On a theorem of Ramanan
On a theorem of Ramanan
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关于拉马南定理
DOI：
10.1017/s0027763000017992
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发表时间：
1978
影响因子：
0.8
通讯作者：
H. Umemura
中科院分区：
文献类型：
--
作者：
H. Umemura
设G为单连通李群，P为无单因子的抛物子群。P的有限维不可约表示定义了齐次空间G/P上的齐次向量束E。Ramanan[2]证明，如果G/P的第二Betti数b2为1，则定义（2.3）中的不等式在F局部自由的情况下成立。由于当时h稳定性的概念还没有建立，所以必然会假设b2 = 1， F是局部自由的。在本文中，我们推广了Ramanan的思想，证明了E对于任何充足的线束h都是h稳定的。我们的证明和Ramanan的证明一样依赖于borell - weil定理。如果我们回想一下Borel-Weil定理在特征p >中失效，那么有趣的是，我们的定理在特征p >中是否仍然成立。
Let G be a simply connected Lie group and P a parabolic subgroup without simple factor. A finite dimensional irreducible representation of P defines a homogeneous vector bundle E over the homogeneous space G/P. Ramanan [2] proved that, if the second Betti number b2 of G/P is 1, the inequality in Definition (2.3) holds provided F is locally free. Since the notion of the H-stability was not established at that time, it was inevitable to assume that b2 = 1 and F is locally free. In this paper, pushing Ramanan’s idea through, we prove that E is H-stable for any ample line bundle H. Our proof as well as Ramanan’s depends on the Borel-Weil theorem. If we recall that the Borel-Weil theorem fails in characteristic p > 0, it is interesting to ask whether our theorem remains true in characteristic p > 0.