Chen's conjecture on smooth long-time existence of the Calabi flow

Theorem 1.1: there is a smooth $U(10)$-invariant Kahler form $\omega_0$ on $\mathbb{CP}^{10}$ in the Fubini-Study class whose maximal smooth Calabi flow exists only on $[0,T_*)$ with $0<T_*<\infty$, and at a point $p$ the scalar curvature blows up like $a(T_*-t)^{-1/2}$. Since the class contains the cscK Fubini-Study metric, smooth global existence fails even there. Corollary 1.2 gives finite-time singularities in every complex dimension $n\ge10$ on $\mathbb{CP}^{10}\times\mathbb{CP}^{n-10}$. Not shown: anything in complex dimensions 2 to 9, or about weak (Streets, Berman-Darvas-Lu) flows, which continue to exist.

Theorem 1.1: there is a smooth $U(10)$-invariant Kahler form $\omega_0$ on $\mathbb{CP}^{10}$ in the Fubini-Study class whose maximal smooth Calabi flow exists only on $[0,T_*)$ with $0<T_*<\infty$, and at a point $p$ the scalar curvature blows up like $a(T_*-t)^{-1/2}$. Since the class contains the cscK Fubini-Study metric, smooth global existence fails even there. Corollary 1.2 gives finite-time singularities in every complex dimension $n\ge10$ on $\mathbb{CP}^{10}\times\mathbb{CP}^{n-10}$. Not shown: anything in complex dimensions 2 to 9, or about weak (Streets, Berman-Darvas-Lu) flows, which continue to exist.