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Complex and real hyperbolic moduli spaces Domingo Toledo

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Niveau: Supérieur, Doctorat, Bac+8
Complex and real hyperbolic moduli spaces Domingo Toledo ? June 30, 2004 The purpose of these lectures is to study a number of moduli spaces that have complex hyperbolic structures. Real forms of these moduli spaces then have real hyperbolic structures. One interest of these moduli spaces is that they provide explicit examples of complex or real hyperbolic orbifolds. Another interest is that the topology of the moduli spaces is closely related to the topology of discriminant complements. The classical example of a complex hyperbolic moduli space is the moduli of cubic curves, which is uniformized by the real hyperbolic plane, or complex hyperbolic line. We will be interested in higher dimensional spaces. The first examples of two-dimensional complex hyperbolic moduli spaces were given by Picard [14], [15] and in several other papers. These examples have been devel- oped further by others, most recently by Terada [18], Deligne and Mostow [5], [13], and Thurston [19]. Only finitely many examples of complex hyperbolic moduli spaces are known, and there may be good reasons to believe that (with proper interpretation) only finitely many examples are possible. The known examples include: 1. Moduli of points on the complex projective line (or Riemann sphere) P1, with certain weights assigned to the points. The number of points must be at most 11, and only certain weights are possible.

  • group

  • examples include

  • components corresponding

  • structure has

  • singular cubic

  • see examples

  • real hyperbolic

  • complex hyperbolic

  • dimensional


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Complex and real hyperbolic moduli spaces
DomingoToledo June 30, 2004
The purpose of these lectures is to study a number of moduli spaces that have complex hyperbolic structures.Real forms of these moduli spaces then have real hyperbolic structures.One interest of these moduli spaces is that they provide explicit examples of complex or real hyperbolic orbifolds.Another interest is that the topology of the moduli spaces is closely related to the topology of discriminant complements. The classical example of a complex hyperbolic moduli space is the moduli of cubic curves, which is uniformized by the real hyperbolic plane, or complex hyperbolicline.Wewillbeinterestedinhigherdimensionalspaces.The rst examples of two-dimensional complex hyperbolic moduli spaces were given by Picard [14], [15] and in several other papers.These examples have been devel-oped further by others, most recently by Terada [18], Deligne and Mostow [5], [13], and Thurston [19]. Only nitelymanyexamplesofcomplexhyperbolicmodulispacesareknown, and there may be good reasons to believe that (with proper interpretation) only nitelymanyexamplesarepossible.Theknownexamplesinclude: 1 1. Moduliof points on the complex projective line (or Riemann sphere)P, with certain weights assigned to the points.The number of points must be at most 11, and only certain weights are possible.For a list, see [5], [13],[19]. 2. Moduliof Del Pezzo surfaces (complex surfaces with positive anti-canonical class).Thesigni cantcasesare (a) Cubicsurfaces: seemy joint work with Allcock and Carlson, [1]. (b) Del Pezzosurfaces of degree 2, which have the same moduli space is plane quartic curves.See the work of Kondo, [9].(See also the related paper [10] where yet another moduli space gets a complex hyperbolic structure.) (c) Del Pezzo surfaces of degree one:See the work of Heckman and Looijenga [6].  Author partially supported by NSF grant DMS 0200877.
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