Abstract:We extend the application of the Guillemin--Abreu formalism to holomorphic sectional and bisectional curvature on $\mathbb C^n$, $\mathcal O(-\ell)$, and Hirzebruch manifolds $M_{n,\ell}$, and further apply it to the total spaces of certain higher-rank vector bundles. The resulting formulas recover known positivity criteria and we show that, when the slope is sufficiently close to $1$, the extremal metrics on $M_{n,\ell}$ have positive holomorphic sectional curvature. We construct complete scalar-flat Kähler metrics on $ \operatorname{Tot}\bigl(\mathcal O(-k)\oplus\mathcal O(-k)\to\mathbb{CP}^n\bigr), $ and identify the Ricci-flat case, which occurs precisely when $2k=n+1$.
From: Xingluan Wang [view email]
[v1]
Thu, 11 Jun 2026 04:08:03 UTC (58 KB)
[v2]
Wed, 17 Jun 2026 14:05:28 UTC (27 KB)
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