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We use the momentum construction of Calabi to study the conical Kähler–Ricci flow on Hirzebruch surfaces with cone angle along the exceptional curve, and show that either the flow Gromov–Hausdorff converges to the Riemann sphere or a single point in finite time, or the flow contracts only the cone divisor to a single point and Gromov–Hausdorff converges to a two dimensional projective orbifold. The limiting behaviour depends only on the cone angle, numerical properties of the initial Kähler class, and the degree of the Hirzebruch surface. This gives the first example of the conical Kähler–Ricci flow contracting the cone divisor to a single point, and shows that the conical flow may contract curves of self-intersection less than \((-\,1)\), as opposed to the smooth Kähler–Ricci flow. At the end, we introduce a conjectural picture of the geometry of finite time non-collapsing singularities of the flow on Kähler surfaces in general.

Metric contraction of the cone divisor by the conical Kähler–Ricci Flow