 Home

Codex Wiki

Mathematics

Area of mathematics

Geometry and topology

Algebraic topology

Complex projective space

{c}
{wiki=Euler_sequence}

The Euler sequence is the short exact sequence of holomorphic vector bundles
$$
0\longrightarrow\mathcal O
\longrightarrow\mathcal O(1)^{\oplus(n+1)}
\longrightarrow T\mathbb{CP}^n
\longrightarrow0.
$$
Taking top exterior powers gives $\det T\mathbb{CP}^n\cong\mathcal O(n+1)$.


Euler sequence on complex projective space

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

Euler sequence on complex projective space

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

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

 Ancestors (6)

 Incoming links (2)

 Discussion (0)

 Articles by others on the same topic (0)

Euler sequence on complex projective space

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

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

 Ancestors (6)

 Incoming links (2)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

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

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

 Discussion (0)