I'm convinced I've read a paper where the authors prove that the blowup of the projective plane in a single point admits a metric of positive holomorphic sectional curvature. This was not the main focus of the paper. The proof was relegated to an appendix, and was by brute force calculations made by realizing the blowup as a subvariety of $\mathbb{C}^2 \times \overline{\mathbb{P^1}}$. However I can't find this paper again now, does this ring a bell to anyone?
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asked Jun 7, 2023 at 10:56
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2$\begingroup$ I do not know what article you read, but this is Theorem 1.3 in the following article, "Hirzebruch manifolds and positive holomorphic sectional curvature", by Bo Yang and Fangyang Zheng: numdam.org/item/10.5802/aif.3303.pdf $\endgroup$– Jason StarrCommented Jun 7, 2023 at 11:03
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1$\begingroup$ Actually, they attribute the result for Hirzebruch surfaces to Hitchin (they prove the result also for Hirzebruch manifolds of dimension greater than two). $\endgroup$– Jason StarrCommented Jun 7, 2023 at 11:04
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1$\begingroup$ The article of Hitchin is the following: “On the curvature of rational surfaces”, in Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, 1973), American Mathematical Society, 1975, p.65-80. $\endgroup$– Jason StarrCommented Jun 7, 2023 at 11:06
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2$\begingroup$ @JasonStarr Thank you. I have actually read those. In the paper I'm looking for the authors worked with the standard local image of a blowup, and not that the first Hirzebruch surface can be realized as a projectivized bundle. I'm interested in their technique, not in the result itself. $\endgroup$– Gunnar Þór MagnússonCommented Jun 7, 2023 at 11:25
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