Abstract
Using the twistor correspondence, this article gives a one-to-one correspondence between germs of toric anti-self-dual conformal classes and certain holomorphic data determined by the induced action on twistor space. Recovering the metric from the holomorphic data leads to the classical problem of prescribing the Čech coboundary of 0-cochains on an elliptic curve covered by two annuli. The classes admitting Kähler representatives are described; each such class contains a circle of Kähler metrics. This gives new local examples of scalar flat Kähler surfaces and generalises work of Joyce [Duke. Math. J. 77(3), 519–552 (1995)] who considered the case where the distribution orthogonal to the torus action is integrable.
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Atiyah M.F., Hitchin N.J., Singer I.M. (1978) Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. Ser. A 362(1711): 425–461
Boyer C.P. (1988) A note on hyper-Hermitian four-manifolds. Proc. Am. Math. Soc. 102(1): 157–164
Calderbank, D.M.J., Mason, L.J.: Spinor-vortex geometry and microtwistor theory
Calderbank D.M.J., Pedersen H. (2002) Selfdual Einstein metrics with torus symmetry. J. Diff. Geom. 60(3): 485–521
Gibbons G.W., Hawking S.W. (1978) Gravitational multi-instantons. Phys. Lett. B 78, 430–432
Hitchin N.J. (1980) Linear field equations on self-dual spaces. Proc. R. Soc. Lond. Ser. A 370(1741): 173–191
Joyce D.D. (1995) Explicit construction of self-dual 4-manifolds. Duke Math. J. 77(3): 519–552
Pontecorvo M. (1992) On twistor spaces of anti-self-dual Hermitian surfaces. Trans. Am. Math. Soc. 331(2): 653–661
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Donaldson, S., Fine, J. Toric Anti-self-dual 4-manifolds Via Complex Geometry. Math. Ann. 336, 281–309 (2006). https://doi.org/10.1007/s00208-006-0003-0
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DOI: https://doi.org/10.1007/s00208-006-0003-0