Calabi-Yau structures on cotangent bundles

Abstract

Starting with a orientable compact real-analytic Riemannian manifold (L, g) with χ(L) = 0, we show that a small neighbourhood Op(L) of the zero section in the cotangent bundle T * L carries a Calabi-Yau structure such that the zero section is an isometrically embedded special Lagrangian submanifold.

The branched deformations of the special Lagrangian submanifolds

“…Theorem 9.3. [4,7] Let (L, g L ) be a real analytic Riemannian manifold, suppose χ(L) = 0, then L admits a Calabi-Yau neighborhood (U L , J, ω, Ω). In addition, let R be the canonical involution on T ⋆ L, then R is an anti-holomorphic involution on U L .…”

“…According to the work of Bryant [4] and Doice [7], the condition (ii) holds when χ(L) = 0. For the condition (iii), in general, it is very hard to check whether a Lagrangian is unobstructed or not.…”