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2021 The space of almost calibrated (1,1)–forms on a compact Kähler manifold
Jianchun Chu, Tristan C Collins, Man-Chun Lee
Geom. Topol. 25(5): 2573-2619 (2021). DOI: 10.2140/gt.2021.25.2573

Abstract

The space 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 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 . We show that is an infinite-dimensional Riemannian manifold with nonpositive sectional curvature. In the hypercritical phase case we show that has a well-defined metric structure, and that its completion is a CAT(0) geodesic metric space, and hence has an intrinsically defined ideal boundary. Finally, we show that in the hypercritical phase case  admits C1,1 geodesics, improving a result of Collins and Yau (2018). Using results of Darvas and Lempert (2012) we show that this result is sharp.

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Jianchun Chu. Tristan C Collins. Man-Chun Lee. "The space of almost calibrated (1,1)–forms on a compact Kähler manifold." Geom. Topol. 25 (5) 2573 - 2619, 2021. https://doi.org/10.2140/gt.2021.25.2573

Information

Received: 24 February 2020; Accepted: 5 July 2020; Published: 2021
First available in Project Euclid: 12 October 2021

Digital Object Identifier: 10.2140/gt.2021.25.2573

Subjects:
Primary: 32Q15
Secondary: 53C22, 53D05, 53D37

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.25 • No. 5 • 2021
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