Hi everybody.

Let $G/P$ be a complex projective homogeneous variety with $G$ a simple Lie group and $P$ a parabolic subgroup.

I believe that it is possible to

• (1) describe ${\rm Pic}(G/P)$
• (2) characterize the ample line bundles and
• (3) express the canonical class of $G/P$

in terms of the nodes corresponding to P in the Dynkin diagram of $G$.

Are they some canonical (or a least some good) references where this is explained? This is certainly "well-known by the experts" but I'm not one of them...

Thanks to those who will answer.

Hi everybody.

Let $G/P$ be a complex projective homogeneous variety with $G$ a simple Lie group and $P$ a parabolic subgroup.

I believe that it is possible to

• (1) describe ${\rm Pic}(G/P)$
• (2) characterize the ample line bundles and
• (3) express the canonical class of $G/P$

in terms of the nodes corresponding to P in the Dynkin diagram of $G$.

Are there some canonical (or a least some good) references where this is explained? This is certainly "well-known by the experts" but I'm not one of them...

Thanks to those who will answer.