Bibliography (with some comments)
Jeremy Daniel
Here is a guide to a selected bibliography for the lectures on Kähler groups.
1
Lecture 1
On complex geometry (algebraic and differential) : [GH78], [Huy05], [Voi02a] and [Voi02b]. I also recall some facts in my master’s thesis [Dan11] ; beware there can be some mistakes (and the english is bad).
On the fundamental groups of compact Kähler manifolds : [ABC+96]. See also the survey [Ara95].
The algebraic proof of the Hodge decomposition is in [DI87]. There is also the survey [Ill02] for more on these methods using characteristic p.
In [CCE14], it is proved that any linear Kähler group is virtually the fundamental group of a smooth projective manifold. In [Cla18], the word
virtually is removed.
On the homotopy types of compact Kähler manifolds : [Voi04] and [Voi06].
On CW-complexes (and algebraic topology in general) : [Spa81].
On the cohomology of discrete groups (with a topological viewpoint) : [AM04]. See also [Dan11] for a quick introduction to Eilenberg-MacLane spaces.
2
Lecture 2
On Morse Theory : [Mil63]
The Andreotti-Frankel theorem is originally proved in [AF59] (cohomo-logical statement). The proof is also given in [Voi02b]. The proof of the statement for homotopy groups is in [Mil63].
The difficulties of construction quotients in the realm of algebraic ge-ometry led to a lot of work in the past 50 years : Mumford’s geometric
invariant theory [MFK94], Artin’s introduction of algebraic spaces, and then stacks [Sta18]. For finite groups, Hironaka’s example [Hir62] can be used to construct aZ/2Z-action on a scheme such that the quotient is an algebraic
space butnot a scheme.
For the quotients of an affine variety by a finite group : [Sha94a]. For the projectivity of the quotient of a projective manifold, see Proposition IV.1.5. of [Knu71] for a handy proof ; a modern treatment can be found in section 30.17 on norms of [Sta18], especially Proposition 30.17.9.
The realization of any finite group as the fundamental group of a smooth projective manifold is due to Serre [Ser58] (it also contains a proof of the projectivity statement of the quotient). I have followed closely the presen-tation given in [Sha94b]; the last three chapters of this book contain a lot of material on the topology of complex algebraic varieties and transcendental methods in complex algebraic geometry.
3
Lecture 3
The Massey triple products were introduced in [Mas58]. The paper [Mil54] defines subtle invariants of embedded links.
The use of differential graded algebras in algebraic topology is very re-lated to rational homotopy theory. Foundational papers are [Qui69] and [Sul77]. A comprehensive introduction is given in [GM13] and the reference book is [FHT01].
The formality of compact Kähler manifolds is proved in [DGMS75]. This paper can also be read as a survey on Sullivan minimal models in rational homotopy theory.
The proof of the theorem of Nomizu [Nom54] about compact nilmanifolds is by induction. The idea is to show that any such manifold fibers over another manifold of this type, with fiber a torus. The induction proof works by using an analysis of the corresponding Leray-Serre spectral sequence.
4
Lecture 4
It is normal to feel quite uncomfortable at the beginning, when dealing with several complex variables. One can consult the books of R. C. Gunning, especially [Gun90] for the local theory of meromorphic functions. A good reference for the Stein factorization in an analytic setting is [GR84].
I have followed closely the chapter 2 of [ABC+96] for the proof of the Siu-Beauville theorem: Castelnuovo-de Franchis→Catanese→Siu-Beauville. I
could not find a complete reference for the Castelnuovo-de Franchis theorem; see the Theorem 2.7 and Lemma 4.32 in [ABC+96] and Lemma 2.2 in [NR97]. However, Siu proved this theorem in [Siu87] some years before Catanese’s paper [Cat91]. Siu uses a totally different approach that has been very fruitful in the study of Kähler groups: construction of harmonic maps and factorization of these maps to produce holomorphic maps.
The proof by Beauville of the same theorem is given in an appendix to [Cat91]: he uses that both the existence of a surjective morphism π1(X)→ π1(Cg)and of a fibrationX →Cg are related to the existence of line bundles
Lon X whose first cohomology does not vanish.
5
Lecture 5
There is a lot of textbooks on Lia algebras, Lie groups and symmetric spaces. The standard reference on Lie algebras and Lie groups is [Kna02] but there is no treatment on symmetric spaces. Chapter 2 of [CS09] summarizes all the important properties.
For Lie groups and symmetric spaces from a differential geometry view-point, see [Hel01]. The reader in a hurry can at least consult the classification tables at the end of the book.
The book [Mor15] is also useful, especially to make the connection from the Lie group viewpoint to the algebraic group one.
I have also used english Wikipedia pages, especially List of simple Lie groups and Hermitian symmetric space.
6
Lecture 6
The book [Bor69] (in french) was the first reference on arithmetic groups. More recent sources are [PR93] and [Mor15] ; the former focuses on the number-theoretic aspects, whereas the latter deals with ergodic and group-theoretic aspects, with a lot on rigidity theorems. The fact that arithmetic groups are always of finite covolume was proved in the paper [BHC62]. Proofs are given in all the references above.
Existence and properties of the Baily-Borel compactification were estab-lished in [BB66]. A lot more on compactifications of symmetric spaces and locally symmetric spaces can be found in the book [BL06].
7
Lecture 7
Any reference book on differentiable manifolds treats the parallel transport of a connection on a vector bundle. We are particularly interested in the zero curvature case, where the monodromy representation of the connection is defined.
The Riemann-Hilbert correspondence is an equivalence between the cat-egory of complex linear representations of the fundamental group and the category of complex vector bundles with flat connections. Nowadays, it is also natural to consider the equivalent category of local systems modeled on a complex vector space. This approached is followed at the beginning of the book [Del70] or in [Voi02a], pages 228–230.
8
Lectures 8-10
The foundational paper in non-abelian Hodge theory is [Sim92]. Introduc-tions to this paper can be found in [Wen16] or [Pot91] (in french).
The whole story began with [NS65], which gives the correspondence be-tween unitary flat bundles and stable holomorphic bundles. The existence theorem for flat bundles was proved in [Don87] in some cases and in general in [Cor98]. The existence theorem for Higgs bundles was proved in [Hit86] for rank2 bundles on Riemann surfaces and in [Sim88] in general.
A proof of the properties of the Hodge filtration with respect to the Gauss-Manin connection is in [Voi02a]. The abstract aspects of variations of Hodge structures, and their relation with non-abelian Hodge theory, are in [Sim92].
Another description of harmonic bundles, as generalized variations of Hodge structures is given in [Dan17].
References
[ABC+96] J. Amoros, M. Burger, K. Corlette, D. Kotschick, and D. Toledo.
Fundamental Groups of Compact Kähler Manifolds, volume 44 of
Mathematical Surveys and Monographs. 1996.
[AF59] A. Andreotti and T. Frankel. The Lefschetz theorem on hyper-plane sections. Annals of Mathematics, 69(3):713–717, 1959.
[AM04] A. Adem and R. J. Milgram. Cohomology of finite groups, vol-ume 309 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, 2004.
[Ara95] D. Arapura. Current Topics in Complex Algebraic Geometry, volume 28 ofMSRI Publications, chapter Fundamental groups of smooth projective varieties, pages 1–16. 1995.
[BB66] W. L. Baily and A. Borel. Compactification of arithmetic quo-tients of bounded symmetric domains. Annals of Mathematics, 84(3):442–528, 1966.
[BHC62] A. Borel and Harish-Chandra. Arithmetic subgroups of algebraic groups. Annals of Mathematics, 75(3):485–353, 1962.
[BL06] A. Borel and J. Li. Compactifications of Symmetric and Locally Symmetric Spaces. Birkhäuser, 2006.
[Bor69] A. Borel. Introduction aux groupes arithmétiques. Hermann, Paris, 1969.
[Cat91] F. Catanese. Moduli and classification of irregular Kähler mani-folds (and algebraic varieties) with Albanese general type fibra-tions. Inventiones Math., 104:263–289, 1991.
[CCE14] F. Campana, B. Claudon, and P. Eyssidieux. Représentations linéaires des groupes kählériens et de leurs analogues projectifs.
Journal de l’École Polytechnique, 1:331–342, 2014.
[Cla18] B. Claudon. Smooth families of tori and linear Kähler groups.
Annales de la Faculté des Sciences de Toulouse, 2018.
[Cor98] K. Corlette. Flat g-bundles with canonical metrics. Journal of Differential Geometry, 28:361–382, 1998.
[CS09] A. Cap and J. Slovak. Parabolic Geometries I, Background and General Theory. Number 154 in Mathematical Surveys and Monographs. American Mathematical Society, 2009.
[Dan11] J. Daniel. Kähler groups and the geometry of Kähler manifolds. Master’s thesis, Université Paris Diderot, http://guests.mpim-bonn.mpg.de/jdaniel/, 2011.
[Dan17] J. Daniel. Loop Hodge structures and harmonic bundles. Alge-braic geometry, 4(5):603–643, 2017.
[Del70] P. Deligne. Èquations différentielles à points singuliers réguliers, volume 163. Springer-Verlag, 1970.
[DGMS75] P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan. Real homo-topy theory of kähler manifolds. Inventiones Math., 29:245–274, 1975.
[DI87] P. Deligne and L. Illusie. Relèvements modulo p2 et décom-position du complexe de de Rham. Inventiones Mathematicae, 89:247–270, 1987.
[Don87] S. K. Donaldson. Twisted harmonic maps and the self-duality equations. Proceedings of the London Mathematical Society, 55(3):127–131, 1987.
[FHT01] Y. Felix, S. Halperin, and J.-C. Thomas. Rational Homotopy Theory, volume 205 ofGraduate Texts in Mathematics. Springer-Verlag, 2001.
[GH78] P. Griffiths and J. Harris. Principles of algebraic geometry. 1978.
[GM13] P. Griffiths and J. Morgan. Rational Homotopy Theory and Differential Forms, volume 16 of Progress in mathematics. Birkhäuser, second edition, 2013.
[GR84] H. Grauert and R. Remmert. Coherent Analytic Sheaves. Springer-Verlag, 1984.
[Gun90] R. C. Gunning. Introduction to holomorphic functions of sever-able varisever-ables, volume 2, Local Theory of Mathematics Series. Wadsworth & Brooks/Cole, 1990.
[Hel01] S. Helgason. Differential Geometry, Lie Groups and Symmetric Spaces. Oxford University Press, 2001.
[Hir62] H. Hironaka. An example of a non-kahlerian complex-analytic deformation of kahlerian complex structures. Annals of Mathe-matics, 75(1):190–208, 1962.
[Hit86] N. J. Hitchin. The self-duality equations on a riemann surface.
Proceedings of the London Mathematical Society, 55(3):59–126, 1986.
[Huy05] D. Huybrechts. Complex geometry, an introduction. Springer-Verlag Berlin Heidelberg, 2005.
[Ill02] L. Illusie. Introduction to Hodge Theory, chapter Frobenius and Hodge degeneration, pages 99–149. 2002.
[Kna02] A. W. Knapp. Lie Groups Beyond an Introduction, volume 140 of Progress in Mathematics. Birkhäuser Basel, 2002.
[Knu71] D. Knutson. Algebraic spaces, volume 203 of Lecture Notes in Mathematics. Springer-Verlag, 1971.
[Mas58] W. S. Massey. Some higher order cohomology operations. In
Symposium International de Topologia Algebraica, Mexico City, pages 145–154, 1958.
[MFK94] D. Mumford, J. Fogarty, and F. Kirwan. Geometric Invariant Theory, volume 34 ofErgebnisse der Mathematik und ihrer Gren-zgebiete. 2. Folge. Springer-Verlag, third edition, 1994.
[Mil54] J. Milnor. Link groups. Annals of Mathematics, 59(2):177–195, 1954.
[Mil63] J. Milnor. Morse Theory. Annals of Mathematical Studies. Princeton, 1963.
[Mor15] D. W. Morris. Introduction to Arithmetic Groups. Deductive Press, 2015.
[Nom54] K. Nomizu. On the cohomology of compact homogeneous spaces of nilpotent lie groups. Annals of Mathematics, 59(3):531–538, 1954.
[NR97] T. Napier and M. Ramachandran. The Bochner-Hartogs di-chotomy for weakly 1-complete Kähler manifolds. Annales de l’Institut Fourier, 47(5):1345–1365, 1997.
[NS65] M. S. Narasimhan and C.S. Seshadri. Stable and unitary vector bundles on a compact riemann surface. Annals of Mathematics, 82(3):540–567, 1965.
[Pot91] J. Le Potier. Fibrés de higgs et systèmes locaux. Séminaire Bourbaki, 33:221–268, 1990-1991.
[PR93] V. Platonov and A. Rapinchuk. Algebraic Groups and Number Theory, volume 139 ofPure and Applied Mathematics. Academic Press, 1993.
[Qui69] D. Quillen. Rational homotopy theory. Annals of Mathematics, 90(2):205–295, 1969.
[Ser58] J. P. Serre. Sur la topologie des variétés algébriques en caractéris-tique p. In Symposium Internacional de Topologica Algebraica, 1958.
[Sha94a] I. R. Shafarevich. Basic algebraic geometry I. Springer, second edition, 1994.
[Sha94b] I. R. Shafarevich. Basic algebraic geometry II. Springer- Verlag, second edition, 1994.
[Sim88] C. T. Simpson. Constructing variations of hodge structures using yang-mills theory and applications to uniformization. Journal of the AMS, 1:867–918, 1988.
[Sim92] C. T. Simpson. Higgs bundles and local systems. Publications mathématiques de l’IHÉS, 75:5–95, 1992.
[Siu87] Y.-T. Siu. Discrete Groups in Geometry and Analysis. Papers in Honor of G. D. Mostow on His Sixtieth Birthday, volume 67 of
Progress in Mathematics, chapter 5. Strong Rigidity for Kaehler Manifolds and the Construction of Bounded Holomorphie Func-tions, pages 124–151. Springer, 1987.
[Spa81] E. H. Spanier. Algebraic Topology. Springer-Verlag, 1981.
[Sta18] The Stacks Project Authors. Stacks Project. http://stacks.math.columbia.edu, 2018.
[Sul77] D. Sullivan. Infinitesimal computations in topology. Publications mathématiques de l’IHÉS, 47:269–331, 1977.
[Tol90] D. Toledo. Examples of fundamental groups of compact Kähler manifolds. Bulletin of the London Mathematical Society, 22:339– 343, 1990.
[Voi02a] C. Voisin. Hodge Theory and Complex Algebraic Geometry I, volume 76 of Cambridge studies in advances mathematics. 2002.
[Voi02b] C. Voisin. Hodge Theory and Complex Algebraic Geometry II, volume 77 of Cambridge studies in advances mathematics. 2002.
[Voi04] C. Voisin. On the homotopy types of compact Kähler and com-plex projective manifolds. Inventiones Math., 157(2):329–343, 2004.
[Voi06] C. Voisin. On the homotopy type of Kähler manifolds and the birational Kodaira problem. Journal of Differential Geometry, 72(1):43–71, 2006.
[Wen16] R. Wentworth. Geometry and Quantization of moduli spaces, chapter Higgs bundles and local systems on Riemann sur-faces, pages 165–219. CRM Advances Courses, Barcelona. Birkauser/Springer, 2016.
