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arXiv · 2609.33656

RC-positivity of complex surfaces

You-Cheng Chou·Kuang-Ru Wu
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Abstract

Given a compact complex surface with a nonflat Ricci-flat Kähler metric, we show that its holomorphic tangent bundle admits an RC-positive Hermitian metric. The proof relies on a characterization of RC-positivity through the anti-self-dual part of the Weyl operator, the Weitzenböck formula on the four dimensional Einstein manifold, and a conformal perturbation on the Ricci-flat metric. As a consequence, we prove that RC-positivity is not preserved after taking tensor, exterior, or symmetric power. Moreover, we show that RC-positivity is a strictly weaker notion than uniform RC-positivity, and that RC-positivity of holomorphic tangent bundle does not necessarily imply rational connectedness of the base manifold.

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You-Cheng Chou, Kuang-Ru Wu. 2026-09-27. RC-positivity of complex surfaces. https://arxiv.org/abs/2609.33656

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