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I was recently looking back at an old question of mine, where I asked about the classification of the line bundles over a general complex flag manifold. Pavel Etingof gave the following excellent answer:

The partial flag manifold X which you mentioned is the set of flags $0 ⊂ V_1 ⊂ \ldots ⊂ V_m=ℂ^n$, such that $\dim(V_j/V_{j−1})=k_j$. So we have vector bundles $V_j$ on $X$ and line bundles $L_j$ on $X$ which are the top exterior powers of the bundles $V_j$, $j = 1,...,m−1$. Any line bundle on $X$ is a tensor product of powers of these line bundles, and the powers are uniquely determined. This is why line bundles on $M$ are labeled by a set of $m−1$ integers.

I then began to wonder how this relates to a more recent question of mine regarding the definition of a globally generated holomorphic vector bundle. So my question is: With respect to the above line bundle classification, and the obvious choice of holomorphic structure for these bundles (I am assuming that there is just one obvious choice here), which of these bundles are globally generated?

For the simplest case of complex projective space, we have of course that the line bundles are classified by the integers, and the positive ones are globally generated (or the negative, depending on convention). Now for the Grassmannians the line bundles are again classified by the integers, and I would again guess that the globally generated ones are exactly those of positive charge. So how does this extend?

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Why do you say "vector bundle" in your title if you are only asking about "line bundles"? –  Jason Starr Feb 20 '13 at 15:30
Good point! I've changed it. –  Jean Delinez Feb 20 '13 at 15:54

1 Answer 1

If I am not mistaken, the criterion is: such a line bundle is globally generated if and only if the space of its holomorphic sections is non-zero. To wit, both the flag variety and these bundles are homogeneous w.r.t. the natural action of $GL_n(\mathbb C)$. Thus if a section of such a bundle $L$ does not vanish at a point $x$, then the section $g^*s$ does not vanish at the point $y=g^{-1}x$, where $g\in GL_n(\mathbb C)$; the point $y$ may be made arbitrary.

The criterion for the existence of a non-zero section may be found in Chapter 1 of the book http://www.amazon.com/Complex-Geometry-Grundlehren-mathematischen-Wissenschaften/dp/3540613781

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I'm used to the (mostly equivalent) language of algebraic geometry for arbitrary connected reductive linear algebraic groups. Here the homogeneous line bundles correspond to characters of a maximal torus $T \subset B$ for the (full) flag variety $G/B$. where nonzero global sections occur just for dominant characters. For partial flag varieties $G/P$, fewer dominant characters of $T$ qualify: those zero at simple (co)roots defining a Levi subgroup of $P$. This relies on Borel-Weil and Bott, or Kempf in positive characteristic. –  Jim Humphreys Feb 24 '13 at 13:34
@Jim Great that seems to answer my question. It's the full flag manifolds I'm really interested in any way. Please put your comment as an answer so I can accept it. –  Jean Delinez Feb 26 '13 at 13:49

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