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Moment Maps, Nonlinear PDE and Stability in Mirror Symmetry, I: Geodesics

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Abstract

In this paper, the first in a series, we study the deformed Hermitian–Yang–Mills (dHYM) equation from the variational point of view as an infinite dimensional GIT problem. The dHYM equation is mirror to the special Lagrangian equation, and our infinite dimensional GIT problem is mirror to Thomas’ GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold \({\mathcal {H}}\) closely related to Solomon’s space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with \(C^{1,\alpha }\) regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on collapsing manifolds with boundary. As an application of our techniques we give a simplified proof of Chen’s theorem on the existence of \(C^{1,\alpha }\) geodesics in the space of Kähler metrics. In two follow up papers, these results will be used to examine algebraic obstructions to the existence of solutions to dHYM [26] and special Lagrangians in Landau–Ginzburg models [27].

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Acknowledgements

T.C.C. would like to thank A. Jacob, J. Ross, and B. Berndtsson for many helpful conversations, as well as J. Solomon, A. Hanlon, and P. Seidel for helpful conversations concerning special Lagrangians and the Fukaya category. T.C.C. is grateful to the European Research Council, and the Knut and Alice Wallenberg Foundation who supported a visiting semester at Chalmers University, where this work was initiated. T.C.C. would also like to thank R. Berman, D. Persson, D. Witt Nyström and the rest of the complex geometry group at Chalmers for providing a stimulating research environment. The authors are grateful to the referees for helpful comments and corrections. T.C.C. is supported in part by NSF Grant DMS-1506652, DMS-1810924 and an Alfred P. Sloan Fellowship.

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  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Tristan C. Collins

  2. Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA, 02138, USA

    Shing-Tung Yau

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Collins, T.C., Yau, ST. Moment Maps, Nonlinear PDE and Stability in Mirror Symmetry, I: Geodesics. Ann. PDE 7, 11 (2021). https://doi.org/10.1007/s40818-021-00100-7

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