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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.

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It all does seem well-known to experts (I'm also not one), but I've never found the ultimate concise reference. Anyway, I've made minor edits and provided a few references. –  Jim Humphreys Oct 11 '12 at 20:47
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3 Answers

Michel Brion's Lectures on the Geometry of Flag Varieties answers all of your questions in the special case $G = SL_n$ and $P = B$ (the Borel subgroup of upper triangular matrices). See Section 1.4. If you are not so familiar with this particular field, you may find the entire first section quite helpful.

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Much (but not quite all) of what you want is written down clearly in the mostly equivalent language of algebraic groups and homogeneous spaces by Birger Iversen and Robert Fossum. I'll give the journal references, which might not be readily available online. Some of their references are also worth following up.

B.Iversen, The geometry of algebraic groups, Adv. in Math. 20 (1976), 57-85. [This works mostly just with Borel subgroups and flag varieties, but many of the ideas are general.]

R. Fossum and B. Iversen, On the Picard groups of algebraic fiber spaces, J. Pure Appl. Algebra 3 (1973), 269-280. [This is fairly general, but they specialize it to varieties $G/P$.]

If you are intrepid enough to try J.C. Jantzen's large and sometimes technical book Representations of Algebraic Groups (2nd ed., Amer. Math. Soc., 2003), he has a careful discussion of ample line bundles in this algebraic framework. But the book is not directed toward algebraic or differential geometry as such. See especially his section II.4.4 for ampleness of homogeneous line bundles on flag varieties, with a remark on the easy adaptation to varieties $G/P$. The criterion here is just that the weight defining the line bundle be strictly dominant (strictly positive at all simple coroots or just positive relative to those simple roots not defining a parabolic subgroup). Jantzen also gives useful references back to work of Kempf. (Ampleness came up in the somewhat related older question on MO here.)

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In addition to the references already mentioned, let me recommend Dennis Snow's excellent notes on homogeneous vector bundles. They cover everything you ask for (and much more):

  1. For a description of ${\rm Pic}(G/P)$ in terms of weights, see Theorem 6.4.
  2. Ample line bundles are characterized in Theorem 6.5.2.
  3. The canonical class is described on page 37, right above Definition 10.2.
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