A special Lagrangian type equation for holomorphic line bundles

TL;DR

This paper investigates a special Lagrangian type equation for holomorphic line bundles over Kähler manifolds, establishing existence criteria, uniqueness, and a flow method for solutions, motivated by mirror symmetry and extending geometric analysis techniques.

Contribution

It introduces a line bundle analogue of the special Lagrangian equation, proves its variational nature, and develops a flow approach for higher dimensions with convergence results.

Findings

Solutions are unique global minimizers of a positive functional.

A necessary and sufficient existence criterion is provided for Kähler surfaces.

Convergence of the line bundle Lagrangian mean curvature flow is proven under certain curvature conditions.

Abstract

Let $L$ be a holomorphic line bundle over a compact K\"ahler manifold $X$. Motivated by mirror symmetry, we study the deformed Hermitian-Yang-Mills equation on $L$, which is the line bundle analogue of the special Lagrangian equation in the case that $X$ is Calabi-Yau. We show that this equation is the Euler-Lagrange equation for a positive functional, and that solutions are unique global minimizers. We provide a necessary and sufficient criterion for existence in the case that $X$ is a K\"ahler surface. For the higher dimensional cases, we introduce a line bundle version of the Lagrangian mean curvature flow, and prove convergence when $L$ is ample and $X$ has non-negative orthogonal bisectional curvature.