Rank 2 vector bundles on higher dimensional projective manifolds (Q1366625)

A well-known theorem of D. Gieseker and J. Li asserts that the moduli space of stable rank 2 vector bundles is reducible if ``the second Chern number'' is large enough and in particular the number of its irreducible components is bounded when \(c_2\) goes to infinity. An old result of Ein shows that this is not true on \(\mathbb{P}^3\). Here we use Ein's result to show that the same is true for a large class of higher-dimensional varieties. Fix a smooth complex projective \(n\)-fold \(X\), \(n\geq 5\), with \(H^1(X,{\mathcal O}_X)=0\). Then there is a smooth projective \(n\)-fold \(Y\) birational to \(X\) and \(H\in\text{Pic}(Y)\), \(H\) ample, with the following property: For any \(d\in\mathbb{Z}\) let \(M(Y,2{\mathcal O},d,H)\) be the moduli scheme of \(H\)-stable rank 2 vector bundles, \(E\), on \(Y\) with \(\text{det}(E)\cong{\mathcal O}_Y\) and \(c_2(E)\cdot H^{n- 2}=d\). Let \(m(Y,2,{\mathcal O},d,H)\) be the number of its irreducible components. Then \(\limsup_{d\to\infty} m(Y,2,{\mathcal O},d,H)=+\infty\).