Geometric structures on the complement of a projective arrangement
Publications Mathématiques de l'IHÉS, Tome 101 (2005), pp. 69-161.

Consider a complex projective space with its Fubini-Study metric. We study certain one parameter deformations of this metric on the complement of an arrangement (= finite union of hyperplanes) whose Levi-Civita connection is of Dunkl type. Interesting examples are obtained from the arrangements defined by finite complex reflection groups. We determine a parameter interval for which the metric is locally of Fubini-Study type, flat, or complex-hyperbolic. We find a finite subset of this interval for which we get a complete orbifold or at least a Zariski open subset thereof, and we analyze these cases in some detail (e.g., we determine their orbifold fundamental group). In this set-up, the principal results of Deligne-Mostow on the Lauricella hypergeometric differential equation and work of Barthel-Hirzebruch-Höfer on arrangements in a projective plane appear as special cases. Along the way we produce in a geometric manner all the pairs of complex reflection groups with isomorphic discriminants, thus providing a uniform approach to work of Orlik-Solomon.

@article{PMIHES_2005__101__69_0,
     author = {Couwenberg, Wim and Heckman, Gert and Looijenga, Eduard},
     title = {Geometric structures on the complement of a projective arrangement},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {69--161},
     publisher = {Springer},
     volume = {101},
     year = {2005},
     doi = {10.1007/s10240-005-0032-3},
     mrnumber = {2217047},
     zbl = {1083.14039},
     language = {en},
     url = {http://www.numdam.org/articles/10.1007/s10240-005-0032-3/}
}
TY  - JOUR
AU  - Couwenberg, Wim
AU  - Heckman, Gert
AU  - Looijenga, Eduard
TI  - Geometric structures on the complement of a projective arrangement
JO  - Publications Mathématiques de l'IHÉS
PY  - 2005
SP  - 69
EP  - 161
VL  - 101
PB  - Springer
UR  - http://www.numdam.org/articles/10.1007/s10240-005-0032-3/
DO  - 10.1007/s10240-005-0032-3
LA  - en
ID  - PMIHES_2005__101__69_0
ER  - 
%0 Journal Article
%A Couwenberg, Wim
%A Heckman, Gert
%A Looijenga, Eduard
%T Geometric structures on the complement of a projective arrangement
%J Publications Mathématiques de l'IHÉS
%D 2005
%P 69-161
%V 101
%I Springer
%U http://www.numdam.org/articles/10.1007/s10240-005-0032-3/
%R 10.1007/s10240-005-0032-3
%G en
%F PMIHES_2005__101__69_0
Couwenberg, Wim; Heckman, Gert; Looijenga, Eduard. Geometric structures on the complement of a projective arrangement. Publications Mathématiques de l'IHÉS, Tome 101 (2005), pp. 69-161. doi : 10.1007/s10240-005-0032-3. http://www.numdam.org/articles/10.1007/s10240-005-0032-3/

1. G. Barthel, F. Hirzebruch, T. Höfer, Geradenkonfigurationen und algebraische Flächen, Aspects of Mathematics, Vieweg, Braunschweig-Wiesbaden 1987. | MR | Zbl

2. D. Bessis, Zariski theorems and diagrams for braid groups, Invent. Math., 145 (2001), 487-507, also available at arXiv math.GR/0010323. | MR | Zbl

3. N. Bourbaki, Groupes et algèbres de Lie, Ch. 4, 5 et 6 Actualités Scientifiques et industrielles, vol. 1337, Hermann, Paris 1968. | MR | Zbl

4. E. Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer komplexen Spiegelungsgruppe, Invent. Math., 12 (1971), 57-61. | MR | Zbl

5. W. Casselman, Families of curves and automorphic forms, Thesis, Princeton University, 1966 (unpublished).

6. A. M. Cohen, Finite complex reflection groups, Ann. Sci. Éc. Norm. Super., 9 (1976), 379-446. | Numdam | MR | Zbl

7. P. B. Cohen, F. Hirzebruch, Review of Commensurabilities among lattices in PU(1,n) by Deligne and Mostow, Bull. Am. Math. Soc., 32 (1995), 88-105. | MR

8. P. B. Cohen, G. Wüstholz, Applications of the André-Oort Conjecture to some questions in transcendency, in: A Panorama in Number Theory, a view from Baker's garden, Cambridge University Press, London New York 2002, 89-106. | Zbl

9. W. Couwenberg, Complex Reflection Groups and Hypergeometric Functions, Thesis (123 p.), University of Nijmegen, 1994, also available at http://members.chello.nl/ w.couwenberg/.

10. H. S. M. Coxeter, Regular complex polytopes, Cambridge University Press, London New York 1974. | MR | Zbl

11. C. W. Curtis, N. Iwahori, R. Kilmoyer, Hecke algebras and characters of parabolic type of finite groups with (B,N)-pairs, Publ. Math., Inst. Hautes Étud. Sci., 40 (1971), 81-116. | Numdam | MR | Zbl

12. P. Deligne, Équations Différentielles à Points Singuliers Réguliers, Lect. Notes Math., vol. 163, Springer, Berlin etc. 1970. | MR | Zbl

13. P. Deligne, Les immeubles de groupes de tresses généralisées, Invent. Math., 17 (1972), 273-302. | MR | Zbl

14. P. Deligne, G. D. Mostow, Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math., Inst. Hautes Étud. Sci., 63 (1986), 5-89. | Numdam | MR | Zbl

15. P. Deligne, G. D. Mostow, Commensurabilities among lattices in PU(1,n), Ann. of Math. Studies, vol. 132, Princeton University Press, Princeton 1993. | MR | Zbl

16. B. R. Doran, Intersection Homology, Hypergeometric Functions, and Moduli Spaces as Ball Quotients, Thesis, Princeton University (93 p.), 2003.

17. H. Grauert, R. Remmert, Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften, vol. 265, Springer, Berlin, 1984. | MR | Zbl

18. B. Hunt, The Geometry of some special Arithmetic Quotients, Springer Lect. Notes Math., vol. 1637, 1996. | MR | Zbl

19. B. Hunt, S. Weintraub, Janus-like algebraic varieties, J. Differ. Geom., 39 (1994), 507-557. | MR | Zbl

20. R.-P. Holzapfel, Chern Numbers of Algebraic Surfaces, Hirzebruch's Examples are Picard Modular Surfaces, Math. Nachr., 126 (1986), 255-273. | Zbl

21. R.-P. Holzapfel, Transcendental Ball Points of Algebraic Picard Integrals, Math. Nachr., 161 (1993), 7-25. | MR | Zbl

22. E. Looijenga, Arrangements, KZ systems and Lie algebra homology, in: Singularity Theory, B. Bruce and D. Mond, eds., London Math. Soc. Lecture Note Series 263, Cambridge University Press, London New York 1999, 109-130. | MR | Zbl

23. E. Looijenga, Compactifications defined by arrangements I: the ball quotient case, Duke Math. J., 118 (2003), 151-187, also available at arXiv math.AG/0106228. | MR | Zbl

24. B. Malgrange, Sur les points singuliers des équations differentielles, Enseign. Math., 20 (1974), 147-176. | MR | Zbl

25. J. I. Manin, Moduli fuchsiani, Ann. Sc. Norm. Super. Pisa, 19 (1965), 113-126. | Numdam | MR | Zbl

26. G. D. Mostow, Generalized Picard lattices arising from half-integral conditions, Publ. Math., Inst. Hautes Étud. Sci., 63 (1986), 91-106. | Numdam | MR | Zbl

27. P. Orlik, L. Solomon, Discriminants in the invariant theory of reflection groups, Nagoya Math. J., 109 (1988), 23-45. | MR | Zbl

28. P. Orlik, H. Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften, vol. 300, Springer, Berlin, 1992. | MR | Zbl

29. H. A. Schwarz, Über diejenigen Fälle in welchen die Gaussische hypergeometrische Reihe eine algebraische Funktion ihres vierten Elementes darstellt, J. Reine Angew. Math., 75 (1873), 292-335. | JFM

30. G. C. Shephard, J. A. Todd, Finite unitary reflection groups, Can. J. Math., 6 (1954), 274-304. | MR | Zbl

31. G. Shimura, On purely transcendental fields of automorphic functions of several variables, Osaka J. Math., 1 (1964), 1-14. | MR | Zbl

32. L. Solomon, Invariants of finite reflection groups, Nagoya Math. J., 22 (1963), 57-64. | MR | Zbl

33. W. P. Thurston, Three-Dimensional Geometry and Topology, vol. I, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton 1997. | MR | Zbl

34. W. P. Thurston, Shapes of polyhedra and triangulations of the sphere, Geom. Topol. Monogr., 1 (1998), 511-549. | MR | Zbl

35. M. Yoshida, Orbifold-uniformizing differential equations. III. Arrangements defined by 3-dimensional primitive unitary reflection groups, Math. Ann., 274 (1986), 319-334. | MR | Zbl

Cité par Sources :