An obstructed bundle on a Calabi-Yau 3-fold (Q5929507)

According to Kontsevich's homological mirror symmetry proposal, mirror symmetry is an equivalence between an \(A_\infty\)-category constructed from the derived category of coherent sheaves (naturally associated to Yang-Mills on Kähler manifolds) and an \(A_\infty\) category proposed by Fukaya, associated to isotopy classes of Lagrangian submanifolds with flat \(U(1)\)-bundles. NEWLINENEWLINENEWLINEOne could then naively think that degree 0 stable bundles on a Calabi-Yau threefold should correspond to special Lagrangian cycles endowed with flat line bundles on the mirror Calabi-Yau. If that were so, then moduli spaces of degree 0 stable bundles on a Calabi-Yau should be smooth since \textit{D. C. McLean} [Commun. Anal. Geom. 6, No. 4, 705-747 (1998; Zbl 0929.53027)] has proven that deformations of smoothly embedded special Lagrangians are unobstructed. NEWLINENEWLINENEWLINEThis note proves that it is not so by providing an example of a completely obstructed rank two stable vector bundle \(A\) with vanishing first Chern class on a particular Calabi-Yau threefold \(X\). In the example \(X\) is a smooth \((3,3)\)-divisor on \(\mathbb P^2\times\mathbb P^2\) and the component of the moduli space containing \(A\) is proven to be isomorphic with a double point in the affine plane.