A cohomological interpretation of Bogomolov's instability

We give a new proof of Bogomolov's instability theorem. Furthermore we prove that it is equivalent to a statement which characterizes when the first cohomology group of a suitable divisor does not vanish.


Introduction
In the theory of stable vector bundles on surfaces the following theorem, known as Bogomolov's instability theorem, plays a central role: (Bogomolov). Let X be a smooth projective surface and V be a rank 2 vector bundle on X. If c 1 (V ) 2 > 4c 2 (V ) then V is unstable.
For the original proof we refer to [1], see also [9]. This theorem was later proved by quite different techniques in [5] and [8]. Furthermore Reider used Theorem 1.1 to study adjoint linear series on surfaces and to derive his famous theorem, [10].
The first cohomological proof of Reider's theorem was given by Sakai in [11]. His proof uses ideas of Serrano [12], and generalizes Reider's theorem to normal surfaces.
The key point in Sakai's proof is the following theorem. Theorem 1.2 (Sakai). Let D be a big divisor with D 2 > 0 on a smooth projective surface X. If H 1 (X, O X (K X + D)) = 0 then there exists an effective divisor E such that As shown in [11] [2]. Based on these new techniques Fernández del Busto gave an elegant proof of Bogomolov's inequality which uses only the Kawamata-Viehweg theorem, see [3].
For a survey on these results we refer to [6].  On the other hand, Mumford shows that we can use Bogomolov's theorem for rank 2 vector bundles to give a short proof of a generalized Kodaira vanishing for surfaces, see [4]. This vanishing theorem is a little less general than the theorem of Kawamata-Viehweg. These results suggest that there should be a connection between Bogomolov's instability and some vanishing theorem.
In this note we prove Theorem 1.3. Bogomolov's instability theorem is equivalent to Theorem 1.2.
Furthermore, using Sakai's proof of Theorem 1.2, one gets a new proof of Bogomolov's instability theorem which is entirely cohomological.
We now outline the proof of Theorem 1.3. After twisting the vector bundle V with a line bundle we can assume that V has a global section. Using this section we have that the extension class of the vector bundle is a nontrivial since V is locally free. The first step of our proof follows Fernández del Busto's argument [3]. At this point we follow a different strategy. The numerical condition of Bogomolov's inequality allows us to apply Theorem 1.2 and we show directly that the divisor E gives the destabilizing subsheaf.

Preliminaries
For the convenience of the reader we sketch the proof of Theorem 1.2.
Proof. Let D = P + N be the Zariski decomposition of D and write N = α j E j with each α j positive and rational. By Sakai's lemma, Example 9.4.12 in [7], we know that H 1 (X, O X (K X + D − N )) = 0 so N > 0. Consider the following sequence of divisors: If D k ·E j k > 0 for any k, we get the vanishing of H 1 (X, O X (K X +D)). Thus we can collect all the E j k 's with positive intersection to construct a sequence D 0 , . . . , D k Corollary 2.1. Let D and E be as above then Proof. By the above construction Since D 0 = D − N and D k = D − E, the result follows from Sakai's lemma.

A COHOMOLOGICAL INTERPRETATION OF BOGOMOLOV'S INSTABILITY 3
In conclusion we recall two results which will be used in the proof of the main theorem. Proof. See Proposition 2 in [11].

Main Theorem
We can now prove the main result of the paper.
We now want to show that Theorem 1. Since V is locally free, the above extension is nontrivial and then Let π : Y → X be the blow up of X at all points in Z. Let E j be the exceptional curve over x j ∈ Z, then Define L := π * L − 2 j E j . Thus, we have so L is big by Lemma 2.2.
By applying Theorem 1.2 we get an effective divisor E s such that Note that E s depends on the section s that we choose at the beginning. Let E s := π * E s . We want to show that, for any s, E s passes through at least one point of Z.
Let E s := π * E s + a i E i , where E i are the exceptional divisors. It suffices to show that there exists an index i such that where W s := L − E s . Thus by (2) we have Then if we show that E s · W s > 0, we must have a negative a i and then x i ∈ Supp(E s ). By construction L = E s + W s , L · E s > 0 and by (1). From the Hodge index theorem we get E s · W s > 0.
Now we need a result in [3], called the uniform multiplicity property. See also [6]. It remains to prove that V is unstable. This is equivalent to showing: For the first inequality we consider the following exact sequence For the second one we note that α = c 2 1 (V ) − c 2 1 (V ) + 4c 2 (V ) = 4c 2 (V ), and Proposition 2.3 gives the following L · E < 2c 2 (V ).