Moment Maps, Nonlinear PDE and Stability in Mirror Symmetry, I: Geodesics
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Author(s)
Collins, Tristan C
Yau, Shing-Tung
Date Issued
April 8, 2021
Publisher
Springer International Publishing
Citation
Annals of PDE. 2021 Apr 08;7(1):11
Version
Author's final manuscript
Abstract
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
$${\mathcal {H}}$$
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
$$C^{1,\alpha }$$
C
1
,
α
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
$$C^{1,\alpha }$$
C
1
,
α
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].
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s40818-021-00100-7
