AI 中文总结
本文通过粘合构造解决一类6阶几何PDE，应用于Bismut Hermitian Einstein流形和超引力模型，得到去奇点与点爆破定理，并构造出多个新流形上的正则结构及无限拓扑类型的$Y_7$解。
AI 中文摘要
我们提出了一类在Kähler流形或orbifold上出现的6阶几何非线性偏微分方程的粘合构造，这类方程源于Bismut Hermitian Einstein复$3$-流形的研究、物理中IIB型和$D=11$超引力模型的超对称${\rm AdS}_3 \times Y_7$和${\rm AdS}_2 \times Y_9$解，以及Dervan的$Z$-临界Kähler度量。作为应用，我们得到了具有crepant奇点的orbifold在允许该PDE解时的去奇点定理，以及允许此类解的光滑复$3$-流形上的点爆破定理。这些结果在光滑流形$3(S^2\times S^4)\sharp 4(S^3\times S^3)$、$5(S^2\times S^4)\sharp 6(S^3\times S^3)$和$S^1\times 9(S^2\times S^3)$上产生了正则Bismut Hermitian Einstein结构的新例子，以及无限拓扑类型的紧致光滑$7$维流形$Y_7$，为超对称${\rm AdS}_3 \times Y_7$型IIB和${\rm AdS}_2 \times T^2 \times Y^7$型$D=11$超引力模型提供了新解。
英文摘要
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 ${\rm AdS}_3 \times Y_7$ and ${\rm AdS}_2 \times Y_9$ 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 $3(S^2\times S^4)\sharp 4(S^3\times S^3)$, $5(S^2\times S^4)\sharp 6(S^3\times S^3)$ 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 ${\rm AdS}_3 \times Y_7$ type IIB and ${\rm AdS}_2 \times T^2 \times Y^7$ type $D=11$ supergravity models.
