Stable vector bundles on fibered threefolds over a surface (Q6618808)

In this paper, the author investigates the existence of stable vector bundles on fibered threefolds. Let \(X\) be a smooth projective threefold over \(\mathbb C\). Let \(H\) be an ample divisor on \(X\), and \(D\) be a divisor on \(X\) which satisfies \(D \cdot H^2 = 0\) and is not numerically trivial. Put \(H_\epsilon = H + \epsilon D\) for sufficiently small \(\epsilon > 0\). When \(X\) is a Fano proper standard conic bundle over \(\mathbb P^2\), \(H\) is the anti-canonical divisor \(-K_X\) and \(D = -(K_X^2 \cdot \pi^*L) K_X + K_X^3 \cdot \pi^*L\) where \(\pi: X \to \mathbb P^2\) is the projection and \(L\) is a line in \(\mathbb P^2\), the author proves that for \(r = 3\) or \(4\), there exists a family \(\{ E_n \}\) of \(H_\epsilon\)-stable rank-\(r\) vector bundles on \(X\) with \(c_1(E_n) = 0\) and \(c_2(E_n) \cdot H \to \infty\) as \(n \to \infty\). Similar result is obtained when the Fano threefold \(X\) is a certain \(\mathbb P^1\) bundle over \(\mathbb P^1 \times \mathbb P^1\) and \(r \ge 3\). The main idea in the proofs is to use inductively non-trivial extensions of stable vector bundles with lower ranks.