Abstract
We show the existence of strictly almost-Kähler anti-self-dual (ASD) metrics on certain 4-manifolds by deforming scalar-flat Kähler metrics. On the other hand, we prove the non-existence of such metrics on certain other 4-manifolds by means of Seiberg–Witten theory. In the process, we provide a simple new proof of the fact that any almost-Kähler ASD 4-manifold must have a non-trivial Seiberg–Witten invariant.
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Acknowledgments
This article is based on the author’s Ph.D. thesis at Stony Brook University [11]. The author would like to thank Prof. Claude LeBrun for suggesting the problem, as well as for his constant help and encouragement. The author would like to thank the referee for pointing out ambiguities and giving helpful comments. This article is finally revised at The Center for Geometry and its Applications at Postech, to whom the author would like to thank.
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Kim, I. Almost-Kähler anti-self-dual metrics. Ann Glob Anal Geom 49, 369–391 (2016). https://doi.org/10.1007/s10455-016-9497-1
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DOI: https://doi.org/10.1007/s10455-016-9497-1