# Why only the first two Chern classes in the BMY and KL inequalities?

The Bogomolov-Miyaoka-Yau inequality for compact complex manifolds with ample canonical bundle and the Kobayashi-Lubke inequality for holomorphic stable vector bundles involve the first two Chern classes.

Naively, I would expect that a more involved stability condition (corresponding to solving a complicated PDE akin to the Hermite-Einstein/Kahler-Einstein equations) would lead to an inequality involving the higher Chern classes. Is there a reason why this has not been studied ? Even a meta reason as to why such universal inequalities are not expected would be nice.

• "Is there a reason why this has not been studied?" It has been studied. S.-T. Yau gave a lecture about this at Bogomolov's birthday conference at Courant about one year ago. – Jason Starr Nov 29 '17 at 11:45
• I looked it up. His talk only refers to the higher ones in the case of line bundles (the deformed HYM equation). – Vamsi Nov 29 '17 at 15:18
• Higher Chern classes of a line bundle are zero... – YangMills Apr 11 '18 at 22:57
• Obviously one is referring to the Chern character classes (which are non trivial) in the line bundles case ! – Vamsi Apr 12 '18 at 3:15