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I have a naive question:(I saw that it is related to relative K-theory of Hodge-Deligne and also Nadel-Chern-Weil theory )

Let $\mathcal M (r, d)$ be the moduli space of stable vector bundles of rank $r$ with a fixed determinant of degree $d$ on a projective variety $X$. Then when the Albanese map $\mathcal M (r, d)\to Alb(\mathcal M (r, d))$ is surjective.?

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    $\begingroup$ Is $X$ a curve? If not, what is the degree? $\endgroup$
    – abx
    Commented Jan 1, 2018 at 6:11
  • $\begingroup$ @abx , many thank for your interesting comment, "however" we can define the degree for a reflexive sheaf, vector bundles,.. see mathoverflow.net/questions/69096/… $\endgroup$
    – 1984
    Commented Jan 1, 2018 at 6:18
  • $\begingroup$ Then you need to choose an ample class on $X$. And what is the Albanese map? Your $\mathcal{M}(r,d)$ is neither projective, nor normal in general. $\endgroup$
    – abx
    Commented Jan 1, 2018 at 6:21
  • $\begingroup$ It is quasi-projective $\endgroup$
    – 1984
    Commented Jan 1, 2018 at 10:41

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