A quantitative sharpening of Moriwaki’s arithmetic Bogomolov inequality
نویسندگان
چکیده
A. Moriwaki proved the following arithmetic analogue of the Bogomolov unstability theorem. If a torsion-free hermitian coherent sheaf on an arithmetic surface has negative discriminant then it admits an arithmetically destabilising subsheaf. In the geometric situation it is known that such a subsheaf can be found subject to an additional numerical constraint and here we prove the arithmetic analogue. We then apply this result to slightly simplify a part of C. Soulé’s proof of a vanishing theorem on arithmetic surfaces.
منابع مشابه
Inequality of Bogomolov-gieseker’s Type on Arithmetic Surfaces
Let K be an algebraic number field, OK the ring of integers of K, and f : X → Spec(OK) an arithmetic surface. Let (E, h) be a rank r Hermitian vector bundle on X such that E Q is semistable on the geometric generic fiber X Q of f . In this paper, we will prove an arithmetic analogy of Bogomolov-Gieseker’s inequality: ĉ2(E, h)− r − 1 2r ĉ1(E, h) 2 ≥ 0. Table of
متن کاملBogomolov Unstability on Arithmetic Surfaces
In this paper, we will consider an arithmetic analogue of Bogomolov unstability theorem, i.e. if (E, h) is a torsion free Hermitian sheaf on an arithmetic surface X and d̂eg ( (rkE − 1)ĉ1(E, h) − (2 rkE)ĉ2(E, h) ) > 0, then there is a non-zero saturated subsheaf F of E such that ĉ1(F, h|F )/rkF − ĉ1(E, h)/rkE lies in the positive cone of X. 0. Introduction In [Bo], Bogomolov proved unstability t...
متن کاملSurfaces Violating Bogomolov-miyaoka-yau in Positive Characteristic
The Bogomolov-Miyaoka-Yau inequality asserts that the Chern numbers of a surface X of general type in characteristic 0 satisfy the inequality c1 ≤ 3c2, a consequence of which is K2 X χ(OX ) ≤ 9. This inequality fails in characteristic p, and here we produce infinite families of counterexamples for large p. Our method parallels a construction of Hirzebruch, and relies on a construction of abelia...
متن کاملSemi-stable extensions on arithmetic surfaces
Let S be a smooth projective curve over the complex numbers and X → S a semi-stable projective family of curves. Assume that both S and the generic fiber of X over S have genus at least two. Then the sheaf of absolute differentials ΩX defines a vector bundle on X which is semi-stable in the sense of Mumford-Nakano with respect to the canonical line bundle on X . The Bogomolov inequality c1(Ω 1 ...
متن کاملArithmetic Bogomolov-gieseker’s Inequality
Let f : X → Spec(Z) be an arithmetic variety of dimension d ≥ 2 and (H, k) an arithmetically ample Hermitian line bundle on X, that is, a Hermitian line bundle with the following properties: (1) H is f -ample. (2) The Chern form c1(H∞, k) gives a Kähler form on X∞. (3) For every irreducible horizontal subvariety Y (i.e. Y is flat over Spec(Z)), the height ĉ1( (H, k)|Y ) dim Y of Y is positive. ...
متن کامل