In this paper we review the moduli theory of polarized CY manifolds.
We briefly sketched Kodaira-Spencer-Kuranishi local deformation
theory developed by the author and G. Tian. We also construct the
Teichm\"{u}ller space of polarized CY manifolds following the ideas of I.
R. Shafarevich and I. I. Piatetski-Shapiro. We review the fundamental
result of E. Viehweg about the existence of the course moduli space
of polarized CY manifolds as a quasi-projective variety. Recently S.
Donaldson computed the moment map for the action of the group of
symplectic diffeomorphisms on the space of K\"{a}hler metrics with fixed
class of cohomology. Combining this results with the solution of
Calabi conjecture by Yau one obtain a very conceptual proof of the
existence of the coarse moduli space for a large class of varieties.
We follow the approach developed in \cite{LTYZI} to study the global
properties of the moduli of polarized CY manifolds. We discuss the
latest results connecting the discriminant locus in the moduli space
of polarized odd dimensional CY manifolds with the
Bismut-Gillet-Soule-Quillen-Donaldson Theory of Determinant line
bundles.
Publié le : 2003-09-14
Classification:
Calabi-Yau manifold,
Hilbert schemes,
Teichmüller space,
moduli space of polarized algebraic variety,
Weil-Petersson metric,
Hodge metric,
14C30,
14E30,
32J27,
32Q57
@article{1063050171,
author = {Todorov, Andrey},
title = {Local and Global Theory of the Moduli of Polarized Calabi-Yau Manifolds},
journal = {Rev. Mat. Iberoamericana},
volume = {19},
number = {2},
year = {2003},
pages = { 687-730},
language = {en},
url = {http://dml.mathdoc.fr/item/1063050171}
}
Todorov, Andrey. Local and Global Theory of the Moduli of Polarized Calabi-Yau Manifolds. Rev. Mat. Iberoamericana, Tome 19 (2003) no. 2, pp. 687-730. http://gdmltest.u-ga.fr/item/1063050171/