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## On Plurigenera

Algebraic Geometry
Tsakanikas Nickos
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### On Plurigenera

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 $1 \leq k \leq m$.