Sequences of stable bundles over compact complex surfaces (Q1589352)

Here and in a companion paper to appear in Pac. J. Math. 196, 69-111 (2000; Zbl 1073.32506) the author studies sequences of irreducible Hermitian-Einstein connections and sequences of stable holomorphic bundles of fixed topological type and bounded degree on a compact complex surface \(X\) equipped with a Gauduchon metric. He proves the following result. Theorem. Let \(X\) be a compact complex surface with \(b_1(X)\) even equipped with a \(\overline\partial\partial\)-closed (1,1) form \(\omega\). Let \(\{A_i\}\) be a sequence of Hermite-Einstein connections on a fixed unitary rank 2 bundle \(E_{\text{top}}\) such that the corresponding holomorphic bundles are stable and have uniformly bounded degree. Suppose that \(E_i\) converges weakly to \(E\) of a finite set \(S\). Then there is a subsequence \(\{E_{i_j}\} \subset\{E_i\}\) such that: 1. There is a sequence of blow-ups \(\pi_{i_j}: \widetilde X_{i_j}\to X\) of \(X\) consisting of at most \(2C(E_{\text{top}})- 2C(E)-1\) individual blow-ups converging to a blow-up \(\pi:\widetilde X\to X\) of \(X\); here \(C(E):= (c_2-c_1^2/4)(E)\) is the charge; 2. \(\pi^*_{i_j} (E_{i_j})\) is stable with respect to a suitable \(\overline\partial \partial\)-closed positive (1,1) form constructed using \(\omega\) and the corresponding sequence of Hermitian-Einstein connections converges strongly on \(X\) to define a stable bundle \(\widetilde E\) on \(\widetilde X\); 3. \(\det(\widetilde E)\cong \pi^*(\det(E))\), \(\pi^*(E)\) is semistable and there are non-zero homomorphisms \((\pi_* (\widetilde E))^{**} \to E\) and \(E\to(\pi_* (\widetilde E))^{**}\). The corresponding statement for arbitrary compact complex surfaces and for bundles of rank \(r>2\) is too long to be stated here. The proofs use a study of sequences of bundles on the blowing up of \(\mathbb{C}^2\) at 0. A key point is to prove the corresponding compactness for sequences of instantons on \(S^4\) viewed as holomorphic bundles on \(\mathbb{C}\mathbb{P}^2\) trivial on the line at infinity and with a fixed such trivialization. The Atiyah-Ward correspondence translates the problem concerning instantons on \(S^4\) to a problem of monads of holomorphic vector bundles on \(\mathbb{C}\mathbb{P}^3\). Several pages are devoted to the study of convergence for monads and this part should have an independent interest.