论文标题
紧凑型kähler歧管上几乎校准$（1,1）$的空间
The space of almost calibrated $(1,1)$ forms on a compact Kähler manifold
论文作者
论文摘要
紧凑的Kähler歧管上的“几乎校准” $（1,1）$的空间$ \ Mathcal {h} $在对镜像对称性变形的荒野 - 扬·米尔斯方程的研究中起着重要作用，这是由第二作者的最新作品和Yau的最新作品强调的，并且由镜子对称性与镜面对称性相关，并由Lagrangians与Lagrang of loagrangians相关。本文启动了$ \ Mathcal {H} $的几何形状的研究。我们表明，$ \ Mathcal {H} $是具有非阳性截面曲率的无限尺寸riemannian歧管。在超临界相的情况下，我们表明$ \ MATHCAL {H} $具有定义明确的度量结构，并且其完成是$ {\ rm cat}（0）$ Geodesic Metric Space，因此具有本质上定义的理想边界。最后，我们表明，在超临界相案例中$ \ MATHCAL {H} $允许$ C^{1,1} $ Geodesics，改善了第二作者和Yau的结果。使用darvas-lempert的结果，我们表明该结果很清晰。
The space $\mathcal{H}$ of "almost calibrated" $(1,1)$ forms on a compact Kähler 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.
