Calabi-Yau锥的crepant消解上的dHYM方程
The dHYM equation on crepant resolutions of Calabi-Yau cones
- Universidade Estadual de Campinas（坎皮纳斯州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过Calabi ansatz将非紧Calabi-Yau流形上的dHYM方程化为ODE，确定了全局解存在的相位条件，并构造了首个非toric流形上非dHYM的Hermitian-Einstein联络例子。
AI中文摘要：
本文研究了非紧Calabi-Yau流形$Z = {\ m{Tot}}({\f{K}}_{X})$上的形变Hermitian-Yang-Mills（dHYM）方程，其中$X$是有理齐次簇。由于$X$是Fano簇，当锥由反典范极化${\f{L}} = {\f{K}}_{X}^{-1}$构造时，$Z$是仿射锥${\ m{Aff}}(X)$顶点处奇点的消解。利用通过Calabi ansatz在$Z$上得到的Ricci-flat Kähler度量的单同余对称性，我们将dHYM方程背后的完全非线性偏微分方程约化为一个标量的、渐近自治的常微分方程（ODE）。由此，我们确定了保证解全局存在的拓扑相位的精确条件。作为应用，我们证明：若总相位位于由Lie理论数据确定的显式开区间内，则$Z$上的每个全纯线丛都容许一个光滑的、全局定义的Hermitian联络，该联络求解dHYM方程。此外，我们证明了一个刚性结果，分类了dHYM解退化为经典Hermitian-Yang-Mills（HYM）联络的精确几何条件。所建立的结果推广了先前的构造，并提供了一类重要的新例子。此外，所提出的方法允许通过ODE方法研究dHYM解的行为。利用这种方法，我们构造了非toric Calabi-Yau流形上线丛上第一个显式的非平凡Hermitian-Einstein联络例子，该联络不是dHYM。
英文摘要：
In this paper, we study the deformed Hermitian-Yang-Mills (dHYM) equation on the non-compact Calabi-Yau manifold $Z = {\rm{Tot}}({\bf{K}}_{X})$, where $X$ is a rational homogeneous variety. Since $X$ is a Fano variety, $Z$ is a resolution of the singularity at the vertex of the affine cone ${\rm{Aff}}(X)$, provided the cone is built using the anticanonical polarization ${\bf{L}} = {\bf{K}}_{X}^{-1}$. Using the cohomogeneity-one symmetry of the Ricci-flat Kähler metric obtained via the Calabi ansatz on $Z$, we reduce the fully nonlinear PDE underlying the dHYM equation to a scalar, asymptotically autonomous ordinary differential equation (ODE). From this, we determine the exact condition on the topological phase that guarantees global existence of solutions. As an application, we show that every holomorphic line bundle over $Z$ admits a smooth, globally defined Hermitian connection solving the dHYM equation, provided the total phase lies in an explicit open interval determined by the Lie-theoretic data. Also, we prove a rigidity result classifying the exact geometric conditions under which the dHYM solution collapses into a classical Hermitian-Yang-Mills (HYM) connection. The results established generalize previous constructions and provide a substantial new class of examples. Furthermore, the approach presented allows one to study the behavior of the dHYM solutions through ODE methods. Using this approach, we construct the first explicit non-trivial example of a Hermitian-Einstein connection on a line bundle over a non-toric Calabi-Yau manifold which is not dHYM.
