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The space of almost calibrated \((1,1)\)-forms on a compact Kähler manifold. (English) Zbl 1487.32097

Summary: 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 et al. [in: Geometry and physics. A festschrift in honour of Nigel Hitchin. Volume 1. Oxford: Oxford University Press. 69–90 (2018; Zbl 1421.35300)], and is related by mirror symmetry to the space of positive Lagrangians studied by J. P. Solomon [Math. Ann. 357, No. 4, 1389–1424 (2013; Zbl 1282.53067); Geom. Funct. Anal. 24, No. 2, 670–689 (2014; Zbl 1296.53157)]. This paper initiates the study of the geometry of \(\mathcal{H}\). We show that \(\mathcal{H}\) is an infinite-dimensional Riemannian manifold with nonpositive 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 [Zbl 1421.35300)]. Using results of T. Darvas and L. Lempert [Math. Res. Lett. 19, No. 5, 1127–1135 (2012; Zbl 1275.58008)] we show that this result is sharp.

MSC:

32J27 Compact Kähler manifolds: generalizations, classification
53C22 Geodesics in global differential geometry
53D05 Symplectic manifolds (general theory)
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
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References:

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