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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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