Moment maps, nonlinear PDE, and stability in mirror symmetry

Abstract

We study the deformed Hermitian-Yang-Mills (dHYM) equation, which is mirror to the special Lagrangian equation, from the variational point of view via an infinite dimensional GIT problem mirror to Thomas' GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold $\mathcal{H}$ mirror 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. We apply these results to the infinite dimensional GIT problem for deformed Hermitian-Yang-Mills. We associate algebraic invariants to certain birational models of $X\times \Delta$, where $\Delta \subset \mathbb{C}$ is a disk. Using the existence of regular weak geodesics we prove that these invariants give rise to obstructions to the existence of solutions to the dHYM equation. Furthermore, we show that these invariants fit into a stability framework closely related to Bridgeland stability. Finally, we use a Fourier-Mukai transform on toric Kähler manifolds to describe degenerations of Lagrangian sections of SYZ torus fibrations of Landau-Ginzburg models $(Y,W)$. We speculate on the resulting algebraic invariants, and discuss the implications for relating Bridgeland stability to the existence of special Lagrangian sections of $(Y,W)$.

94 pages, 3 figures

References in corpus (1)

• Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties

Cited by in corpus (15)

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• Deformed Hermitian Yang-Mills connections, extended gauge group and scalar curvature
• A class of curvature type equations
• The space of almost calibrated $(1,1)$ forms on a compact Kähler manifold
• Collapsing of the line bundle mean curvature flow on Kähler surfaces
• Stability and the deformed Hermitian-Yang-Mills equation
• Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow
• Regularity of fully non-linear elliptic equations on Hermitian manifolds. II
• Deformed Hermitian-Yang-Mills Equation on Compact Hermitian Manifolds
• The deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds
• Chern number inequalities of deformed Hermitian-Yang-Mills metrics on four dimensional Kaehler manifolds
• Geodesics of positive Lagrangians from special Lagrangians with boundary
• Examples of dHYM connections in a variable background
• Second order estimates for fully nonlinear elliptic equations with gradient terms on Hermitian manifolds
• Second order estimates for a class of complex Hessian equations on Hermitian manifolds