Complex Sectional Curvature Blow-Up under Circle Cheeger Collapse

An AIM problem asks whether complex sectional curvatures remain uniformly bounded below in a collapsing circle Cheeger deformation. At a fixed point whose normal circle representation contains rotation blocks of speeds \(a,b>0\) , we derive an exact fixed-point formula. A totally isotropic complex two-plane has \[ \KC(g_\varepsilon)=\KC(g)-\frac{ab}{\varepsilon^2}. \] Thus every fixed component of real codimension at least four forces complex sectional curvature to diverge to \(-\infty\) . For the standard weight- \((1,1)\) action on the unit round \(S^4\) , the value at either fixed pole is \(1-\varepsilon^{-2}\) . With only one rotation block, the singular curvature operator is instead rank one and positive semidefinite. The AIM workshop report records the corresponding negative curvature-operator phenomenon but does not identify a complex decomposable direction; the observation here is that its two real negative directions combine into a decomposable totally isotropic bivector. The note is unrefereed and makes no priority claim for this elementary calculation.