The Moduli Space of Asymptotically Cylindrical Calabi–Yau Manifolds

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 Kahler. By proving a local families L 2-index theorem, we exhibit its Kahler form as a multiple of the curvature of a certain determinant line bundle.