Ricci-flat Deformations of Asymptotically Cylindrical Calabi–yau Manifolds

We study a class of asymptotically cylindrical Ricci-flat Kähler metrics arising on quasiprojective manifolds. Using the Calabi–Yau geometry and analysis and the Kodaira–Kuranishi–Spencer theory and building up on results of N.Koiso, we show that under rather general hypotheses any local asymptotically cylindrical Ricci-flat deformations of such metrics are again Kähler, possibly with respect to a perturbed complex structure. We also find the dimension of the moduli space for these local deformations. In the class of asymptotically cylindrical Ricci-flat metrics on 2n-manifolds, the holonomy reduction to SU(n) is an open condition. Let M be a compact smooth manifold with integrable complex structure J and g a Ricci-flat Kähler metric with respect to J . A theorem due to N.Koiso [10] asserts that if the deformations of the complex structure of M are unobstructed then the Ricci-flat Kähler metrics corresponding to the nearby complex structures and Kähler classes fill in an open neighbourhood in the moduli space of Ricci-flat metrics onM . The proof of this result relies on Hodge theory and Kodaira–Spencer– Kuranishi theory and Koiso also found the dimension of the moduli space. The purpose of this paper is to extend the above result to a class of complete Ricci-flat Kähler manifolds with asymptotically cylindrical ends (see §1 for precise definitions). A suitable version of Hodge theory was developed as part of elliptic theory for asymptotically cylindrical manifolds in [13, 14, 15, 16]. A complex manifold underlying an asymptotically cylindrical Ricci-flat Kähler manifold admits a compactification by adding a ‘divisor at infinity’. There is an extension of Kodaira– Spencer–Kuranishi theory for this class of non-compact complex manifolds using the cohomology of logarithmic sheaves [8]. On the other hand, manifolds with asymptotically cylindrical ends appear as an essential step in the gluing constructions of compact manifolds endowed with special Riemannian structures. In particular, the Ricci-flat Kähler asymptotically cylindrical manifolds were prominent in [11] in the construction of compact 7-dimensional Ricci-flat manifolds with special holonomy G2. We introduce the class of Ricci-flat Kähler asymptotically cylindrical manifolds in §1, where we also state our first main Theorem 1.3 and give interpretation in terms of special holonomy. We review basic facts about the Ricci-flat deformations in §2. §§3–5 contain the proof of Theorem 1.3 and our second main result Theorem 5.1 on the dimension of the moduli space for the Ricci-flat asymptotically cylindrical deformations of a Ricci-flat Kähler asymptotically cylindrical manifold. Some examples (motivated by [11]) are considered in §6. In S. Akbulut, T. Önder, and R.J. Stern, editors, Proceedings of Gökova Geometry-Topology Conference, pages 137–153. International Press, 2006.