Corollary 11.27.5`9A2FC61929F1`

Corollary 11.27.5.9A2FC61929F1 For a connected compact complex surface $X$ admitting a nonflat Ricci-flat Kähler metric, the complement

\[ \mathbb P(\mathcal O_ X\oplus T^{1,0}X)\setminus \mathbb P(\mathcal O_ X) \]

is $3$-convex. The tangent bundle admits no Hermitian metric with positive definite mean curvature relative to any Hermitian form.

Proof. By theorem 11.27.4, the tangent bundle has an RC-positive Hermitian metric. Its directional curvature form has a positive eigenvalue for every nonzero fiber vector. Apply theorem 11.18.2 with $n=r=2$ and $s=1$. The mean-curvature obstruction is the second assertion of theorem 11.27.4. $\square$

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BibTeX citation

@misc{pome-tag-9A2FC61929F1,
  author = {{Pome contributors}},
  title = {Corollary 11.27.5},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/9A2FC61929F1/}
}

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Corollary 11.27.59A2FC61929F1

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Corollary 11.27.5`9A2FC61929F1`