Local and global theory of the moduli of polarized Calabi-Yau manifolds. (English) Zbl 1058.32019

Todorov, Andrey

Rev. Mat. Iberoam. 19, No. 2, 687-730 (2003).

Given a complex manifold \(M\), its Teichmüller space is defined as the quotient of all integrable complex structures on \(M\) by the action of the group of its diffeomorphisms isotopic to the identity.
The author shows that for a Calabi-Yau manifold \(M\) its Teichmüller space has a finite number of irreducible components, each one a non-singular complex manifold. The modular group of a compact complex manifold is defined as the quotient of the orientation preserving diffeomorphisms by its normal subgroup of diffeomorphisms isotopic to the identity. The moduli space of a polarized algebraic manifold is the quotient of its Teichmüller space by the action of its modular group. If \(M\) is a Calabi-Yau manifold, then it is proved the existence of a subgroup of finite index in its modular group acting freely so that the action of the Teichmüller space by this subgroup is a non-singular quasi-projective variety.
The author reviews the construction of the coarse moduli space of polarized Calabi-Yau manifolds as quasi-projective varieties due to E. Viehweg and gives a conceptual proof of the existence of the coarse moduli space for a large class of varieties. Also discussed are latest results connecting the discriminant locus in moduli space of polarized odd dimensional Calabi-Yau manifolds with the Bismut-Gillet-Soule-Quillen-Donaldson theory of determinant line bundles.

Reviewer: Ruben A. Hidalgo (Valparaiso)

MSC:

32Q25	Calabi-Yau theory (complex-analytic aspects)
14D20	Algebraic moduli problems, moduli of vector bundles
32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Keywords:

Calabi-Yau manifolds; Teichmüller spaces; moduli spaces

References:

[1]	Abramovich, D., J.-F. Burnol, J. Kramer and C. Soulé: Lectures on Arakelov Geometry. Cambridge Studies In Advanced Mathematics 33, Cambridge University Press, Cambridge, 1992. · Zbl 0812.14015
[2]	Aspinwall, P. and D. Morrison: Chiral Rings do not Suffice: N = (2, 2) theories with non zero fundamental group, Phys. Lett. B 334 (1994), 79-86.
[3]	Berline, N., E. Getzler and M. Vergne: Heat Kernels and Dirac Operators, Springer Verlag, 1991. · Zbl 1037.58015
[4]	Bershadsky, M., S. Cecotti, H. Ooguri and C. Vafa: Kodaira- Spencer Theory of Gravity and Exact Results for Quantum String Ampli- tude, Comm. Math. Phys. 165 (1994), 311-428. · Zbl 0815.53082 · doi:10.1007/BF02099774
[5]	Bismut, J.-M. and D. Freed: The Analysis of Elliptic Families I. Metrics and Connections on Determinant Line Bundles, Comm. Math. Phys. 106 (1986), 159-176. · Zbl 0657.58037 · doi:10.1007/BF01210930
[6]	Bismut, J.-M., H. Gillet and C. Soulé: Analytic Torsion and Holo- morphic Determinant Bundles I. Bott-Chern Forms and Analytic Torsion, Comm. Math. Phys. 115 (1998), 49-78.
[7]	Bismut, J.-M., H. Gillet and C. Soulé: Analytic Torsion and Holo- morphic Determinant Bundles II. Direct Images and Bott-Chern Forms, Comm. Math. Phys. 115 (1988), 79-126. · Zbl 0651.32017
[8]	Bryant, R. and P. Griffiths: Some Observations on the Infinitesimal Period Relations for Regular Threefolds with Trivial Canonical Bundle. In Arithmetic and geometry, Vol. II, 77-102. Progr. Math. 36, Birkhäuser Boston, Boston, MA, 1983. · Zbl 0543.14005
[9]	Delaroche, Cl. and A. A. Kirillov: Sur les Relations entre l’espace dual d’un groupe et la structure de ses sous-groupes fermés” (d’apr‘ es D. A. Kazhdan). Séminaire Bourbaki 10, Exp. No. 343, 507-528. Soc. Math. France, Paris, 1995.
[10]	Demailly, J.-P.: L2 vanishing theorems for positive line bundles and ad- junction theory. In Transcendental methods in algebraic geometry (Cetraro, 1994), 1-97. Lecture Notes in Math. 1646, Springer, Berlin, 1996. · Zbl 0883.14005
[11]	Grothendieck, A.: Éléments de Géométrie Algébrique. Inst. Hautes Études Sci. Publ. Math. 8, 1961. · Zbl 0203.23301
[12]	Grothendieck, A.: Séminaire de Géométrie Algébrique. Inst. Hautes Études Sci., Paris, 1960-1961.
[13]	Griffiths, Ph.: Periods of Integrals on Algebraic Manifolds I and II. Amer. J. Math. 90 (1968), 568-626 and 805-865. · Zbl 0183.25501 · doi:10.2307/2373485
[14]	Griffiths, Ph.: Periods of Integrals on Algebraic Manifolds III. Inst. Hautes Études Sci. Publ. Math. 38 (1970), 125-180. · Zbl 0212.53503