Vanishing first betti number of kaehler manifold with global SU(m) holonomy

[Ygor]

Hi,

I have the following question. In "Joyce D.D. Compact manifolds with special holonomy" I read on page 125 that a compact Kaehler manifold with global holonomy group equal to SU(m), has vanishing first betti number, or more specifically vanishing Hodge numbers h^(1,0)= h^(0,1) = 0. However, in "Candelas, Lectures on complex manifolds" the constrains h^1 = 0 is not automatic if the holonomy equals SU(m).

So, I wonder, which case is the correct one?

I hope anyone could help me out,

Ygor

 

[jvicens]

Dear Ygor,

Thank you for your question. The answer is that both cases are correct, but they refer to slightly different situations.

In "Joyce D.D. Compact manifolds with special holonomy," the statement refers to a compact Kähler manifold with global holonomy group equal to SU(m). In this case, the vanishing of Hodge numbers h^(1,0) and h^(0,1) is a consequence of the fact that the manifold is Kähler and has a special holonomy group. This is a specific case, and the statement holds true in this situation.

On the other hand, in "Candelas, Lectures on complex manifolds," the statement refers to a more general case, where the holonomy group is not necessarily SU(m). In this case, the vanishing of h^1 is not automatic and may depend on other properties of the manifold. So, while the statement in "Joyce D.D." is a specific case, the one in "Candelas" is a general statement that may or may not hold true depending on the specific properties of the manifold.

I hope this clarifies the difference between the two statements. If you have any further questions, please do not hesitate to ask.



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