Moment Maps, Nonlinear PDE and Stability in Mirror Symmetry, I: Geodesics
Moment Maps, Nonlinear PDE and Stability in Mirror Symmetry, I: Geodesics
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DOI：
10.1007/s40818-021-00100-7
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发表时间：
2018-11
期刊：
影响因子：
2.8
通讯作者：
Tristan C. Collins;S. Yau
中科院分区：
文献类型：
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作者：
Tristan C. Collins;S. Yau
本文首先从变分的角度研究了变形的Hermitian-Yang-Mills(DHYM)方程，它是一个无限维GIT问题。DHYM方程是特殊拉格朗日方程的镜像，我们的无限维GIT问题是Thomas的特殊拉格朗日方程的镜像。这就产生了与所罗门正拉格朗日空间密切相关的无限维流形。在超临界相情形下，我们证明了光滑近似测地线和具有正则性的弱测地线的存在性。这是通过证明关于带边界的折叠流形上的拉格朗日相算子的尺度估计的锐性来实现的。作为技巧的一个应用，我们给出了Kähler度量空间中关于测地线存在定理的一个简化证明。在两篇后续论文中，这些结果将被用来检验在Landau-Ginzburg模型中dHYM[26]和特殊拉格朗日解[27]的解的存在的代数障碍。
In this paper, the first in a series, we study the deformed Hermitian–Yang–Mills (dHYM) equation from the variational point of view as an infinite dimensional GIT problem. The dHYM equation is mirror to the special Lagrangian equation, and our infinite dimensional GIT problem is mirror to Thomas’ GIT picture for special Lagrangians. This gives rise to infinite dimensional manifoldclosely related to Solomon’s space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics withregularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on collapsing manifolds with boundary. As an application of our techniques we give a simplified proof of Chen’s theorem on the existence ofgeodesics in the space of Kähler metrics. In two follow up papers, these results will be used to examine algebraic obstructions to the existence of solutions to dHYM [26] and special Lagrangians in Landau–Ginzburg models [27].