The complex hyperbolic geometry of the moduli space of cubic surfaces
Authors:
Daniel Allcock, James A. Carlson and Domingo Toledo
Journal:
J. Algebraic Geom. 11 (2002), 659-724
DOI:
https://doi.org/10.1090/S1056-3911-02-00314-4
Published electronically:
July 2, 2002
MathSciNet review:
1910264
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References |
Additional Information
Abstract: We prove that the moduli space of semistable cubic surfaces over the complex numbers is biholomorphic to the Satake compactification of the quotient of the four-ball by the projective unitary group of the standard Hermitian form of signature $(4,1)$ with coefficients in the ring of integers of $\mathbb {Q}(\sqrt {-3})$. We also explain the precise relation between the orbifold structures on the moduli space of stable cubic surfaces and on the quotient of the ball.
Allcock D. Allcock, New complex and quaternion-hyperbolic reflection groups, Duke Math. Jour., 103 (2000), 303–333.
Allcockaspherical D. Allcock, Asphericity of Moduli Spaces via Curvature. J. Diff. Geom. 55 (2001), 441–451.
ACT D. Allcock, J. Carlson, and D. Toledo, A complex hyperbolic structure for moduli of cubic surfaces, C. R. Acad. Sci. Paris , t. 326, Série I (1998), 49–54.
ACTfundgp D. Allcock, J. Carlson, and D. Toledo, Orthogonal complex hyperbolic arrangements, Symposium in Honor of C. H. Clemens, a special volume of Contemp. Math., A. Bertram, J. Carlson and H. Kley, eds. (to appear).
AF D. Allcock and E. Freitag, Cubic surfaces and Borcherds products, Comm. Math. Helv. (to appear).
AGV V.I. Arnold, S.M. Gusein-Zade, and A.N. Varchenko, Singularities of Differentiable Maps, vol II, Birkhauser, Boston, 1988.
BB W. L. Baily, Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. Math., 84 (1966) 442–528.
Bourbaki N. Bourbaki, Elements de Mathématiques, vol XXXIV: Groupes et Algèbres de Lie, chapitres 4, 5 et 6, Hermann et Cie, Paris 1968, pp 288.
Bruce J. W. Bruce, A stratification of the space of cubic surfaces. Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 3, 427–441.
BruceWall J. W. Bruce, C. T. C. Wall, On the classification of cubic surfaces. J. London Math. Soc. (2) 19 (1979), no. 2, 245–256.
CarlsonGriffiths J. Carlson and P. Griffiths, Infinitesimal variations of Hodge structure and the global Torelli problem. Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, pp. 51–76, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980.
CTDiscriminant J. Carlson and D. Toledo, Discriminant complements and kernels of monodromy representations, Duke Math. Jour. 97 (1999), 621–648.
Clemens C. H. Clemens, Double Solids, Advances in Math. 47 (1983) 107–230
ClemensGriffiths C. H. Clemens and P. A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. 95 (1972), 281–356.
CKM H. Clemens, J. Kollár, S. Mori, Higher Dimensional Complex Geometry , Astérisque 166 (1988), Soc. Math. de France.
atlas J. H. Conway, R. T. Curtis, S. P, Norton, R. A. Parker, R. A. Wilson, ATLAS of Finite Groups, Clarendon Press, Oxford, 1985.
SPLAG J. H. Conway, N. J. A. Sloane. Sphere Packings, Lattices and Groups, second edition. Springer-Verlag, 1993.
DeligneSGA P. Deligne, Comparaison avec la théorie transcendente, exposé XIV in Groupes de Monodromie en Géometrie Algebrique, pp 116-164, LNM 340, Springer Verlag, Berlin, 1973.
DeligneMostow P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math. I.H.E.S. 63 (1986) 5–89.
Feit W. Feit, Some lattices over ${Q}(\surd -3)$, J. Algebra 52 (1978), no. 1, 248–263.
Fox R. H. Fox, Covering spaces with singularities. Algebraic Geometry and Topology; A symposium in honor of S. Lefschetz, pp. 243–257, Princeton University Press, 1957.
Garside F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser.(2) 20 (1969), 235–254.
GoreskyMacPherson M. Goresky and R. MacPherson, Stratified Morse Theory, Springer Verlag, Berlin, 1988.
Griffiths P.A. Griffiths, On the periods of certain rational integrals: I and II, Ann. of Math. 90 (1969), 460-541.
GriffithsHarris P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, New York, 1978.
HeckmanLooijenga G. Heckman and E. Looijenga, The moduli space of rational elliptic surfaces, math.AG/0010020.
Hilbert D. Hilbert, Über die volle Invariantensysteme, Math. Annalen 42 (1893), 313-370. English transl. in Hilbert’s invariant theory papers; Lie Groups: History, Frontiers and Applications, VIII. Math Sci Press, Brookline, Mass., 1978.
Hunt B. Hunt. Higher-dimensional ball quotients, the invariant quintic fourfold and moduli spaces of cubic surfaces, preprint 1998, Max-Plank-Institut, Leipzig. http://personal-homepages.mis.mpg.de/bhunt/preprints.html
HuntWeintraub B. Hunt and S. H. Weintraub, Janus-like algebraic varieties, Jour. Diff. Geom. 39, (1994), 509–557.
Kondothree S. Kondo, A complex hyperbolic structure for the moduli space of curves of genus $3$, J. Reine Angew. Math. 525 (2000), 219–232.
Kondofour S. Kondo, The moduli space of curves of genus $4$ and Deligne-Mostow’s complex reflection groups, Advanced Studies in Pure Mathematics , Proc. Algebraic Geometry, Azumino 2000.
Libgober A. Libgober, On the fundamental group of the space of cubic surfaces, Math. Zeit. 162 (1978), 63–67.
LooijengaLondon E. Looijenga, Isolated Singular Points on Complete Intersections, Cambridge University Press, Cambridge, 1984.
Looijenga E. Looijenga, Affine Artin groups and the fundamental groups of some moduli spaces, math.AG/ 9801117.
Manin Yu. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic, North-Holland, Amsterdam (1974), pp. 292.
Matsumoto K. Matsumoto, T. Sasaki and M. Yoshida, The monodromy of the period map of a $4$-parameter family of $K3$-surfaces and the hypergeometric function of type $(3,6)$, Inter. Jour. Math. 3 (1992), 1-164.
MatsumotoTerasoma K. Matsumoto and T. Terasoma, Theta constants associated to cubic threefolds, math.AG0008024.
Mess G. Mess, The Torelli group of genus $2$ and $3$ surfaces, Topology 31 (1992), 775-790.
Milnor J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies, Princeton University Press 1968, p. 122.
milnor J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Springer-Verlag, 1973.
mostow Mostow, G. D., Generalized Picard lattices arising from half-integral conditions. Inst. Hautes Études Sci. Publ. Math. 63 (1986), 91–106.
Mumford D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, Springer-Verlag, Berlin, 1994, p. 292.
MumfordEnseignement D. Mumford, Stability of projective varieties, Enseignement Mathématique 23 (1977), pp. 39–110.
Naruki I. Naruki, Cross ratio variety as a moduli space of cubic surfaces, Proc. London Math. Soc. (3) 45 (1982), 1-30.
OMeara O. T. O’Meara, Introduction to Quadratic Forms, Springer-Verlag, 1963.
Picard E. Picard, Sur les fonctions de deux variables indépendantes analogues aux fonctions modulaires, Acta Math. 2 (1883) 114–135.
SasakiYoshida T. Sasaki and M. Yoshida, The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces, Proc. Japan Acad. 75, Ser. A, No. 7 (1999) 129–133.
SasakiYoshidatwo T. Sasaki and M. Yoshida, The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces II, math.AG/0008026
Segre B. Segre, The Non-singular Cubic Surfaces, Clarendon Press, Oxford, 1942.
Siegel C. L. Siegel, Symplectic Geometry, Am. Jour. Math. 65 (1943), 1–86.
ST M. Sebastiani and R. Thom, Un résultat sur la monodromie, Invent. Math. 13 (1971), 90–96.
Thurston W. P. Thurston, Shapes of polyhedra, Research Report GCG 7, University of Minnesota, math.GT/9801088.
Toledo D. Toledo, Projective varieties with non-residually finite fundamental group, Pub. Math. IHES 77 (1993) 103-119.
Zeeman E. C. Zeeman, Dihomology III, A generalization of the Poincaré duality for manifolds, Proc. London Math. Soc. (3) 13 (1963), 155–183.
Allcock D. Allcock, New complex and quaternion-hyperbolic reflection groups, Duke Math. Jour., 103 (2000), 303–333.
Allcockaspherical D. Allcock, Asphericity of Moduli Spaces via Curvature. J. Diff. Geom. 55 (2001), 441–451.
ACT D. Allcock, J. Carlson, and D. Toledo, A complex hyperbolic structure for moduli of cubic surfaces, C. R. Acad. Sci. Paris , t. 326, Série I (1998), 49–54.
ACTfundgp D. Allcock, J. Carlson, and D. Toledo, Orthogonal complex hyperbolic arrangements, Symposium in Honor of C. H. Clemens, a special volume of Contemp. Math., A. Bertram, J. Carlson and H. Kley, eds. (to appear).
AF D. Allcock and E. Freitag, Cubic surfaces and Borcherds products, Comm. Math. Helv. (to appear).
AGV V.I. Arnold, S.M. Gusein-Zade, and A.N. Varchenko, Singularities of Differentiable Maps, vol II, Birkhauser, Boston, 1988.
BB W. L. Baily, Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. Math., 84 (1966) 442–528.
Bourbaki N. Bourbaki, Elements de Mathématiques, vol XXXIV: Groupes et Algèbres de Lie, chapitres 4, 5 et 6, Hermann et Cie, Paris 1968, pp 288.
Bruce J. W. Bruce, A stratification of the space of cubic surfaces. Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 3, 427–441.
BruceWall J. W. Bruce, C. T. C. Wall, On the classification of cubic surfaces. J. London Math. Soc. (2) 19 (1979), no. 2, 245–256.
CarlsonGriffiths J. Carlson and P. Griffiths, Infinitesimal variations of Hodge structure and the global Torelli problem. Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, pp. 51–76, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980.
CTDiscriminant J. Carlson and D. Toledo, Discriminant complements and kernels of monodromy representations, Duke Math. Jour. 97 (1999), 621–648.
Clemens C. H. Clemens, Double Solids, Advances in Math. 47 (1983) 107–230
ClemensGriffiths C. H. Clemens and P. A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. 95 (1972), 281–356.
CKM H. Clemens, J. Kollár, S. Mori, Higher Dimensional Complex Geometry , Astérisque 166 (1988), Soc. Math. de France.
atlas J. H. Conway, R. T. Curtis, S. P, Norton, R. A. Parker, R. A. Wilson, ATLAS of Finite Groups, Clarendon Press, Oxford, 1985.
SPLAG J. H. Conway, N. J. A. Sloane. Sphere Packings, Lattices and Groups, second edition. Springer-Verlag, 1993.
DeligneSGA P. Deligne, Comparaison avec la théorie transcendente, exposé XIV in Groupes de Monodromie en Géometrie Algebrique, pp 116-164, LNM 340, Springer Verlag, Berlin, 1973.
DeligneMostow P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math. I.H.E.S. 63 (1986) 5–89.
Feit W. Feit, Some lattices over ${Q}(\surd -3)$, J. Algebra 52 (1978), no. 1, 248–263.
Fox R. H. Fox, Covering spaces with singularities. Algebraic Geometry and Topology; A symposium in honor of S. Lefschetz, pp. 243–257, Princeton University Press, 1957.
Garside F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser.(2) 20 (1969), 235–254.
GoreskyMacPherson M. Goresky and R. MacPherson, Stratified Morse Theory, Springer Verlag, Berlin, 1988.
Griffiths P.A. Griffiths, On the periods of certain rational integrals: I and II, Ann. of Math. 90 (1969), 460-541.
GriffithsHarris P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, New York, 1978.
HeckmanLooijenga G. Heckman and E. Looijenga, The moduli space of rational elliptic surfaces, math.AG/0010020.
Hilbert D. Hilbert, Über die volle Invariantensysteme, Math. Annalen 42 (1893), 313-370. English transl. in Hilbert’s invariant theory papers; Lie Groups: History, Frontiers and Applications, VIII. Math Sci Press, Brookline, Mass., 1978.
Hunt B. Hunt. Higher-dimensional ball quotients, the invariant quintic fourfold and moduli spaces of cubic surfaces, preprint 1998, Max-Plank-Institut, Leipzig. http://personal-homepages.mis.mpg.de/bhunt/preprints.html
HuntWeintraub B. Hunt and S. H. Weintraub, Janus-like algebraic varieties, Jour. Diff. Geom. 39, (1994), 509–557.
Kondothree S. Kondo, A complex hyperbolic structure for the moduli space of curves of genus $3$, J. Reine Angew. Math. 525 (2000), 219–232.
Kondofour S. Kondo, The moduli space of curves of genus $4$ and Deligne-Mostow’s complex reflection groups, Advanced Studies in Pure Mathematics , Proc. Algebraic Geometry, Azumino 2000.
Libgober A. Libgober, On the fundamental group of the space of cubic surfaces, Math. Zeit. 162 (1978), 63–67.
LooijengaLondon E. Looijenga, Isolated Singular Points on Complete Intersections, Cambridge University Press, Cambridge, 1984.
Looijenga E. Looijenga, Affine Artin groups and the fundamental groups of some moduli spaces, math.AG/ 9801117.
Manin Yu. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic, North-Holland, Amsterdam (1974), pp. 292.
Matsumoto K. Matsumoto, T. Sasaki and M. Yoshida, The monodromy of the period map of a $4$-parameter family of $K3$-surfaces and the hypergeometric function of type $(3,6)$, Inter. Jour. Math. 3 (1992), 1-164.
MatsumotoTerasoma K. Matsumoto and T. Terasoma, Theta constants associated to cubic threefolds, math.AG0008024.
Mess G. Mess, The Torelli group of genus $2$ and $3$ surfaces, Topology 31 (1992), 775-790.
Milnor J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies, Princeton University Press 1968, p. 122.
milnor J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Springer-Verlag, 1973.
mostow Mostow, G. D., Generalized Picard lattices arising from half-integral conditions. Inst. Hautes Études Sci. Publ. Math. 63 (1986), 91–106.
Mumford D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, Springer-Verlag, Berlin, 1994, p. 292.
MumfordEnseignement D. Mumford, Stability of projective varieties, Enseignement Mathématique 23 (1977), pp. 39–110.
Naruki I. Naruki, Cross ratio variety as a moduli space of cubic surfaces, Proc. London Math. Soc. (3) 45 (1982), 1-30.
OMeara O. T. O’Meara, Introduction to Quadratic Forms, Springer-Verlag, 1963.
Picard E. Picard, Sur les fonctions de deux variables indépendantes analogues aux fonctions modulaires, Acta Math. 2 (1883) 114–135.
SasakiYoshida T. Sasaki and M. Yoshida, The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces, Proc. Japan Acad. 75, Ser. A, No. 7 (1999) 129–133.
SasakiYoshidatwo T. Sasaki and M. Yoshida, The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces II, math.AG/0008026
Segre B. Segre, The Non-singular Cubic Surfaces, Clarendon Press, Oxford, 1942.
Siegel C. L. Siegel, Symplectic Geometry, Am. Jour. Math. 65 (1943), 1–86.
ST M. Sebastiani and R. Thom, Un résultat sur la monodromie, Invent. Math. 13 (1971), 90–96.
Thurston W. P. Thurston, Shapes of polyhedra, Research Report GCG 7, University of Minnesota, math.GT/9801088.
Toledo D. Toledo, Projective varieties with non-residually finite fundamental group, Pub. Math. IHES 77 (1993) 103-119.
Zeeman E. C. Zeeman, Dihomology III, A generalization of the Poincaré duality for manifolds, Proc. London Math. Soc. (3) 13 (1963), 155–183.
Additional Information
Daniel Allcock
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
MR Author ID:
620316
Email:
allcock@math.utexas.edu
James A. Carlson
Affiliation:
Department of Mathematics, University of Utah, 155 S 1400 E JWB 233, Salt Lake City, Utah 84112-0090
Email:
carlson@math.utah.edu
Domingo Toledo
Affiliation:
Department of Mathematics, University of Utah, 155 S 1400 E, Room 233, Salt Lake City, Utah 84112-0090
Email:
toledo@math.utah.edu
Received by editor(s):
July 9, 2000
Received by editor(s) in revised form:
February 9, 2001
Published electronically:
July 2, 2002
Additional Notes:
The first author was partially supported by an NSF postdoctoral fellowship. The second and third authors [1] were partially supported by NSF grants DMS 9625463 and DMS 9900543. The third author was partially supported by the IHES
Dedicated:
To Herb Clemens on his 60th birthday
