Asymptotically cylindrical Calabi–Yau and special Lagrangian geometry
We study asymptotically cylindrical Calabi–Yau manifolds and their asymptotically cylindrical special Lagrangian submanifolds. As a prototype problem, we also consider an extension of Hodge theory to general asymptotically cylindrical manifolds. For our study of asymptotically cylindrical Calabi–Yau manifolds, we restrict to complex dimension three. We regard a Calabi–Yau structure as a pair of closed forms ($\Omega$, $\omega$); the assumption that the structure is asymptotically cylindrical gives an asymptotic condition on ($\Omega$, $\omega$). Regarding the Riemannian products of Calabi-Yau threefolds with S$^{1}$ as G$_{2}$ manifolds, we show that the asymptotically cylindrical deformations of a Calabi–Yau structure (with possibly varying asymptotic limit) are unobstructed. Locally, the spaces of deformations are given by appropriate spaces of harmonic forms. We then show that we can glue asymptotically cylindrical Calabi–Yau manifolds, and that if we do so the “gluing map” of moduli spaces is essentially a local diffeomorphism. In particular, it is an open mapping. In the case of asymptotically cylindrical special Lagrangian submanifolds, we no longer explicitly restrict to dimension three; we assume only that we have a gluing theorem for Calabi–Yau manifolds of the kind obtained in dimension three. McLean and others have constructed deformation spaces of special Lagrangian submanifolds; we show that gluing of asymptotically cylindrical special Lagrangian submanifolds is again unobstructed. As in the Calabi–Yau case, we can define a “gluing map” and this map is a local diffeomorphism of moduli spaces. In both cases, the local diffeomorphism property gives a “local Mayer–Vietoris principle” for deformations. In the special Lagrangian case, the linearisation of the “ungluing” map so defined is just the map of harmonic forms induced by Hodge theory from the natural map of de Rham cohomology; in the Calabi–Yau case it is only slightly more involved.
Asymptotically cylindrical Calabi–Yau manifolds, Asymptotically cylindrical special Lagrangian submanifolds
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