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On the Curvature Tensor of the Hodge Metric of Moduli Space of Polarized Calabi-Yau Threefolds

Journal of Geometric Analysis (Impact Factor: 0.87). 06/2005; DOI: 10.1007/BF02930760
Source: arXiv

ABSTRACT We give an expression and some estimates of the curvature tensor of the Hode metric over the moduli space of a polarized Calabi-Yau threefold. The symmetricity of the Yukawa coupling is also studied. In the last section of this article, an extra restriction of the limiting Hodge structure for the degeneration of Calabi-Yau threefolds is given. This article is the continuation of the article [Z. Lu, ibid. 11, No. 1, 103-118 (2001; Zbl 0986.32010)] of our study of the moduli space of polarized Calabi-Yau threefolds.

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    • "There should be one more layer of convexity in the Weil-Petersson geometry (Conjecture 1). By the computation in [20], the conjecture can be interpreted as the property of the underlying Calabi-Yau manifolds. We will include results in this direction in a separate paper. "
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    ABSTRACT: In this paper, we prove a Gauss–Bonnet–Chern type theorem in full generality for the Chern–Weil forms of Hodge bundles. That is, the Chern–Weil forms compute the corresponding Chern classes. This settles a long standing problem. Second, we apply the result to Calabi–Yau moduli, and proved the corresponding Gauss–Bonnet–Chern type theorem in the setting of Weil–Petersson geometry. As an application of our results in string theory, we prove that the number of flux vacua of type II string compactified on a Calabi–Yau manifold is finite, and their number is bounded by an intrinsic geometric quantity.
    Mathematische Annalen 10/2013; 357(2). DOI:10.1007/s00208-013-0907-4 · 1.20 Impact Factor
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    • "By the Schwarz lemma of Yau [37], the Hodge metric must be Poincaré bounded. In [11], the results in [20] were generalized to degenerate cases, and the generalized Hodge metrics were defined. Again, the generalized Hodge metrics are Poincaré bounded. "
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    • "Mathematically, this paper is a continuation of the previous works in [20] [22] [21] [23] [24] [15] [16], on the local and global geometry of the moduli space and the BCOV torsion of Calabi-Yau moduli. "
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    ABSTRACT: In this paper, we study the Chern classes on the moduli space of polarized Calabi-Yau manifolds. We prove that the integrations of the invariants of the curvature of the Weil-Petersson metric are finite. In some special cases, they are even rational numbers.
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