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Gauss–Bonnet–Chern theorem on moduli space

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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.

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Notes

  1. The result is different from that of Griffiths–Schmid, because the negativity of the curvature of a submanifold, instead of the whole classifying space, was proved.

  2. The inequality is a refinement of  [28, (4.12)].

  3. The Hodge metrics are generally referred to the natural Hermitian metrics on Hodge bundles. After [20, 22], the metrics in Definition 5.1 are also called the Hodge metrics or the generalized Hodge metrics, because they have the similar curvature properties as the original ones on Hodge bundles.

  4. It may be more accurate to say that, up to a finite cover, \(\bar{M}\) can be chosen to be a manifold.

  5. During the process, it is possible that some divisors are added. However, along these divisors, the monodromy operators are the identity operator.

  6. Since in this paper we use a large set of notations, for simplicity, we will try to keep them the same as in [4]. Thus we have to sacrifice the uniqueness of the notations. For example, \(\alpha \) is used in \(U=U_\alpha \) and also as the subscripts of the set \(I\), etc. It should be clear from the context.

  7. That is, if \(h_{PH^k}=(h_{PH^k})_{i\bar{j}} dt_i\otimes d\bar{t}_j\), then \(\omega _{PH^k}=\frac{\sqrt{-1}}{2\pi }(h_{PH^k})_{i\bar{j}} dt_i\wedge d\bar{t}_j\).

  8. This is essentially due to [15]. See also [13, page 34].

  9. The metric is equivalent to the partial Hodge metric in [23, Section 4]. But we don’t need this fact here.

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Acknowledgments

During the last four years, we have discussed the topic with many mathematicians and string theorists. We thank them all. The first author particularly thanks J. Kollár and W. Schmid for stimulating discussions during the preparation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, UC Irvine, Irvine, CA, 92697, USA

    Zhiqin Lu

  2. Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY, 11794, USA

    Michael R. Douglas

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  1. Zhiqin Lu
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  2. Michael R. Douglas
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Correspondence to Zhiqin Lu.

Additional information

The first author is supported by the NSF grant DMS-12-06748, and the second author is supported by DOE grant DE-FG02-96ER40959.

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Lu, Z., Douglas, M.R. Gauss–Bonnet–Chern theorem on moduli space. Math. Ann. 357, 469–511 (2013). https://doi.org/10.1007/s00208-013-0907-4

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