Simultaneous desingularizations of Calabi-Yau and special Lagrangian 3-folds with conical singularities. I: the gluing construction
The author extends a previous result on desingularizing conically singular Calabi-Yau manifolds. Let \(M_{0}\) be a Calabi-Yau 3-fold with isolated concial singularities at \(\{x_{i} : 1 \leq i \leq n \}\), and let \(N_{0}\) be a special Langrangian 3-manifold in \(M_{0}\) with conical singularities at the same points \(\{x_{i} : 1 \leq i \leq n \}\). Let \(Y_{i}\) be an asymptotically canonical Calabi-Yau 3-fold , and Let \(L_{i}\) be an asymptotically canonical special Lagrangian 3-fold in \(Y_{i}\). The author glues in rescaled copies of \(Y_{i}\) and \(L_{i}\) simultaneously to \(M_{0}\) and \(N_{0}\) at \(x_{i}\) respectively and deforms both together to achieve a smooth Calabi-Yau manifold. He expects that this method will produce new examples of compact special Langrangian 3-folds in compact Calabi-Yau 3-folds. In a sequel to his paper with the same main title the author applies the results of this paper to obtain examples. This paper fits into a comprehensive program developed by D. Joyce.
scientific article; zbMATH DE number 5577697
scientific article