Surveys in Differential Geometry

Volume 24 (2019)

Stability and the deformed Hermitian–Yang–Mills equation

Pages: 1 – 38

DOI:  https://dx.doi.org/10.4310/SDG.2019.v24.n1.a1

Authors

Tristan C. Collins (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Yun Shi (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Mass., U.S.A.)

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 [3], 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.

Tristan C. Collins was supported in part by NSF grant DMS-1810924, NSF CAREER grant DMS-1944952, and by an Alfred P. Sloan Fellowship.

Published 29 December 2021