1.1 Background and motivation

Let $(X,\omega _0)$ be a compact Kähler manifold, and let $\omega ^{\bullet }(t),t\in [0,T)$ be a family of Kähler metrics on X which solve the Kähler-Ricci flow

(1.1) $$ \begin{align} \partial_t\omega^{\bullet} (t)=-\textrm{Ric}(\omega^{\bullet} (t))-\omega^{\bullet}(t), \quad \omega^{\bullet} (0)=\omega_0 \end{align} $$

for some $0<T\leqslant +\infty $ . In this paper, we are interested in the case when $\omega ^{\bullet }(t)$ is an immortal solution (i.e., when $T=+\infty $ ). Thanks to a result of Tian-Zhang [Reference Tian and Zhang37] (see also [Reference Tsuji45]), we know that the solution $\omega ^{\bullet }(t)$ is immortal if and only if the canonical bundle $K_X$ is nef, which means that $c_1(K_X)$ lies in the closure of the cone of Kähler classes in $H^{1,1}(X,\mathbb {R})$ . This condition does not depend on $\omega _0$ , and manifolds with $K_X$ nef are also known as smooth minimal models.

The Abundance Conjecture in birational geometry, and its natural extension to Kähler manifolds, predicts that if the canonical bundle of a compact Kähler manifold is nef, then it must be semiample, which means that $K_X^p$ is base-point-free for some $p\geqslant 1$ . This conjecture is known when $\dim X\leqslant 3$ by [Reference Campana, Höring and Peternell1, Reference Das and Ou6, Reference Das and Ou7].

Throughout the rest of the paper, we will assume that $K_X$ is semiample. It is then known (see, for example, [Reference Lazarsfeld25, Theorem 2.1.27]) that global sections of $K_X^p$ for $p\geqslant 1$ sufficiently divisible define a surjective holomorphic map $f:X\to B\subset \mathbb {CP}^N$ (the Iitaka fibration of X) with connected fibers onto a normal projective variety B (known as the canonical model of X), of dimension m equal to the Kodaira dimension of X (in particular, we have $0\leqslant m\leqslant \dim X$ ). The smooth fibers of f are then Calabi-Yau manifolds, of dimension $n:=\dim X-m$ , which are pairwise diffeomorphic but in general are not pairwise biholomorphic.

In the two extreme cases when $m=0$ or $m=\dim X$ , the behavior of the flow is completely understood thanks to the work of many people (see, for example, the recent survey [Reference Tosatti42] and references therein), so we will furthermore assume from now on that $0<m<\dim X$ , which is known as ‘intermediate Kodaira dimension’. Thus, we have $\dim X=m+n$ , and both the fibers and the base of f are positive-dimensional.

The simplest examples of this setup arise when $m=n=1$ , where X is a minimal properly elliptic surface, B is a compact Riemann surface, and $f:X\to B$ is an elliptic fibration. In this case, the behavior of the Kähler-Ricci flow (1.1) was first studied by Song-Tian [Reference Song and Tian31], who shortly afterwards also considered the case of general $m,n$ in [Reference Song and Tian32]. A major difficulty in this setting is that the total volume of $(X,\omega ^{\bullet }(t))$ is easily seen to converge to zero as $t\to +\infty $ , and this ‘collapsing’ behavior makes it extremely hard to analyze the flow. As we will now explain, in [Reference Song and Tian31, Reference Song and Tian32], it was shown that the metrics $\omega ^{\bullet }(t)$ collapse in the weak topology to the pullback of a canonical metric on B, and our main goal is to obtain higher order regularity and a uniform Ricci curvature bound for $\omega ^{\bullet }(t)$ (away from the singular fibers of f) and thus prove two conjectures of Song-Tian.

When X is projective, the condition that $K_X$ be nef means that X is a smooth minimal model. The connection between the Minimal Model Program in birational geometry and the behavior of the Kähler-Ricci flow was first discovered independently by Cascini-La Nave [Reference Cascini and La Nave3] and Song-Tian [Reference Song and Tian31], and remains an area of current research. These works outlined a conjectural picture for the behavior of the Kähler-Ricci flow on any projective (or more generally compact Kähler) manifold. When $K_X$ is not nef, singularities must develop in finite time, and the flow should implement the corresponding birational contractions or collapse the fibers of a Mori fiber space. The case when $K_X$ is nef (so the manifold is a smooth minimal model) is the topic of our paper.

1.2 Setup

We now describe our setup in more detail. As mentioned above, we have a compact Kähler manifold $(X^{m+n},\omega _0)$ with semiample canonical bundle and intermediate Kodaira dimension m (so $m,n>0$ ), and $\omega ^{\bullet }(t)$ denotes the immortal solution of the Kähler-Ricci flow (1.1). Let $f:X\to B$ be the Iitaka fibration of X, and let $S\subset X$ be the preimage of the union of the set of singular values of f and the singular set of B. Thus, by construction, $f:X\backslash S\to B\backslash f(S)$ is a proper holomorphic submersion with n-dimensional connected Calabi-Yau fibers $X_{z}=f^{-1}(z), z\in B\backslash f(S)$ . By Ehresmann’s Lemma (and the connectedness of $B\backslash f(S)$ ), the fibers $X_z$ are pairwise diffeomorphic, but, in general, their complex structure varies with z, and this variation can be encoded in a smooth semipositive Weil-Petersson form $\omega _{\mathrm {WP}}\geqslant 0$ on $B\backslash f(S)$ , defined in [Reference Song and Tian31] (see also [Reference Tosatti39, §5.6]).

By [Reference Song and Tian32], there exists a smooth Kähler metric $\omega _{\mathrm {can}}$ on $B\backslash f(S)$ satisfying the twisted Kähler-Einstein equation

(1.2) $$ \begin{align} \textrm{Ric}(\omega_{\mathrm{can}})=-\omega_{\mathrm{can}}+\omega_{\mathrm{WP}}. \end{align} $$

The pullback of $\omega _{\mathrm {can}}$ to $X\backslash S$ will also be denoted by the same symbol, for convenience. The metric $\omega _{\mathrm {can}}$ extends to a closed positive current on B, and in [Reference Song and Tian31, Reference Song and Tian32] it is shown that as $t\to +\infty $ , we have

(1.3) $$ \begin{align} \omega^{\bullet}(t)\to \omega_{\mathrm{can}}, \end{align} $$

weakly as currents on X as well as in the $C^{0}_{\mathrm {loc}}(X\backslash S)$ topology of Kähler potentials. Motivated by this, in [Reference Song and Tian31, p.612], [Reference Song and Tian32, p.306], [Reference Tian35, Conjecture 4.5.7], [Reference Tian36, p.258], Song-Tian posed the following:

Explicitly, Conjecture 1.1 asks to show that given any $K\Subset X\backslash S$ and $k\in \mathbb {N}$ , we have

(1.4) $$ \begin{align} \|\omega^{\bullet}(t)-\omega_{\mathrm{can}}\|_{C^k(K,g_0)}\to 0. \end{align} $$

There have been a number of partial results towards Conjecture 1.1, often using techniques that were first developed for a family of elliptic PDEs that describe the collapsing of families of Ricci-flat Kähler metrics on a Calabi-Yau manifold with a fibration structure, and which share some of the features of (1.1) (see, for example, the survey [Reference Tosatti40]). Indeed, Fong-Zhang [Reference Fong and Zhang12] adapted work of the third-named author [Reference Tosatti38] to prove that (1.3) holds in the $C^{1,\alpha }_{\mathrm { loc}}(X\backslash S)$ topology of Kähler potentials ( $\alpha <1$ ), and the works [Reference Fong and Zhang12, Reference Hein and Tosatti19, Reference Tosatti and Zhang44] proved Conjecture 1.1 when the smooth fibers of f are tori or finite quotients of tori (see also [Reference Gill13] and [Reference Tosatti39, §5.14]), using and improving a method of Gross-Tosatti-Zhang [Reference Gross, Tosatti and Zhang14]. Later, Tosatti-Weinkove-Yang proved that (1.3) holds in $C^0_{\mathrm {loc}}(X\backslash S)$ , and this was improved to $C^{\alpha }_{\mathrm {loc}}(X\backslash S),\alpha <1$ by Chu-Lee [Reference Chu and Lee4] adapting the techniques of Hein-Tosatti [Reference Hein and Tosatti20], which also allowed Fong-Lee [Reference Fong and Lee11] to prove Conjecture 1.1 when all smooth fibers are pairwise biholomorphic.

In a later work [Reference Song and Tian33], Song-Tian proved that the scalar curvature of $\omega ^{\bullet }(t)$ remains uniformly bounded on X, independent of $t\geqslant 0$ . They then conjectured a similar statement for the Ricci curvature, away from the singular fibers of f (see [Reference Tian36, Conjecture 4.7]):

This is only known when the smooth fibers of f are tori, or finite quotients of tori [Reference Fong and Lee11] (hence, in particular, it holds on minimal properly elliptic surfaces), or when the smooth fibers are pairwise biholomorphic [Reference Chu and Lee4]. It is known that, in general, the conjectural Ricci bound cannot be improved to a full Riemann curvature bound (on compact subsets of $X\backslash S$ ): by [Reference Tosatti and Zhang44], this holds if and only if the smooth fibers are tori or finite quotients.

It is well known that the Kähler-Ricci flow (1.1) reduces to a scalar PDE, of parabolic complex Monge-Ampère type, for a family of evolving Kähler potentials. Following [Reference Song and Tian32], we construct a closed real $(1,1)$ -form $\omega _F$ on $X\backslash S$ , which is of the form $\omega _F=\omega _0+{i\partial \bar \partial }\rho $ , such that for every $z\in B\backslash f(S)$ , we have that $\omega _F|_{X_z}$ is the unique Ricci-flat Kähler metric on $X_z$ cohomologous to $\omega _0|_{X_z}$ . While $\omega _F$ is not semipositive definite in general (see [Reference Cao, Guenancia and Păun2] for a counterexample), given any compact set $K\Subset X\backslash S$ , we can find $t_0$ such that for all $t\geqslant t_0$ ,

(1.5) $$ \begin{align} \omega^\natural(t):=(1-e^{-t})\omega_{\mathrm{can}}+e^{-t}\omega_F \end{align} $$

is a Kähler metric on K, with fibers of size $\approx e^{-t/2}$ and base of size $\approx 1$ . On $X\backslash S$ , we can then write $\omega ^{\bullet }(t)=\omega ^\natural (t)+{i\partial \bar \partial }\varphi (t)$ , where the potentials $\varphi (t)$ satisfy

(1.6) $$ \begin{align} \left\{ \begin{aligned} &\frac{\partial}{\partial t}\varphi (t)=\log\frac{e^{nt}(\omega^\natural(t)+{i\partial\bar\partial}\varphi (t))^n}{\binom{m+n}{n}\omega_{\mathrm{can}}^m\wedge\omega_F^n}-\varphi (t),\\ &\varphi (0)=-\rho,\\ &\omega^{\bullet}(t)=\omega^\natural(t)+{i\partial\bar\partial}\varphi (t)>0, \end{aligned} \right. \end{align} $$

for $t\geqslant 0$ (see, for example, [Reference Tosatti39, §5.7] and [Reference Tosatti, Weinkove and Yang43, §3.1]. Then, since we know the weak convergence in (1.3), Conjecture 1.1 is equivalent to the a priori estimates

(1.7) $$ \begin{align} \|\omega^{\bullet}(t)\|_{C^k(K,g_0)}\leqslant C_{K,k} \end{align} $$

for all $k\in \mathbb {N}$ and all $t\geqslant 0$ . Furthermore, since $\varphi (t)$ is uniformly bounded in $L^\infty (X)$ by [Reference Song and Tian32] (which uses [Reference Demailly and Pali8, Reference Eyssidieux, Guedj and Zeriahi9] – see also [Reference Guo, Phong and Tong17] for a new proof), these estimates are also equivalent to

(1.8) $$ \begin{align} \|\varphi (t)\|_{C^k(K,g_0)}\leqslant C_{K,k}, \end{align} $$

for all $k\in \mathbb {N}$ and all $t\geqslant 0$ .

1.3 Main result

The main result of this paper gives a complete solution of Conjectures 1.1 and 1.2:

In fact, in both conjectures, we prove much more precise statements. The higher order estimates for $\omega ^{\bullet }(t)$ are derived as consequences of a very detailed asymptotic expansion for $\omega ^{\bullet }(t)$ , which is in the same spirit as the expansion recently obtained in [Reference Hein and Tosatti21] by Hein-Tosatti for collapsing Ricci-flat metrics on Calabi-Yau manifolds. As for the Ricci curvature bound, we show that on $X\backslash S$ we have

(1.9) $$ \begin{align} \textrm{Ric}(\omega^{\bullet}(t))=-\omega_{\mathrm{can}}+\mathrm{Err}, \end{align} $$

where on any fixed compact subset of $X\backslash S$ we have $|\mathrm {Err}|_{g^{\bullet }(t)}\to 0$ , as $t\to +\infty $ . Thus, in a strong sense, the Ricci curvature of the evolving metrics $\omega ^{\bullet }(t)$ is asymptotic to $-\omega _{\mathrm {can}}$ . Furthermore, our bound on the Ricci curvature (and on all of the pieces of the asymptotic expansion of the metric) is an a priori bound: it only depends on the uniform constants in Lemma 4.1, which are due to [Reference Fong and Zhang12, Reference Song and Tian33, Reference Tosatti, Weinkove and Yang43].

The starting point of our analysis, which was proved in [Reference Fong and Zhang12] by adapting [Reference Tosatti38] in the elliptic setting, is the following estimate: given $K\Subset X\backslash S$ , there is $C>0$ such that on K we have

(1.10) $$ \begin{align} C^{-1}\omega^\natural(t)\leqslant \omega^{\bullet}(t)\leqslant C\omega^\natural(t) \end{align} $$

for all $t\geqslant t_0$ . In other words, $\omega ^{\bullet }(t)$ is shrinking in the fiber directions and remains of bounded size in the base directions. Since the linearized operator of (1.6) is the time-dependent heat operator of $\omega ^{\bullet }(t)$ , we see from (1.10) that the ellipticity is degenerating in the fiber directions as $t\to +\infty $ , and so there is no clear way to approach the a priori estimates (1.8). Indeed, the local analog of such estimates are false; see the discussion in [Reference Hein and Tosatti20] in the elliptic case.

However, it turns out that we can work locally on the base (but using crucially that the fibers are compact without boundary), and since f is differentiably a locally trivial fiber bundle over $B\backslash f(S)$ , we may without loss assume that our base B is now simply the Euclidean unit ball in $\mathbb {C}^m$ , and $f:B\times Y\to B$ is just the projection onto the first factor, where Y is a closed manifold and $B\times Y$ is equipped with a complex structure J (not necessarily a product) such that f is $(J,J_{\mathbb {C}^m})$ holomorphic. The fibers $\{z\}\times Y, z\in B$ are then compact n-dimensional Calabi-Yau manifolds diffeomorphic to Y. Under this trivialization, the Ricci-flat Kähler metric $\omega _F|_{X_z}$ defines a Riemannian metric $g_{Y,z}$ on $\{z\}\times Y$ , which we extend trivially to $B\times Y$ , and we use these to define a family of shrinking Riemannian product metrics

(1.11) $$ \begin{align} g_z(t)=g_{\mathbb{C}^m}+e^{-t}g_{Y,z}, \end{align} $$

on $B\times Y$ , which are uniformly equivalent to $\omega ^\natural (t)$ and hence to $\omega ^{\bullet }(t)$ . We will also denote by $g(t):=g_0(t)$ the shrinking product metrics with z equal to the origin in B.

1.4 Overview of the proof

As in [Reference Hein and Tosatti20, Reference Hein and Tosatti21], the first attempt to overcome the issue of degenerating ellipticity is to try to prove much more – namely, try to prove uniform bounds for $\varphi (t)$ or $\omega ^{\bullet }(t)$ in the shrinking norms $C^k(K,g(t))$ , since $g^{\bullet }(t)$ is uniformly equivalent to $g(t)$ . This, however, cannot be proved in general since we know from [Reference Tosatti, Weinkove and Yang43] that $e^t\omega ^{\bullet }(t)|_{X_z}$ converge smoothly to $\omega _F|_{X_z}$ , and since $g_{Y,z}$ and $g_{Y,z'}$ are not in general parallel with respect to each other, the shrinking $C^k$ norms of $g_{z}(t)$ and $g_{z'}(t)$ are not uniformly equivalent as $t\to +\infty $ . To address this issue, the first and third-named authors defined in [Reference Hein and Tosatti21] a connection ${\mathbb {D}}$ on $B\times Y$ which on each fiber $\{z\}\times Y$ acts like the Levi-Civita connection of $g_z(t)$ , and using its parallel transport operator, they defined new shrinking $C^{k,\alpha }$ norms, $0<\alpha <1$ . We will consider the natural parabolic extension of these norms to space-time derivatives in Section 2 below. Since parabolic Hölder seminorms behave differently according to the parity of k, we will only work with $k=2j$ even (cf. Remark 2.5).

The hope would then be to show that $\omega ^{\bullet }(t)-\omega ^\natural (t)={i\partial \bar \partial }\varphi (t)$ is uniformly bounded in these shrinking $C^{2j+\alpha ,j+{\alpha }/2}$ norms. This turns out to be true when $j=0$ , but false starting from $j=1$ . This phenomenon, which was discovered in [Reference Hein and Tosatti21] in the elliptic setting, manifests itself only when the complex structure J is not a product and the fibers are not tori or quotients. In a nutshell, the variation of complex structures, and the non-flatness of $g_{z}(t)$ , destroy these desired shrinking norm bounds. However, with much work, we are able to construct a collection of ‘obstruction functions’ on $B\times Y$ (up to shrinking B) and decompose the solution ${i\partial \bar \partial }\varphi (t)$ into a sum of finitely many terms $\gamma _1(t),\dots ,\gamma _j(t)$ (constructed roughly speaking using the fiberwise $L^2$ projections of $\Delta ^{g^\natural (t)}\varphi (t)$ onto the space of obstructions), and a remainder $\eta _j(t)$ . We then show via a contradiction and blowup argument that the remainder $\eta _j(t)$ is bounded in the shrinking $C^{2j+\alpha ,j+{\alpha }/2}$ norm, while the terms $\gamma _1(t),\dots ,\gamma _j(t)$ are not, but they satisfy strong enough estimates which guarantee that they are bounded in the $C^{2j+\alpha ,j+{\alpha }/2}$ norm of a fixed metric $\omega _X$ on X. As mentioned earlier, the higher order estimates on all these pieces depend only on the constant in the $C^0$ estimate (1.10), and on the other constants that appear in Lemma 4.1 (including the uniform bound on the scalar curvature of $\omega ^{\bullet }(t)$ from [Reference Song and Tian33]), and thus ultimately, they depend only on the geometry of X and on the initial metric $\omega _0$ .

This procedure is iterated by replacing j with $j+1$ , and new obstruction functions are constructed by measuring the failure of the remainder $\eta _j(t)$ to be bounded in the shrinking $C^{2(j+1)+\alpha ,j+1+{\alpha }/2}$ norm. This way, we can split $\eta _j(t)=\gamma _{j+1}(t)+\eta _{j+1}(t)$ and obtain the next order in the expansion. As in [Reference Hein and Tosatti21], there is an extra technical difficulty, which arises from the fact that the terms $\gamma _j(t)$ are constructed by plugging in $\eta _{j-1}(t)$ and the obstruction functions into an approximate elliptic Green operator, which has an extra parameter $k\in \mathbb {N}$ that measures the quality of the approximation. Thus, all the terms in the expansion also end up depending on k, which is large and chosen a priori, and the procedure works for $j\leqslant k$ .

The resulting asymptotic expansion of $\omega ^{\bullet }(t)$ is described in detail in Theorem 4.2 below, which is the main technical result of the paper. It is the parabolic analog of [Reference Hein and Tosatti21, Theorem 4.1], and its proof follows the same overall method via blowup and contradiction, but there are some new key difficulties. First, as mentioned earlier, the (shrinking) parabolic Hölder norms that we use are better behaved when the order of derivatives is even, which compels us to use $C^{2j+\alpha ,j+{\alpha }/2}$ norms instead of $C^{j+\alpha ,(j+{\alpha })/2}$ (see, for example, Lemma 2.4 and Remark 2.5). More importantly, since the approximate Green operator that we use in this paper is the same as in [Reference Hein and Tosatti21], it provides an approximate parametrix for the Laplacian of $\omega ^\natural (t)$ (in a rough sense) but not for the heat operator (it seems far from clear that a similar strategy could be implemented with an approximate heat kernel construction). Because of this, to obtain a contradiction at the end of the blowup argument (which is divided into 3 cases, with the last case itself divided into 3 subcases A, B and C), we now have to deal with new terms that come from taking time derivatives of the solution, which are not taken care of by construction, unlike [Reference Hein and Tosatti21]. To make matters worse, in the blowup argument, the evolving Kähler potential has $L^\infty $ norm that is blowing up, so it cannot be passed to a limit to obtain a contradiction. Dealing with these issues requires substantial work.

Another new difficulty, compared to [Reference Hein and Tosatti21], is that the case $j=0$ (i.e., where we prove $C^{{\alpha },{\alpha }/2}$ estimates) does not behave in the same way as the cases $j\geqslant 1$ because the parabolic complex Monge-Ampère equation also involves $\varphi (t)$ without derivatives landing on it, unlike the elliptic complex Monge-Ampère equation where only ${i\partial \bar \partial }\varphi $ enters. To deal with this issue, we employ a different blowup quantity for $j=0$ , which is closer in spirit to our earlier works [Reference Hein and Tosatti20, Reference Fong and Lee11]. As a result, different ideas will be required to close the blowup argument, according to whether $j=0$ or $j\geqslant 1$ . Furthermore, when $j\geqslant 1$ , we are forced to add one new term to the main blowup quantity (when compared to [Reference Hein and Tosatti21]), to gain better control on the fiber average of the Kähler potential and its time derivative, and we later have to show that this new term can be dealt with in the blowup argument. Next, in subcase A, dealing with these terms forces us to refine the Selection Theorem 3.1 where the obstruction functions are chosen, and when $j=0$ , we need a whole new argument. In subcase B, we employ an energy argument inspired by [Reference Fong and Lee11, Claim 3.2], and in subcase C a different energy argument has to be applied fiber by fiber.

Once the asymptotic expansion is established, the smooth convergence of Conjecture 1.1 follows easily. However, proving the Ricci curvature bound for $\omega ^{\bullet }(t)$ in Conjecture 1.2 requires substantial work by plugging in the expansion with $j=1$ into the formula for the Ricci curvature (as two derivatives of the logarithm of the volume form) and using our explicit a priori estimates for the terms of the expansion to deduce boundedness of Ricci. Here again, we encounter a new difficulty compared to [Reference Hein and Tosatti21], which arises from the fact that one of the estimates in Theorem 4.2 is weaker than the corresponding one in the elliptic setting, because of the fact that we can only work with even order norms. During the course of the proof of the Ricci bound, we also prove a fact of independent interest in Proposition 5.1, by showing that $\varphi +\dot {\varphi }$ minus its fiberwise average decays to zero (away from the singular fibers) faster than $e^{-t}$ (see (5.33)). This improves on earlier work of Fong-Zhang [Reference Fong and Zhang12, p.110] (see also [Reference Tosatti39, Lemma 5.13]) and Tosatti-Weinkove-Yang [Reference Tosatti, Weinkove and Yang43, Lemma 3.1 (iv)].

1.5 Organization of the paper

In Section 2 we introduce our parabolic shrinking norms and seminorms and prove an interpolation inequality, the crucial Proposition 2.6 and a Schauder estimate. Section 3 contains the proof of the Selection Theorem 3.1 where the obstruction functions are selected. Section 4 is the main part of the paper and is where the asymptotic expansion is proved in Theorem 4.2. Lastly, in Section 5, we give the proof of our main Theorem 1.3.