On the Geometry of Moduli Space of Polarized Calabi-Yau manifolds (original) (raw)

Generalized Hodge metrics and BCOV torsion on Calabi-Yau moduli

Journal für die reine und angewandte Mathematik (Crelles Journal), 2005

We establish an unexpected relation among the Weil-Petersson metric, the generalized Hodge metrics and the BCOV torsion. Using this relation, we prove that certain kind of moduli spaces of polarized Calabi-Yau manifolds do not admit complete subvarieties. That is, there is no complete family for certain class of polarized Calabi-Yau manifolds. We also give an estimate of the complex Hessian of the BCOV torsion using the relation. After establishing a degenerate version of the Schwarz Lemma of Yau, we prove that the complex Hessian of the BCOV torsion is bounded by the Poincaré metric. theory. For the general reference of Mirror Symmetry and related topics, see the book of Cox and Katz [9] and the recent survey paper of Todorov .

Analytic torsion for Calabi-Yau threefolds

Journal of Differential Geometry, 2008

After Bershadsky-Cecotti-Ooguri-Vafa, we introduce an invariant of Calabi-Yau threefolds, which we call the BCOV invariant and which we obtain using analytic torsion. We give an explicit formula for the BCOV invariant as a function on the compactified moduli space, when it is isomorphic to a projective line. As a corollary, we prove the formula for the BCOV invariant of quintic mirror threefolds conjectured by Bershadsky-Cecotti-Ooguri-Vafa. Contents 1. Introduction 2. Calabi-Yau varieties with at most one ordinary double point 3. Quillen metrics 4. The BCOV invariant of Calabi-Yau manifolds 5. The singularity of the Quillen metric on the BCOV bundle 6. The cotangent sheaf of the Kuranishi space 7. Behaviors of the Weil-Petersson metric and the Hodge metric 8. The singularity of the BCOV invariant I -the case of ODP 9. The singularity of the BCOV invariant II -general degenerations 10. The curvature current of the BCOV invariant 11. The BCOV invariant of Calabi-Yau threefolds with h 1,2 = 1 12. The BCOV invariant of quintic mirror threefolds 13. The BCOV invariant of FHSV threefolds ANALYTIC TORSION FOR CALABI-YAU THREEFOLDS 5 an arbitrary Calabi-Yau manifold of arbitrary dimension, which we obtain using determinants of cohomologies [28], Quillen metrics [11], [44], and a Bott-Chern class like A(·). Then the BCOV Hermitian line of a Calabi-Yau manifold depends only on the complex structure of the manifold. The Hodge diamond of Calabi-Yau threefolds are so simple that the BCOV Hermitian line reduces to the scalar invariant τ BCOV in the case of threefolds. Hence Eq. (1.1) on P 1 \ D is deduced from the curvature formula for the BCOV Hermitian line bundles. (See Sect. 4).

Calabi-Yau manifolds of some special forms

Letters in Mathematical Physics, 1988

Let M = M L x 9 9-x M m be a product of K~ihler C-spaces with second Betti numbers b2(g t) = 1 (1 ~t~ m). The work establishes that the complete intersections X of M produce a finite number of N-dimensxonal Calabi-Yau manifolds. Moreover, if b4(M,) = 1, then the complete intersections with vanishing first Pontrjagin classes are finitely many, as well. On the other hand, we consider hypersurfaces of weighted projective spaces and give an explicit formula for their Euler characteristics. As in the previous case, it turns out that only a finite number of these are Calabi-Yau mamfolds.

Calabi-Yau Manifolds -Motivations and Constructions, Tristan HÜBSCH, 1987

The possible ways of compactifϊcation of thezyxwvutsrqponmlkjihgfedcbaZYX E 8 ®E 8 Superstring theory to four dimensions are reviewed. The phenomenological need for N = 1 supersymmetry is argued (on quite general grounds) to favour the choice of a Calabi-Yau manifold for the compact internal manifold. The massless spectrum after compactification is derived in full detail revealing, beside the usual particles, others that may have great phenomenological impact. The technical aspects of the construction of such manifolds are examined and the methods of calculation of the relevant topological properties are given. A big family of such constructions, giving rise to many new Calabi-Yau manifolds, is presented and its relevance to the search of a phenomenologically acceptable solution is discussed.

Calabi-Yau manifolds — Motivations and constructions

Communications in Mathematical Physics, 1987

The possible ways of compactifϊcation of the E 8 ®E 8 Superstring theory to four dimensions are reviewed. The phenomenological need for N = 1 supersymmetry is argued (on quite general grounds) to favour the choice of a Calabi-Yau manifold for the compact internal manifold. The massless spectrum after compactification is derived in full detail revealing, beside the usual particles, others that may have great phenomenological impact. The technical aspects of the construction of such manifolds are examined and the methods of calculation of the relevant topological properties are given. A big family of such constructions, giving rise to many new Calabi-Yau manifolds, is presented and its relevance to the search of a phenomenologically acceptable solution is discussed.

Geometry and Arithmetic of Certain Double Octic Calabi–Yau Manifolds

Canadian Mathematical Bulletin, 2005

We study Calabi–Yau manifolds constructed as double coverings of ℙ3 branched along an octic surface. We give a list of 87 examples corresponding to arrangements of eight planes defined over ℚ. The Hodge numbers are computed for all examples. There are 10 rigid Calabi–Yau manifolds and 14 families with h1,2 = 1. The modularity conjecture is verified for all the rigid examples.