I will always consider a Grassmannian $G=\mathbb{G}(k,\mathbb{C}^n)$ as an embedded manifold of $\mathbb{P}(\bigwedge^k \mathbb{C}^n)$ throught the Plucker embedding. To me the canonical bundle of an embedded manifold $X$ is $\omega_X=\bigwedge^{dim X}\Omega_X$, $\Omega_X$ being the dual bundle of the tangent bundle of $X$.
I would like to compute a local section of the canonical bundle of the Grassmannian on an affine open set, but I don't know how to do such explicit computations.
I know how to use Poincaré residue and adjunction formula to find a section in the case of a hypersurface or a complete intersection, but while I can write the Grassmannian as an intersection of quadrics, it isn't a complete intersection, so I'm stuck here.
I image that maybe I could use the proof of the fact that $\omega_G=\mathcal{O}_G(-n)$, but I'm not so able to handle it. In case it can helps, I'm mostly interested in the case of the Grassmannian $G(2,\mathbb{C}^5)$.
Any help will be appreciated. Thank you!
Best Answer
Here is a non-so-explicit answer using linear algebra : it is a standard fact that the tangent space of a Grassmannian $X = G(k,V)$ at $W$ is canonically identified with $Hom(W,V/W)$, so the cotangent bundle at $W$ is $Hom(V/W,W)$.
Let us pick a supplementary $W' \oplus W = V$ and identify $V/W \cong W'$. Let $p : V = W \oplus W' \to W'$ and $q : V = W \oplus W' \to W$ be the projections. We get an isomorphism $$ Hom(V/W,W) \cong \{Z \subset V : \dim Z = k, p_{|Z} : Z \to W' \text{ is an isomorphism } \}$$
given by $f \mapsto Graph(f)$ and $ Z \mapsto (p_{Z})^{-1} \circ q$. So picking generic $Z_1, \dots, Z_{n(n-k)}$ gives you an element of $\bigwedge^{top}T_W^*X$. We take them generic because I assume you don't want the zero section :-)
Now let $$U_i = \{W \in X : p_{|Z} : Z \to W' \text{ is an isomorphism}\}$$ $$U = \{W \in X : W \oplus W' = V \}$$ Clearly on $\tilde U = U \cap U_1 \cap \dots \cap U_{k(n-k)}$ each of the $Z_i$ is a well defined section of $T^*X$ using $W'$ so $Z_1 \wedge \dots \wedge Z_{k(n-k)}$ is a local section of $\omega_X$ as expected.