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Let $M$ be a complex manifold and $E$ a holomorphic vector bundle over $M$. The discriminant $\Delta(E)$ of $E$ is then defined to be $$\Delta(E)=c_2(\text{End}(E))=2rc_1(E)-(r-1)c^2_1(E).$$

This class pops up quite a bit when one considers the stability of bundles and I'm wondering why this ends up being relatively important? If anyone has some historical notes or heuristics on why this particular class is of importance I would be glad to hear them.

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  • $\begingroup$ The determinant of a vector bundle is a line bundle, which is the top exterior power of it. Maybe your formula is related to its first Chern class. $\endgroup$
    – Z. M
    Commented Jun 24 at 10:15

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