Stability of Einstein-Hermitian vector bundles

Variations on stability

“…Fortunately it is sufficient to carry out the check for only so-called reflexive sheaves [14], and the only such sheaves of rank one are line bundles. More generally, it is shown in [20] that HYM implies stability even in the stronger sense (that is, including subsheaves). One however needs stability with subsheaves in order to show that stability implies the existence of a HYM [14].…”

Perspectives on geometric analysis

“…(That polystability of bundle is a consequence of existence of Hermitian Yang-Mills connection was first observed by Lüber [482]. Donaldson [196] was able to make use of the theorem of Mehta-Ramanathan [505] and ideas of above two papers to prove the theorem for projective manifold).…”

Book Review: Differential geometry of complex vector bundles

“…Kobayashi arrived at the definition of an Einstein-Hermitian structure when he was working on vanishing theorems for holomorphic tensor fields, but he realized immediately that there is a close relation between his Einstein condition and the concept of a stable vector bundle in algebraic geometry, a notion that ultimately goes back to Mumford. In fact, he and independently Lübke [6] proved that irreducible Einstein-Hermitian bundles are stable in the sense of Mumford and Takemoto [8]. The question whether the converse of this is true has been posed by Hitchin and Kobayashi.…”