Harder-Narasimhan filtration/Strong version/Tight closure/Section

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In general, there exists an exact criterion for the affineness of the torsor depending on and the strong Harder-Narasimhan filtration of . For this we give the definition of the Harder-Narasimhan filtration.

Let be a vector bundle on a smooth projective curve over an algebraically closed field . Then the (uniquely determined) filtration

of subbundles such that all quotient bundles are semistable with decreasing slopes ,

is called the Harder-Narasimhan filtration of .

The Harder-Narasimhan filtration exists uniquely (by a Theorem of Harder and Narasimhan). A Harder-Narasimhan filtration is called strong if all the quotients are strongly semistable. A Harder-Narasimhan filtration is not strong in general, however, by a Theorem of A. Langer, there exists some Frobenius pull-back such that its Harder-Narasimhan filtration is strong.

Let denote a smooth projective curve over an algebraically closed field and let be a vector bundle over together with a cohomology class . Let

be a strong Harder-Narasimhan filtration. We choose such that has degree and that has degree . We set

. Then the following are equivalent.
  1. The torsor is not an affine scheme.
  2. Some Frobenius power of the image of inside is .