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Semi-stability and reduction $$\bmod p$$. (English) Zbl 0926.14021
From the text: Miyaoka has raised the following question [Y. Miyaoka, in: Algebraic Geometry, Proc. Symp., Sendai 1985, Adv. Stud. Pure Math. 10, 449-476 (1987; Zbl 0648.14006); section 5]: Suppose that $$(X,H)$$ is a smooth polarized complex projective variety and that $${\mathcal E}$$ is a vector bundle on $$X$$ that is semi-stable with respect to $$H$$. Then is it true that the set $$\Sigma= \Sigma(X, {\mathcal E},H)$$ of primes $$p$$ modulo which $${\mathcal E}$$ is not strongly semi-stable (i.e. every Frobenius pull-back of $${\mathcal E}$$ is semi-stable) is finite, or at least of density zero?
This paper provides some evidence for believing the answer to be affirmative. For example, it is shown that any example of such a bundle of rank 2 must have flat adjoint bundle, so that if also $$X$$ is simply connected then $$\Sigma$$ is finite.
Theorem. Suppose that $$\text{rank} {\mathcal E}=2$$, that $$\dim X=n\geq 2$$ and that $$\Sigma$$ is infinite. Then the following statements hold:
(1) $${\mathcal E}$$ is stable;
(2) there is an invertible subsheaf $${\mathcal O}(L)$$ of $$\Omega^1_X$$ such that $$H^{n-1}$$. $$L<0$$;
(3) $$\delta ({\mathcal E})=0$$, where $$\delta ({\mathcal E})= c^2_1({\mathcal E})-4c_2 ({\mathcal E})$$ by definition;
(4) $${\mathcal E}\otimes {\mathcal E}^\vee$$ is flat.

##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14G20 Local ground fields in algebraic geometry
##### Keywords:
semi-stable vector bundle; Chern class; reduction mod $$p$$
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