About: Why Yau's theorem implies the existence of hyperkähler metric on complex symplectic manifolds   Sponge Permalink

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Every expository article on hyperkähler manifolds that I have read states without detailed proof the following fact: I have trouble trying to see this. Indeed, for the metric to be hyperkähler we require the holonomy group to be in (where ), so a least the holonomy should be in , which means that the Levi-Civita covariant derivation of is . This is a condition stronger than Ricci-flatness at first glance and I can't see how to realize this only knowing the existence of Ricci-flat metric.

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  • Why Yau's theorem implies the existence of hyperkähler metric on complex symplectic manifolds
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  • Every expository article on hyperkähler manifolds that I have read states without detailed proof the following fact: I have trouble trying to see this. Indeed, for the metric to be hyperkähler we require the holonomy group to be in (where ), so a least the holonomy should be in , which means that the Levi-Civita covariant derivation of is . This is a condition stronger than Ricci-flatness at first glance and I can't see how to realize this only knowing the existence of Ricci-flat metric.
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  • Xin Nie
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  • Maths Overflow
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  • Robert Bryant
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  • Every expository article on hyperkähler manifolds that I have read states without detailed proof the following fact: I have trouble trying to see this. Indeed, for the metric to be hyperkähler we require the holonomy group to be in (where ), so a least the holonomy should be in , which means that the Levi-Civita covariant derivation of is . This is a condition stronger than Ricci-flatness at first glance and I can't see how to realize this only knowing the existence of Ricci-flat metric.
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