Stable vector bundles on fibered threefolds over a surface

Let X be a smooth projective threefold and let H be an ample line bundle on X. We investigate the existence of vector bundles on X which are \(\mu \)-stable with respect to an ample divisor \(H_{\epsilon }=H+\epsilon D\) for sufficiiently small \(\epsilon >0\) where D is a divisor with \(D\cdot H^2=0\). In particular, when X is a Fano conic bundle over a rational surface, we show that there exists a family \(\{E_n\}\) of \(H_{\epsilon }\)-stable vector bundles with \(c_1(E_n)=0\) and \(c_2(E_n)\cdot H\) becomes arbitrarily large as n goes to infinity.