Duke Mathematical Journal

Compactifications defined by arrangements, II: Locally symmetric varieties of type IV

Eduard Looijenga

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We define a new class of completions of locally symmetric varieties of type IV which interpolates between the Baily-Borel compactification and Mumford's toric compactifications. An arithmetic arrangement in a locally symmetric variety of type IV determines such a completion canonically. This completion admits a natural contraction that leaves the complement of the arrangement untouched. The resulting completion of the arrangement complement is very much like a Baily-Borel compactification: it is the Proj of an algebra of meromorphic automorphic forms. When that complement has a moduli-space interpretation, then what we get is often a compactification obtained by means of geometric invariant theory. We illustrate this with several examples: moduli spaces of polarized K3 and Enriques surfaces and the semiuniversal deformation of a triangle singularity.

We also discuss the question of when a type IV arrangement is definable by an automorphic form.

Article information

Duke Math. J. Volume 119, Number 3 (2003), 527-588.

First available in Project Euclid: 23 April 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J15: Moduli, classification: analytic theory; relations with modular forms [See also 32G13]
Secondary: 32S22: Relations with arrangements of hyperplanes [See also 52C35]


Looijenga, Eduard. Compactifications defined by arrangements, II: Locally symmetric varieties of type IV. Duke Math. J. 119 (2003), no. 3, 527--588. doi:10.1215/S0012-7094-03-11933-X. http://projecteuclid.org/euclid.dmj/1082744772.

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  • A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth Compactification of Locally Symmetric Varieties, Lie Groups: Hist., Frontiers and Appl. 4, Math. Sci. Press, Brookline, Mass., 1975.
  • W. L. Baily Jr. and A. Borel, Compactifiation of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442--528.
  • R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405--444.
  • --. --. --. --., The moduli space of Enriques surfaces and the fake Monster Lie superalgebra, Topology 35 (1996), 699--710.
  • --. --. --. --., Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), 491--562.
  • R. E. Borcherds, L. Katzarkov, T. Pantev, and N. I. Shepherd-Barron, Families of $K3$ surfaces, J. Algebraic Geom. 7 (1998), 183--193.
  • A. Borel, Introduction aux groupes arithmétiques, Publ. Inst. Math. Univ. Strasbourg 15, Actualités Sci. Indust. 1341, Hermann, Paris, 1969.
  • J. H. Bruinier, Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors, Lecture Notes in Math. 1780, Springer, New York, 2002.
  • J. H. Bruinier and E. Freitag, Local Borcherds products, Ann. Inst. Fourier 51 (Grenoble) (2001), 1--26.
  • J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3d ed., Grundlehren Math. Wiss. 290, Springer, New York, 1999.
  • V. A. Gritsenko and V. V. Nikulin, Automorphic forms and Lorentzian Kac-Moody Algebras, I, Internat. J. Math. 9 (1998), 201--275.
  • --------, On classification of Lorentzian Kac-Moody algebras, preprint.
  • E. Looijenga, Invariant theory for generalized root systems, Invent. Math. 61 (1980), 1--32.
  • --. --. --. --., The smoothing components of a triangle singularity, II, Math. Ann. 269 (1984), 357--387.
  • --. --. --. --., "New compactifications of locally symmetric varieties" in Proceedings of the 1984 Conference in Algebraic Geometry, ed. J. Carrell, A. V. Geramita, and P. Russell, CMS Conf. Proc. 6, Amer. Math. Soc., Providence, 1986, 341--364.
  • --. --. --. --., Compactifications defined by arrangements, I: The ball quotient case, Duke Math. J. 118 (2003), 151--187. \CMP1 978 885
  • --------, Semi-toric partial compactifications, I, preprint, 1985, report 8520, University of Nijmegen, Netherlands.
  • A. L. Mayer, Families of $K$-3 surfaces, Nagoya Math. J. 48 (1972), 1--17.
  • S. Mukai, "Curves, $K3$ surfaces and Fano $3$-folds of genus $\le 10$" in Algebraic Geometry and Commutative Algebra, Vol. 1, ed. H. Hijikata, H. Hironaka, M. Maruyama, H. Matsumura, M. Miyanishi, T. Oda, and K. Ueno, Kinokuniya, Tokyo, 1988, 357--377.
  • V. V. Nikulin, "A remark on discriminants for moduli of $K3$ surfaces as sets of zeros of automorphic forms" in Algebraic Geometry, Vol 4: Dedicated to the Sixtieth Birthday of Professor Yuri Ivanovich Manin, ed. E. S. Golod and A. N. Tyurin, J. Math. Sci. 81 (1996), 2738--2743.
  • I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, Torelli's theorem for algebraic surfaces of type $\mathrmK$ (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530--572.
  • B. Saint-Donat, Projective models of $K$-3 surfaces, Amer. J. Math. 96 (1974), 602--639.
  • J. Shah, A complete moduli space for $K3$ surfaces of degree $2$, Ann. of Math. (2) 112 (1980), 485--510.
  • --. --. --. --., Degenerations of $K3$ surfaces of degree $4$, Trans. Amer. Math. Soc. 263 (1981), 271--308.
  • --. --. --. --., Projective degenerations of Enriques' surfaces, Math. Ann. 256 (1981), 475--495.
  • H. Sterk, Compactifications of the period space of Enriques surfaces, I, Math. Z. 207 (1991), 1--36.
  • --. --. --. --., Compactifications of the period space of Enriques surfaces, II, Math. Z. 220 (1995), 427--444.
  • --. --. --. --., Lattices and $K3$ surfaces of degree $6$, Linear Algebra Appl. 226/228 (1995), 297--309.

See also

  • See also: Eduard Looijenga. Compactifications defined by arrangements, I: The ball quotient case. Duke Math. J. Vol. 118, No. 1 (2003), pp. 151-187.