Abstract
We apply Nadel’s method of multiplier ideal sheaves to show that every complex del Pezzo surface of degree at most six whose automorphism group acts without fixed points has a Kähler–Einstein metric. In particular, all del Pezzo surfaces of degree 4, 5, or 6 and certain special del Pezzo surfaces of lower degree are shown to have a Kähler–Einstein metric. These existence statements are not new, but the proofs given in the present paper are less involved than earlier ones by Siu, Tian and Tian–Yau.
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Acknowledgments
The author would like to thank I. Dolgachev for helpful comments concerning the automorphism groups of del Pezzo surfaces of degree at most three.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Heier, G. Existence of Kähler–Einstein metrics and multiplier ideal sheaves on del Pezzo surfaces. Math. Z. 264, 727–743 (2010). https://doi.org/10.1007/s00209-009-0486-y
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DOI: https://doi.org/10.1007/s00209-009-0486-y