Fundamental groups of compact Kähler manifolds.

*(English)*Zbl 0849.32006
Mathematical Surveys and Monographs. 44. Providence, RI: American Mathematical Society (AMS). xi, 140 p. (1996).

The book of 140 p. is based on lectures given in Swiss 1995 Borel Seminar [see also D. Kotschik, Bull. Lond. Math. Soc. 24, No. 4, 377-378 (1992; Zbl 0756.53020) and Topology 31, No. 2, 317-321 (1992; Zbl 0756.32017), K. Corlette, Proc. Symp. Pure Math. 54, Part 2, 125-144 (1993; Zbl 0789.58006) and D. Toledo, Publ. Math., Inst. Hautes Étud. Sci. 77, 103-119 (1993; Zbl 0818.14009)].

Chapter 1. Introduction in Kähler geometry. Some new results on fundamental groups of compact complex surfaces. Chapter 2. Problem to find a holomorphic map inducing a given representation of homomorphism of the fundamental group of a compact Kähler manifold. Two main cases: quotient homomorphism of the fundamental group to its first homology modulo torsion (Albanese map), and representations onto surface groups of genus at least two. Kähler groups divide into fibered and non-fibered groups. Chapter 3. Techniques of real homotopy theory to study Kähler groups (Malcev, Deligne, Griffits, Morgan, Sullivan). Chapter 4. \(L^2\)-cohomology and restrictions on the fundamental groups of Kähler manifolds (Gromov, Arapura, Bressler, Ramachandran). It is shown that Kähler groups have finitely many ends. Chapter 5. Existence theorems for harmonic maps (Eells, Sampson, Corlette, Donaldson, Labourie). Chapter 6. Applications of harmonic maps. Siu-Sampson Bochner formula. Pluriharmonic maps from Kähler manifolds to negatively curved manifolds. Fundamental groups of real hyperbolic manifolds of dimension at least three cannot be fundamental groups of compact Kähler manifolds. Chapter 7. Non-Abelian Hodge theory of Corlette and Simpson. Yang-Mills equations. Higgs bundles. Hyperkähler structures. Bloch conjecture. Chapter 8. Examples of groups which occur as Kähler groups, in fact, as fundamental groups of smooth complex projective varieties. Appendix A: Generalities about projective completions of finitely generated groups. Appendix B: Glossary of Hodge theory. Bibliography: 138 titles.

Chapter 1. Introduction in Kähler geometry. Some new results on fundamental groups of compact complex surfaces. Chapter 2. Problem to find a holomorphic map inducing a given representation of homomorphism of the fundamental group of a compact Kähler manifold. Two main cases: quotient homomorphism of the fundamental group to its first homology modulo torsion (Albanese map), and representations onto surface groups of genus at least two. Kähler groups divide into fibered and non-fibered groups. Chapter 3. Techniques of real homotopy theory to study Kähler groups (Malcev, Deligne, Griffits, Morgan, Sullivan). Chapter 4. \(L^2\)-cohomology and restrictions on the fundamental groups of Kähler manifolds (Gromov, Arapura, Bressler, Ramachandran). It is shown that Kähler groups have finitely many ends. Chapter 5. Existence theorems for harmonic maps (Eells, Sampson, Corlette, Donaldson, Labourie). Chapter 6. Applications of harmonic maps. Siu-Sampson Bochner formula. Pluriharmonic maps from Kähler manifolds to negatively curved manifolds. Fundamental groups of real hyperbolic manifolds of dimension at least three cannot be fundamental groups of compact Kähler manifolds. Chapter 7. Non-Abelian Hodge theory of Corlette and Simpson. Yang-Mills equations. Higgs bundles. Hyperkähler structures. Bloch conjecture. Chapter 8. Examples of groups which occur as Kähler groups, in fact, as fundamental groups of smooth complex projective varieties. Appendix A: Generalities about projective completions of finitely generated groups. Appendix B: Glossary of Hodge theory. Bibliography: 138 titles.

Reviewer: M.Rahula (Tartu)

##### MSC:

32Q15 | Kähler manifolds |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

58E20 | Harmonic maps, etc. |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

32J27 | Compact Kähler manifolds: generalizations, classification |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

14F35 | Homotopy theory and fundamental groups in algebraic geometry |