Tosatti–Weinkove's canonical surgical contraction conjecture for the Chern–Ricci flow

Assume that $X$ and $Y$ are compact Hermitian manifolds and that a holomorphic map

$$\pi:X\rightarrow Y$$

blows down an exceptional divisor $E$ to a point $y_0\in Y$. Suppose that a smooth function $\rho$ on $X$ satisfies

$$\omega_0-T\operatorname{Ric}(\omega_0)+dd^c\rho=\pi^*\omega_Y,$$

where $T<+\infty$. Let $\omega'$ be the push-down to $Y$ of the limiting current induced by the Chern–Ricci flow on $X$, and let $d_T$ be the distance on $Y$ obtained from the limiting metric space.

Tosatti–Weinkove's canonical surgical contraction conjecture. There exists a smooth maximal solution $\omega_t$ of the Chern–Ricci flow on $Y$ for $t\in(T,T_Y)$, with $T<T_Y\leq+\infty$, such that $\omega_t$ converges to $\omega'$ as $t\rightarrow T^+$ in $C^{\infty}_{loc}(Y\setminus\{y_0\})$. Furthermore, $\omega_t$ is uniquely determined by $\omega_0$, and the metric space $(Y,\omega_t)$ converges to $(Y,d_T)$ as $t\rightarrow T^+$ in the Gromov–Hausdorff sense.

This conjecture proposes that the Chern–Ricci flow continues uniquely after contracting the exceptional divisor, with the post-contraction metrics recovering the limiting metric space. The preceding convergence before time $T$ is known under the stated cohomological condition, but the continuation and its Gromov–Hausdorff behavior after the contraction remain open.