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

Jianchun Chu, Tristan C Collins and Man-Chun Lee

Geometry & Topology 25 (2021) 2573–2619

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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mirror symmetry, deformed Hermitian Yang-Mills, special Lagrangian
Mathematical Subject Classification 2010
Primary: 32Q15
Secondary: 53C22, 53D05, 53D37
Received: 24 February 2020
Accepted: 5 July 2020
Published: 3 September 2021
Proposed: Gang Tian
Seconded: Tobias H Colding, Bruce Kleiner
Jianchun Chu
Department of Mathematics
Northwestern University
Evanston, IL
United States
Tristan C Collins
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Man-Chun Lee
Department of Mathematics
Northwestern University
Evanston, IL
United States