Asymptotically cylindrical Calabi–Yau and special Lagrangian geometry
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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 (
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.