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On Plurigenera

Algebraic Geometry
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Tsakanikas Nickos
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On Plurigenera

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Post by Tsakanikas Nickos » Thu Dec 08, 2016 6:43 pm

Definition: Let $X$ be a smooth projective variety. For $m \in \mathbb{N}_{>0}$, the $m$-th plurigenus of $X$ is defined as $ P_{m}(X) = \mathrm{h}^{0}(X, \omega_{X} ) = \dim \mathrm{H}^{0}(X, \omega_{X} ) $, where $\omega_{X}$ is the canonical sheaf of $X$.

Show that if $P_{1}(X) = 1 $ and $ P_{m}(X) = 1 $ for some $ m \in \mathbb{N}_{>0}$, then $P_{k}(X) = 1 $ for all $ 1 \leq k \leq m$.
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