Stability and the deformed Hermitian-Yang-Mills equation

Abstract

We survey some recent progress on the deformed Hermitian-Yang-Mills (dHYM) equation. We discuss the role of geometric invariant theory (GIT) in approaching the solvability of the dHYM equation, following work of the first author and S.-T. Yau. We compare the GIT picture with the conjectural picture for dHYM involving Bridgeland stability. In particular, following Arcara-Miles, we show that on the blow-up of ${\mathbb{P}}^2$ any line bundle admitting a solution of the deformed Hermitian-Yang-Mills equation is Bridgeland stable, but not conversely. Finally, we survey some recent progress on heat flows associated to the dHYM equation.

Survey article,

References in corpus (2)

• The space of almost calibrated $(1,1)$ forms on a compact Kähler manifold
• Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow

Cited by in corpus (5)

• 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
• Openness of uniform K-stability in the Kähler cone
• Examples of dHYM connections in a variable background