The moduli space of asymptotically cylindrical Calabi-Yau manifolds

TL;DR

This paper proves the unobstructed deformation theory of asymptotically cylindrical Calabi-Yau manifolds, describes their metric expansions, and studies the geometry of their moduli space, including its Kähler structure.

Contribution

It establishes the unobstructedness of deformations, analyzes metric expansions at infinity, and characterizes the moduli space's Kähler geometry with a new index theorem.

Findings

Deformation theory is unobstructed for these manifolds.

Calabi-Yau metrics admit polyhomogeneous expansions at infinity.

The moduli space has a Kähler Weil-Petersson metric related to a determinant line bundle.

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 K\"ahler. By proving a local families L^2 index theorem, we exhibit its K\"ahler form as a multiple of the curvature of a certain determinant line bundle.