A finite-time singularity of Calabi flow

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Disproves Chen’s smooth long-time existence conjecture for Calabi flow by constructing a smooth \(U(10)\)-invariant Kähler metric on \(\mathbb{CP}^{10}\) whose flow develops a finite-time singularity. The metric lies in the Fubini–Study class, so the failure occurs even in a class containing a constant-scalar-curvature metric.

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Theorem 1.1

In A finite-time singularity of Calabi flow on projective space, 24 September 2026

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There is a smooth U(10)-invariant Kähler form \omega_0 on \CP^{10}, in the class [\omega_{\mathrm{FS}}], whose maximal smooth Calabi flow has existence interval [0,T_*) with 0<T_*<\infty. There are a point p\in\CP^{10} and a constant a>0 such that

S(\omega(t))(p)\sim a(T_*-t)^{-1/2}\qquad\text{as }t\uparrow T_*.

Corollary 1.2 (Products with stationary factors)

In A finite-time singularity of Calabi flow on projective space, 24 September 2026

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Let \omega(t), T_*, p, and a be as in Theorem 1.1, and let (Y,\eta) be a compact connected Kähler manifold with \eta of constant scalar curvature. With \pi_X and \pi_Y the projections from \CP^{10}\times Y, the maximal smooth Calabi flow from \Omega_0=\pi_X^*\omega_0+\pi_Y^*\eta is

\Omega(t)=\pi_X^*\omega(t)+\pi_Y^*\eta, \qquad 0\leq t<T_*.

For every y\in Y, S(\Omega(t))(p,y)\sim a(T_*-t)^{-1/2}. In particular, finite-time smooth Calabi-flow singularities occur in every complex dimension n\geq10, on \CP^{10}\times\CP^{n-10}, in a class containing a Kähler–Einstein metric.

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A finite-time singularity of Calabi flow on projective space

24 September 2026, current edition.

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We construct a smooth Kähler metric in the Fubini–Study class of \(\mathbb{CP}^{10}\) whose Calabi flow develops unbounded scalar curvature in finite time. This disproves Chen’s smooth long-time existence conjecture, even in a class containing a constant-scalar-curvature metric.

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Proposition A.1 of A finite-time singularity of Calabi flow on projective space, 24 September 2026
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“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“the whole of Proposition cert:shoot (existence on [0,56], 1 + p > 0, |q| <= .25R, ||(Re xi, Im xi) - z*|| <= .13R for every z* in the square), by an independent validated integration; not the tail lemma, the matching or the theorem”
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21 references in A finite-time singularity of Calabi flow on projective space

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