Example 19.9.25`DE28BE657AA4`

Example 19.9.25.DE28BE657AA4 Let $0\leq r\leq n$, let $Y$ be any connected compact complex manifold of dimension $n-r$, and take $X=\mathbb P^ r\times Y$ and $L=\operatorname {pr}_1^*\mathcal O(1)$ with the pulled-back Fubini–Study metric. Its curvature has rank $r$, and

\[ h^0(X,L^ p)=\binom {p+r}{r}. \]

Thus the section-growth exponent cannot be reduced, including when $Y$ is non-Kähler.

Proof. A section, expressed in a line-bundle frame on the projective factor, is a holomorphic function of $Y$ and hence is constant there. Restriction to any slice $\mathbb P^ r\times \{ y_0\} $ is therefore inverse to pullback on sections. Counting degree-$p$ homogeneous monomials in $r+1$ variables gives the formula; for $r=0$ it equals one. $\square$

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@misc{pome-tag-DE28BE657AA4,
  author = {{Pome contributors}},
  title = {Example 19.9.25},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/DE28BE657AA4/}
}

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Example 19.9.25DE28BE657AA4

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Example 19.9.25`DE28BE657AA4`