be a smooth projective variety of general type, \(X\) having a generically large algebraic fundamental group. Then \(\dim H^0 (X,K_X) \geq 1\).

Conjecture. Let \(X\) be as in the theorem above. Then \(\chi (X,K_X) \geq 0\).

The methods used in the book come from very diverse sources, e.g. Atiyah’s \(L^2\)-index theorem, Gromov theory of Poincaré series and recents generalizations of the Kodaira vanishing theorem by Demailly, Esnault-Viehweg, Kawamata, Kollár et Nadel.

The exposition is clear and very attractive, with a continuous care to explain what is hiding behind the various formalisms that one is forced to introduce along the way.

The author makes a systematic use of the Shafarevich maps, a non-abelian generalization of the classical Albanese morphisms. Let \(X\) be a normal proper variety. Then a normal proper variety \(\text{Sh} (X)\) and a morphism \(\text{sh}_X: X\to \text{Sh} (X)\) are called the Shafarevich variety of \(X\) and, respectively, the Shafarevich morphism of \(X\) if \(\text{sh}_X\) has connected fibers and for any closed subvariety \(Z\subset X\) either \(\text{sh}_X (Z)\) is a point, or \(\text{image} (\pi_1(Z) \to\pi_1 (X))\) is infinite. – The existence of Shafarevich morphisms in general depends on an open conjecture by Shafarevich going back to 1972. What the author shows is that the rational version of a Shafarevich morphism (called a Shafarevich map) always exists.

The following notion describes roughly the varieties \(X\) for which the Shafarevich map is birational. One says that a proper variety \(X\) has a generically large fundamental group if \(\text{image} (\widehat \pi_1 (Z)\to \widehat \pi_1(X))\) is infinite for “most” subvarieties \(Z\subset X\), where \(\widehat \pi_1\) denotes the algebraic fundamental group. – Here are an example of a main result and an example of an open question from this book:

Theorem. Let \(X\) be a smooth projective variety of general type, \(X\) having a generically large algebraic fundamental group. Then \(\dim H^0 (X,K_X) \geq 1\).

Conjecture. Let \(X\) be as in the theorem above. Then \(\chi (X,K_X) \geq 0\).

The methods used in the book come from very diverse sources, e.g. Atiyah’s \(L^2\)-index theorem, Gromov theory of Poincaré series and recents generalizations of the Kodaira vanishing theorem by Demailly, Esnault-Viehweg, Kawamata, Kollár et Nadel.

The exposition is clear and very attractive, with a continuous care to explain what is hiding behind the various formalisms that one is forced to introduce along the way.

Reviewer: A.Dimca (Bordeaux)