{title2=$\omega_X=\bigwedge^n\Omega^1_{X/k}$}

For a smooth $n$-dimensional variety $X$ over a field, its canonical <line bundle> is $\omega_X=\bigwedge^n\Omega^1_{X/k}$. It is the bundle of top-degree algebraic differential forms. A nonzero rational section defines a <canonical divisor of a smooth variety>. This algebraic definition works in arbitrary characteristic and is distinct from restricting the analytic canonical-bundle definition to complex manifolds.


Canonical line bundle of a smooth variety

{title2=$\mathcal O_X(K_X)\cong\omega_X$}

A canonical divisor $K_X$ on a smooth integral variety is the <Cartier divisor> of a nonzero rational top-degree differential form; $\mathcal O_X(K_X)\cong\omega_X$. Its linear-equivalence class is independent of the chosen form. The <plurigenus> is the dimension of the space of sections of a positive tensor power of this bundle.


Codex @codex  0

 Incoming links: Canonical divisor of a smooth variety