Canonical sections of Hodge bundles on moduli spaces

Document Type: Original Article


1 Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095-1555, USA

2 Department of Mathematics, Nanjing University, Nanjing 210093, P.R.China


We review recent works in [K. Liu, S. Rao, and X. Yang, Quasi-isometry and deformations of Calabi Yau manifolds, Inventiones mathematicae, 199(2) (2015),  423–453] and [K. Liu and Y. Shen, Moduli spaces as ball quotients I, local theory, preprint] on geometry of sections of Hodge bundles and their applications to moduli spaces.


D. Allcock, J. Carlson and D. Toledo, The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Alg. Geom. 11 (2002), no. 4, 659-724.
D. Allcock, J. Carlson and D. Toledo, The Moduli Space of Cubic Threefolds as a Ball Quotient, Mem. Amer. Math. Soc. 209 (2011), no. 985, xii+70.
A. Beauville, Moduli of cubic surfaces and Hodge theory (after Allcock, Carlson, Toledo), G_eom_etriesa courbure negative ou nulle, groupes discrets et rigidit_es, S_eminaires et Congres 18 Soc. Math. France, Paris, 2009.
C. H. Clemens, Degenerations of Kahler manifolds, Duke Math J. 44 (1977), no. 2, 215--290.
C. H. Clemens, Geometry of formal Kuranishi theory, Adv. Math. 198 (2005), no. 1, 311--365.
P. Deligne, Th_eorie de Hodge II, Publ. Math. IHES 40 (1971) 5--57.
P. Deligne and G. W. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Publ. Math. IHES 63 (1972) 5--89.
I. V. Dolgachev and S. Kondo, Moduli of K3 surfaces and complex ball quotients, Arithmetic and Geometry Around
Hypergeometric Functions, Progress in Mathematics 260, 2007, pp. 43-100.
P. Griffiths, Periods of integrals on algebraic manifolds I, Construction and properties of the modular varieties, Amer. J. Math. 90 (1968) 568--626.
P. Griffiths, Periods of integrals on algebraic manifolds II, Amer. J. Math. 90 (1968) 805--865.
P. Griffiths, On the Periods of Certain Rational Integrals: I, II, Ann. of Math. (2) 90 (1969), no. 2, 460--495 and 496--541.
P. Griffiths, Periods of integrals on algebraic manifolds, III, Some global differential-geometric properties of the period mapping, Publ. Math. IHES 38 (1970) 125--180.
P. Griffiths, Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems, Bull. Amer. Math. Soc. 76 (1970), no.2, 228--296.
P. Griffiths, Topics in transcendental algebraic geometry, Proceedings of a seminar held at the Institute for Advanced
Study, Princeton, N.J., during the academic year 1981/1982. Edited by Phillip Griffiths, Annals of Mathematics
Studies, 106. Princeton University Press, Princeton, NJ, 1984.
P. Griffiths and W. Schmid, Locally homogeneous complex manifolds, Acta Math. 123 (1969) 253--302.
P. Griffiths and W. Schmid, Recent developments in Hodge theory: a discussion of techniques and results, Discrete
subgroups of Lie groups and applicatons to moduli (Internat. Colloq., Bombay, 1973), pp. 31--127. Oxford Univ. Press, Bombay, 1975.
P. Griffiths and J. Wolf, Complete maps and differentiable coverings, Michigan Math. J. 10 (1963) 253--255.
K. Kodaira and D. C. Spencer, On Deformations of Complex Analytic Structures, III, Ann. of Math. (2) 71 (1960) 43--76.
K. Liu, S. Rao and X. Yang, Quasi-isometry and deformations of CalabiYau manifolds, Invent. Math. 199 (2015),
no. 2, 423--453.
K. Liu and Y. Shen, Global Torelli theorem for projective manifolds of Calabi--Yau type, arXiv:1205.4207, (2012).
K. Liu and Y. Shen, Simultaneous normalization of period map and affine structures on moduli spaces, arXiv:1910.06767, (2019).
K. Liu and Y. Shen, From local Torelli to global Torelli, arXiv: 1512.08384, (2015).
K. Liu and Y. Shen, Moduli spaces as ball quotients I, local theory. preprint.
W. Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973) 211--319.
A. Sommese, On the rationality of the period mapping, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (1978), no. 4, 683--717.
B. Szendroi, Some finiteness results for Calabi-Yau threefolds, J. London Math. Soc. (2) 60 (1999), no. 3, 689--699.
G. Tian, Smoothness of the universal deformation space of compact Calabi--Yau manifolds and its PeterssonWeil metric, Mathematical aspects of string theory, In: Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987, pp. 629-646.
A. Todorov, The Weil-Petersson geometry of the moduli space of SU(n_3) (Calabi-Yau) manifolds. I, Commun. Math. Phys. 126 (1989), no. 2, 325--346.