Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows

TL;DR

The paper proves that for a compact Kähler manifold with nef canonical bundle, the long-time singularity type of the normalised Kähler-Ricci flow is independent of the initial metric: if one solution is Type III, all are Type III, and similarly for Type IIb. The authors develop a framework that uses a Monge–Ampère reduction to compare an arbitrary solution with a Type III reference flow, establishing metric equivalence and transferring curvature bounds. They first analyze the special case of flows starting from a scaled initial metric, then extend to general higher-order estimates via a maximum-principle argument on a tensor measuring the difference of connections, ensuring uniform curvature control. These results generalize prior semi-ample cases and bolster the understanding of the infinite-time behavior of Kähler-Ricci flows on nef manifolds, with implications for the abundance conjecture and collapsing phenomena in related geometric settings.

Abstract

In this paper, we show that the singularity type of solutions to the Käher-Ricci flow on a numerically effective manifold does not depend on the initial metric. More precisely if there exists a type III solution to the Kähler-Ricci flow, then any other solution starting from a different initial metric will also be Type III. This generalises previous results by Y. Zhang for the semi-ample case.

Figures (1)

• Figure 1: Scaling of Initial Metric

Theorems & Definitions (10)

• Definition 1.1
• Conjecture 1.2: Tosatti T15Kawa
• Theorem 1.3: Tosatti-YG. Zhang TygZ15
• Theorem 1.4
• Remark 1.5
• Lemma 2.1
• proof : Proof of Lemma \ref{['2L']}
• Lemma 2.2
• proof
• Proposition 2.3: [Y. Zhang ysZ20] Let $\widetilde{\omega}(t)$ be a type III solution to the normalised Kähler-Ricci flow and $\omega(t)$ be a arbitrary solution such that for some $C_0>0$, $C_0^{-1}\widetilde{\omega}(t) \leqslant \omega(t) \leqslant C_0 \widetilde{\omega}(t),$ for all $t \geqslant 0$. Then there exists $C>0$ such that $|\text{Rm}(\omega (t))|_{\omega(t)} \leqslant C,$ for all $t \geqslant 0$. Let $\Psi=\left(\Psi_{i j}^k\right)$ where $\Psi_{i j}^k:=\Gamma_{i j}^k-\widetilde{\Gamma}_{i j}^k$ and $S = |\Psi|_\omega^2$. In local coordinates, we have $S=g^{\bar{j} i} g^{\bar{l} k} g^{\bar{q} p} \widetilde{\nabla}_i g_{k \bar{q}} \overline{\widetilde{\nabla}_j g_{l \bar{p}}} .$ For the last term in \ref{['2traceev']}, we have $g^{\bar{j} i} g^{\bar{q} p} \widetilde{g}^{\bar{b} a} \nabla_i \widetilde{g}_{p \bar{b}} \nabla_{\bar{j}} \widetilde{g}_{a \bar{q}}=g^{\bar{j} i} g^{\bar{q} p} \widetilde{g}^{\bar{b} a}\left(\nabla_i-\widetilde{\nabla}_i\right) \widetilde{g}_{p \bar{b}}\left(\nabla_{\bar{j}}-\widetilde{\nabla}_{\bar{j}}\right) \widetilde{g}_{a \bar{q}}=g^{\bar{j} i} g^{\bar{q} p} \widetilde{g}^{\bar{b} a}\left(-\Psi_{i p}^d\right) \widetilde{g}_{d \bar{b}}\left(-\overline{\Psi_{j q}^e}\right) \widetilde{g}_{a \bar{e}}\geqslant C^{-1} S .$ Hence, we have the estimate $\left(\pdv{}{t}-\Delta_\omega\right) t r_\omega \widetilde{\omega} \leqslant C-C^{-1} S.$ The evolution of the tensor $S$ is given by $\left(\partial_t-\Delta_\omega\right) S=S-|\nabla \Psi|_\omega^2-|\overline{\nabla} \Psi|_\omega^2 + \langle \widetilde{\nabla}_i \widetilde{R}_p^k - \Psi * R m(\widetilde{\omega})- g^{\bar{b} a} \widetilde{\nabla}_a \widetilde{R}_{i \bar{b} p}^k, \Psi \rangle_{\omega},$ where $\langle \cdot , \cdot \rangle_\omega$ is the tensor inner product induced by the metric $g(t)$ associated with $\omega(t)$. Following Hamilton's argument in H82 (or see SW13Intro), the assumption \ref{['2L2As']} on the curvature tensor of $\widetilde{\omega}$ implies $\left|\nabla_{\widetilde{\omega}(t)} R m(\widetilde{\omega}(t))\right|_{\widetilde{\omega}(t)} \leqslant C,$ for some $C>0$. In light of this, we estimate \ref{['2S']} by $\left(\pdv{}{t}-\Delta_\omega\right) S \leqslant C S+C-|\nabla \Psi|_\omega^2-|\overline{\nabla} \Psi|_\omega^2.$ We now have all the pieces to set up a maximum principle argument. Let $Q:=S + A \tr_{\omega} \widetilde{\omega}$ for some sufficiently large $A>0$. Using \ref{['2traceest']} and \ref{['2Sev']}, there exists $C>0$ such that $\left(\pdv{}{t}-\Delta_\omega\right)Q \leqslant -S+C .$ By a standard maximum principle argument and noting that $\tr_{\omega(t)} \widetilde{\omega}(t) \leqslant C$, we find a constant $C \geqslant 1$ such that $S \leqslant C.$ The last term in \ref{['2Sev']} can be estimated by $|\overline{\nabla} \Psi|_\omega^2=\left|\widetilde{R}_{i \overline{b} p}^k-R_{i \overline{b} p}^k\right|_\omega^2 \geqslant \frac{1}{2}|R m(\omega)|_\omega^2-C,$ thus $\left(\pdv{}{t}-\Delta_\omega\right) S \leqslant C-\frac{1}{2}|R m(\omega)|_\omega^2 .$ The evolution of curvature can be estimated by $\left(\pdv{}{t}-\Delta_\omega\right)|R m(\omega)|_\omega \leqslant C|R m(\omega)|_\omega^2-\frac{1}{2}|R m(\omega)|_\omega.$ Let $Q := |\text{Rm}(\omega)|_{\omega} + A S$ for some large constant $A>0$, we combine the previous two estimates to obtain $\left(\pdv{}{t}-\Delta_\omega\right)\left(|R m(\omega)|_\omega+A S\right) \leqslant-\frac{1}{2}|R m(\omega)|_\omega+C.$ Then by the maximum principle, there exists $C>0$ such that $\sup _{X \times[0, \infty)}|R m(\omega)|_\omega \leqslant C.$ Using Lemma \ref{['2L']}, we scale a Type III solution $\widetilde{\omega}(t)$ twice to $\omega^+(t)$ and $\omega^-(t)$, such that $\omega_0^- := \lambda_0^- \widetilde{\omega_0} \leqslant \widetilde{\omega}_0 \leqslant \lambda_0^+ \widetilde{\omega}_0 := \omega_0^+$. We then choose $\lambda_0^-$ small enough and $\lambda_0^+$ large enough, such that $\omega_0^{-} \leqslant \omega_0 \leqslant \omega_0^+$. Furthermore, these solutions starting from these scaled metrics are type III. We treat these new solutions act as reference metrics, where either $\omega^+(t)$ or $\omega^-(t)$ is chosen to obtain a favourable sign when carrying out maximum principle arguments. Scaling of Type III solutions Let $\chi \in [\omega_\infty]$ be a representative of the limiting class under the Kähler-Ricci flow. Using the usual reference metrics, we have $\omega(t)= e^{-t} \omega_0 + (1-e^{-t}) \chi + \sqrt{-1} \partial \overline{\partial} \varphi,\omega^+(t)= e^{-t} \omega^+_0 + (1-e^{-t}) \chi + \sqrt{-1} \partial \overline{\partial} \varphi^+, \quad \text{ and}\omega^-(t)= e^{-t} \omega^-_0 + (1-e^{-t}) \chi + \sqrt{-1} \partial \overline{\partial} \varphi^-.$ In our proof of Theorem \ref{['T1']}, we require three Monge-Ampère equations relating pairs of $\omega(t)$, $\omega^+(t)$ and $\omega^-(t)$. Taking a time derivative of $\omega(t)$ and $\omega^-(t)$ in \ref{['3ref']}, and using the flow equation, we arrive at a Monge-Ampère equation for $u:= \varphi - \varphi^-$ given by $\pdv{u}{t} = \log\left( \frac{\omega^n}{(\omega^-)^n} \right) - u$ where $\omega(t) = \omega^-(t) + e^{-t}(\omega_0 - \omega^-_0) + \sqrt{-1} \partial \overline{\partial} u.$ Equation \ref{['3MAu']} will be the primary Monge-Ampère utilised in the proof of the main Theorem. Note that the construction of $\omega^-_0$ ensures $\omega_* := \omega_0 - \omega^-_0 >0.$ Similarly for $\psi := \varphi^+ - \varphi^-$ we have $\pdv{\psi}{t} = \log \left( \frac{(\omega^+)^n}{(\omega^-)^n} \right) - \psi.$ The function $\psi$ relates the two evolving metrics by $\omega^+(t) = \omega^-(t) + e^{-t} (\omega_0^+ - \omega^-_0) + \sqrt{-1} \partial \overline{\partial} \psi.$ Finally, for $v:= \varphi - \varphi^+$, we have the following Monge-Ampere equation $\pdv{v}{t} = \log\left( \frac{\omega^n}{(\omega^+)^n} \right) - v$ where $\omega(t) = \omega^+(t) + e^{-t}(\omega_0 - \omega^+_0) + \sqrt{-1} \partial \overline{\partial} v.$ Our aim is to derive potential estimates for $u$ and its time derivative. We begin by calculating the evolution of $u$ and $\pdv{u}{t}$. As before, we drop the $t$ dependence on the metrics and potential functions for ease of notation. To denote geometric quantities associated with $\omega^+(t)$ and $\omega^-(t)$ with their respective symbol in superscripts. For $\omega_* := \omega_0 - \omega^-_0 >0$, any solution $u$ to \ref{['3MAu']} satisfies $\left(\pdv{}{t} - \Delta_{\omega(t)}\right)u = \pdv{u}{t} - n + \tr_{\omega} \omega^- + e^{-t} \tr_\omega \omega_*,$ and $\left(\pdv{}{t} - \Delta_{\omega(t)}\right)\pdv{u}{t} = R^- - \tr_{\omega} \text{Ric}(\omega^-) - \pdv{u}{t} - \tr_{\omega} \omega^- - e^{-t} \tr_\omega \omega_* + n.$ The first equation, \ref{['3lemev1']} follows from \ref{['3ref']}; $\left(\pdv{}{t} - \Delta_{\omega(t)}\right)u = \pdv{u}{t} - \tr_{\omega} (\omega - \omega^- - e^{-t}\omega_*) = \pdv{u}{t} - n + \tr_{\omega} \omega^- + e^{-t} \tr_\omega \omega_*.$ Equation \ref{['3lemev2']} is derived using \ref{['3MAu']}; $\left(\pdv{}{t} - \Delta_{\omega(t)}\right) \pdv{u}{t}= \left(\pdv{}{t} - \Delta_{\omega(t)}\right) \log \left( \frac{\omega^n}{{\omega^-}^n}\right) - \left(\pdv{}{t} - \Delta_{\omega(t)}\right)u= R^- - \tr_{\omega} \text{Ric}(\omega^-) - \pdv{u}{t} + n - \tr_{\omega} \omega^- - e^{-t} \tr_{\omega}\omega_* .$ Let $u$ be a solution to \ref{['3MAu']}. Then there exists a uniform $C>0$ such that $-C e^{-t} \leqslant u(t) \leqslant C$ for all $t \geqslant 0$. The lower bound follows from the maximum principle. Indeed, at a minimum point of $u$, $\pdv{}{t} \left( e^t u \right) = e^t \log \left( \frac{(\omega^-(t) + e^{-t}(\omega_0-\omega_0^-)+\sqrt{-1}\partial \overline{\partial} u)^n}{(\omega^-)^n}\right) \geqslant e^t\log(1) = 0.$ Integrating yields $u(t) \geqslant - Ce^{-t}.$ To derive an upper bound for $u$, we first observe that at a maximum for $v$ in \ref{['3MAv']}, we have $\pdv{v_{\max}}{t} \leqslant \log \left( \frac{(\omega_+ + e^{-t}(\omega_0 - \omega_0^+))^n}{(\omega^+)^n} \right) - v_{\max} \leqslant - v_{\max}.$ Then by the maximum principle, we deduce that $v \leqslant 0$. On the other hand, we have $C>0$ such that $|\psi| \leqslant C$ for all $t>0$. Indeed, the initial metrics $\omega^+_0, \omega^-_0$ and $\widetilde{\omega}_0$ are scalar multiples of each other, Lemma \ref{['2L']} and \ref{['3MApsi']} imply $\pdv{}{t} \left( e^t \psi \right) = e^t \log \left( \frac{(\omega^+)^n}{(\omega^-)^n} \right) \leqslant Ce^{t}$ for some $C>0$. Integrating the above gives us \ref{['psibdd']}. It now follows that $u(t) = \varphi(t) - \varphi^-(t) = v(t) + \psi(t) \leqslant C$ where $C>0$ is independent of time. Let $u$ be a solution to \ref{['3MAu']}. Then there exists $C>0$ such that $\left|\pdv{u}{t}\right| \leqslant C,$ for all $t \geqslant 0$. We first note that Lemma \ref{['2L']} implies that for some $C>0$, we have $C^{-1}\widetilde{\omega} (t) \leqslant \omega^- (t) \leqslant C \widetilde{\omega} (t),$ and $|\text{Rm}(\omega^-(t))|_{\omega^-(t)}^2 \leqslant C,$ for all $t \geqslant 0$. Let $Q:= \pdv{u}{u} - Au,$ where $A>0$ a constant that will be set later. By combining \ref{['3lemev1']} and \ref{['3lemev2']}, and using \ref{['3curbound']}, we obtain $\left( \pdv{}{t} - \Delta_{\omega} \right) Q= R^- - \tr_{\omega} \text{Ric} (\omega^-) -(A+1) \tr_{\omega} \omega^- - (A+1)e^{-t}\tr_{\omega} \omega_* - (A+1)\pdv{u}{t}+ (A+1)n\leqslant -(A+1-C_0) \tr_{\omega} \omega^- - (A+1)e^{-t}\tr_{\omega} \omega_* - (A+1)\pdv{u}{t}+ (A+1+C_0)n.$ We choose $A = 2+C_0$ to obtain the estimate $\left( \pdv{}{t} - \Delta_{\omega} \right) Q \leqslant - C \pdv{u}{t} +C.$ Applying the parabolic maximum principle yields the upper bound $\pdv{u}{t} \leqslant C,$ for some $C>0$. For the lower bound, we instead consider the quantity $Q:= \pdv{u}{t} +Au,$ for some $A >0$ to be determined later. Once again, we use \ref{['3lemev1']}, \ref{['3lemev2']} and \ref{['3curbound']} to set up a maximum principle argument: $\left( \pdv{}{t} - \Delta_{\omega} \right) Q= R^-(t) - \tr_{\omega} \text{Ric} (\omega^-) + (A-1) \tr_{\omega} \widetilde{\omega} +(A-1)e^{-t}\tr_{\omega}\omega_* + (A-1)\pdv{u}{t}+ n \left( 1 - A \right)\geqslant (C_0+1-A)n +(A-1)\pdv{u}{t}+(A -C_0-1)\tr_{\omega}\omega^- +(A-1)e^{-t}\tr_{\omega}\omega_*.$ Applying a similar argument to before, we choose $A = 2+C_0$ to obtain an estimate $\left( \pdv{}{t} - \Delta_{\omega} \right) Q\geqslant -C + C\pdv{u}{t} + \tr_{\omega}\omega^-\geqslant -C + C \pdv{u}{t} + n \left( \frac{(\omega^-)^n}{\omega^n} \right)^\frac{1}{n}= -C + C\pdv{u}{t} +n e^{- \frac{1}{n}\left( \pdv{u}{t} + u \right)}$ Then applying the maximum principle, using that $u$ is uniformly bounded, shows that there exists a $C>0$ independent of time such that $\pdv{u}{t} \geqslant -C.$ We now derive a metric equivalence between the Type III solution $\widetilde{\omega}$ and the arbitrary solution $\omega$. From a standard calculation, using \ref{['3curbound']}, we have for some $C>0$ and $C_0>0$ $\left(\pdv{}{t} - \Delta_{\omega} \right) \log \operatorname{tr}_\omega \omega^- \leqslant C_0 \operatorname{tr}_\omega \omega^- +C,$ for all times $t \geqslant 0$. The uniform bound on $\pdv{u}{t}$ from Lemma \ref{['3pot2']} implies that $\left(\pdv{}{t} - \Delta_{\omega} \right) u = \pdv{u}{t} -n + \tr_{\omega} \omega^- + e^{-t} \tr_{\omega} \left( \omega_0-\omega_0^- \right) \geqslant -C + \tr_{\omega} \omega^-.$ Following a similar argument as before, we define $Q: = \log \tr_{\omega} \omega^- - Au$, then using the two inequalities, we choose $A$ large enough such that $\left(\pdv{}{t} - \Delta_{\omega} \right)Q \leqslant C - \tr_{\omega} \omega^-.$ Applying the maximum principle, and again using Lemmas \ref{['3pot1']} and \ref{['3pot2']}, we obtain some $C>0$ such that $\tr_{\omega} \omega^- \leqslant C,$ for all $t \geqslant 0$. Combining the potential estimates from Lemma \ref{['3pot1']} and \ref{['3pot2']}, we derive from the Monge–Ampère equation \ref{['3MAu']} the volume bounds $C^{-1} \leqslant \frac{\omega^n}{(\omega^-)^n} \leqslant C ,$ for some $C>0$ for all $t \geqslant 0$. Similar to before, we employ standard eigenvalue estimates to obtain $\tr_{\omega^{-}} \omega \leqslant n \left( \frac{\omega^n}{(\omega^{-})^n}\right) \tr_{\omega} \omega^{-} \leqslant C.$ Then the following metric equivalence follows from the two trace bounds; $C^{-1} \omega^{-}(t) \leqslant \omega(t) \leqslant C \omega^{-}(t).$ Then the above inequality and \ref{['3meq1']} yields \ref{['T1equiv']}. To obtain bounds on the curvature tensor, we apply Proposition \ref{['2propcurvbdd']} with $\omega^{-}(t)$ in place of $\widetilde{\omega}$. This completes the proof of Theorem \ref{['T1']}. We can now apply Theorem \ref{['T1']} to answer Conjecture \ref{['conj']}. Let $X$ be a Kähler manifold with numerically effective canonical line bundle. Then the singularity type of solutions to the Kähler-Ricci flow is independent of the initial metric. Suppose a solution $\widetilde{\omega}$ is Type III. Then by Theorem \ref{['T1']}, any other solution starting from a different initial metric must also be Type III. On the other hand, if a type IIb solution exists, then any other solution must be Type IIb as otherwise, Theorem \ref{['T1']} implies that all solutions must be Type III, a contradiction. Finally, we note that Theorem \ref{['T1']} can be interpreted as evidence for the Kähler extension of the abundance conjecture. Indeed, if a counter example to Theorem \ref{['T1']} could be constructed, then Y. Zhang's result ysZ20 implies that the manifold cannot be semiample. Our results and techniques have applications and generalisations in many settings. We start with some examples where an explicit solution to the normalised Kähler-Ricci Flow is known. This allows us to deduce the singularity type for an arbitrary initial metric for these manifolds. These examples are obtained from ysZ19. Calabi-Yau Metrics: Suppose that $X$ admits a Calabi-Yau metric. Then under the Kähler-Ricci flow, we check that $\omega(t) = e^{-t} \omega_{CY}$ is a solution with $\omega(0) = \omega_{CY}$. For such solutions, the curvature evolves as $|\text{Rm}(\omega(t))|_{\omega(t)}=e^t\left|\text{Rm}\left(\omega_{C Y}\right)\right|_{\omega_{C Y}},$ and therefore develops a type III singularity if and only if $\omega_{CY}$ is a flat metric, which is only possible if and only if $X$ is a finite quotient of a torus. Therefore, we conclude from Theorem \ref{['T1']} that the Kähler-Ricci Flow develops a type III solution on a numerically effective manifold $X$ if and only if it is a finite quotient of a torus. Furthermore, this demonstrates Item (1) of Theorem \ref{['T0']} without assuming $K_X$ is semiample. Item (2) in Theorem \ref{['T0']} assumes $K_X$ is big. Since nef and big imply semiample, the generalisation to nef is immediate.Product Manifold: Consider the product manifold $X:= B \times Y$ where $Y$ is a Calabi-Yau manifold. This is a special case of item (3). Suppose that $\omega$ is a Kähler-Einstein metric on $B$, then we check that $\widetilde{\omega}(t) = e^{-t} \omega_{CY} + (1-e^{-t})\omega_B$ is a solution to the Kähler-Ricci flow. The curvature is given by $|\text{Rm}(\omega(t)|_{\omega(t)}^2 = e^{2t}|\text{Rm}(\omega_{CY})|_{\omega_{CY}}^2 + \frac{1}{(1-e^{-t})^2}|\text{Rm}(\omega_B)|_{\omega_B}^2.$ Thus, we have a type III singularity if and only if $\omega_{CY}$ is flat which occurs if $Y$ is a finite quotient of a torus. Thus, a type III singularity develops if and only if $Y$ is a finite quotient of a torus. Thus, using Theorem \ref{['T1']}, any initial metric $\omega_0$, there exists $C>0$ such that $C^{-1} \widetilde{\omega}(t) \leqslant \omega (t)\leqslant C \widetilde{\omega}(t)$ and $|\text{Rm}(\omega(t))|_{\omega(t)} \leqslant C,$ for all $t \geqslant 0$. Furthermore, our method can be applied to other flows, for example, the modified Kähler-Ricci flow as studied in zZ09modyY11WzZ22. 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