Stable Approximations of Certain Vector Bundles

0 −−−−→ O s −−−−→ E0 s∗ −−−−→ I −−−−→ 0. As usual, s is the distinguished nonzero global section of E0. The determinant line bundle of E0 is trivial, while the second Chern class is equal to the integer k. If X is in particular an algebraic surface, and H a fixed ample line bundle, then we can talk about slope-stability with respect to H. The bundle E0 is only slopesemistable, of slope 0, but there will be stable bundles among small deformations of E0. By a theorem of Donaldson, these stable approximations admit (essentially unique) ASD metrics; this note will answer the question what happens to the curvature of these ASD metrics as the approximations get closer to E0. The following result says that the curvature becomes more and more concentrated at the points x1, . . . , xk. Theorem (from The Geometry of Four-Manifolds, p. 230). Let Et be a family of holomorphic vector bundles, parametrized by t ∈ C, such that Et is slope-stable for t 6= 0, and with E0 as above. Let At be ASD-connections corresponding to Et for t 6= 0. Then the sequence [At] converges weakly to the ideal connection ( [θ], (x1, . . . , xk) ) as t→ 0.