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I recently started the study of vector bundles on $\mathbb{P}^n$, and started to read Rao's article 'A family of vector bundles on $\mathbb{P}^3$'. There, there is a notion of spectrum of a vector bundle. I searched this in some other articles, even in the excellent book "Vector Bundles on Complex Projective Space" by Okonek, Schneider and Spindler, but I cannot find the definition of spectrum.

Does anyone has any reference where I can find this?

Thank you so much.

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It seems that the notion of spectrum of vector bundles used in Rao's article is defined in Section 3 of

  • C. Okonek and H. Spindler. Reflexive Garben vom Rang $r>2$ auf $\mathbb{P}^n$. Crelle Journal Reine Angew. Math. 344 (1983), pp. 38-64.

If you are having trouble reading German, I could give the definition here, but refrain from it now because it is rather lengthy. The basic idea of the definition is that for a reflexive sheaf with specified splitting type, the cohomology of the restriction of the sheaf to a generic line can be described in terms of cohomology of explicitly given vector bundles on that line. The spectrum is a series of numbers telling you exactly which vector bundle on the line you have to take.

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  • $\begingroup$ Thank you @Matthias Wendt ! I don`t know German, but I guess I can try to infer the things from the context. Thank you again! $\endgroup$
    – User43029
    Mar 19, 2015 at 11:02

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