Let $X$ be a smooth projective variety over $\mathbb{C}$. For an ample divisor H, we can define the slop of vector bundle with respect to $H$, then we can define stablilty of vector bundle with respect to $H$. However it might be possible that an $H$-stable vector bundle is no longer stable with respect to another ample divisor $H'$.
On the other hand the famous theory of Uhlenbeck-Yau about irreducible representation and stable bundle implies that if a vector bundle has trivial Chern classes, then its stablility does not depend on the ample divisor. I wonder is there a direct proof of this fact without using Uhlenbeck-Yau thoery?

A natural question to ask : does the statement hold in charateristic p case ( one may need some condition like algebraic closed field, the variety has $W_2$-lifting, etc.)?


1 Answer 1


See the following paper by Adrian Langer: http://arxiv.org/abs/0905.4600 and more precisely section 4.

I quote:

In this section we show that strongly semistable torsion free sheaves with vanishing Chern classes are locally free and that they are strongly semistable with respect to all polarizations

(A torsion free sheaf $\mathcal{E}$ on $X$ is called strongly semistable if it is semistable and if all ${F^e}^\ast \mathcal{E}$ remain semistable, where $F: X \to X$ is the absolute Frobenius.)

More precisely Proposition 4.5 of that paper should prove what you want. Langer works in arbitrary characteristics (over an algebraically closed field).

  • $\begingroup$ Thanks a lot. One more question, is there any counterexample of the statement for semistable but not strongly semistable? $\endgroup$
    – Lan
    Jul 3, 2014 at 15:29
  • $\begingroup$ I don't know of a counterexample in this case. But I'd expect the statement to fail. $\endgroup$ Jul 4, 2014 at 5:53

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