The space of almost calibrated (1,1)–forms on a compact Kähler manifold
The space of almost calibrated (1,1)–forms on a
compact Kähler manifold
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几乎校准的 (1,1) 空间 – 形成在
作者：
Jianchun Chu;Tristan C. Collins;Man
紧化Kahler流形上的“几乎校准”$(1,1)$形式的$\数学{H}$在第二作者和Yau最近的工作中强调的镜面对称变形Hermitian-Yang-Mills方程的研究中起着重要作用，并且通过镜面对称与Solomon研究的正拉格朗日空间相联系。本文研究了$\mathcal{H}$的几何性质。我们证明$\mathcal{H}$是一个非正截面曲率的无限维黎曼流形。在临界相情况下，我们证明了$\mathcal{H}$具有一个定义良好的度量结构，并且它的补全是${\rm CAT}(0)$测地度量空间，因此具有一个内在定义的理想边界。最后，我们证明了在超临界相情况下$\mathcal{H}$允许$C^{1,1}$测地线，改进了第二作者和Yau的结果。利用Darvas-Lempert的结果，我们证明了这个结果是尖锐的。
The space $\mathcal{H}$ of "almost calibrated" $(1,1)$ forms on a compact Kahler manifold plays an important role in the study of the deformed Hermitian-Yang-Mills equation of mirror symmetry as emphasized by recent work of the second author and Yau, and is related by mirror symmetry to the space of positive Lagrangians studied by Solomon. This paper initiates the study of the geometry of $\mathcal{H}$. We show that $\mathcal{H}$ is an infinite dimensional Riemannian manifold with non-positive sectional curvature. In the hypercritical phase case we show that $\mathcal{H}$ has a well-defined metric structure, and that its completion is a ${\rm CAT}(0)$ geodesic metric space, and hence has an intrinsically defined ideal boundary. Finally, we show that in the hypercritical phase case $\mathcal{H}$ admits $C^{1,1}$ geodesics, improving a result of the second author and Yau. Using results of Darvas-Lempert we show that this result is sharp.