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The Moduli Space of Asymptotically Cylindrical Calabi–Yau Manifolds

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Abstract

We prove that the deformation theory of compactifiable asymptotically cylindrical Calabi–Yau manifolds is unobstructed. This relies on a detailed study of the Dolbeault–Hodge theory and its description in terms of the cohomology of the compactification. We also show that these Calabi–Yau metrics admit a polyhomogeneous expansion at infinity, a result that we extend to asymptotically conical Calabi–Yau metrics as well. We then study the moduli space of Calabi–Yau deformations that fix the complex structure at infinity. There is a Weil–Petersson metric on this space, which we show is Kähler. By proving a local families L 2-index theorem, we exhibit its Kähler form as a multiple of the curvature of a certain determinant line bundle.

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Authors and Affiliations

  1. Département de Mathématiques, Université du Québec à Montréal, Montréal, Canada

    Ronan J. Conlon & Frédéric Rochon

  2. Department of Mathematics, Stanford University, Stanford, USA

    Rafe Mazzeo

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  1. Ronan J. Conlon
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  2. Rafe Mazzeo
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  3. Frédéric Rochon
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Correspondence to Frédéric Rochon.

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Communicated by S. Zelditch

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Conlon, R.J., Mazzeo, R. & Rochon, F. The Moduli Space of Asymptotically Cylindrical Calabi–Yau Manifolds. Commun. Math. Phys. 338, 953–1009 (2015). https://doi.org/10.1007/s00220-015-2383-z

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  • DOI: https://doi.org/10.1007/s00220-015-2383-z

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