zbMATH — the first resource for mathematics

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.

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14G20 Local ground fields in algebraic geometry
Full Text: DOI