Stability and the deformed Hermitian-Yang-Mills equation

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.

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