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

Abstract

The space $\mathcal{\mathscr{H}}$ of “almost calibrated” $\left(1,1\right)$–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 Collins and Yau (2018), and is related by mirror symmetry to the space of positive Lagrangians studied by Solomon (2013, 2014). This paper initiates the study of the geometry of $\mathcal{\mathscr{H}}$. We show that $\mathcal{\mathscr{H}}$ is an infinite-dimensional Riemannian manifold with nonpositive sectional curvature. In the hypercritical phase case we show that $\mathcal{\mathscr{H}}$ has a well-defined metric structure, and that its completion is a $\text{CAT}\left(0\right)$ geodesic metric space, and hence has an intrinsically defined ideal boundary. Finally, we show that in the hypercritical phase case $\mathcal{\mathscr{H}}$ admits $C^{1,1}$ geodesics, improving a result of Collins and Yau (2018). Using results of Darvas and Lempert (2012) we show that this result is sharp.