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The Kähler-Einstein metric on the third Del Pezzo surface
Abstract
The compact four-dimensional manifold ℂℙ2# 3ℂℙ2̅ is known to admit a toric Kähler-Einstein metric gKE, but the metric is not known in closed form, which makes it difficult to draw conclusions about its geometry. In this article, we use a combination of analytic and computer-assisted techniques to produce an approximate Einstein metric g described explicitly herein, and also prove that the true Einstein metric gKE is close to g, where both the closeness and the topology are described explicitly. As an application, we prove bounds on the first invariant eigenvalue of the Laplace-Beltrami operator, and prove that this Kähler-Einstein metric does not have positive holomorphic sectional curvature everywhere.
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BibTeX
@misc{buttsworth2026kählereinstein,
title = {{The Kähler-Einstein metric on the third Del Pezzo surface}},
author = {Timothy Buttsworth and William Hadden and Elli Heyes and Daniel Platt and Toby Wiseman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32655},
eprint = {2609.32655},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32655},
}
RIS
TY - UNPB
TI - The Kähler-Einstein metric on the third Del Pezzo surface
AU - Timothy Buttsworth
AU - William Hadden
AU - Elli Heyes
AU - Daniel Platt
AU - Toby Wiseman
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32655
UR - https://arxiv.org/abs/2609.32655
AB - The compact four-dimensional manifold ℂℙ_2# 3ℂℙ_2̅ is known to admit a toric Kähler-Einstein metric g_KE, but the metric is not known in closed form, which makes it difficult to draw conclusions about its geometry. In this article, we use a combination of analytic and computer-assisted techniques to produce an approximate Einstein metric g described explicitly herein, and also prove that the true Einstein metric g_KE is close to g, where both the closeness and the topology are described explicitly. As an application, we prove bounds on the first invariant eigenvalue of the Laplace-Beltrami operator, and prove that this Kähler-Einstein metric does not have positive holomorphic sectional curvature everywhere.
ER -