Holonomy of compact Ricci-flat Kaehler manifold

  1. Hi,

    I have come across the following apparent contradiction in the literature. In "Joyce D.D., Compact manifolds with special holonomy" I find on page 125 the claim that if M is a compact Ricci-flat Kaehler manifold, then the global holonomy group of M is contained in SU(m) if and only if the canonical bundle of M is trivial.

    In "Candelas, Lectures on Complex manifolds" however, on page 61 I read that any Ricci-flat Kaehler manifold has global holonomy in SU(m) and there is no mentioning of any condition on the the canonical bundle.

    Note that I am assuming M to be multiply connected and I am talking about the global holonomy group, i.e. not the restricted holonomy group.

    So my questions is, what is going on ? Who is right? (Both somehow provide proofs of their claims).

    I hope anyone can help me out.

    Thanks!

    Ygor
     
  2. jcsd
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