IIB系统的余齐一解
Cohomogeneity one solutions of the IIB system
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- Sapienza Università di Roma（罗马大学）
- Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)（数学科学研究所）
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中文总结 AI 辅助
本文构造了IIB系统的余齐一解，在conifold光滑化上得到渐近锥形的完整非紧非Kähler度量族，并证明crepant resolution上无解，同时产生无穷多个全和乐Bismut--Hermitian--Einstein度量。
中文摘要 AI 辅助
IIB系统是关于6维流形上SU(3)结构与正函数的偏微分方程组。其解描述了具有全纯平凡典范线丛的复三维流形上的共形平衡的pluriclosed Hermitian度量。特别地，IIB系统的解是广义Ricci流和Bismut--Hermitian--Einstein度量的稳态孤子。本文研究了IIB系统的余齐一解。我们在conifold的光滑化上构造了具有无穷远受控几何的完整非紧非Kähler解的1参数族（模去缩放对称性）。该族的典型成员是渐近锥形的，其无穷远切锥为conifold上的Calabi--Yau锥度量。作为该族的极限，我们恢复了物理文献中称为Chamseddine--Volkov/Maldacena--Nuñez解的显式解，该解具有奇异的渐近几何。我们证明了在conifold的crepant resolution（或其商）上不存在完整的余齐一解，但我们也建立了在conifold的外部区域上定义的类似1参数族的前向完整解的存在性，这些解沿内边界具有指定的不完全行为，并在无穷远具有类似的渐近行为。作为这些存在性结果的副产品，我们在复维度3中构造了无穷多个具有Bismut连接全和乐的完整非紧非Kähler Bismut--Hermitian--Einstein度量，这与紧致例子中被迫的和乐约化形成对比。我们还通过重新审视物理文献中Callan--Harvey--Strominger的构造，在复维度2中找到了无穷多个这样的具有至少两个端的完整非紧全和乐非Kähler例子。
英文摘要
The IIB system is a system of PDEs for an SU(3)--structure and a positive function on a 6-manifold. Its solutions describe conformally balanced pluriclosed Hemitian metrics on complex 3-folds with holomorphically trivial canonical line bundle. In particular, solutions of the IIB system are steady solitons for the generalized Ricci-flow and Bismut--Hermitian--Einstein metrics. In this paper we study cohomogeneity one solutions of the IIB system. We construct 1-parameter families (up to scaling symmetries) of complete non-compact non-Kähler solutions on the smoothing of the conifold with controlled geometry at infinity. The generic member of the family is asymptotically conical with tangent cone at infinity the Calabi--Yau cone metric on the conifold. As a limit of the family we recover an explicit solutions known in the physics literature as the Chamseddine--Volkov/Maldacena--Nuñez solution, which has an exotic asymptotic geometry. We show that there are no complete cohomogeneity one solutions on crepant resolutions of the conifold (or its quotient), but we also establish the existence of an analogous 1-parameter family of forward complete solutions defined on exterior domains of the conifold with prescribed incomplete behaviour along the interior boundary and similar asymptotic behaviour at infinity. As a byproduct of these existence results, we produce infinitely many complete non-compact non-Kähler Bismut--Hermitian--Einstein metrics in complex dimension 3 with full holonomy of the Bismut connection, in contrast to the holonomy reduction forced upon compact examples. We also find infinitely many such complete non-compact full-holonomy non-Kähler examples (with at least two ends) in complex dimension 2 by revisiting a construction due to Callan--Harvey--Strominger in the physics literature.
