Revista Matemática Iberoamericana

Local and Global Theory of the Moduli of Polarized Calabi-Yau Manifolds

Andrey Todorov

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Abstract

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.

Article information

Source
Rev. Mat. Iberoamericana, Volume 19, Number 2 (2003), 687-730.

Dates
First available in Project Euclid: 8 September 2003

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1063050171

Mathematical Reviews number (MathSciNet)
MR2023203

Zentralblatt MATH identifier
1058.32019

Subjects
Primary: 14C30: Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture 14E30: Minimal model program (Mori theory, extremal rays) 32J27: Compact Kähler manifolds: generalizations, classification 32Q57: Classification theorems

Keywords
Calabi-Yau manifold Hilbert schemes Teichmüller space moduli space of polarized algebraic variety Weil-Petersson metric Hodge metric

Citation

Todorov, Andrey. Local and Global Theory of the Moduli of Polarized Calabi-Yau Manifolds. Rev. Mat. Iberoamericana 19 (2003), no. 2, 687--730. https://projecteuclid.org/euclid.rmi/1063050171


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