The Hermitean Yang-Mills connection or Hermite-Einstein connection is a connection on a vector bundle on a complex vector bundle over a base space with almost complex structure and almost Hermitean structure such that its curvature 2-form is of Dolbeault-type $(1,1)$ and, finally, the contraction of the almost Hermitean structure with the curvature 2-form is proportional to the identity bundle endomorphism (see e.g. Popov 09 (2.5)).
The Donaldson-Uhlenbeck-Yau theorem (Uhlenbeck-Yau 86) relates moduli of Hermite-Einstein connections over compact Kähler manifolds to (semi-)stable vector bundles.
The Kobayashi-Hitchin correspondence generalizes this to more general complex manifolds.
Karen Uhlenbeck, Shing-Tung Yau, On the existence of Hermitean Yang-Mills-connections on stable bundles over Kähler manifolds, Comm. Pure Appl. Math. 39 (1986) 257-293
Ignacio Sols Lucía, Tomás L. Gómez, The Hermite-Einstein equation and stable principal bundles (an updated survey) Geometriae Dedicata, 139 (1). 83-98. (2009) (web)
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