Corollary 11.11.3`EB345AEE34BC`

Corollary 11.11.3.EB345AEE34BC Let $F$ carry a holomorphic structure in the neighborhood of theorem 11.9.4. If $F^*$ is ample, there is no positive continuous complex homogeneous function of degree two on $F$ that is plurisubharmonic off zero and locally strongly convex in the fibers.

Proof. By proposition 11.11.2, ampleness of $F^*$ gives the function $H$ required in proposition 11.10.5. A function $Q$ as in the statement would therefore give a smooth complex norm on $F$ with strictly plurisubharmonic square, contradicting theorem 11.9.4. $\square$

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@misc{pome-tag-EB345AEE34BC,
  author = {{Pome contributors}},
  title = {Corollary 11.11.3},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/EB345AEE34BC/}
}

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Corollary 11.11.3EB345AEE34BC

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Corollary 11.11.3`EB345AEE34BC`