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Friday, November 9, 2018 — 1:30 PM EST

Artour Tomberg, Western University

"Metrics on twistor spaces of hypercomplex manifolds"

A smooth manifold M is said to be hypercomplex if it admits a triple of integrable almost complex structures (I, J, K) that satisfy quaternionic relations I^2 = J^2 = K^2 = -1, IJ = -JI = K. If in addition there is a metric which is Kähler with respect to I, J and K, M is called hyperkähler. In addition to I, J, K, a hypercomplex manifold M has a whole 2-sphere of complex structures. These are parametrized by the twistor space Tw(M), which is topologically a cartesian product of M and S^2. Tw(M) has a natural structure of a complex manifold, and in case the original manifold M had a hyperkähler metric, there is also a natural Hermitian metric on Tw(M), which however fails to be Kähler. I will show that this metric on Tw(M) satisfies the weaker condition of being balanced, and also show that the twistor space Tw(M) of a general compact hypercomplex manifold M with no metric assumptions whatsoever admits a balanced metric as well.

MC 5403

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