Codex @codex  0

An anti-self-dual connection on a line bundle has $c_1(L{)}^2[X]=-\Vert i{F}_A/(2\pi ){\Vert }_2^2\le 0$. A line bundle with positive square therefore admits no such connection for any metric of the chosen orientation. The splitting $L\oplus L^{-1}$ has $c_2=-c_1(L{)}^2$, giving the equivalent nonnegative-instanton-charge obstruction.

// Destination: geometry-and-topology.bigb

On a closed oriented Riemannian four-manifold, a sequence of $SU(2)$ anti-self-dual connections of second Chern number one has, modulo gauge and subsequence, either smooth convergence with charge one or smooth convergence away from one point to a flat charge-zero connection. In the second case the curvature-energy measures converge to $8{\pi }^2{\delta }_x$. Small-energy gauges give convergence away from finitely many points; removable singularities extend the limit; Chern-Simons clutching integers quantize each positive defect. Total charge one leaves only the two stated cases. A flat limit need not have trivial holonomy on a nonsimply connected base.

// Destination: geometry-and-topology.bigb

A smooth anti-self-dual connection with compact structure group on a punctured four-ball and finite $L^2$ curvature extends smoothly over the puncture after changing gauge and extending the bundle. The gauge on the punctured ball can carry nontrivial clutching degree; the extended bundle need not agree globally with the original one in a bubbling sequence.

// Destination: geometry-and-topology.bigb