Holonomy of compact Ricci-flat Kaehler manifold

In summary, holonomy refers to the group of transformations that preserve the metric and other geometric structures of a manifold. For compact Ricci-flat Kaehler manifolds, the holonomy group is the special unitary group SU(n). If a compact Ricci-flat Kaehler manifold has holonomy preserved, it means that it retains its geometric structures and properties after being transformed by an element of the holonomy group. These manifolds have a rich symmetry structure that is preserved by the holonomy group, making them significant in mathematics and physics. The holonomy group can also be used to classify different types of compact Ricci-flat Kaehler manifolds based on their geometric properties.
  • #1
Ygor
4
0
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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  • #2



Dear Ygor,

Thank you for bringing this apparent contradiction to our attention. As scientists, it is important to carefully examine and analyze any discrepancies in the literature. After reviewing the sources you mentioned, I believe I have found a possible explanation for the discrepancy.

In "Joyce D.D., Compact manifolds with special holonomy," the author is specifically discussing compact Ricci-flat Kähler manifolds, while in "Candelas, Lectures on Complex manifolds," the author is discussing any Ricci-flat Kähler manifold. The difference lies in the compactness of the manifold.

A compact manifold is one that is closed and bounded, while a non-compact manifold is one that is not closed and may extend to infinity. In the case of a compact Ricci-flat Kähler manifold, the global holonomy group will be contained in SU(m) if and only if the canonical bundle is trivial. This is because the compactness of the manifold allows for a more specific and restrictive condition to be applied.

However, in the case of a non-compact Ricci-flat Kähler manifold, the global holonomy group will always be contained in SU(m), regardless of the triviality of the canonical bundle. This is because the non-compactness of the manifold allows for more freedom in the structure of the manifold, and the global holonomy group can be larger.

I hope this explanation helps to clarify the apparent contradiction and answer your question. As always, further research and analysis may be needed to fully understand the intricacies of these concepts.



 

1. What is holonomy in the context of compact Ricci-flat Kaehler manifolds?

Holonomy refers to the group of transformations that preserve the metric and other geometric structures of a manifold. In the case of compact Ricci-flat Kaehler manifolds, the holonomy group is the special unitary group SU(n).

2. What does it mean for a compact Ricci-flat Kaehler manifold to be holonomy preserved?

If a compact Ricci-flat Kaehler manifold has holonomy preserved, it means that the manifold has the same geometric structures and properties after being transformed by an element of the holonomy group. This includes preserving the curvature and other metric properties.

3. How do compact Ricci-flat Kaehler manifolds relate to holonomy?

Compact Ricci-flat Kaehler manifolds have a special property where the holonomy group is a subgroup of the full isometry group. This means that the manifold has a rich symmetry structure that is preserved by the holonomy group.

4. What is the significance of compact Ricci-flat Kaehler manifolds with holonomy preserved?

Compact Ricci-flat Kaehler manifolds with holonomy preserved are often studied in mathematics and physics due to their rich symmetry structure. They have been used in the study of string theory and other areas of theoretical physics.

5. Can holonomy be used to classify compact Ricci-flat Kaehler manifolds?

Yes, holonomy is an important tool for classifying compact Ricci-flat Kaehler manifolds. The holonomy group can give information about the geometric properties of the manifold and can help distinguish between different types of compact Ricci-flat Kaehler manifolds.

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