Convergence of scalar curvature of Kähler-Ricci flow on manifolds of positive Kodaira dimension

Abstract

In this paper, we consider Kähler-Ricci flow on n-dimensional Kähler manifold with semi-ample canonical line bundle and $0<m:=\mathrm{\text{Kod}}(X)<n$. Such manifolds admit a Calabi-Yau fibration over its canonical model. We prove that the scalar curvature of the Kähler metrics along the normalized Kähler-Ricci flow converge to −m outside the singular set of this fibration.

Introduction

The “Analytic Minimal Model Program” starts with the discovery of Tian-Zhang [15], based on previous work of Tsuji [21], that on a minimal model (i.e. a projective manifold X with $K_X$ nef), the Kähler-Ricci flow always has a long-time solution, and on a general Kähler manifold, the Kähler-Ricci flow encounters a finite-time singularity if and only if the canonical bundle is not nef, and the solution exists as long as the Kähler class remains in the Kähler cone. This matches perfectly with the beginning step of the minimal model program in higher dimensional algebraic geometry. Motivated by Perelman's solution of the geometrization conjecture via Ricci flow, J. Song and G. Tian suggest in [8] that one can use Kähler-Ricci flow to study the classification problem in birational geometry. This program is called “Analytic Minimal Model Program”.

There are many important works from then on. In [12], Song-Weinkove gave a criterion under which a solution of the Kähler-Ricci flow contracts exceptional divisors on a compact manifold and can be uniquely continued on a new manifold. Then in [13], they investigate the case of the Kähler-Ricci flow blowing down disjoint exceptional divisors with normal bundle $\mathcal{O}(-k)$ to orbifold points.

The analytic minimal model program with Ricci flow was laid out in [8], [9], [10], [11] to study the formation of singularities for the Kähler-Ricci flow on algebraic varieties. It is conjectured in [11] that the Kähler-Ricci flow will deform a projective variety X of nonnegative Kodaira dimension, to its minimal model via finitely many divisorial metric contractions and metric flips in Gromov-Hausdorff topology, then eventually converge to a unique canonical metric of Einstein type on its unique canonical model. The existence and uniqueness is proved in [11] for the analytic solutions of the Kähler-Ricci flow on algebraic varieties with log terminal singularities.

In [24], it is shown that if the Kähler-Ricci flow develops finite time singularity, the scalar curvature blows up at most of rate ${(T-t)}^{-2}$ if X is projective and if the initial Kähler class lies in $H^2(X,\mathbb{Q})$. In [2], Collins-Tosatti proved that the non-Kähler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kähler manifold equals its null locus. Also see Song-Weinkove's note [14] and Tosatti's note [18] for clearer and more unified discussions.

In this paper, we shall study Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical bundles. Following Song-Tian, let $(X^n,{\omega }_0)$ be a compact Kähler manifold with canonical line bundle $K_X$ being semi-ample and $0<m:=\mathrm{\text{Kod}}(X)<n$. Therefore the canonical ring $R(X,K_X)$ is finitely generated, and so the pluricanonical system $|\ell K_X|$ for sufficiently large and sufficiently divisible $\ell \in {\mathbb{Z}}^{+}$ induces a holomorphic map

$$f:X\to B\subset \mathbb{C}{\mathbb{P}}^N:=\mathbb{P}{H}^0(X,K_X^{\otimes \ell }),$$

where B is the canonical model of X. We have $dimB=m$.

Let $S^{\prime }$ be the singular set of B together with the set of critical values of f, and we define $S=f^{-1}(S^{\prime })\subset X$.

Now let $\omega (t)$ be the smooth global solution of the normalized Kähler-Ricci flow

$$\frac{\partial \omega }{\partial t}=-Ric(\omega )-\omega ,\omega {|}_{t=0}={\omega }_0.$$

It's well-known [15], [21] that the flow has a global solution on $X\times [0,\infty )$. It's shown by Song-Tian [8], [9] that $\omega (t)$ collapses nonsingular Calabi-Yau fibers and the flow converges weakly to a generalized Kähler-Einstein metric ${\omega }_B$ on its canonical model B, with ${\omega }_B$ is smooth and satisfies the generalized Einstein equation on $B\backslash S^{\prime }$

$$Ric({\omega }_B)=-{\omega }_B+{\omega }_{\mathrm{WP}},$$

where ${\omega }_{\mathrm{WP}}$ is the Weil-Petersson metric induced by the Calabi-Yau fibration f. They also proved the $C^0$-convergence on the potential level and in the case when X is an elliptic surface the $C_{loc}^{1,\alpha }$-convergence of potentials on $X\backslash S$ for any $\alpha <1$. In [10], Song-Tian showed that the scalar curvature is uniformly bounded on $X\times [0,\infty )$ along the normalized flow. The case when X is of general type is given by Z. Zhang in [23]. The case for conical Kähler-Ricci flow is given by G. Edwards in [3].

In [4], Fong-Zhang proved the $C^{1,\alpha }$-convergence of potentials when X is a global submersion over B and showed the Gromov-Hausdorff convergence in the special case. In [19] Tosatti-Weinkove-Yang improved the estimate and showed that the metric $\omega (t)$ converges to $f^{⁎}{\omega }_B$ in the $C^0$ local-topology on $X\backslash S$. Moreover, Tosatti-Weinkove-Yang [19] proved that the restricted metric $\omega (t){|}_{X_y}$ converges (up to scalings) in the $C^0$-topology to the unique Ricci flat metric in the class $[{\omega }_0{|}_{X_y}]$ on the fibre $X_y$ for any regular value y; this result is improved to be smooth convergence by Tosatti-Zhang in [20]. Also see Tosatti's note [18] for clearer and more unified discussions.

In fact, Tosatti-Weinkove-Yang [19] obtained in their proof that $|\omega (t)-\tilde{\omega }(t){|}_{\omega (t)}\to 0$ as $t\to \infty$ on $X\backslash S$, where $\tilde{\omega }(t)=e^{-t}{\omega }_{\mathrm{SRF}}+(1-e^{-t}){\omega }_B$ (see Section 2 for definition of ${\omega }_{\mathrm{SRF}}$). This estimate describes very precisely how the metrics deform when approaching the infinity time since here we use the collapsing metrics to measure the norm. With the help of this good estimate, we are able to prove that $|{\mathrm{tr}}_{\omega (t)}{\omega }_B-m|+|{\Vert {\omega }_B\Vert }_{\omega (t)}^2-m|\to 0$ as $t\to \infty$ on $X\backslash S$. With these preparations, we can prove the convergence of the scalar curvature along the normalized flow on the regular part $X\backslash S$. More precisely, we prove that

Theorem 1.1

Let $(X,{\omega }_0)$ be given as above, let $\omega (t)$ be the smooth global solution of the normalized Kähler-Ricci flow (1.2). Then we have

$$\underset{t\to \infty }{\lim }R(t)=-m,\mathit{\text{on}}X\backslash S\times [0,\infty ).$$

In particular, if $S=\varnothing$, then f is a holomorphic submersion and we have

$$|R(t)+m|⩽C{e}^{-\eta t},\mathit{\text{on}}X\times [0,\infty ),$$

for some constants $\eta ,C>0$ depending on $(X,{\omega }_0)$.

After rescaling time and space simultaneously, we have the following immediately corollary from Theorem 1.1 of the unnormalized Kähler-Ricci flow.

Corollary 1.2

Let $(X,{\omega }_0)$ be given as above, let $\omega (t)$ be the smooth global solution of the unnormalized Kähler-Ricci flow

$$\frac{\partial \omega }{\partial t}=-Ric(\omega ),\omega {|}_{t=0}={\omega }_0.$$

Then we have

$$\underset{t\to \infty }{\lim }(1+t)R(t)=-m,\mathit{\text{on}}X\backslash S\times [0,\infty ).$$

In particular, if $S=\varnothing$, then f is a holomorphic submersion and we have

$$|(1+t)R(t)+m|⩽\frac{C}{{(1+t)}^{\eta }},\mathit{\text{on}}X\times [0,\infty ),$$

for some constants $\eta ,C>0$ depending on $(X,{\omega }_0)$.

Note that in Theorem 1.1, the limiting behavior of scalar curvature on the singular set S is unknown. A recent result of the author and two other authors [6] says that: If the canonical bundle $K_X$ is semi-ample, then for any Kähler class $[\omega ]$ on X, there exists ${\delta }_{X,[\omega ]}>0$ such that for any $0<\delta <{\delta }_{X,[\omega ]}$, there exists a unique cscK metric in the Kähler class $[K_X]+\delta [\omega ]$. Hence we can propose the following conjecture.

Conjecture 1.3

Let X be an n-dimensional Kähler manifold with nef canonical bundle $K_X$ and positive Kodaira dimension. Then for any initial Kähler metric ${\omega }_0$, the solution $\omega (t)$ of the normalized Kähler-Ricci flow

$$\frac{\partial }{\partial t}\omega =-Ric(\omega )-\omega ,\omega (0)={\omega }_0$$

converges in Gromov-Hausdorff topology to the metric completion of $(B\backslash S^{\prime },{\omega }_B)$ and the scalar curvature $R(t)$ converges to $-\mathit{\text{Kod}}(X)$ in $C^0(X)$, where $\mathit{\text{Kod}}(X)$ is the Kodaira dimension of X.

In general, it is natural to ask if the following holds for the maximal solution of the unnormalized Kähler-Ricci flow on $X\times [0,T)$, where X is a Kähler manifold and $T>0$ is the maximal existence time.

• If $T<\infty$, then there exists $C>0$ such that

  $$-C⩽R(t)⩽C{(T-t)}^{-1}.$$

• If $T=\infty$, then there exists $C>0$ such that

  $$|R(t)|⩽C{(1+t)}^{-1}.$$

In [7], the answer to the first question is affirmative due to Perelman for the Kähler-Ricci flow on Fano manifolds with finite time extinction. For further developments, see [5], [16], [17], [25], [26], [27].

Acknowledgments. The author would like to thank his advisor Gang Tian for leading him to study Kähler-Ricci flow, constant encouragement and support. The author also would like to thank Jian Song, Yalong Shi and Dongyi Wei for helpful discussions. This work was carried out while the author was visiting Jian Song at the Department of Mathematics of Rutgers University, supported by the China Scholarship Council (File No. 201706010022). The author would like to thank the China Scholarship Council for supporting this visit. The author also would like to thank Jian Song and the Department of Mathematics of Rutgers University for hospitality and support. The author is grateful to the referees for valuable comments and suggestions.