Abstract
Using the twistor correspondence, we give a classification of toric anti-self-dual Einstein metrics: each such metric is essentially determined by an odd holomorphic function. This explains how the Einstein metrics fit into the classification of general toric anti-self-dual metrics given in an earlier paper (Donaldson and Fine in Math Ann 336(2):281–309, 2006). The results complement the work of Calderbank–Pedersen (J Differential Geom 60(3):485–521, 2002), who describe where the Einstein metrics appear amongst the Joyce spaces, leading to a different classification. Taking the twistor transform of our result gives a new proof of their theorem.
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Fine, J. Toric anti-self-dual Einstein metrics via complex geometry. Math. Ann. 340, 143–157 (2008). https://doi.org/10.1007/s00208-007-0141-z
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DOI: https://doi.org/10.1007/s00208-007-0141-z