A cohomological interpretation of Bogomolov's instability (Q2845454)

\textit{F. A. Bogomolov}'s instability theorem [Math. USSR, Izv. 13, 499--555 (1979); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 42, 1227--1287 (1978; Zbl 0439.14002)] states that every rank \(2\) vector bundle \(E\) with \(c_1(E)^2 > 4c_2\) on a smooth projective surface is unstable. It is known [\textit{F. Sakai}, Lect. Notes Math. 1417, 307 (1990)] that Bogomolov's theorem implies the following theorem: If \(D\) is a big divisor with \(D^2 > 0\) on a smooth projective surface \(X\) such that \(H^1(X,(O)_X(K_X+D)) \neq 0\), then there is an effective divisor \(E\) such that (1) \(D-2E\) is big, (2) \((D-E)\cdot E \leq 0\). This paper gives a proof of the following result: Bogomolov's and Sakai's theorems are equivalent.