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Complex projective space

Euler sequence on complex projective space

{title2=$K_{\mathbb{CP}^n}\cong\mathcal O(-n-1)$}

Dualizing the determinant identity in the <Euler sequence on complex projective space> gives
$$
K_{\mathbb{CP}^n}\cong\mathcal O(-n-1)\cong[-(n+1)H],
$$
where $H$ is a <hyperplane divisor>.

Canonical bundle of complex projective space

Canonical bundle of a smooth projective hypersurface

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 118 / 2 / Solution

Canonical bundle of complex projective space ( $K_{CP^{n}} ≅ O (- n - 1)$ )

Canonical bundle of a smooth projective hypersurface ( $K_{X_{d}} ≅ O_{X_{d}} (d - n - 1)$ )

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Canonical bundle of complex projective space (KCPn​≅O(−n−1))

Canonical bundle of a smooth projective hypersurface (KXd​​≅OXd​​(d−n−1))