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Holonomy of compact Ricci-flat Kaehler manifold

by Ygor
Tags: holonomy, kaehler, ricci-flat, su(m)
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Ygor
#1
Jul2-09, 07:27 AM
P: 4
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
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