= Global holomorphic sections of the complex tautological line bundle vanish
{title2=$H^0(\mathbb{CP}^n,\mathcal O(-1))=0\quad(n\geq1)$}
The <complex tautological line bundle> is a <holomorphic subbundle> of the trivial bundle with fibre $\mathbb C^{n+1}$. A global <holomorphic section> therefore gives $n+1$ global <holomorphic functions> on compact connected <Complex projective space>. The maximum modulus principle makes them constant. The constant vector must lie in every projective line through the origin, whose intersection is zero for $n\geq1$. For $n=0$, the base is one point and the section space is instead $\mathbb C$.
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