The space of almost calibrated $(1,1)$ forms on a compact Kähler manifold

Abstract

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 $\mathrm{CAT}(0)$ 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.

50 pages

Cited by in corpus (5)

• Stability and the deformed Hermitian-Yang-Mills equation
• Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow
• The deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds
• A new flow solving the LYZ equation in Kähler geometry
• Geodesics of positive Lagrangians from special Lagrangians with boundary