A gluing construction of $Z$-extremal Kähler metrics and new examples of Bismut Hermitian-Einstein $3$-folds

We present a gluing construction for a class of 6th order geometric non-linear PDE’s on a Kähler manifold or orbifold, arising in the study of Bismut Hermitian Einstein complex $3$-folds, the supersymmetric ${\mathrm{A}\mathrm{d}\mathrm{S}}_3\times Y_7$ and ${\mathrm{A}\mathrm{d}\mathrm{S}}_2\times Y_9$ solutions of type IIB and $D=11$ supergravity models in physics, and in Dervan’s Z-critical Kähler metrics. As applications, we obtain a desingularization theorem for orbifolds with crepant singularities admitting a solution of the PDE, as well as a blowing-up of points theorem for smooth complex $3$-folds admitting such solutions. These results yield new examples of regular Bismut Hermitian Einstein structures on the smooth manifolds $3(S^2\times S^4)\mathrm{♯}4(S^3\times S^3)$, $5(S^2\times S^4)\mathrm{♯}6(S^3\times S^3)$ and $S^1\times 9(S^2\times S^3)$, as well as infinite topological types of smooth compact $7$-dimensional manifolds $Y_7$, providing new solutions to the supersymmetric ${\mathrm{A}\mathrm{d}\mathrm{S}}_3\times Y_7$ type IIB and ${\mathrm{A}\mathrm{d}\mathrm{S}}_2\times T^2\times Y^7$ type $D=11$ supergravity models.