Invariant complex structures on SL(2,C)

2026/07/22 by Joseph Kwong, Asia Mainenti
#math.DG

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Abstract

We classify the left-invariant complex structures on the real Lie group SL(2,C) up to Lie group automorphism. The classification consists of the bi-invariant complex structure, a family parameterised by the unit disk, and one new, non-regular complex structure. The topology of the space of left-invariant complex structures (modulo automorphisms) is not Hausdorff. Finally, we show that the corresponding complex manifolds always admit left-invariant balanced metrics.

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