New Results on the Singularity Analysis of the Kahler-Ricci¨ Flow

NEW RESULTS ON THE SINGULARITY ANALYSIS OF THE KAHLER-RICCI¨ FLOW

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Tsz-Ho Fong May 2012

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/ch375kk8984

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Richard Schoen, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Simon Brendle

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Brian White

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

In this thesis, we study the singularity development of the Kähler-Ricci flow on holomorphic fibrations, and classify the singularity models of some classes of fibrations using parabolic rescaling. We first study the collapsing behavior of Calabi-Yau fibrations under the Kähler-Ricciflow. A compact Kählermanifold with semi-ample canonical line bundle admits a fibration of Calabi-Yau manifolds with possibly singular fibers. The convergence behavior for this class of manifolds under the Kähler-Ricciflow was first studied by Song and Tian in [73] and [74]. These authors establish the metric convergence in the sense of currents. In this thesis, we obtain the optimal collapsing rate of the nonsingular Calabi-Yau fibers, thus improving Song and Tian’s work in analytic and geometric aspects. 1 Secondly, we focus on a specific type of holomorphic fibration, namely the CP -bundles over Kähler-Einsteinmanifolds. Fiber collapsing in the sense of Gromov-Hausdorff convergence was shown to occur in this case by Song, Székelyhidi and Weinkove. We study the finite-time singularities for these manifolds using parabolic rescaling and dilation procedures adapted from Hamilton and Perelman in their works of the Ricci flow on 3-manifolds. We prove that when the flow metric has cohomogeneity-1 symmetry the collapsing occurs as a Type I singularity and we show that the 1 singularity is modelled by Cn × CP .

iv Acknowledgement

This thesis marks the end of a period and opens a new chapter of my academic life. I would like to take this opportunity to express my appreciation to all persons who fruitfully helped me along my journey of academic pursuit. First and foremost, I would like to express my heartfelt gratitude to my advisor Professor Richard Schoen for all his encouragement, care and support throughout my five years in Stanford. His pedagogy not only opens my eyes to the beautiful landscape of differential geometry but also brings tremendous impact on my critical thinking in research mathematics. Without his insightful advice and generous sharing of mathematical ideas, I would not have completed this thesis today. It is truly my great honor to study under such a world-renowned scholar, and I am deeply influenced by his perpetual enthusiasm and brilliance in both research and teaching. I would like to thank sincerely Professor Simon Brendle, Professor Leon Simon and Professor Brian White for agreeing to be on my thesis committee and for their valuable comments on my research work. Throughout the years, I am very much benefited by their inspiring teaching, as well as from classes taught by Professor Yakov Eliashberg, Professor Eleny Ionel, Professor Rafe Mazzeo and Professor AndrásVasy which undeniably broadened my background in differential geometry and partial differential equations. I would also like to thank Yanir Rubinstein for guiding me through my learning in complex geometry, and purposing directions for my research. I am very much benefited by the weekly informal geometry and analysis seminar. I would like to thank the generous contributions by Eric Bahuaud, Richard Bamler, Jacob Bernstein, Brian Clarke and Yanir Rubinstein whom I learned a lot about various research areas in geometry and analysis, and also thank the participations by Dean Baskin, Alessandro Carlotto, Otis Chodosh, Zachary Cohn, Jeff Danciger, Nick Haber, Chris Henderson, Lan-Hsuan Huang, Brian Krummel, Martin Li, Jesse Gell-Redman, David Sher and Xin Zhou, who all stimulated my interest in different topics in geometry and analysis.

v I wish to thank the Department of Mathematics in Stanford which provides me with a perfect environment for research, and offers me opportunities and support in teaching from which I gained a lot of rewarding experience. I would like thank especially for the constant supplies of coffee and tea which energize me and keep me awake after each sleepless night of struggling in my research work. Hailing from Hong Kong, I am deeply indebted to many respectable teachers there. My special thank goes to Professor Shiu-Yuen Cheng who enlightened me a lot back in my days in the Hong Kong University of Science and Technology. His patient guidance in the past was indispensable for my scholastic development nowadays. I would like to thank him sincerely for all his great help in paving my way to realize my childhood dream of becoming a mathematician. I also like to thank Professor Kin-Yin Li, Professor Wei-Ping Li, Professor Guo-Wu Meng, Professor Ngai-Ming Mok, and Professor Min Yan whom I learned a lot in their classes in Hong Kong which built up my mathematical foundation for my current research areas. I would also like to express my gratitude to my high school mathematics teacher Kwong-Ki Chan who arouse my interest in mathematics back in my teenage. I learned a lot from the experts in the Ricci flow with whom I have collaborated or communicated. I thank Professor John Morgan and Professor Zhou Zhang for offering me opportunities to collaborate with them in two recent publications from which I gained a lot of mathematical insights. I also thank Professor Jian Song, Professor Valentino Tosatti and Professor Ben Weinkove for all productive discussions and their interest in my works. My life in both Stanford and Hong Kong is very much enriched by the company of colleagues and classmates. In Stanford, I enjoy very much the wonderful evenings in the dining hall with Peter Diao, John Jiang, Seung-Ki Kim, Pak-Hin Lee, Calvin Lin, Khoa Lu Nguyen, Amy Pang, Fernando Shao and many others, where we converse on various non-mathematical topics ranging from classical music to politics. I also miss the undergraduate years at HKUST with my comrades Chris Chan, Jane Dai, Ivan Ip, Vic Law, Herman Lee, Joshua Lui, Kin-Fai Mak, Alvin Wong, Geo Tam, Boris Yim and many others who share common enthusiasm in academic pursuit. This acknowledgement cannot be ended without mentioning my mum, Miss Lily Wai-Yee Li, who brings me to this world and nurtures me in all aspects of life. I appreciate her understanding of the tough life in the academic career trek, and thank very much for all her support and care for keeping me tough in this journey.

vi Contents

Abstract iv

Acknowledgement v

1 Introduction 1 1.1 Overview ...... 1 1.2 Motivations ...... 3 1.3 Summary of new results ...... 7

2 Background 13 2.1 Kählergeometry ...... 13 2.1.1 Complex manifolds ...... 13 2.1.2 Kählermanifolds ...... 15 2.1.3 Characteristic classes ...... 16 2.2 Kähler-Ricciflow ...... 17 2.2.1 Complex Monge-Ampèreequations ...... 18 2.2.2 Maximal existence time ...... 19

3 Collapsing of Holomorphic Fibrations 24 3.1 Some classic estimates ...... 24 3.1.1 Parabolic Schwarz’s estimate ...... 25 3.1.2 Yau’s L∞-estimate ...... 27 3.2 Collapsing of Calabi-Yau ﬁbrations ...... 30 3.2.1 Regular inﬁnite-time singularity ...... 32 3.2.2 Pointwise estimates ...... 33

vii 3.2.3 Collapsing rate of Calabi-Yau fibers ...... 37 3.2.4 Long-time convergence ...... 40 3.2.5 Toric fibration ...... 43 3.3 Results on Fano fibrations ...... 51 3.3.1 Pointwise estimates ...... 53 3.3.2 Collapsing rate of Fano fibers ...... 57 3.3.3 Blow-up rate of curvature ...... 60

4 Singularity Model of Projective Bundles 69 4.1 Hirzebruch surfaces and projective bundles ...... 70 4.2 Calabi’s ansatz ...... 71 4.3 Estimates under Calabi’s ansatz ...... 76 4.4 Singularity analysis ...... 84 4.4.1 Splitting lemma ...... 84 4.4.2 Analysis of kRmk ...... 91 4.4.3 Type I singularity ...... 95 4.4.4 Type II singularity ...... 98

Bibliography 103

viii Chapter 1

Introduction

1.1 Overview

The study of geometric flows is one of the most central topics in modern differential geometry. The most notable ones include the Ricci flow, Mean Curvature flow, Yamabe flow, Yang-Mills flow, etc. They have been successfully applied to resolve a variety of problems in differential geometry as well as in other areas such as topology and algebraic geometry. The Ricci flow equation, ∂g(t) = −Ric(g(t)) ∂t is one of the most extensively studied geometric evolution equations in recent decades. It was introduced by Hamilton in his seminal paper [39] in 1982, proving the existence of constant sectional curvature metrics on any closed 3-manifold with positive Ricci curvature. Several breakthroughs were made in the last decade in resolving several long-standing conjectures. Just to name a few, a complete proof of the Poincaréand Thurston’s Geometrization Conjectures was given by Perelman [56–58] around 2003 using the Ricci flow with Surgery based on a program developed by Hamilton. Detail expositions can be found in the works by Cao-Zhu [8], Kleiner-Lott [44], Morgan-Tian [51], etc. In 2007, the Differentiable Sphere Theorem was proved by Brendle-Schoen [4] and gave an affirmative answer to a conjecture about differential structures of quarter-pinched manifolds proposed by Hopf in 1940s. All the above-mentioned achievements involve seeking canonical metrics by geometric deformations. The analysis of singularity formations and convergence behaviors plays a crucial role in this regard.

1 CHAPTER 1. INTRODUCTION 2

This thesis focuses on the study of the Ricci flow on Kählermanifolds, an important class of manifolds in complex differential geometry. Kählergeometry is the meeting place of Riemannian geometry, complex geometry and symplectic geometry, and the study of Kählermanifolds brings together the techniques developed in a wide range of mathematical disciplines including partial differential equations (notably the complex Monge-Ampèreequations), algebraic geometry and cohomology theory. It is a well-established fact that the Ricci flow preserves the Kählercondition, meaning that if the initial metric is Kähler,it will remain so as long as the Ricci flow solution exists. As such, the Ricci flow on Kählermanifolds is commonly called the Kähler-Ricci flow. The use of the Ricci flow on studying Kählermanifolds was initiated by Cao, Bando, Mok and Shi in the 1980s. In particular, Cao showed in [6] that the Kähler-Ricciflow gives an alternative proof to the Calabi Conjecture (proved by Yau in [85]) and to the existence of Kähler-Einstein metrics on manifolds with negative and zero first Chern classes. The works by Bando, Mok and Shi concern about the Ricci flow on Kählermanifolds with non-negative holomorphic bisectional curvature. It was proved by Bando, Mok and Shi that the non-negativity of holomorphic bisectional curvature is preserved along the flow. With the use of de Rham splitting, strong maximum principle and Siu-Yau’s proof of the Frankel Conjecture1, a classification theorem of closed Kählermanifolds with non-negative bisectional curvature (commonly called the generalized Frankel conjecture) was given by Mok [50]. The Kähler-Ricciflow has tremendous developments in 2000s and becomes a rapidly growing topic in geometry. There are several major research directions for the Kähler-Ricciflow. The refer- ences listed below are far from being complete: (1) the existence problem of Kähler-Einsteinmetrics on manifolds with positive first Chern class (commonly called Fano manifolds), and its relation with geometric invariant theory (GIT) and algebraic-geometric stability. A “folklore” conjecture relating polystability and existence of Kähler-Einstein metrics is now called the Yau-Tian-Donaldson’s Con- jecture. Recent works along this avenue using the Kähler-Ricci flow include [14,15,20,59,61–63,66] etc.; (2) the classification problem of Kähler manifolds (both compact and non-compact) with nonnegative holomorphic bisectional curvature, e.g. [9–13, 54, 55, 60]; (3) using the Kähler-Ricciflow to give a geometric classification of algebraic varieties. This is motivated by the Mori’s minimal model program in algebraic geometry, e.g. [73–75, 77–79, 82] which suggested that the Kähler-Ricciflow can perform certain operations which are closely related to the minimal model program, such as contracting exceptional divisors, and collapsing of fibrations to arrive at a canonical model.

1The Frankel Conjecture asserted that every compact K¨ahlermanifold with positive bisectional curvature must be n biholomorphic to CP . It was proved independent by Mori and Siu-Yau. CHAPTER 1. INTRODUCTION 3

The research conducted in this thesis is motivated by (3) in the above paragraph. The general idea of classification of algebraic varieties is inspired by the works of Hamilton and Perelman on the Poincaréand Geometrization Conjectures where the Ricci flow with Surgery plays the most crucial role. As the Ricci flow of Riemannian 3-manifolds may encounter singularities (i.e. the flow stops after some finite time), a crucial ingredient of Hamilton-Perelman’s works is to understanding possible singularity developments of the flow. Once a solid understanding of singularities is in place, the next issue is to understand the long-time and convergence behavior. The ultimate goal of (3) in the above paragraph is to investigate whether an analogous surgery program can be carried out on algebraic varieties using the Kähler-Ricciflow. In this regard, understanding singularity formations and convergence behaviors is fundamental. The organization of this thesis is as follows. In Chapter 1, we first survey over the past results which motivate the research of this thesis. Then, we present a summary of new results and explain how they extend the existing literature. Chapter 2 presents the background material in Kähler geometry, and introduces notations and concepts relevant to this thesis. Chapter 3 investigates the collapsing behavior of some holomorphic fibrations (Calabi-Yau and Fano fibrations) under the Kähler-Ricciflow. Finally, Chapter 4 studies a special case of holomorphic fibrations, namely projective bundles, and aims at classifying the singularity models of fiber collapsing using the techniques developed by Hamilton and Perelman.

1.2 Motivations

This section gives a survey of past results of the Kähler-Ricciflow which motivated the conducted research in this thesis. The common theme shared by the works mentioned below is that characteristic classes and their evolutions under the flow can predict the singular and convergence behaviors.

Problem 1.1. Let T ∈ (0, ∞] be the maximal existence time of the Kähler-Ricci flow. Given that we know the limiting Kählerclass [ωT ], what can we say about the convergence of the Kählerform

ωt as t → T ?

We start by stating part of the results in Cao’s paper in 1985. In [6], Cao studied the Kähler- Ricci flow on compact Kählermanifolds whose first Chern class has a definite sign. In particular, when c1(M) < 0 and the initial Kählerclass [ω0] is chosen to be −c1(M), the following result was obtained:

Theorem 1.1 (Cao [6]). Let M be a compact K¨ahlermanifold with c1(M) < 0. Suppose ω0 is a CHAPTER 1. INTRODUCTION 4

Kählermetric on M such that [ω0] = −c1(M), then along the normalized flow ∂tωt = −Ric(ωt)−ωt, we have ωt → ω∞ smoothly where ω∞ is a Kähler-Einsteinmetric with Ric(ω∞) = −ω∞.

As we will see in Chapter 2, the Kählerclass [ωt] under the normalized flow ∂tωt = −Ric(ωt)−ωt is given by −t [ωt] = −c1(M) + e (c1(M) + [ω0]) = −c1(M) and the flow solution ωt exists for all t ∈ [0, ∞). Clearly, as t → ∞,

[ωt] → −c1(M).

It predicts that ωt might converge to a form representing −c1(M). Some delicate higher-order estimates developed by Aubin [2] and Yau [85] give an affirmative answer to this prediction and also prove that the limiting metric ω∞ is in fact a Kähler-Einsteinmetric. The upshot of Cao’s Theorem is to rewrite the Kähler-Ricciflow into a complex Monge-Ampèreequation √ ∂ϕ (ω + −1∂∂ϕ¯ )n t = log 0 t − ϕ , ϕ| = 0, ∂t Ω t t=0 √ ¯ where the flow metric is given by ωt = ω0 + −1∂∂ϕt. The non-degeneacy of ωt can be proved by the parabolic Schwarz’s estimate stated in Chapter 3. The most crucial argument is to show

0 ∂ϕt the C -estimate for both ϕt and the decaying estimate for ∂t using the maximum principle. Once these estimates are in place, the higher-order estimates developed by Aubin [2] and Yau [85] give the smooth convergence of ϕt to a function ϕ∞, and hence one has √ (ω + −1∂∂ϕ¯ )n log 0 ∞ = ϕ Ω ∞ √ ¯ from which it is not diﬃcult to see ω∞ = ω0 + −1∂∂ϕ∞ is a K¨ahler-Einsteinmetric.

Compact K¨ahlermanifolds with c1(M) < 0 is equivalent to saying the canonical bundle KM is ample by the Kodaira Embedding Theorem. One direction for further generalizing this result is to 2 consider the case where KM is semi-ample, or more generally, numerically eﬀective (nef) . In this regard, there are two sides of a story: c1(M) is big versus c1(M) is not big, or heuristically speaking, non-collapsing versus collapsing. The nef and big (i.e. non-collapsing) case was studied by Tsuji [84] and Tian-Zhang [82], and a

2 A long-standing conjecture in algebraic geometry, called the Abundance Conjecture, claimed that if KM is nef and M is projective, then KM is semi-ample. CHAPTER 1. INTRODUCTION 5

K¨ahlermanifold in such case is usually called a minimal model of general type. The following result was obtained by

Theorem 1.2 (Tian-Zhang [82]). Let M be a smooth minimal model of general type (i.e. KM is nef and big), then under the normalized Kähler-Ricci flow ∂tωt = −Ric(ωt) − ωt starting with any Kählermetric ω0, the flow ωt exists for all time. Also, there exists a codimension 1 analytic ∞ subvariety S ⊂ M such that as t → ∞, ωt converges in Cloc(M\S) to a Kähler-Einsteinmetric ω∞ on M\S.

The semi-ample and non-big (i.e. collapsing) case was studied by Song-Tian in [73,74] respectively for elliptic and Calabi-Yau fibrations. The degree of the collapsing is characterised by the Kodaira dimension Kod(M) which is strictly less than dimC M in the non-big case. The case Kod(M) = 0 corresponds to Calabi-Yau manifolds, and if in addition dimC M = 2, M must be covered by either a K3 surface or a complex 2-torus. The flow behavior of Calabi-Yau manifold the Ricci flow is well understood in [6].

The case studied by Song-Tian in [73,74] is 0 < Kod(M) < dimC M. The semi-positivity of KM N gives rise to a holomorphic map π into a complex projective space CP and Kod(M) is the dimension of the image of π. In this case, the image of π is a Kählermanifold Σ with dimC Σ = Kod(M), and π : M → Σ can be regarded as a fibration map with possibly singular fibers. By pulling back the N ∗ Fubini-Study metric of CP to M one can find a Kählermetric ωΣ on Σ such that c1(M) = −π [ωΣ].

As before, the Kählerclass [ωt] under the normalized flow converges to −c1(M), which in case it ∗ is π [ωΣ]. Song-Tian proved the following result which confirmed that the convergence behavior in this case is as predicted by the limiting Kählerclass.

Theorem 1.3 (Song-Tian [73,74]). Suppose M is a compact Kählermanifold with semi-ample and non-big canonical line bundle KM . Let π : M → Σ be the above associated fibration map such ∗ that c1(KM ) = π α for some Kählerclass α on Σ. Denote Σreg to be the regular part of Σ, i.e. −1 π (z) is regular for any z ∈ Σreg. Then for any initial Kählermetric ω0 on M, the normalized ∗ Kähler-Ricci flow ∂tωt = −Ric(ωt) − ωt has a long-time solution ωt and ωt → π ω∞ ∈ −c1(M) as currents for a positive current ω∞ on Σ. Moreover, on the regular part Σreg, ω∞ is smooth and

Ric(ω∞) = −ω∞ + ωWP where ωWP is the Weil-Petersson metric.

An interesting investigation of the above ﬂows is the study of blow-up or boundedness of curvatures. All the above results concern K¨ahlermanifold M with semi-ample c1(KM ). In this case, the following results concerning about the scalar curvature are obtained: CHAPTER 1. INTRODUCTION 6

Theorem 1.4 (Zhang [86]; Song-Tian [76]). Let M be a compact Kählermanifold with c1(KX ) being semi-ample, then along the normalized Kähler-Ricci flow ∂tωt = −Ric(ωt) − ωt starting with any initial metric ω0, the scalar curvature R(ωt) is uniformly bounded along the flow defined on t ∈ [0, ∞).

r Let us change the gear to survey over the results on Fano fibrations, with emphasis on CP - bundles. From the evolutions of Kählerclasses, finite-time singularity is expected in both normalized and unnormalized flows if the fibers are Fano manifolds. It is more desirable to consider the unnormalized Kähler-Ricciflow in this case as the Kählerclass [ωt] simply evolves linearly: under the unnormalized Kähler-Ricciflow ∂ω t = −Ric(ω ), ∂t t the Kählerclass is given by:

[ωt] = [ω0] − tc1(M).

Unlike the normalized flow with infinite-time singularity, the limiting Kählerclass is not solely determined by c1(M) but also depends on the initial Kählerclass [ω0]. Different limiting Kähler classes exhibit distinct convergence and singular behaviors of the flow. This interesting phenomena can be observed in the following work by Song-Weinkove:

Theorem 1.5 (Song-Weinkove [77]). Let M be a Hirzebruch surface/manifold, i.e. M = P(O ⊕ π n 1 n O(−k)) −→ CP which is a CP -bundle over CP with twisting number k ∈ N. Suppose [ω0] is con- 3 structed by Calabi’s ansatz , then the unnormalized K¨ahler-Ricci ﬂow ∂tωt = −Ric(ωt) encounters

ﬁnite-time singularity for some T < ∞, and depending on the triple (n, k, [ω0]), one of the following holds:

1. (M, ωt) converges to (Morb, ωorb) in Gromov-Hausdorﬀ topology, where Morb is an orbifold

obtained by the blow-down of M, and ωorb is an orbifold metric.

2. (M, ωt) converges to a point in Gromov-Hausdorﬀ topology. n 3. (M, ωt) converges to (CP , ωFS) in Gromov-Hausdorﬀ topology, where ωFS is the Fubini-Study n metric of CP .

The above result employs the Calabi’s ansatz which is preserved under the flow and allows us to reduce the Kähler-Ricciflow into an ODE. This cohomogeneity-1 assumption is later removed and the result is further generalized by Song-Weinkove [78, 79] for case (1) and Song-Székelyhidi- Weinkove [72] for case (3). Case 2 is the canonical class case which is addressed by an unpublished

3See Chapter 4 for the deﬁnition and discussion of Calabi’s ansatz CHAPTER 1. INTRODUCTION 7

work due to Perelman (see [66]). As our further development is motivated by case (3) above, let us state the result in [72]:

π r Theorem 1.6 (Song-Székelyhidi-Weinkove [72]). Let M = P(Er+1) −→ Σ be a CP -bundle over a smooth projective variety Σ. Here Er+1 is a holomorphic vector bundle over Σ and P denotes the projectivization. Then the Kähler-Ricci flow ∂tωt = −Ric(ωt) will encounter finite-time singularity at some time T < ∞. Assume further that [ω0] satisfies the condition

∗ [ω0] − T c1(M) = [π ωΣ]

for some K¨ahlermetric ωΣ on Σ, then we have

−1 −1 1/3 • the ﬁbers π (z), z ∈ Σ collapse at the rate diamt(π ) = O((T − t) ).

• (M, ωt) converges subsequentially in Gromov-Hausdorﬀ sense to (Σ, dΣ) for some metric dΣ on Σ.

1.3 Summary of new results

The research conducted in this thesis is motivated by the works surveyed in the previous section. Broadly speaking, we are interested in exploring how the cohomology data, such as K¨ahlerclasses and Chern classes, determine the convergence and singular behavior of the ﬂow metric. We focus on the collapsing case. The major results established in this thesis concern about the following aspects of singularity analysis:

• rate of collapsing; • Gromov-Hausdorﬀ convergence; • blow-up rate or uniform boundedness of curvatures; and • singularity model obtained by rescaling analysis.

We first investigate the collapsing results under the normalized Kähler-Ricciflow further based on the set-up in Song-Tian’s [73, 74]. Recall that Theorem 1.3 asserted the convergence behavior of metrics on Calabi-Yau fibrations, which is predicted by the semi-ampleness of the canonical line bundle KM . We give an extension to Song-Tian’s collapsing result and prove that the fibers collapse in the optimal rate. This is a joint work with Zhang. Precisely, we proved CHAPTER 1. INTRODUCTION 8

Theorem 1.7 (F-Zhang [32]; Theorem 3.4 of Chapter 3). Let M −→π Σ be a holomorphic submersion ∗ for some Kählermanifold Σ. Suppose c1(M) = −π α for some Kählerclass α on Σ, then under the normalized Kähler-Ricci flow ∂ω t = −Ric(ω ) − ω ∂t t t starting with any initial metric ω0, then the flow has long-time solution and along the fiber directions, −t −1 the metric ωt is uniformly equivalent to e ω0 and hence the fibers π (z), z ∈ Σ collapse at the optimal rate, i.e. there exists a constant C > 0 depending only on n, r, ω0, α such that

−1 − t −1 − t C e 2 ≤ diamt(π (z)) ≤ Ce 2 , for any z ∈ Σ.

The estimates on the flow metric ωt in Theorem 1.7 lead to several consequences of convergence and boundedness of curvatures. For Gromov-Hausdorff convergence, one can follow the standard argument as in [36, 72, 77] etc. to show that the above Kählermanifold converges subsequentially −1 r to the base manifold. Furthermore, when the fibers π (z) all complex tori C /Λz with possibly varying complex structures, the Riemann curvature tensor kRmkωt can be shown to be uniformly bounded along the flow by a similar argument as in Gross-Tosatti-Zhang’s work [36] which makes use of Evans-Krylov’s theory. The upshot is that there exists a semi-flat metric with good rescaling properties which allows us to rewrite, at least locally, the Kähler-Ricciflow into a non-degenerated complex Monge-Ampèreequation. The boundedness of kRmkωt can then be established by standard higher order estimates and Evans-Krylov theory. Secondly, we study the Fano fibrations under the unnormalized Kähler-Ricciflow

∂ω t = −Ric(ω ). ∂t t

r Recall that the results concerning the collapsing behavior of CP -bundles were established by Song, Székelyhidi and Weinkove in [72,77] (see Theorems 1.5 and 1.6). The diameter of fibers collapses at a rate of at most (T − t)1/3 where T < ∞ is the singular time. While this collapsing rate is sufficient for proving the Gromov-Hausdorff convergence, it will be more desirable to improve this estimate to ' (T − t)1/2, i.e. the optimal rate, especially when one would like to study their singularity models.

In this regard, we show that it suffices to prove the Ricci curvature Ric(ωt) is uniformly bounded with respect to a fixed background metric, say ω0, along the flow. The allowable type of fibrations is also widen. Precisely, we proved CHAPTER 1. INTRODUCTION 9

Theorem 1.8 (F [31]; Theorem 1.8 of Chapter 3). Let M −→π Σ be a surjective holomorphic submersion where Σ is a K¨ahlermanifold with dimC Σ < dimC M. Suppose the (unnormalized) K¨ahler-Ricci

flow ∂tωt = −Ric(ωt) on X encounters finite-time singularity at T < ∞ and the initial Kählerclass

[ω0] satisﬁes ∗ [ω0] − T c1(M) = π α for some K¨ahlerclass α of Σ, we have

Ric(ωt) ≤ Bω0 for some uniform constant B > 0

−1 1/2 −1 1/2 ⇒ C (T − t) ≤ diamωt π (z) ≤ C(T − t) for any t ∈ [0,T ), z ∈ Σ,

where C is a uniform constant depending only on dimensions of M and Σ, ω0, α and B.

Theorems 1.7 and 1.8 and their proofs constitute most part of Chapter 3. In Chapter 4, we will study a specific type of fibrations which yields stronger results and leads to further understanding 1 of singularities. Namely, we focus on CP -bundles over Kähler-Einsteinmanifolds and construct the initial Kählermetric by Calabi’s ansatz, a cohomogeneity-1 symmetry. Under such a construction, the Gromov-Hausdorff convergence can be obtained as in [77] without taking subsequences. 1 The CP -bundles to be studied in Chapter 4 are constructed in the following way: Let Σ be a π Kähler-Einsteinmanifold with metric ωΣ such that Ric(ωΣ) = νωΣ and L −→ Σ is a holomorphic line bundle over Σ which admits a Hermitian metric h such that the Chern curvature is given by √ ¯ −1∂∂ log h = λωΣ for some λ > 0. We consider the projective bundle

M = P(OΣ ⊕ L)

1 whose ﬁbers are CP ’s with transition functions inherited from L.

Under this construction, it is possible to construct cohomogeneity-1 initial K¨ahlermetrics ω0 using the Hermitian metric h such that the cohomogeneity-1 symmetry is preserved along the ﬂow.

Detail about this symmetry is given in Chapter 4. The K¨ahlerclass [ω0] is in the form

[ω0] = λb0[Σ∞] − λa0[Σ0]

for some 0 < a0 < b0. Here we denote [Σ∞] and [Σ0] to be the Poincaréduals of the ∞- and 0-sections respectively. The first result concerning the Kähler-Ricciflow on these bundles is about the Gromov-Hausdorff convergence. Precisely, we have CHAPTER 1. INTRODUCTION 10

Theorem 1.9 (F [30]; Theorem 4.6 of Chapter 4). Let M = P(OΣ ⊕ L) be a projective bundle described above and ω0 be a Kählermetric on M constructed by the Calabi’s ansatz with Kähler class [ω0] = λb0[Σ∞] − λa0[Σ0]. Suppose the triple (Σ, L, [ω0]) satisfies the following conditions

ν ≤ λ, or

ν > λ and (ν − λ)b0 < (ν + λ)a0,

then along the Kähler-Ricci flow ∂tωt = −Ric(ωt), t ∈ [0,T ), (M, ωt) converges in Gromov-Hausdorff sense to (Σ, ωΣ)..

The idea of proof of Theorem 1.9 is a variation on the theme of Song-Weinkove’s work [77] on

Hirzebruch surfaces (see Theorem 1.5). The key ingredient is to show the degeneration of ωt along the fiber direction by applying the maximum principles on the ODE obtained from the Calabi’s ansatz. A more interesting question about the Kähler-Ricciflow on these bundles is classifying the singularity model. Singularity analysis of various geometric evolution equations, including the Ricci flow and the Mean Curvature flow, involves some appropriate rescaling procedures. For the Ricci

flow, the finite-time singularity is due to the blow-up of the Riemann curvature tensor kRmkg(t). In other words, some regions of the manifold have extremely high curvature when approaching to the singular time. By a suitable enlargement, the Riemann curvature can be scaled down to a smaller and bounded magnitude. One would like to understand the convergence of this rescaled sequence of manifolds and classify the limit manifold if it exists. Precisely, a generic rescaling procedure is given by the following: Let T < ∞ be the singular time of the Ricci flow (M, g(t)), t ∈ [0,T ) and take a sequence of times ti → T as i → ∞. For each i ∈ N, let xi ∈ M be a point at which maxM kRmkg(ti) is achieved.

Denote Ki = maxM kRmkg(ti), we deﬁne the following rescaled and dilated sequence of Ricci ﬂows:

−1 gi(t) := Kig(ti + Ki t), t ∈ [−tiKi, (T − ti)Ki).

By Hamilton-Cheeger-Gromov’s Compactness and Perelman’s local non-collapsing theorem [56], one can extract a subsequence of ﬂows, still denoted by gi(t), such that (M, gi(t), xi) converges in pointed Cheeger-Gromov’s sense to a limit (M∞, g∞(t), x∞) which is a complete solution of the

Ricci ﬂow and may be topologically distinct from M. We say (M∞, g∞(t)) the singularity model of the Ricci ﬂow (M, g(t)). CHAPTER 1. INTRODUCTION 11

Classifying the singularity model (M∞, g∞(t)) gives rise to a finer and concrete understanding of the singularity development of (M, g(t)) as t approaches T . In many occasions, the solution g∞(t) is a self-similar solution. A recent result due to Enders, Müllerand Topping [25] (see also [52]) asserts −1 that if the Ricci flow encounters Type I singularity, i.e. maxM kRmkg(t) = O((T − t) ) as t → T , then one can obtain a self-similar solution as the singularity model. The next few results to be described concern about the singularity development of the projective bundles P(OΣ ⊕ L) constructed as in before. We will first consider the case where the Kähler-Ricci 1 flow encounters Type I singularity, and show that the singularity model is Cn × CP . Precisely, we have

Theorem 1.10 (F [30]; Theorem 4.13 of Chapter 4). Let M = P(OΣ ⊕ L) be the projective bundle with (ν, λ) and a cohomogeneity-1 metric ω0 satisﬁed as in Theorem 1.9. Let (M, ωt) be the K¨ahler-

Ricci flow ∂tωt = −Ric(ωt), t ∈ [0,T ) with initial Kählerclass [ω0]. Denote g(t) to be the Kähler metric associated to the form ωt. Suppose the flow encounters Type I singularity, we choose (xi, ti) in space-time such that Ki := kRm(xi, ti)kg(ti) = maxM kRmkg(ti) and ti → T . Consider the rescaled dilated sequence of metrics

−1 gi(t) := Kig(ti + Ki t), t ∈ [−tiKi, (T − ti)Ki).

Then the pointed sequence (M, gi(t), xi) converges smoothly, after passing to a subsequence, in pointed

Cheeger-Gromov sense to an ancient κ-solution (M∞, g∞(t), x∞), whose universal cover splits isometrically as n 1 (C × P , δ ⊕ (1 − t)ωFS), where δ is the Euclidean metric and ωFS denotes the Fubini-Study metric.

It is an open conjecture whether the Kähler-Ricciflow with finite-time singularity is always of

Type I. Indeed, a weaker conjecture on whether the scalar curvature R(ωt) must blow-up at the rate O((T − t)−1) is still open. In case of the projective bundles being studied, we are able to show that the K¨ahler-Ricciﬂow must encounter Type I singularity. Precisely, we have

Theorem 1.11 (F [30]; Theorem 4.14 of Chapter 4). Let M = P(OΣ ⊕ L) be the projective bundle with Σ,L and [ω0] satisfying all the conditions stated in Theorem 1.9 and ω0 is constructed by the

Calabi’s ansatz. Let (M, ωt) be the Kähler-Ricci flow ∂tωt = −Ric(ωt), t ∈ [0,T ) with initial Kähler class [ω0]. Then (M, g(t)) must be of Type I, i.e. Type II singularity is not possible. CHAPTER 1. INTRODUCTION 12

The idea of proofs of both Theorems 1.10 and 1.11 makes use of the de Rham splitting theorem and some Hamilton’s classiﬁcation results of ancient and eternal solutions. Perelman’s Non- collapsing Theorem is also used in Theorem 1.11. The upshot is that if Type II singularity occurs, one can carefully pick (xi, ti) ∈ M × [0,T ) such that the pointed sequence of rescaled metrics converges to the product of a ﬂat space and the cigar soliton. However, such a singularity model is ruled out by Perelman’s result. Chapter 2

Background

In this chapter, we introduce deﬁnitions and concepts relevant to this thesis.

2.1 K¨ahlergeometry

This section introduces the notions of complex and Kählermanifolds which is the class of manifolds under the spotlight of this thesis. We will outline the local expressions of connections and curvatures, and define characteristic classes which are fundamental in the study of the Kähler-Ricci flow.

2.1.1 Complex manifolds

We begin by defining complex manifolds. Heuristically speaking, a differentiable manifold is a topological space whose every point admits a local Rn-coordinate chart, and on the overlap of the charts the transition is a C∞-function (i.e. infinitely differentiable). A complex manifold is a natural complex analogue of a differentiable manifold. At every point of a complex manifold there is a local

Cn-coordinate chart, and on the overlap of the charts the transition function is biholomorphic. A complex manifold M necessarily has even dimension as a diﬀerentiable manifold. Each Cn- √ coordinate chart gives a local holomorphic coordinate zi = xi + −1yi, i = 1, 2, . . . , n and deﬁne an endomorphism J : TM → TM by decreeing

∂ ∂ J = ∂xi ∂yi ∂ ∂ J = − ∂yi ∂xi

13 CHAPTER 2. BACKGROUND 14

2 for any i = 1, 2, . . . , n. It is easy to see that J = −idTM . This endomorphism J is called the complex structure of M. We will give a relatively abstract and succinct deﬁnition for complex manifolds which is characterized by a given endomorphism J : TM → TM.

Deﬁnition 2.1 (Complex manifold). A diﬀerential manifold M is called a complex manifold if there 2 exists an endomorphism J : TM → TM such that J = −idTM and the following Nijenhuis tensor

NJ vanishes, i.e.

NJ (X,Y ) = [X,Y ] + J[JX,Y ] + J[X,JY ] − [JX,JY ] = 0 for any X,Y ∈ TM.

The vanishing of the Nijenhuis tensor implies the integrability of the endomorphism J by a classical theorem due to Newlander-Nirenberg [53]. An integrable complex structure J on M gives a well-defined complex multiplication on the √ complexified tangent space TM ⊗ C, namely (a + b −1) · X = aX + bJX for any a, b ∈ R and X ∈ TM, and analogously on the complexified cotangent space T ∗M ⊗ C. There is a pair of naturally defined dual bases over C for T ∗M ⊗ C and TM ⊗ C, respectively

√ √ dzi = dxi + −1dyi, dz¯i = dxi − −1dyi;

∂ 1 ∂ √ ∂ ∂ 1 ∂ √ ∂ = − −1 , = + −1 ∂zi 2 ∂xi ∂yi ∂z¯i 2 ∂xi ∂yi for i = 1, . . . , n. We say a diﬀerential form ω a (p, q)-form if it can be locally expressed as a linear combinations of basic forms of the following type:

dzi1 ∧ ... ∧ dzip ∧ dz¯j1 ∧ ... ∧ dz¯jq .

For each k ∈ N, the space of diﬀerential k-forms on a complex manifold admits the following decomposition: M ∧kM = ∧p,qM p+q=k where ∧p,qM denotes the space of (p, q)-forms on M. CHAPTER 2. BACKGROUND 15

2.1.2 K¨ahlermanifolds

This thesis focuses on Kählermanifolds which form an important class of complex manifolds and draw attention to many mathematical fields including symplectic topology and algebraic geometry. There are several equivalent definitions of Kählermanifolds. Here we give the one which has the greatest impact on the discussion that follows:

Definition 2.2 (Kählermanifold). A complex manifold (M,J) is called a Kählermanifold if it admits a Riemannian metric g such that (1) g is J-invariant1, i.e. g(JX,JY ) = g(X,Y ) for any X,Y ∈ TM and (2) the following associated (1, 1)-form ω is closed, i.e. dω = 0

ω(X,Y ) := g(JX,Y ).

We call g the K¨ahlermetric and ω the K¨ahlerform of (M, J, g, ω).

In holomorphic coordinates {z1, . . . , zn}, the K¨ahlerform ω has the following local expression:

√ i j ω = −1gi¯jdz ∧ dz¯

∂ ∂ where g ¯ = g( , ). ij ∂zi ∂z¯j The J-invariant condition guarantees that g( ∂ , ∂ ) = g( ∂ , ∂ ) = 0, and the K¨ahler condition ∂zi ∂zj ∂z¯i ∂z¯j dω = 0 is equivalent to saying ∂g ¯ ∂g ¯ ij = kj , for any i, j, k. (2.1.1) ∂zk ∂zi

C Using (2.1.1), it is not diﬃcult to observe that the Christoﬀel symbols ΓAB vanish unless A, B and C are either all (1, 0)- or all (0, 1)-type, and the non-vanishing ones admit the following local expression: k k¯l ∂ Γij = g gj¯l. ∂zi The Riemann (3, 1)-tensor2 has the following local expression:

l ∂ l Ri¯jk = − Γik, ∂z¯j

1a J-invariant metric is also called a Hermitian metric 2Please do not confuse with (3, 1)-forms. CHAPTER 2. BACKGROUND 16

and hence the Ricci tensor Ric(ω) is given by

X k ∂ kq¯ ∂ Rici¯j = Ri¯jk = − g gkq¯ . ∂z¯j ∂zi k

kq¯ ∂ ∂ By the diﬀerentiation formula for log det(g), we have g gkq¯ = log det(g), and so the Ricci ∂zi ∂zi tensor has a very nice local expression:

∂2 Rici¯j = − log det(g). (2.1.2) ∂zi∂z¯j

With abuse of notations, we will also denote the Ricci form by Ric(ω):

√ i j Ric(ω) = −1Rici¯jdz ∧ dz¯ .

By (2.1.2), it is clear that the Ricci form Ric is a closed (1, 1)-form. Here is an important example of a K¨ahlermetric.

n n Example 2.3 (Fubini-Study metric on CP ). The complex projective space CP is a quotient space of Cn+1\{0} under the equivalence relations

(Z0,...,Zn) ∼ (λZ0, . . . , λZn)

n for some λ ∈ C. The Fubini-Study metric ωFS on CP is deﬁned by

√ ¯ 2 2 ωFS = −1∂∂ log(|Z0| + ... + |Zn| ).

It is clear that this deﬁnition factors through the equivalence relations (Z0,...,Zn) ∼ (λZ0, . . . , λZn).

2.1.3 Characteristic classes

The de Rham cohomology classes represented by closed forms, such as the Kählerform and the Ricci form, play crucial roles in Kählergeometry. We will see throughout the rest of the thesis that these characteristic classes often predict the limiting and singular behavior of the Kähler-Ricciflow. A fundamental result of characteristic classes is the following ∂∂¯-lemma stemmed from Hodge theory.

Lemma 2.4 (∂∂¯-Lemma). Let (M, J, ω) be a compact K¨ahlermanifold. Given any two closed real CHAPTER 2. BACKGROUND 17

(1, 1)-forms α and β, if α − β is d-exact (i.e. α − β = dγ for some 1-form γ), they must also be ∂∂¯-exact, i.e. there exists a smooth real-valued function f such that

√ α − β = −1∂∂f.¯

Two important characteristic classes in Kählergeometry are the Kählerclass and the first Chern class.

Definition 2.5 (Kählerclass). Given a compact Kählermanifold (M, J, ω), the Kählerclass [ω] is the de Rham cohomology class represented by ω. Hence, by the ∂∂¯-lemma, it is given by

√ ∞ [ω] = {ω + −1∂∂ϕ¯ : ϕ ∈ C (M, R)}.

Definition 2.6 (First Chern class). Given a compact Kählermanifold (M, J, ω), the first Chern class c1(M) is the de Rham cohomology class represented by the Ricci form Ric(ω), i.e.

c1(M) = [Ric(ω)].

It is a fundamental fact that the first Chern class c1(M) depends only on (M,J) but not on the Kählermetric3 ω. To see this, we letω ˜ be another Kählermetric on M. Then by (2.1.2), we have

√ Ric(ω) − Ric(˜ω) = − −1∂∂¯(log det(ω) − log det(˜ω) √ det(ω) = − −1∂∂¯log . det(˜ω)

det(ω) As det(˜ω) deﬁnes a global function on M, it shows that Ric(ω) and Ric(˜ω) represent the same de Rham cohomology class. It does depend on J because the ∂∂¯-operator depends on the holomorphic coordinates induced by the integrable complex structure J.

2.2 K¨ahler-Ricciﬂow

We can proceed to the Kähler-Ricciflow equation. In this thesis, both normalized and unnormalized flows will be investigated. We will first show in this section that the Kähler-Ricciflow equation is equivalent to a parabolic complex Monge-Ampèreequation of its scalar potential. We then state

3From now on, we will use the terms “K¨ahlermetric” and “K¨ahler form” interchangeably. CHAPTER 2. BACKGROUND 18

and prove the result due to Tian-Zhang [82] on the maximal existence time of the K¨ahler-Ricciﬂow. Some examples will be supplied to illustrate the use of Tian-Zhang’s theorem.

Given a compact Kählermanifold (M, J, ω0), the (unnormalized) Kähler-Ricciflow equation is defined as ∂ω t = −Ric(ω ), ω | = ω . (2.2.1) ∂t t t t=0 0

Throughout the thesis, unless otherwise specified, we will refer the “Kähler-Ricciflow” as the unnormalized one (2.2.1). In Chapter 3, we will also deal with the normalized Kähler-Ricciflow which is more natural when dealing with semi-ample c1(M). The normalized Kähler-Ricciflow is defined as

∂ω t = −Ric(ω ) − ω , ω | = ω . (2.2.2) ∂t t t t t=0 0

It can be easily verified that if ω(t) = ωt satisfies the unnormalized Kähler-Ricciflow (2.2.1), −t t then the family of metricsω ˜t = e ω(e − 1) will satisfy the normalized Kähler-Ricciflow (2.2.2).

2.2.1 Complex Monge-Amp`ereequations

We will derive a scalar version of (2.2.1) and (2.2.2) in this subsection. Firstly, recall that the ﬁrst

Chern class c1(M) = [Ric(ω0)] depends only on J which is fixed under the flow. Therefore, passing to the cohomology, the Kählerclass [ωt] under the Kähler-Ricciflow (2.2.1) satisfies the following ODE:

d [ω ] = −c (M) dt t 1

⇒ [ωt] = [ω0] − tc1(M).

One central theme shared with many current literature of the Kähler-Ricciflow is that the limiting behavior of the flow can be read off and predicted by the evolution of the Kählerclass [ωt], which is explicitly given once the first Chern class c1(M) is known.

Pick a representative ρ ∈ c1(M), deﬁne the following reference metric

ωˆt = ω0 − tρ.

¯ Thenω ˆt is in the same K¨ahlerclass as the ﬂow metric ωt. By Lemma 2.4 (i.e. the ∂∂-lemma), CHAPTER 2. BACKGROUND 19

there exists a family of smooth functions ϕt on M such that

√ ¯ ωt =ω ˆt + −1∂∂ϕt.

√ Let Ω be a time-independent volume form on M such that −1∂∂¯log Ω = −ρ. It is straight- forward to check that the Kähler-Ricciflow (2.2.1) can be rewritten as the following parabolic complex Monge-Ampèreequation for ϕt: √ ∂ϕ (ˆω + −1∂∂ϕ¯ )n t = log t t , ϕ = 0. (2.2.3) ∂t Ω 0

Remark 2.7. In later chapters we will replace Ω by f(t)Ω for some function f(t) depending only on t. This is for the purpose of giving uniform bounds for ϕt. One can easily check that this adjustment √ ¯ only changes the solution ϕt to (2.2.3) but not the ﬂow metric ωt =ω ˆt + −1∂∂ϕt.

Similarly, one can derive the Kählerclass [ωt] of the normalized Kähler-Ricciflow (2.2.2):

d [ω ] = −c (M) − [ω ] dt t 1 t −t ⇒ [ωt] = −c1(M) + e ([ω0] + c1(M)).

Pick a representative ρ ∈ c1(M), we deﬁne a reference metricω ˜t by:

−t ω˜t = −ρ + e (ω0 + ρ).

Similar to the unnormalized flow, the normalized Kähler-Ricciflow (2.2.2) is equivalent to the following scalar equation: √ ∂ϕ (˜ω + −1∂∂ϕ¯ )n t = log t t − ϕ , ϕ = 0, (2.2.4) ∂t Ω t 0 √ ¯ where Ω is a volume form such that −1∂∂ log Ω = −ρ, and the flow metric is given by ωt = √ ¯ ωˆt + −1∂∂ϕt.

2.2.2 Maximal existence time

The short-time existence of the Ricci ﬂow on a general Riemannian manifold is proved by Hamilton in [39] and DeTurck in [23]. A Ricci ﬂow solution may not exist for all time t. We say the Ricci CHAPTER 2. BACKGROUND 20

flow g(t) encounters finite-time singularity if the maximal time of existence T is finite. By a result of Hamilton, the occurrence of finite-time singularity is due to the blow-up of curvature. Precisely, we have

Theorem 2.8 (Hamilton). If (M, g(t)) is a solution of the Ricci ﬂow ∂tg(t) = −Ric(g(t)) on a closed manifold and T < ∞ is the maximal time, then we have

sup kRmkg(t) = ∞ (2.2.5) M×[0,T )

Proof. The gist of the proof is by contradiction. Assume on the contrary that

kRmkg(t) ≤ C, t ∈ [0,T ) (2.2.6) where C is independent of t. One can show g(t) extends to a continuous metric g(T ) by the following argument: Given Vx ∈ TxM and t ∈ [0,T ), we have

2 t 2 kVxk Z ∂τ kVxk log g(t) = g(τ) dτ (2.2.7) 2 2 kVxkg(0) 0 kVxkg(τ) Z t Vx Vx ≤ Ric , dτ (2.2.8) 0 kVxkg(τ) kVxkg(τ) Z t ˜ ≤ kRicxkg(τ)dτ ≤ CT, (2.2.9) 0 where C˜ depends on C and n. This implies

e−CT˜ g(x, 0) ≤ g(x, t) ≤ eCT˜ g(x, 0). (2.2.10)

Therefore, g(t) extends to a continuous metric g(T ). By Shi’s derivative estimate in [68], the uniform k bound on kRmkg(t) implies uniform bounds on all k∇ Rmkg(t), k > 1. The smooth convergence of g(t) → g(T ) as t → T hence follows, and so the Ricci ﬂow extends beyond [0,T ) which leads to a contradiction.

Theorem 2.8 applies to all Ricci flows on any closed Riemannian manifold with finite-time singularity. In general, the exact maximal time T of a Riemannian Ricci flow may not be easy to find. However, for Kähler-Ricci flows, the maximal time of existence T is explicitly determined by the initial Kählerclass [ω0] and the first Chern class. CHAPTER 2. BACKGROUND 21

Theorem 2.9 (Tian-Zhang, [82]). Let (M, ω(t)) be an (unnormalized) K¨ahler-Ricci ﬂow ∂tωt =

−Ric(ωt) on a compact K¨ahlermanifold M with dimC = n. Then the maximal existence time T is given by

T = sup{t :[ω0] − tc1(M) > 0}.

Proof. For each suﬃciently small ε > 0, pick ρε ∈ c1(M) such that ω − (T − ε)ρε > 0. The

Kähler-Ricciflow ∂tωt = −Ric(ωt) can be rewritten as the complex Monge-Ampèreequation, c.f.

(2.2.3): √ ¯ n ∂ϕt (ω0 − tρε + −1∂∂ϕt) = log , ϕ0 = 0, t ∈ [0,T − ε] ∂t Ωε √ ¯ where Ωε is a volume form such that −1∂∂ log Ωε = −ρε and the ﬂow metric ωt is given by √ ¯ ωt = ω0 − tρε + −1∂∂ϕt. −1 n Using maximum principle and the fact that Cε Ωε ≤ (ω0 − tρε) ≤ CεΩε for some Cε > 0 depending only on ε, one can easily see that |ϕt| ≤ Cε for t ∈ [0,T − ε]. By straight-forward computations, we have

∂ ∂ϕ − ∆ t t − ϕ − nt = −Tr ω , ∂t ωt ∂t t ωt 0

where ∆ωt denotes the Laplacian with respect to ωt. Here we have used the fact that n = Trωt ωt =

Trωt ω0 − tTrωt ρε + ∆ωt ϕt. Hence by maximum principle, we have

∂ϕ t t − ϕ − nt ≤ C t ∈ [0,T − ε], ∂t t ε

∂ϕt and so ∂t ≤ Cε for any t ∈ [0,T − ε]. Similarly, applying maximum principle to the following evolution equation

∂ ∂ϕ − ∆ (T − ε − t) t + ϕ = −n + Tr (ω − (T − ε)ρ ) ∂t ωt ∂t t ωt 0 ε

∂ϕt gives ∂t ≥ −Cε for t ∈ [0,T − 2ε].

∂ϕt 0 The uniform bounds on ϕt and ∂t give C -estimate to the complex Monge-Amp`ereequation

√ ∂ϕt ¯ n ∂t (ω0 − tρε + −1∂∂ϕt) = e Ωε, t ∈ [0,T − 2ε].

By standard higher-order estimates due to Aubin [2] and Yau [85], one can get the existence of a CHAPTER 2. BACKGROUND 22

√ ¯ smooth solution ϕε,t for t ∈ [0,T − 2ε], and hence a smooth solution ωt = ω0 − tρε + −1∂∂ϕε,t to the Kähler-Ricciflow (2.2.1) up to [0,T − 2ε]. 4 + By the uniqueness of the Ricci flow , one can extend ωt to [0,T ) by taking ε → 0 . It completes the proof of the theorem.

Remark 2.10. Recall that [ω0] − tc1(M) is the Kählerclass [ωt] of the flow metric. Tian-Zhang’s result essentially asserts that the Kähler-Ricciflow stops exactly at the time that the Kählerclass degenerates.

By the rescaling correspondence between the normalized ﬂow (2.2.2) and the unnormalized ﬂow (2.2.1), we also have

Theorem 2.11 (Tian-Zhang, [82]). Let (M, ω(t)) be a normalized K¨ahler-Ricci ﬂow ∂tωt = −Ric(ωt)−

ωt on a compact K¨ahlermanifold M with dimC = n. Then the maximal existence time T is given by −t T = sup{t : −c1(M) + e ([ω0] + c1(M)) > 0}.

Throughout the thesis, T will be denoted as the maximal existence time unless otherwise specified. If T < ∞, we say the flow encounter finite-time singularity and T is often called the singular time. There are several interesting special cases which illustrate the use of Tian-Zhang’s result.

Example 2.12. When c1(KM ) is semi-ample, in particular c1(M) ≤ 0, both the normalized and unnormalized Kähler-Ricciflows have global solutions, i.e. defined on t ∈ [0, ∞).

Example 2.13. When [ω0] = c1(M), i.e. canonical class, then the unnormalized K¨ahler-Ricciﬂow (2.2.1) has maximal time interval t ∈ [0, 1).

2 2 2 Example 2.14 (to be revisited in Chapter 4). Let M be CP #(−CP ), i.e. CP blow-up at a point. Then

c1(M) = −[Σ0] + 3[Σ∞]

5 where Σ0 and Σ∞ are respectively the zero- and inﬁnity- sections . Any K¨ahlerclass [ω0] can be written as

[ω0] = −a[Σ0] + b[Σ∞],

4It is also possible to simply use the uniqueness of the complex Monge-Amp`ereequation. For detail, please see [82]. 5 2 1 The CP blow-up at a point can also be thought as a compactiﬁed O(−1)-bundle over CP . CHAPTER 2. BACKGROUND 23

and hence the K¨ahlerclass [ωt] under the unnormalized ﬂow ∂tωt = −Ric(ωt) is given by:

[ωt] = −(a − t)[Σ0] + (b − 3t)[Σ∞].

To see the maximal time T , we need to ﬁgure out when [ωt] degenerates. There are three possibilities:

(1) 3a < b:[ωt] degenerates to c[Σ∞] for some c > 0, and T = a; (2) 3a = b:[ωt] degenerates to 0 b−a and T = a; (3) 3a > b:[ωt] degenerates to c(−[Σ0] + [Σ∞]) and T = 2 .

As we shall see in later chapters, the limiting K¨ahlerclass [ωT ] predict the convergence and singular behavior of the ﬂow. Chapter 3

Collapsing of Holomorphic Fibrations

This chapter investigates the collapsing behavior of some holomorphic fibrations and gives the proofs of Theorems 1.7 and 1.8 which concern about Calabi-Yau fibrations and Fano fibrations respectively. In Section 3.1, we state and give the proofs of two classic estimates, namely the parabolic Schwarz’s estimate and Yau’s L∞-estimates which will be used frequently in the whole chapter. Section 3.2 is devoted to the collapsing of non-singular Calabi-Yau fibrations under the normalized Kähler-Ricci flow. We prove that the collapsing rate is optimal and gives an extension of the non-singular case of Song-Tian’s works [73, 74] in some analytic and geometric aspects. Section 3.3 discusses the collapsing behavior of more general types of fibrations with emphasis on Fano fibrations under the unnormalized Kähler-Ricciflow.

3.1 Some classic estimates

In this section we give the proofs of two crucial estimates, namely the parabolic Schwarz’s estimate and Yau’s L∞-estimate, which are frequently used in this chapter as well as in many literature in K¨ahlergeometry.

24 CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 25

3.1.1 Parabolic Schwarz’s estimate

An overriding goal of this thesis is to investigate how the metric g(t) degenerates along the K¨ahler-

Ricci Flow when the total volume collapses to zero. It is done by comparing the eigenvalues of ωt with those of a family of reference metricsω ˆt, which is not necessarily a Ricci flow solution. In doing so, we need to estimate Trωˆt ωt and Trωt ωˆt using maximum principles. This type of estimates is often called the parabolic Schwarz’s estimate. Since this chapter deals with both normalized and unnormalized Kähler-Ricciflows, we give a form of the parabolic Schwarz’s estimate which is applicable to both flows.

For the sake of simplicity, we may suppress the subscript t for the Kählerforms ωt andω ˆt, and denote g = g(t) as the Kählermetric of ωt and similarly,g ˆ =g ˆ(t) the Kählermetric ofω ˆt.

Lemma 3.1 (Parabolic Schwarz’s estimate). Suppose ωt is a solution to the K¨ahler-Ricci ﬂow

∂tωt = −Ric(ωt) − µωt where µ ∈ R, and ωˆt is a family of K¨ahlermetrics such that ∂tωˆt = ηt.

Denote = ∂t − ∆ωt , then we have

log Trωˆ ω ¯ 2 1 iq¯ p¯j i¯j ˆ kl i¯j k¯l pq¯ ˆ ˆ |∇Trωˆ ω| = −gˆ gˆ ηpq¯gi¯j − g Rmi¯j gk¯l − g gˆ g ∇igkq¯∇¯jgp¯l + − µ Trωˆ ω Trωˆ ω ¯ 1 iq¯ p¯j i¯j ˆ kl ≤ −gˆ gˆ ηpq¯gi¯j − g Rmi¯j gk¯l − µ. Trωˆ ω

Proof.

∂ 1 ∂ i¯j log Trωˆ ω = (ˆg gi¯j) ∂t Trωˆ ω ∂t 1 iq¯ p¯j ∂ i¯j ∂ = −gˆ gˆ gˆpq¯gi¯j +g ˆ gi¯j Trωˆ ω ∂t ∂t

1 iq¯ p¯j i¯j = −gˆ gˆ ηpq¯gi¯j − gˆ Rici¯j − µ. Trωˆ ω CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 26

To compute ∆ log Trωˆ ω, we use normal coordinate with respect toω ˆ at a ﬁxed (x0, t0):

i¯j k¯l ∆ log Trωˆ ω = g ∂i∂¯j logg ˆ gk¯l i¯j 1 kq¯ p¯l k¯l = g ∂i (−gˆ gˆ gˆpq,¯ ¯j +g ˆ gk¯l,¯j) Trωˆ ω

i¯j 1 kq¯ p¯l k¯l = g ∂i(−gˆ gˆ gˆpq,¯ ¯j +g ˆ gk¯l,¯j) Trωˆ ω i¯j 1 kq¯ p¯l k¯l αβ¯ − g 2 (−gˆ gˆ gˆpq,¯ ¯j +g ˆ gk¯l,¯j) · ∂i(ˆg gαβ¯) (Trωˆ ω)

i¯j 1 kq¯ p¯l k¯l = g −gˆ gˆ gˆpq,i¯ ¯jgk¯l +g ˆ gk¯l,i¯j Trωˆ ω i¯j 1 k¯l αβ¯ − g 2 gˆ gk¯l,¯j · gˆ gαβ,i¯ (Trωˆ ω)

The Riemann curvature tensor for K¨ahlermanifolds is given by

αβ¯ Rmi¯jk¯l = −gi¯j,k¯l + g ∂igα¯l · ∂¯jgkβ¯

Therefore, we have

i¯j 1 kq¯ p¯l ˆ k¯l k¯l αβ¯ ∆ log Trωˆ ω = g (ˆg gˆ Rmpqi¯ ¯jgk¯l − gˆ Rmk¯li¯j +g ˆ g ∂igα¯l · ∂¯jgkβ¯) Trωˆ ω i¯j 1 k¯l αβ¯ − g 2 gˆ gk¯l,¯j · gˆ gαβ,i¯ (Trωˆ ω) ¯ 2 1 i¯j ˆ kl k¯l i¯j αβ¯ k¯l ˆ ˆ |∇Trωˆ ω| = g Rmi¯j gk¯l − gˆ Rick¯l + g g gˆ ∇igα¯l · ∇¯jgkβ¯ − 2 Trωˆ ω (Trωˆ ω)

Hence we have

log Trωˆ ωt ¯ 2 1 iq¯ p¯j i¯j ˆ kl i¯j k¯l pq¯ ˆ ˆ |∇Trωˆ ω| = −gˆ gˆ ηpq¯gi¯j − g Rmi¯j gk¯l − g gˆ g ∇igkq¯∇¯jgp¯l + − µ. Trωˆ ω Trωˆ ω

To prove the stated inequality, we will show

2 i¯j k¯l pq¯ ˆ ˆ |∇Trωˆ ω| −g gˆ g ∇igkq¯∇¯jgp¯l + ≤ 0. (3.1.1) Trωˆ ω

To see this, we choose holomorphic normal coordinate at a ﬁxed point (x0, t0) such thatg ˆi¯j = δi¯j, dgˆ = 0 and gi¯j = λiδij. CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 27

It proves (3.1.1) and completes the proof of the lemma.

We also need to bound ωt from below, one needs the following lemma whose proof is similar to that of Lemma 3.1 and hence is omitted.

Lemma 3.2. Let ωt be the solution to the K¨ahler-Ricci ﬂow ∂tωt = −Ric(ωt) − µωt where µ ∈ R.

∂ωˆt Let ωˆt be a family of non-negative closed (1, 1)-forms such that ∂t = η. Denote = ∂t − ∆ωt , we have

log Trωωˆ 2 1 i¯j i¯j k¯l ˆ i¯j k¯l pq¯ |∇Trωωˆ| = g ηi¯j + g g Rmi¯jk¯l − g g gˆ ∇igˆkq¯∇¯jgˆp¯l + + µ Trωωˆ Trωωˆ 1 i¯j i¯j k¯l ˆ ≤ g ηi¯j + g g Rmi¯jk¯l + µ. Trωωˆ

3.1.2 Yau’s L∞-estimate

Yau’s L∞-estimate, proved in [85], is fundamental in the study of complex Monge-Ampèreequations. It gives an estimate to the oscillation of the Kählerpotential by the intrinsic geometric quantities of the Kählermanifold. Several variant forms appeared in the works by Ko lodziej [46], Demailly- Pali [22], Eyssidieux-Guedj-Zeriahi [27] and many others. We will present the original form due to Yau: CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 28

Lemma 3.3 (Yau [85]). Let (M, ω0) be a compact K¨ahlermanifold and let ϕ be a smooth function on M satisfying √ Z ¯ n F n n (ω0 + −1∂∂ϕ) = e ω0 , ϕω0 = 0 M for some bounded function F : M → R. Then there exists a uniform constant C > 0 depending on n, supM F, ω0, Vol(M, ω0), the Sobolev and Poincar´econstants of (M, ω0) such that |ϕ| ≤ C.

2 Proof. The proof is a well-known application of the Moser iteration. We ﬁrst claim that kϕkL (ω0) ≤ C. By the Poincar´eInequality, we have

2 2 kϕk 2 ≤ Ck∇ϕk 2 . (3.1.2) L (ω0) L (ω0)

n F n Here C depends on the Poincar´econstant of (M, ω0). Since ω ≤ (supM e )ω0 , we have

Z Z Z n n F n n ϕ(ω0 − ω ) = (1 − e )ω0 ≤ C |ϕ|ω0 , M M M and so

Z Z n−1 n X k n−k−1 C |ϕ|ω0 ≥ ϕ(ω0 − ω) ∧ ω0 ∧ ω M M k=0 Z √ n−1 ¯ X k n−k−1 = − ϕ −1∂∂ϕ ∧ ω0 ∧ ω M k=0 Z √ n−1 ¯ X k n−k−1 = −1∂ϕ ∧ ∂ϕ ∧ ω0 ∧ ω M k=0 Z √ ¯ n−1 ≥ ( −1∂ϕ ∧ ∂ϕ) ∧ ω0 M Z √ ¯ n = Trω0 ( −1∂ϕ ∧ ∂ϕ)ω0 M Z ω0 2 n 2 = k∇ ϕk ω = k∇ϕk 2 ω0 0 L (ω0) M

2 2 Combining with (3.1.2), we have kϕk 2 ≤ Ckϕk 1 . Hence by H¨older’sinequality, we have L (ω0) L (ω0)

2 kϕkL (ω0) ≤ C (3.1.3)

where C also depends on Vol(M, ω0).

p For simplicity, we denote kϕkL (ω0) by kϕkp for any p ≥ 1. Next we derive an inequality relating CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 29

2 kϕk n and kϕk . Similar to the L -estimate, we have for any p ≥ 2, p( n−1 ) p

Z Z p−2 n n p−1 n ϕ|ϕ| (ω0 − ω ) ≤ C |ϕ| ω0 , M M where C depends on supM F . Then, we have

Z Z p−1 n p−2 n n C |ϕ| ω0 ≥ ϕ|ϕ| (ω0 − ω ) M M Z √ n−1 p−2 ¯ X k n−k−1 = − ϕ|ϕ| −1∂∂ϕ ∧ ω0 ∧ ω M k=0 Z √ n−1 p−2 ¯ X k n−k−1 = (p − 1) |ϕ| · −1∂ϕ ∧ ∂ϕ ∧ ω0 ∧ ω . M k=0 (3.1.4)

Here we have used the fact that ∂(ϕ|ϕ|α) = (α + 1)|ϕ|α∂ϕ for any α ≥ 0. Continuing on (3.1.4), we have

Z Z n−1 √ p−2 p−2 X p−1 n 2 2 ¯ k n−k−1 C |ϕ| ω0 ≥ (p − 1) −1|ϕ| ∂ϕ ∧ |ϕ| ∂ϕ ∧ ω0 ∧ ω M M k=0 n−1 p − 1 Z √ X ≥ −1∂(ϕ|ϕ|(p−2)/2) ∧ ∂¯(ϕ|ϕ|(p−2)/2) ∧ ωk ∧ ωn−k−1 p2 0 M k=0 1 Z ≥ k∇ω0 (ϕ|ϕ|(p−2)/2)k2 ωn. ω0 0 2p M

Therefore, we have (p−2)/2 2 p−1 k∇(ϕ|ϕ| )k2 ≤ Cpkϕkp−1. (3.1.5)

By the Sobolev inequality (note that dimR = 2n), we have

(p−2)/2 2 (p−2)/2 2 p−1 kϕ|ϕ| k2n/(n−1) ≤ Ck∇(ϕ|ϕ| )k2 ≤ Cpkϕkp−1. CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 30

Here we have used (3.1.5) in the second inequality. Therefore, we have

n−1 n Z n p−2 p( n−1 ) n 2 2 |ϕ| ω0 = kϕ|ϕ| k 2n M n−1 Z p−1 p−1 n ≤ Cpkϕkp−1 = Cp |ϕ| ω0 M p−1 1 Z p p p n = CpVol(M, ω0) |ϕ| ω0 , M which implies p−1 1 1 1 2 p kϕk n ≤ C p p p Vol(M, ω ) p kϕkp . p( n−1 ) 0

n p Denote Ap = max{1, kϕkp} and γ = n−1 > 1, we have the following L -inequality for the Moser iteration: 1 1 1 2 Apγ ≤ C p p p Vol(M, ω0) p Ap (3.1.6) for any p ≥ 2. By induction and take p = 2, we have for any N ≥ 1,

1 PN 1 1 PN 1 1 PN k 1 PN 1 2 k=1 k 2 k=1 k 2 k=1 k 4 k=1 2k A2γN+1 ≤ C γ 2 γ γ γ Vol(M, ω0) γ A2.

P∞ 1 P∞ k P∞ 1 Since γ > 1, one can easily verify that k=1 γk , k=1 γk and k=1 γ2k are convergent series, therefore by taking N → ∞, we have

kϕk∞ = lim A2γN+1 ≤ CA2 N→∞

From (3.1.3), we get kϕk∞ ≤ C where C depends only on the quantities stated in the theorem.

3.2 Collapsing of Calabi-Yau ﬁbrations

In this section, we first recapitulate Song-Tian’s works [73,74] on the collapsing behavior of Calabi- Yau fibrations with possibly singular fibers. Next, we focus on smooth fibrations and show that the fiber-collapsing rate is optimal. Based on the approaches adopted by [36, 72] etc., we further prove the Gromov-Hausdorff convergence, and in the toric fibration case, the uniform bound on Riemann curvature kRmkωt and smooth convergence along the flow. The fibration structure in this section is characterized by the semi-ampleness of the canonical line bundle KM . Let M be a compact Kählermanifold with dimC M = n with the canonical line CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 31

⊗m bundle KM being semi-ample. The global sections of KM for some m ∈ N induce a holomorphic map N π : M → Σ ⊂ CP

N into some complex projective space CP . The image Σ of the map π is called the canonical model of M and its dimension coincides the Kodaira dimension of M, i.e.

Kod(M) = dimC Σ.

N N Denote ωFS to be the Fubini-Study metric of CP and ι :Σ → CP the inclusion map. The semi-ampleness of KM implies there exists an integer m > 0 such that

⊗m ∗ ∗ c1(KM ) = π ι [ωFS] and hence we have 1 c (K ) = π∗(ι∗ω ). 1 M m FS

∗ Note that ι ωFS is a K¨ahlermetric on the canonical model Σ.

The Kodaira dimension Kod(M) can be any integer between 0 and dimC M = n. When Kod(M) = n, M is birational to Σ and is called a minimal model of general type. Another extreme is when Kod(M) = 0, then M is a Calabi-Yau manifold with c1(M) = 0. When 0 < Kod(M) < n, M −→π Σ is a Calabi-Yau fibration over Σ. For each z ∈ Σ, we say π−1(z) a fiber of M −→π Σ which may −1 −1 be singular. A non-singular fiber π (z) is a Calabi-Yau manifold with dimC π (z) = n − Kod(M). If there is no singular fiber, then M −→π Σ is a smooth Calabi-Yau fiber bundle over Σ. Kähler-Ricciflow on these fibrations was studied by Song-Tian in [73,74,76]. Their main results

(stated in Theorem 1.3) are about (i) the convergence of ωt as currents, (ii) the twisting of the Kähler-Einsteincondition by the Weil-Petersson metric in the limit and (iii) the boundedness of the scalar curvature and decay rate of the volume measure on the regular part. We focus on regular fibrations, i.e. absence of singular fibers, in this whole section. We would like to study the singularity formation of these fibrations further by understanding more about the geometric and analytic aspects on the problem. CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 32

3.2.1 Regular inﬁnite-time singularity

Using the recipe from the semi-ampleness of KM , we can reformulate the smooth Calabi-Yau fibration structure as follows: M is a closed connected Kähler manifold with dimC M = n which admits the π a surjective holomorphic submersion M −→ Σ, where Σ is a Kählermanifold with 0 ≤ dimC Σ = ∗ n−r < n. Furthermore, we assume that the first Chern class c1(M) = −π α for some Kählerclass α on Σ. This submersion gives a smooth fibration structure by classical results due to Ehresmann [24] and Fischer-Grauert [29]. For each z ∈ Σ, we call π−1(z) a fiber based at z, which is a complex

Calabi-Yau submanifold of M with dimC = r. The Kodaira dimension Kod(M) of M in this case is dimC Σ = n − r. The induced complex structure on each fiber may vary, but in the case where the fibers are isomorphic, M is a holomorphic fiber bundle over Σ. We consider the following normalized Kähler-Ricci flow on M, defined as in (2.2.2):

∂ω t = −Ric(ω ) − ω , ω | = ω , ∂t t t t t=0 0 with any K¨ahlermetric ω0 as the initial metric. −t The K¨ahlerclass [ωt] at time t is precisely given by −c1(M) + e ([ω0] + c1(M)). Recall from

Chapter 2 that the first Chern class is defined by c1(M) = [Ric(ω)] for any Kählermetric ω on M. The maximal existence time T of (2.2.2) is uniquely determined by the optimal existence result due to Tian-Zhang in [82], namely

−t T = sup{t : −c1(M) + e ([ω0] + c1(M)) > 0}.

∗ Since c1(M) = −π α ≤ 0 for some Kählerclass α on Σ, the optimal existence result asserts that the flow (2.2.2) has a long-time solution ωt defined on the whole t ∈ [0, ∞). Due to the absence of singular fibers, we say the flow encounters regular infinite-time singularity. ∗ Let ωΣ be a Kählermetric of Σ such that c1(M) = −[π ωΣ]

∗ −t ∗ ωˆt = π ωΣ + e (ω0 − π ωΣ). (3.2.1)

Thenω ˆt is a reference metric in the same Kähler class as the flow metric ωt. Let us restate Theorem 1.7 using the notations defined in this section:

Theorem 3.4 (F-Zhang [32]; Theorem 1.7 of Chapter 1). Let M n −→π Σn−r be a holomorphic ∗ submersion with r = dimC M − dimC Σ > 0 and c1(M) = −[π ωΣ] for some K¨ahlermetric ωΣ on CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 33

Σ. Let ωt be the solution to the normalized Kähler-Ricci flow ∂tωt = −Ric(ωt) − ωt on M starting with any initial Kählermetric ω0. Then, we have

−1 C ωˆt ≤ ωt ≤ Cωˆt

where C is a uniform constant depending only on n, r, ω0 and ωΣ. Here ωˆt is the reference metric deﬁned in (3.2.1). Hence, the diameters of ﬁbers are decay at an optimal exponential rate, precisely we have −1 − t −1 − t C e 2 ≤ diamt(π (z)) ≤ Ce 2 , for any z ∈ Σ.

The proof of the above theorem relies on estimates of the parabolic complex Monge-Ampère equation. An elliptic analogue of the theorem can be found in [83] Tosatti where the collapsing behavior of Ricci-flat metrics on Calabi-Yau manifolds was studied. We rewrite the normalized Kähler-Ricciflow (2.2.2) into the following complex Monge-Amèreequation which is slightly modified from (2.2.4) to allow suitable estimates on the Kählerpotential: √ ∂ϕ (ˆω + −1∂∂ϕ¯ )n t = log t t − ϕ , ϕ | = 0 (3.2.2) ∂t e−rtΩ t t t=0 √ where r > 0 is the dimension of the fibers and Ω is a volume form on M such that −1∂∂¯log Ω = √ ∗ ¯ π ωΣ. The metric solution is given by ωt =ω ˆt + −1∂∂ϕt. One can easily check that it is equivalent to the normalized Kähler-Ricciflow (2.2.2).

3.2.2 Pointwise estimates

The above modiﬁcation on the parabolic complex Monge-Amp`ereequation and the fact that C−1Ω ≤ n ωˆt ≤ CΩ for some uniform constant C > 0 allow us to establish the following zeroth-order estimates.

Lemma 3.5 (c.f. [73, 74, 76, 82, 86]). Consider the complex Monge-Amp`ere equation (3.2.2), there exists a uniform constant C = C(n, r, ω0, ωΣ) such that

∂ϕt |ϕt| ≤ C, ≤ C ∂t

n −rt Proof. Sinceω ˆt ' e Ω, by a standard maximum principle argument, we have |ϕt| ≤ C. CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 34

∂ϕt Next we derive the uniform bound for ∂t . Diﬀerentiating (3.2.2) with respect to t yields

∂ ∂ϕ ∂ϕ ∂ϕ t = ∆ t − e−tTr (ω − π∗ω ) − t + r. ∂t ∂t ∂t ωt 0 Σ ∂t which implies the following two equations:

∂ ∂ϕ ∂ϕ et t = ∆ et t − Tr (ω − π∗ω ) + ret, ∂t ∂t ∂t ωt 0 Σ

∂ ∂ϕ ∂ϕ + ϕ = ∆ t + ϕ − n + r + Tr π∗ω . (3.2.3) ∂t ∂t t ∂t t ωt Σ

Taking their diﬀerence, we get

∂ ∂ϕ ∂ϕ (et − 1) t − ϕ = ∆ (et − 1) t − ϕ − Tr ω + ret + n − r. ∂t ∂t t ∂t t ωt 0

Applying maximum principle and the uniform bound for |ϕt|, we have

∂ϕ (n − r)t + ret + C t ≤ ≤ C. ∂t et − 1

∂ϕt Hence ∂t is uniformly bounded from above. For the lower bound, we ﬁrst derive.

n n −n ωˆt ωˆt − ∂ϕt n Trω ωˆt ≥ = ≥ Ce ∂t . t n ∂ϕt +ϕ −rt ωt e ∂t t Ω

By adding the following two inequalities:

∂ ∂ϕ − ∆ t + ϕ = −n + r + Tr π∗ω ≥ −n + r ∂t ∂t t ωt Σ

∂ ∂ϕt ∂ϕt − ∂ϕt − ∆ ϕ = − n + Tr ωˆ ≥ − n + Ce ∂t , ∂t t ∂t ωt t ∂t we get ∂ ∂ϕt ∂ϕt − ∂ϕt − ∆ + 2ϕ ≥ − C + Ce ∂t . ∂t ∂t t ∂t

Finally, using a maximum principle argument and the uniform bound for |ϕt|, we can conclude

∂ϕt the lower bound for ∂t .

Remark 3.6. In a recent work [76] by Song-Tian, there is a delicate argument to establish the CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 35

uniform bound for ∂ϕt as in Lemma 3.5 when the π is not assumed to be regular, i.e. c (K ) is ∂t 1 M just semi-ample.

We will need the following estimates in the proof of Theorem 3.4. It is an application of the parabolic Schwarz’s estimate (Lemma 3.2).

Lemma 3.7. There exists a uniform constant C depending on only n, r, ω0 and ωΣ such that

−1 ∗ ωt ≥ C π ωΣ for all t ∈ [0, ∞).

Proof. Applying Lemma 3.2, we have

∗ log Trωt π ωΣ (3.2.4)

1 i¯j k¯l ∗ ≤ ∗ (g g Rm(π ωΣ)i¯jk¯l) + 1 Trωt π ωΣ ∗ ≤ CTrωt π ωΣ + 1

∗ where Rm(π ωΣ) is the Riemann curvature tensor of ωΣ which is independent of t. On the other hand, we have

∂ϕ ϕ = t − Tr (ω − ωˆ ) (3.2.5) t ∂t ωt t t ∂ϕ = t − n + Tr (e−tω + (1 − e−t)π∗ω ) ∂t ωt 0 Σ ˜ ∗ ≥ −C + CTrωt π ωΣ.

∗ Let H = log Trωt π ωΣ − Aϕt where A > 0 is a constant to be chosen later. By (3.2.4) and (3.2.5), we have, ˜ ∗ H ≤ (C − AC)Trωt π ωΣ + (AC + 1).

By choosing A large enough such that C − AC˜ < −1, we have

∗ H H ≤ −Trωt π ωΣ + C ≤ −Ce + C.

Here we have used |ϕt| ≤ C from Lemma 3.5. By maximum principle and the uniform bound of ∗ |ϕt|, one can show Trωt π ωΣ ≤ C and it completes the proof of the lemma. CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 36

The zeroth-order estimates in Lemma 3.5 and the Schwarz’s type estimate in Lemma 3.7 together provide suﬃcient ingredients to establish the uniform scalar curvature bound (see [66,73,74,76,87]).

∂ϕt ∂ϕt The upshot is to derive uniform estimates on k∇( ∂t + ϕt)k and ∆( ∂t + ϕt) using maximum principle similar to [17, 48].

Lemma 3.5 tells us that the volume form of ωt has pointwise decay exactly as predicted by the cohomology. Since it is useful for establishing the main theorem, we summarize it in the following lemma:

Lemma 3.8. There exists a uniform constant C = C(n, r, ω0, ωΣ) > 0 such that for any t ∈ [0, ∞), we have −1 −rt n −rt C e Ω ≤ ωt ≤ Ce Ω. (3.2.6)

−t We now show the Kählerpotential ϕt decays at a rate of e after a suitable normalization described below. −1 For each z ∈ Σ and t ∈ [0,T ), we denote ωt,z to be the restriction of ωt on the fiber π (z). For each t ∈ [0,T ), we define a function Φt :Σ → R by

Z 1 r Φt(z) = −1 ϕt ω0,z Volω0,z (π (z)) π−1(z)

−1 ∗ which is the average value of ϕt over each ﬁber π (z). The pull-back π Φt is then a function deﬁned ∗ on M. For simplicity, we also denote π Φt by Φt.

Lemma 3.9. There exists a uniform constant C = C(n, r, ω0, ωΣ) such that for any t ∈ [0, ∞), we have −t |ϕt − Φt| ≤ Ce . (3.2.7)

t −t Proof. Denoteϕ ˜t = e (ϕt − Φt). For each z ∈ Σ, we haveω ˆt,z = e ω0,z, and so

√ −t ¯ ωt,z = e ω0,z + −1∂∂ϕt π−1(z) .

√ ¯ Since Φt depends only on z ∈ Σ, we have −1∂∂Φt π−1(z) = 0. By rearranging, we have

√ t ¯ e ωt,z = ω0,z + −1∂∂ϕ˜t π−1(z) . (3.2.8) CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 37

Regard (3.2.8) to be a metric equation on the manifold π−1(z), and we have

√ r ¯ t r ω0,z + −1∂∂ϕ˜t π−1(z) = e ωt,z (3.2.9)

Using Lemma 3.8, we can see along π−1(z),

r r ∗ n−r ωt,z ωt ∧ (π ωΣ) r = r ∗ n−r (3.2.10) ω0,z ω0 ∧ (π ωΣ) r ∗ n−r n ωt ∧ (π ωΣ) ωt = n · r ∗ n−r ωt ω0 ∧ (π ωΣ) ∗ n−r −rt ≤ C(Trωt π ωΣ) · e .

Combining Lemma 3.7 with (3.2.10), we see that (3.2.9) can be restated as

√ r ¯ r ω0,z + −1∂∂ϕ˜t π−1(z) = Fz(ξ, t)(ω0,z) (3.2.11)

−1 where Fz(ξ, t): π (z) × [0,T ) → R>0 is uniformly bounded from above. R r ∞ Since π−1(z) ϕ˜tω0,z = 0, by applying Yau’s L -estimate (Lemma 3.3) on (3.2.8), we then have

sup |ϕ˜t| ≤ Cz, (3.2.12) π−1(z)×[0,T )

−1 where Cz depends on n, r, ω0, ωΣ, supπ−1(z)×[0,T ) Fz, Volω0,z (π (z)), the Sobolev and Poincar´econ- −1 stants of π (z) with respect to metric ω0,z, all of which can be bounded uniformly independent of z. It completes the proof of the lemma.

−1 Remark 3.10. In our setting, the uniform boundedness of Sobolev and Poincaréconstants of (π (z), ω0,z) follow from the compactness of Σ and the absence of singular fibers. It is also possible to give such uniform bounds by noting that π−1(z)’s are minimal submanifolds of M and hence they can be embedded into some Euclidean space RN with bounded mean curvature. Combining the classical results in [16,48,49] one can obtain uniform bounds on the Sobolev and Poincaréconstants. A detail discussion in this regard can be found in [83] which also dealt with singular fibers.

3.2.3 Collapsing rate of Calabi-Yau ﬁbers

Now we can proceed to the proof of the main result in this section. CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 38

Proof of Theorem 3.4. We apply maximum principle to the following quantity

−t t Q := log(e Trωt ω0) − Ae (ϕt − Φt),

where A is a positive constant to be chosen. Denote = ∂t − ∆ωt . Using Lemma 3.2, we have

−t log(e Trωt ω0) ≤ C + CTrωt ω0 (3.2.13)

where C depends on the curvature of ω0. We also need to compute the evolution equation for the second term in Q.

∂ϕ ∂Φ Aet(ϕ − Φ ) = Aet t − t + Aet(ϕ − Φ ) t t ∂t ∂t t t

t − Ae (∆ϕt − ∆Φt) Z ! t ∂ϕt ∂ϕt r ≥ Ae − ω0,z − CA ∂t π−1(z) ∂t

t − Ae (n − Trωt ωˆt − ∆Φt).

∂ϕt Using the lower bound of ∂t given by Lemma 3.5, we have

t t t e (ϕt − Φt) ≥ −CAe + Ae Trωt ωˆt (3.2.14) Z ! t ∂ϕt r + Ae ∆Φt − ω0,z . π−1(z) ∂t

Combining (3.2.13) and (3.2.14), we have

t t −t −t ∗ Q ≤ CAe + CTrωt ω0 − Ae Trωt e ω0 + (1 − e )π ωΣ (3.2.15) Z ! t ∂ϕt r − Ae ∆Φt − ω0,z π−1(z) ∂t

t ≤ CAe + (C − A)Trωt ω0 Z ! t ∂ϕt r − Ae ∆Φt − ω0,z . π−1(z) ∂t CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 39

∂ϕt By Lemma 3.8, we have ∂t ≤ C for some uniform constant C. It follows that

Z ∂ϕt r ω0,z ≤ C. π−1(z) ∂t

−1 Note that Volω0,z (π (z)) is actually independent of z.

For the Laplacian term of Φt, we have

Z Z √ r ¯ r ∆ ϕtω0,z = Trωt −1∂∂ϕt ∧ ω0,z π−1(z) π−1(z) Z r = Trωt (ωt − ωˆt) ∧ ω0,z π−1(z) Z r ≥ −Trωt ωˆt ∧ ω0,z π−1(z) Z r ∗ r ≥ −Trωt ω0 ∧ ω0,z + π ωΣ ∧ ω0,z . π−1(z)

∗ R r ∗ r Since Trωt π ωΣ ≤ C and π−1(z) ω0 ∧ ω0,z + π ωΣ ∧ ω0,z is a smooth (1, 1)-form on Σ independent of t, we have Z r ∆ ϕtω0,z ≥ −C π−1(z) for some uniform constant C. Back to (3.2.15), we have

t t Q ≤ CAe + (C − A)Trωt ω0 ≤ CAe − Trωt ω0 (3.2.16) if we choose A suﬃciently large such that C − A ≤ −1. Hence, for any S > 0, at the point where Q achieves its maximum over M × [0,S], we have −t Trωt (e ω0) ≤ C for some uniform constant C independent of S. Together with Lemma 3.9, it follows that for any t ∈ [0, ∞) we have,

−1 −t C e ω0 ≤ ωt. (3.2.17)

−1 ∗ Combining this with the fact that ωt ≥ C π ωΣ, we have

−1 C ωˆt ≤ ωt. (3.2.18)

n n Together with ωt ≤ Cωˆt , we also have ωt ≤ Cωˆt for any t ∈ [0, ∞). It completes the proof of the theorem. CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 40

3.2.4 Long-time convergence

This subsection presents applications of the collapsing rate result proved in Theorem 3.4 on the convergence behaviors of the Calabi-Yau fibrations. Two kinds of convergence will be studied, namely the Gromov-Hausdorff convergence and the C1,α-convergence of the Kähler potential. The Gromov-Hausdorff distance between two compact metric spaces measure how far they are from being isometric. There are several equivalent definitions of the Gromov-Hausdorff distance. We use the one adopted in [37, 72, 77] etc. Namely,

Deﬁnition 3.11 (Gromov-Hausdorﬀ distance). Given two compact metric spaces (X, dX ) and

(Y, dY ), the Gromov-Hausdorff distance dGH((X, dX ), (Y, dY )) is defined as the infimum of all ε > 0 such that there exist two set maps Fε : X → Y and Gε : Y → X such that all of the followings hold:

|dX (x1, x2) − dY (Fε(x1),Fε(x2))| < ε for any x1, x2 ∈ X;

dX (x, Gε ◦ Fε(x)) < ε for any x ∈ X;

|dY (y1, y2) − dY (Gε(y1),Gε(y2))| < ε for any y1, y2 ∈ Y ;

dY (y, Fε ◦ Gε(y)) < ε for any y ∈ Y.

We say a sequence of compact metric spaces (Xi, dXi ) converges to (Y, dY ) in Gromov-Hausdorﬀ topology if

lim dGH((Xi, dX ), (Y, dY )) = 0. i→∞ i

−1 −t/2 −1 −t/2 −1 Given that C e ≤ diamt(π (z)) ≤ Ce for each ﬁber π (z), z ∈ Σ, one can follow the argument as in [36, 37, 72] etc. to show that (M, ωt) converges subsequentially to (Σ, dΣ) for some metric dΣ.

Proposition 3.12. Under the same assumptions as in Theorem 3.4, there exists ti → ∞ such that

(M, ωti ) converges to (Σ, dΣ) in Gromov-Hausdorﬀ topology as i → ∞.

Proof. Using Theorem (3.4), we have ωt ≤ Cωˆt ≤ Cω˜ 0. Denote dt be the distance function on M √ induced by the K¨ahlermetric ωt. Then we have dt ≤ Cd0, which implies {dt : M × M → R} is a uniformly bounded family of functions. It is straight-forward using triangle inequalities to check that {dt} is also equicontinuous with respect to d0. The Arzela-Ascoli Theorem asserts there exists a sequence ti → ∞ such that di converges uniformly to a function d∞ on M × M.

Let s :Σ → M be any map (not necessarily continuous) such that π ◦ s = idΣ. We deﬁne CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 41

dΣ :Σ × Σ → R by

dΣ(z1, z2) = d∞(s(z1), s(z2)).

−1 Since diamt(π (z)) → 0 as t → ∞ for any z ∈ Σ, one can show dΣ is independent of the choice of −1 s. Using the fact that diamti (π (z)) → 0 and dti → d∞ both uniformly on M × M, one can show each ε > 0 and suﬃciently large i, we have for any z1, z2, z ∈ Σ and x1, x2, x ∈ M,

|dΣ(z1, z2) − dti (s(z1), s(z2))| = |d∞(s(z1), s(z2)) − dti (s(z1), s(z2))| < ε;

dΣ(z, π ◦ s(z)) = dΣ(z, z) = 0;

|dti (x1, x2) − dΣ(π(x1), π(x2))| = |dti (x1, x2) − d∞(˜s ◦ π(x1), s˜ ◦ π(x2))|

= |dti (x1, x2) − d∞(x1, x2)| < ε

wheres ˜ :Σ → M takes π(xk) to xk;

−1 dti (x, s ◦ π(x)) ≤ diamt(π (π(x))) < ε.

These show, by the deﬁnition of Gromov-Hausdorﬀ distance, that for any ε > 0, we have dGH((X, dti ), (Σ, dΣ)) <

ε for suﬃciently large i (we take Fε = s and Gε = π for any ε > 0). Hence (X, ωti ) converges to

(Σ, dΣ) in Gromov-Hausdorﬀ topology.

Next we discuss the C1,α-convergence of the Kählerpotential. We use the argument adopted by [73] for the elliptic fibrations to show that the Kähler-Ricciflow converges to the generalized Kähler-Einsteinmetric in C1,α-sense on the potential level.

From now on we focus on the non-trivial case dimC Σ ≥ 1. The fibers of the map π : X → Σ √ ¯ are all smooth Calabi-Yau manifolds, and so there is a Ricci-flat metric ω0,z + −1∂∂Ψz for each −1 fiber π (z), z ∈ Σ where Ψz :Σ → R is a smooth function. After normalizing Ψz we may assume R r π−1(z) Ψzω0,z = 0. By pulling back Ψ to X, still denoted by Ψ, we can define a smooth closed (1, 1)-form √ ¯ ωSF = ω0 + −1∂∂Ψ which is a Ricci flat metric when restricted to each fiber. Consider the following function defined on X: Ω F = . n n−r r Cr ωΣ ∧ ωSF √ ¯ ∗ Since −1∂∂ log Ω = π ωΣ and ωSF is a Ricci-flat metric along each fiber, we know that F is constant along each fiber and depends only on z ∈ Σ. CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 42

One can get a unique solution u to the following complex Monge-Amp`ere equation on Σ:

√ ¯ n−r u n−r (ωΣ + −1∂∂u) = F e ωΣ .

√ ¯ We denote ωGKE = ωΣ + −1∂∂u, one can show as in [73] that this metric satisﬁes

Ric(ωGKE) = −ωGKE + ωWP

where ωWP is the Weil-Petersson metric unique determined by the ﬁbration structure π : X → Σ.

With a little abuse of notations, we also denote ωGKE and ωΣ to be their pull-backs on X via π.

We ﬁrst show that u is the limit of the potential ϕt as t → ∞.

Lemma 3.13. The solution ϕt for (3.2.2) converges uniformly to u as t → ∞. √ −t ¯ Proof. Let vt = ϕt − u − e Ψ. We can rewrite the ﬂow metric ωt =ω ˆt + −1∂∂ϕt as

√ −t −t ¯ ωt = (ωGKE − e ωΣ) + e ωSF + −1∂∂vt.

n−r u n−r Ω Meanwhile, since ωGKE = F e ωΣ and F = n n−r r , we have Cr ωΣ ∧ωSF

n n−r r u Cr ωGKE ∧ ωSF = Ωe .

Using this, one can compute the evolution of vt as follows:

∂v ∂ϕ t = t + e−tΨ ∂t ∂t √ n ert (ω − e−tω ) + e−tω + −1∂∂v¯ = log GKE Σ SF t − ϕ + e−tΨ Ω √ n ert (ω − e−tω ) + e−tω + −1∂∂v¯ = log GKE Σ SF t + u − ϕ + e−tΨ n n−r r Cr ωGKE ∧ ωSF √ n ert (ω − e−tω ) + e−tω + −1∂∂v¯ = log GKE Σ SF t − v. n n−r r Cr ωGKE ∧ ωSF

−t −t n By expanding ((ωGKE − e ωΣ) + e ωSF ) , one can show

ert ((ω − e−tω ) + e−tω )n log GKE Σ SF = log(1 ± O(e−t)) = O(e−t) n n−r r Cr ωGKE ∧ ωSF CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 43

as t → ∞. Denote vmax(t) = maxx∈X vt(x), then we have

dv (t) max ≤ Ce−t − v (t) dt max

−t −t −t −t and so vmax(t) ≤ Cte + Ce . Similarly vmin(t) ≥ −Cte − Ce . − t Hence we conclude |ϕt − u| ≤ Ce 2 .

−1 Combining with Theorem 3.4 which asserts C ωˆt ≤ ωt ≤ Cωˆt, we have the following corollary

Corollary 3.14. K¨ahler-Ricci ﬂow ωt converges to ωGKE as t → ∞ in the sense that the metric 1,α potential ϕt → u in C -norm for any α < 1. √ ¯ Proof. Since ωt =ω ˆt + −1∂∂ϕt, taking the trace with respect to ω0 yields

Trω0 ωt = Trω0 ωˆt + ∆ω0 ϕt.

−1 By Theorem 3.4, we have C Trω0 ωˆt ≤ Trω0 ωt ≤ CTrω0 ωˆt, and so

0 |∆ω0 ϕt| ≤ CTrω0 ωˆt ≤ C .

0 The corollary then follows from standard Schauder’s theory and the fact that ϕt → u in C -norm.

3.2.5 Toric ﬁbration

In this subsection, we specialize on one category of Calabi-Yau fibrations, namely toric fibrations, −1 r where all fibers π (z) are complex tori C /Λz. We will provide several geometric and analytic applications of the collapsing rate result proved in Theorem 3.4 when the initial Kählerclass [ω0] is rational. These answer some conjectures in Song-Tian’s works [73, 74] in this special type of fibrations. n π n−r Let M −→ Σ be a holomorphic submersion fibered by complex tori such that c1(M) = ∗ −[π ωΣ] for some Kählermetric ωΣ on Σ. For each point z ∈ Σ, there exists a neighborhood −1 z ∈ B ⊂ Σ such that π (B) ⊂ M is trivialized: i.e. there exists a lattice section Λz varying over r −1 z ∈ B such that (B × C )/Λz is biholomorphic to π (B). 2 From now on we assume the initial Kählerclass [ω0] is rational, i.e. [ω0] ∈ H (M, Q), and −1 hence M must be projective. Then there exists a closed nonnegative semi-flat form ωSF on π (B), −1 with a good rescaling property, such that on each fiber π (z) we have ωSF |π−1(z) cohomologous to CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 44

ω0|π−1(z). A semi-flat form ωSF is a (1, 1)-form such that for each z ∈ B the restriction ωSF |π−1(z) on the fiber π−1(z) is flat.

Lemma 3.15 (Gross-Tosatti-Y.Zhang [36]). Given that M is projective and [ω0] is rational, then one can ﬁnd a closed nonnegative (1, 1)-form ωSF such that there exists a smooth function f : π−1(B) → R with √ ¯ ωSF − ω0 = −1∂∂f

r r and passing to the universal cover p : B × C → B × (C /Λz), we have

√ ∗ ¯ p ωSF = −1∂∂ψ where ψ : B × Cr → R is a smooth function with the following rescaling property:

2 r ψ(z, λξ) = λ ψ(z, ξ) for any (z, ξ) ∈ B × C and λ ∈ R.

r r t/2 Denote λt : B × C → B × C to be the rescaling map (z, ξ) 7→ (z, e ξ). One can easily verify that √ √ √ −t ∗ ∗ −t ∗ ¯ −t ¯ ¯ ∗ e λt p ωSF = e λt −1∂∂ψ = e −1∂∂(ψ ◦ λt) = −1∂∂ψ = p ωSF . (3.2.19)

∂ωt As before, we rewrite the normalized Kähler-Ricciflow ∂t = −Ric(ωt) − ωt as the following complex Monge-Ampèreequation (3.2.2) √ ∂ϕ (ˆω + −1∂∂ϕ¯ )n t = log t t − ϕ , ∂t e−rtΩ t √ −t −t ∗ ¯ whereω ˆt = e ω0 + (1 − e )π ωΣ and ωt =ω ˆt + −1∂∂ϕt. Here Ω is a volume form on M such √ ¯ ∗ that −1∂∂ log Ω = π ωΣ. We first establish the following lemma:

Lemma 3.16. There is a constant C > 0 such that on B × Cr we have

−1 ∗ ∗ ∗ ∗ ∗ ∗ C p (π ωΣ + ωSF ) ≤ λt p ωt ≤ Cp (π ωΣ + ωSF ) for any t ≥ 1.

Proof. First we use the metric equivalence of ωt andω ˆt established in Theorem 3.4:

−1 C ωˆt ≤ ωt ≤ Cωˆt.

For the sake of simplicity, we denote ωt ' ωˆt for the above metric equivalence (and for any other CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 45

pair of metrics). Then,

∗ ∗ ∗ ∗ λt p ωt ' λt p ωˆt

∗ ∗ −t −t ∗ = λt p (e ω0 + (1 − e )π ωΣ

−t ∗ ∗ −t ∗ ∗ = e λt p ω0 + (1 − e )p π ωΣ.

∗ ∗ ∗ ∗ ∗ ∗ Note that λt p π ωΣ = p π ωΣ since λt rescales the ﬁber directions only. As ω0 ' ωSF + π ωΣ, we have

∗ ∗ −t ∗ ∗ −t ∗ ∗ λt p ωt ' e λt p ωSF + (1 − e )p π ωΣ

∗ −t ∗ ∗ ' p ωSF + (1 − e )p π ωΣ

∗ ∗ ' p (ωSF + π ωΣ) for t ≥ 1.

∗ ∗ ∗ r Next we show λt p ωt is locally cohomologous to p (ωΣ + ωSF ) in B × C :

√ ∗ ∗ ∗ ∗ −t −t ∗ ¯ λt p ωt = λt p (e ω0 + (1 − e )π ωΣ) + −1∂∂(ϕt ◦ p ◦ λt) √ −t ∗ ∗ −t ∗ ∗ ¯ = e λt p ω0 + (1 − e )p π ωΣ + −1∂∂(ϕt ◦ p ◦ λt) √ √ −t ∗ ∗ ¯ −t ∗ ∗ ¯ = e λt p (ωSF − −1∂∂f) + (1 − e )p π ωΣ + −1∂∂(ϕt ◦ p ◦ λt) √ √ ∗ ¯ −t −t ∗ ∗ ¯ = p ωSF − −1∂∂(e f ◦ p ◦ λt) + (1 − e )p π ωΣ + −1∂∂(ϕt ◦ p ◦ λt).

√ ¯ On the open ball B ⊂ Σ, the K¨ahlermetric ωΣ can be locally expressed as −1∂∂ζ for some smooth function ζ : B → R. Therefore, we have

√ ∗ ∗ ∗ ∗ ¯ λt p ωt = p (ωSF + π ωΣ) + −1∂∂ut, (3.2.20)

−t −t where ut = ϕt ◦ p ◦ λt − e (f ◦ p ◦ λt) − e (ζ ◦ p). As ϕt is uniformly bounded on M, we have ut uniformly bounded on B × Cr. One can show the following higher-order estimates using Evans-Krylov’s and Schauder’s estimates: CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 46

Lemma 3.17. Given any compact set K ⊂ B × Cr and any k ≥ 0, there exists a constant C = C(K, k) such that ∗ ∗ kλt p ωtkCk(K,δ) ≤ C (3.2.21) where δ is the Euclidean metric of B × Cr.

Proof. We first derive a complex Monge-Ampèreequation for ut: from the Kähler-Ricciflow equation, we have

∂ϕt n ∂t +ϕt−rt ωt = e Ω.

By rescaling, we have

∂ϕt ∗ ∗ n ∗ ∗ n ∂t +ϕt−rt ∗ ∗ (λt p ωt) = λt p ωt = (e ◦ p ◦ λt) · λt p Ω.

√ √ ¯ ∗ ¯ Since −1∂∂ log Ω = π ωΣ and so −1∂∂ log Ω π−1(z) = 0 for each z ∈ B. By the compactness −1 −rt ∗ ∗ ∗ of the toric ﬁbers π (z), we have Ω depends only on z ∈ Σ and hence e λt p Ω = p Ω. Therefore, from (3.2.20) the potential ut satisﬁes the following equation:

√ ∂ϕ log(p∗(ω + π∗ω ) + −1∂∂u¯ )n = t + ϕ ◦ p ◦ λ + log(p∗Ω). (3.2.22) SF Σ t ∂t t t

The following quantities are uniformly bounded according to the gradient and Laplacian estimates due to [17, 48] (see also [64, 66, 73, 74, 86] etc.)

k∇ωt (ϕ ˙ t + ϕt)kωt ≤ C, |∆(ϕ ˙ t + ϕt)| ≤ C.

Hence,

k∇ ∗ ∗ (ϕ ˙ + ϕ ) ◦ p ◦ λ k ∗ ∗ ≤ C, λt p ωt t t t λt p ωt

|∆ ∗ ∗ (ϕ ˙ + ϕ ) ◦ p ◦ λ )| ≤ C. λt p ωt t t t

∗ ∗ ∗ ∗ r By Lemma 3.16, we have λt p ωt ' p (ωSF + π ωΣ) ' δ on K ⊂ B × C . Hence, apply Evans- 2,α Krylov’s theory [26, 47] (see also [70]) on (3.2.22) one can get a uniform C -estimate on ut on a possibly smaller K. Finally, by the Schauder’s estimate (see e.g. [34, 69]) and a bootstrapping argument, one can complete the proof of the lemma. Here we supply the detail of the bootstrapping argument: CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 47

Let D be any first-order differential operator on B × Cr. Differentiating (3.2.22) by D gives

∗ ∗ ∆ ∗ ∗ (Du ) = −Tr ∗ ∗ Dp (ω + π ω ) λt p ωt t λt p ωt SF Σ ∂ϕ + D λ∗p∗ t + ϕ + D log(p∗Ω). (3.2.23) t ∂t t

From (3.2.3), one can show using chain rule that

∂ ∂ϕ ∂ϕ λ∗p∗ t + ϕ = λ∗p∗ ∆ t + ϕ − n + r + Tr π∗ω ∂t t ∂t t t ωt ∂t t ωt Σ r X ∗ ∗ ∂ ∂ϕt 1 t + λ p + ϕ · e 2 ξ t ∂ξ ∂t t 2 j j=1 j r X ∂ ∂ϕt 1 t ∗ ∗ 2 ¯ + λt p + ϕt · e ξj ∂ξ¯ ∂t 2 j=1 j ∗ ∗ ∂ϕt ∗ ∗ = ∆λ∗p∗ω λ p + ϕt − n + r + Trλ∗p∗ω p π ωΣ t t t ∂t t t r 1 X ∂ ∂ϕt + λ∗p∗ + ϕ · ξ 2 ∂ξ t ∂t t j j=1 j r 1 X ∂ ∗ ∗ ∂ϕt ¯ + λt p + ϕt · ξj 2 ∂ξ¯ ∂t j=1 j

∂ϕ Hence, λ∗p∗ t + ϕ satisﬁes the following parabolic equation: t ∂t t

∂ ¯ ∗ ∗ − ∆λ∗p∗ω H = h∂H, ∂ξ + ∂ξiδ − n + r + Trλ∗p∗ω p π ωΣ (3.2.24) ∂t t t t t

r r where ∂ denotes the ﬂat connection on B × C and ∂ξ = (ξ1, . . . , ξr) ∈ C . k,α ∗ ∗ k−2,α Assume that ut ∈ C for some k ≥ 2 and 0 < α < 1. Then by (3.2.20) we have λt p ωt ∈ C . ∗ ∗ ∗ ∗ −1 k−2,α By the uniform bound of λt p ωt, one also has (λt p ωt) ∈ C . Hence applying parabolic Schauder’s estimate on (3.2.24) one get

∂ϕ λ∗p∗ t + ϕ ∈ Ck,α t ∂t t

Hence the coeﬃcients of the elliptic equation (3.2.23) are in Ck−2,α, and applying elliptic k,α k+1,α Schauder’s estimate one has Dut ∈ C and therefore ut ∈ C which is one higher-order 2,α up than our assumption. Since Evans-Krylov’s theory asserts that ut ∈ C , this bootstrapping CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 48

∞ argument implies ut ∈ C which completes the proof of the lemma.

For each point x ∈ M, ﬁnd a compact subset K containing x such that K ⊂ π−1(B) ≡ B × r (C /Λz) for some small open ball B ⊂ Σ. We then get

∗ sup kRmkωt = sup kRmkp ωt K K0 for some K0 ⊂ B × Cr such that p(K0) = K. Therefore,

sup kRmk = sup kRmk ∗ ∗ . ωt λt p ωt K −1 0 λt (K )

−1 0 −t/2 0 −1 0 As λt (K ) = {(z, e ξ):(z, ξ) ∈ K .}, one can easily see ∪t>0λt (K ) is precompact. By

Lemma 3.17, one has sup −1 0 kRmk ∗ ∗ ≤ C where C depends on K. By covering the λ (K ) λt p ωt K K compact manifold M by ﬁnitely many such K’s we have proved:

Proposition 3.18. Suppose π : M → Σ is a smooth holomorphic submersion fibered by complex tori such that the initial Kählerclass [ω0] is a rational. Then along the normalized Kähler-Ricci

ﬂow (2.2.2), we have kRmkωt ≤ C for some constant C > 0 independent of t.

∞ Another consequence of Lemma 3.17 is the C -convergence of ωt to the generalized K¨ahler- Einstein, which strengthened the C1,α-convergence result (on the potential level) in Corollary 3.14.

Recall that ϕt → u as t → ∞ where u :Σ → R is the potential function such that ωGKE = √ ¯ ωΣ + −1∂∂u. Under the setting in this section, we have the following proposition:

Proposition 3.19. Under the same assumption as in Proposition 3.18, we have

∞ (i) ϕt → u in C (ω0)-topology, and

∗ ∞ (ii) ωt → π ωGKE in C (ω0)-topology.

Remark 3.20. From now on all the Ck-norms below are with respect to a time-independent metric. Also, by what we mean uniform bounds on Ck-norms means that they are independent of t but may depend on k.

Proof. First ﬁx a compact set K ⊂ M and ﬁnd K0 ⊂ B × Cr such that K0 and K are biholomorphic 0 0 via p, i.e. p(K ) = K. From Lemma 3.13 we already know that ϕt → u in C -norm, hence to prove

(i) it suﬃces to establish uniform bounds on kϕtkCk(K). Note that

√ ∗ ∗ ¯ p ωt = p ωˆt + −1∂∂(ϕt ◦ p) CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 49

∗ 0 and it is straight-forward to check that kp ωˆtkCk(K0) ≤ C(K , k) for some constant C > 0 depending 0 ∗ only on K and k. We are left to show kp ωtkCk(K0) is uniformly bounded independent of t. r Denote {zi, ξα} to be the base-ﬁber coordinates on B × C , i.e. i = 1, . . . , n − r and α = 1, . . . , r. ∗ ∗ ∗ The local components of p ωt and λt p ωt are related by

∗ ∗ ∗ −t/2 (p ωt)i¯j(z, ξ) = (λt p ωt)i¯j(z, e ξ),

∗ −t/2 ∗ ∗ −t/2 (p ωt)iα¯(z, ξ) = e (λt p ωt)iα¯(z, e ξ),

∗ −t/2 ∗ ∗ −t/2 (p ωt)β¯j(z, ξ) = e (λt p ωt)β¯j(z, e ξ),

∗ −t ∗ ∗ −t/2 (p ωt)αβ¯(z, ξ) = e (λt p ωt)αβ¯(z, e ξ).

∗ ∗ k By Lemma 3.17, the local components of λt p ωt are uniformly bounded in every C -norm. It is ∗ easy to check from the above relations that the local components of p ωt are also uniformly bounded k ∗ in every C -norm. Combining with the uniform bounds on kp ωˆtkCk(K0), we establish the uniform ∗ bounds on kp ϕtkCk(K0) and hence kϕtkCk(K). One can then prove (i) by covering M by ﬁnitely many compact subsets K.

(ii) is a direct consequence of Lemma 3.13 and (i) above. Now we have ϕt → u andω ˆt → ωΣ √ ∞ ¯ ∞ both in C -topology. Hence ωt → ωΣ + −1∂∂u = ωGKE in C -topology as t → ∞.

To finish this section, we prove a result concerning fiber-wise convergence. We establish that the flow metric restricted on each fiber converges smoothly, after a suitable rescaling, to a flat metric on the torus fiber. Precisely, we have

Proposition 3.21. Under the same assumption as in Proposition 3.18, we have

∞ r (i) ut → u ◦ p in Cloc(B × C ) as t → ∞,

t ∞ −1 (ii) e ωt|π−1(z) → ωSF |π−1(z) in C (π (z))-topology.

Proof. By the proof of Lemma 3.17 we have uniform bounds on kutkCk(K) for any compact subset r 0 K ⊂ B × C . Hence for (i) it suﬃces to show ut → u ◦ p in C -norm. Recall that ut is deﬁned by

−t −t ut = ϕt ◦ p ◦ λt − e (f ◦ p ◦ λt) − e (ζ ◦ p) where f and ζ are time-independent functions and hence are bounded on any compact subset of r 0 B × C . It suﬃces to show ϕt ◦ p ◦ λt → u ◦ p in C -norm, it can be established by Lemmas 3.9 and CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 50

3.13 as below:

|ϕt ◦ p ◦ λt(z, ξ) − u ◦ p(z, ξ)|

t/2 ≤ |ϕt(z, e ξ) − ϕt(z, ξ)| ◦ p + |ϕt(z, ξ) − u(z, ξ)| ◦ p

= O(e−t) + O(e−t/2) = O(e−t/2).

Taking t → ∞ completes the proof of (i). To prove (ii), we restrict (3.2.20) to the ﬁbers,

√ ∗ ∗ ∗ ¯ λt p ωt| r = p ωSF | r + −1∂∂ut r . {z}×C {z}×C {z}×C

−t/2 Pulling-back by λ−t deﬁned by (z, ξ) 7→ (z, e ξ) gives

√ ∗ ∗ ∗ ∗ ¯ p ωt| r = λ−tp ωSF r + λ−t −1∂∂ut r . {z}×C {z}×C {z}×C

By the rescaling property of ωSF given by (3.2.19), we have

∗ ∗ −t ∗ λ−tp ωSF = e p ωSF .

Note also that √ √ ∗ ¯ −t ¯ λ−t −1∂∂ut r = e −1∂∂ut r . {z}×C {z}×C

Combining these, we have

√ t ∗ ∗ ¯ e p ωt| r = p ωSF | r + −1∂∂ut r . {z}×C {z}×C {z}×C

∞ r From (i), we have ut → u ◦ p in Cloc(B × C ) and since u ◦ p depends only on z ∈ B, we have

√ −t ¯ e −1∂∂ut r → 0 {z}×C

∞ t ∗ ∗ ∞ r as t → ∞ in C -topology. Hence, we have e p ωt| r → p ωSF | r in C ({z} × )- loc {z}×C {z}×C loc C topology, and so t ∗ ∗ e p ωt|π−1(z) → p ωSF |π−1(z)

∞ −1 ∗ in C (π (z))-topology. It completes the proof of (ii) as p ωSF |π−1(z) is ﬂat for each z ∈ Σ. CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 51

Remark 3.22. Results proved in Propositions 3.18, 3.19 and 3.21 are conjectured by Song-Tian’s [73, 74] (see also [80]) for general non-singular Calabi-Yau fibrations, and analogously on singular Calabi-Yau in local sense away from singular fibers. A recent preprint [35] by Gill gave an affirmative answer to these conjectures when the fibration is in fact a Cartesian product.

3.3 Results on Fano ﬁbrations

In this section, we now change the gear to discuss Fano fibrations under the unnormalized Kähler- Ricci flow (2.2.1): ∂ω t = −Ric(ω ). ∂t t

We consider the following fibration structure: Let M be a compact connected Kählermanifold with π dimC = n which admits a surjective holomorphic submersion M −→ Σ over a Kählermanifold (Σ, ωΣ) −1 with dimC Σ = n − r < n. As in the previous section, for each z ∈ Σ, we call π (z) a fiber based at z, which is a complex submanifold of M with dimC = r.

Let ω0 be the initial K¨ahlermetric on M, the K¨ahler class [ωt] at time t is precisely given by

[ωt] = [ω0] − tc1(M). The maximal existence time T of (2.2.1) is uniquely determined by a result of Tian-Zhang in [82], namely

T = sup{t :[ω0] − tc1(M) > 0}.

We study the Kähler-Ricciflow on M defined as in (2.2.1)

∂ω t = −Ric(ω ), ω | = ω . ∂t t t t=0 0

The K¨ahlerclass [ωt] at time t is precisely given by [ω0] − tc1(M). The maximal existence time T of (2.2.1) is uniquely determined by a result of Tian-Zhang in [82], namely

T = sup{t :[ω0] − tc1(M) > 0}.

1 In particular, if the ﬁbers are Fano manifolds such as CP -bundles which will be studied in r −1 Chapter 4, we have c1(M) · [π (z)] > 0 for some z ∈ Σ, we must have T < ∞. In this article, we only focus on the case where (2.2.1) encounters ﬁnite-time singularity at T < ∞.

As seen in [72,77–79] etc., the limiting behavior of ωt under as t → T depends on the choice of the initial Kähler class [ω0]. This phenomenon is remarkably different from the Calabi-Yau fibrations CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 52

studied in Section 3.2. We focus on the case where the initial K¨ahlerclass [ω0] is chosen such that

∗ [ω0] − T c1(M) = [π ωΣ]. (3.3.1)

∗ [π ωΣ] can be regarded as the limiting Kählerclass of the flow. Similarly to the Calabi-Yau fibrations, we can rewrite the Kähler-Ricciflow (2.2.1) as a parabolic complex Monge-Ampèreequation. We define a family of reference metricsω ˆt in the same Kähler class as ωt by: 1 ωˆ = ((T − t)ω + tπ∗ω ). t T 0 Σ

One can argue that [ωt] = [ˆωt] by observing that [ωt] = [ω0] − tc1(M) which is a linear path ∗ connecting [ω0] and [π ωΣ] as t goes from 0 to T by our assumption (3.3.1). By Lemma 2.4, there √ ¯ exists a family of smooth functions ϕt such that ωt =ω ˆt + −1∂∂ϕt. Let Ω be a volume form on M such that √ ∂ωˆ 1 −1∂∂¯log Ω = t = (π∗ω − ω ). ∂t T Σ 0

Then it is easy to check that the Kähler-Ricciflow (2.2.1) is equivalent to the following complex Monge-Ampèreequation which is slightly different from (2.2.3): √ ∂ϕ (ˆω + −1∂∂ϕ¯ )n t = log t t , ϕ = 0. (3.3.2) ∂t (T − t)rΩ 0

1 n When M is a CP -bundle over CP , it was studied in by Song-Weinkove in [77] with the assump- 1 tion that the initial metric ω0 is constructed by Calabi’s ansatz . In this case, the Kählerclass [ω0] can be expressed as −a0[Σ0] + b0[Σ∞] with 0 < a0 < b0, where [Σ0] and [Σ∞] denote the Poincaré duals of the zero and infinity sections respectively, and (3.3.1) can be achieved by a suitable choice of a0 and b0. It was shown that under the condition (3.3.1) the Kähler-Ricciflow collapses the 1 CP -fiber. r For projectivized vector bundles, i.e. the fibers are CP ’s, over projective manifolds, it was proved by Song-Székelyhidi-Weinkove in [72] (see Theorem 1.6) that the fibers collapse at a rate of −1 1/3 at most diamt(π (z)) = O((T − t) ) without using any symmetry assumption. This collapsing rate is sufficient to establish the Gromov-Hausdorff convergence towards the base manifold. It is more desirable for the diameters of fibers decay at a rate ' (T − t)1/2 as far as singularity analysis is concerned. Singularity analysis by rescaling has been fundamental in the study of singularity

1The Calabi’s ansatz is a cohomogeneity-1 symmetry to be discussed in Chapter 4. CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 53

formations of the Ricci flow and other geometric deformations. The Cheeger-Gromov limit obtained from a suitably rescaled and dilated sequence of the Ricci flow encodes crucial geometric information of the singularity development near the singular time. In case of Type I singularity (see definition in p91), the rescaling factor of the metric tensor is uniformly equivalent to (T − t)−1. If a fiber −1 −1 1/2 −1 π (z) shrinks at a rate such that diamωt (π (z)) ' (T − t) , then under the (T − t) -rescaling the diameter of the fiber remains bounded away from 0 and ∞. It is conjectured in [72,80] that the diameter decay can be improved to O((T − t)1/2). It is only known to be true by an unpublished result of Perelman (see [66]) in the very special case where Σ is a point. −1 1/2 We will show in this section that the optimal decay rate diamt(π (z)) ' (T − t) is closely related to the Ricci form Ric(ωt). Using the notations introduced in this section, let us restate the main result:

Theorem 3.23 (F [31]; Theorem 1.8 of Chapter 1). Let M −→π Σ be a surjective holomorphic submersion. Suppose the Kähler-Ricci flow (2.2.1) on M encounters finite-time singularity at T < ∞ and the initial Kählerclass [ω0] satisfies (3.3.1), we have

Ric(ωt) ≤ Bω0 for some uniform constant B > 0

−1 ⇒ C ωˆt ≤ ωt ≤ Cωˆt,

where C is a uniform constant depending only on n, r, ω0, ωΣ and B, and hence the ﬁber-collapsing rate is optimal:

−1 1/2 −1 1/2 C (T − t) ≤ diamωt π (z) ≤ C(T − t) for any t ∈ [0,T ), z ∈ Σ.

3.3.1 Pointwise estimates

In this subsection, we give the estimates necessary to establish Theorem 3.23. In all the estimates below, we will denote C > 0 to be a uniform constant which depends only on n, r, ω0, ωΣ and B, and may change from line to line. We ﬁrst prove the following:

Lemma 3.24. Given that Ric(ωt) ≤ Bω0 for some uniform constant B > 0, then along (3.3.2) there exists a uniform constant C = C(n, r, ω0, ωΣ,B) such that

Z n ∂ϕt 1 ωt n ≤ n log r ω0 + C. ∂t [ω0] M (T − t) Ω CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 54

Proof. We let G(x, y): M × M − {x = y} → R be a nonnegative Green’s function with respect to

ω0. Then we have for any x ∈ M,

Z Z ∂ϕt 1 ∂ϕt n 1 ∂ϕt n (x) − n ω0 = − n G(x, ·)∆ω0 ω0 ∂t [ω0] M ∂t [ω0] M ∂t Z n 1 ωt n = − n G(x, ·)∆ω0 log r ω0 [ω0] M (T − t) Ω Z ∂ωˆt n = G(x, ·)Trω0 Ric(ωt) + ω0 M ∂t Z 1 ∗ n = G(x, ·) Trω0 Ric(ωt) + Trω0 (π ωΣ − ω0) ω0 , M T where we have used (3.3) in the third and fourth steps. By our assumption Ric(ωt) ≤ Bω0, it is easy to see that 1 Tr Ric(ω ) + Tr (π∗ω − ω ) ≤ C. ω0 t T ω0 Σ 0 for some uniform constant C = C(n, r, ω0, ωΣ,B) > 0. Therefore, we have for x ∈ M,

Z Z ∂ϕt 1 ∂ϕt n n (x) − n ω0 ≤ C G(x, y)ω0 (y). ∂t [ω0] M ∂t M

R n Note that M G(x, y)ω0 (y) ≡ constant. It completes the proof of the lemma.

∂ϕt Next we derive a uniform upper bound for ∂t . By the previous lemma, it suﬃces to bound the total volume of (M, ωt), which depends only on [ωt]. Let us consider the reference metric 1 ∗ ωˆt = T ((T − t)ω0 + tπ ωΣ). We have

n 1 X ωˆn = Cn(T − t)ktn−kωk ∧ (π∗ω )n−k. t T n k 0 Σ k=0

∗ n−k Since Σ has complex dimension n − r, we have (π ωΣ) = 0 for any k < r. Therefore, we have

n 1 X ωˆn = Cn(T − t)ktn−kωk ∧ (π∗ω )n−k t T n k 0 Σ k=r (T − t)r = Cntn−rωr ∧ (π∗ω )n−r + ... + (T − t)n−rωn . T n r 0 Σ 0

It is not diﬃcult to see that there exists a uniform constant C = C(n, r, ω0, ωΣ) > 0 such that for CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 55

any t ∈ [0,T ), we have −1 r n n r n C (T − t) ω0 ≤ ωˆt ≤ C(T − t) ω0 . (3.3.3)

By Jensen’s Inequality, we have

Z n n Z n n ωt ω0 ωt ω0 log r n ≤ log r n M (T − t) Ω [ω0] M (T − t) Ω [ω0] Z n C ωˆt ≤ log n r [ω0] M (T − t) ≤ log C using (3.3.3).

Combining with Lemma 3.24, we have established the following:

Lemma 3.25. There exists a uniform constant C = C(n, r, ω0, ωΣ,B) > 0 such that for any t ∈ [0,T ), we have n ωt ∂ϕt = e ∂t ≤ C. (3.3.4) (T − t)rΩ

Lemma 3.25 gives a pointwise bound for the volume form, which will be used in Lemma 3.27 to show the K¨ahlerpotential ϕt is decaying at a rate of O(T − t) after a suitable normalization. n r Using the uniform equivalenceω ˆt ' (T − t) Ω, we can derive the following using the parabolic Schwarz’s estimate (Lemma 3.2).

Lemma 3.26. There exists a constant C such that

∗ Trωt π ωt ≤ C.

Proof. These estimates is well-known to hold. By (3.3.3), a straight-forward maximum principle argument shows that |ϕt| ≤ C. Deﬁne

∗ H = log Trωt π ωΣ − Aϕt where A is a large constant to be chosen. CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 56

T Denote = ∂t − ∆ωt as before. Lemma 3.2 gives for any t ≥ 2 ,

∂ϕ H ≤ CTr π∗ω − A t + A∆ϕ ωt Σ ∂t t ωn ≤ CTr π∗ω − A log t + A(n − Tr ωˆ ) ωt Σ (T − t)rΩ ωt t ωn A(T − t) A ≤ CTr π∗ω − A log t + An − Tr ω − Tr π∗ω ωt Σ (T − t)rΩ T ωt 0 2 ωt Σ A ωn 1 ≤ C − Tr π∗ω − A log t + Tr π∗ω + An. 4 ωt Σ (T − t)rΩ 4 ωt Σ

1 1/n Since x 7→ − log x + 4 cnx is uniformly bounded from below, we have

n r r 1/n ωt 1 (T − t) Ω 1 (T − t) Ω log r + Trωt ωˆt ≥ − log n + cn n ≥ −C. (T − t) Ω 4 ωt 4 ωt

A ∗ Choose A such that C − 4 < −1, we have H ≤ −Trωt π ωΣ + A(C + n). Applying maximum principle on H and the uniform boundedness of ϕt completes the proof of the lemma.

−1 For each z ∈ Σ and t ∈ [0,T ), we denote ωt,z to be the restriction of ωt on the ﬁber π (z). For each t ∈ [0,T ), we deﬁne a function Φt :Σ → R by

Z 1 r Φt(z) = −1 ϕtω0,z Volω0,z (π (z)) π−1(z)

−1 ∗ which is the average value of ϕt over each ﬁber π (z). The pull-back π Φt is then a function deﬁned ∗ on M. For simplicity, we also denote π Φt by Φt.

Lemma 3.27. There exists a uniform constant C = C(n, r, ω0, ωΣ,B) such that for any t ∈ [0,T ), we have

ϕt − Φt ≤ C. T − t

ϕt−Φt T −t Proof. Denoteϕ ˜t = T −t . For each z ∈ Σ, we haveω ˆt,z = T ω0,z, and so

√ T − t ¯ ωt,z = ω0,z + −1∂∂ϕt −1 . T π (z) √ ¯ Since Φt depends only on z ∈ Σ, we have −1∂∂Φt π−1(z) = 0. By rearranging, we have

√ 1 1 ¯ ωt,z = ω0,z + −1∂∂ϕ˜t −1 . (3.3.5) T − t T π (z) CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 57

Regard (3.3.5) to be a metric equation on the manifold π−1(z), then we have

√ r r 1 ¯ 1 ω0,z + −1∂∂ϕ˜t −1 = ωt,z (3.3.6) T π (z) T − t

Using Lemma 3.25, we can see

r r ∗ n−r ωt,z ωt ∧ (π ωΣ) r = r ∗ n−r ω0,z ω0 ∧ (π ωΣ) r ∗ n−r n ωt ∧ (π ωΣ) ωt ≤ n · r ∗ n−r ωt ω0 ∧ (π ωΣ) ∗ n−r r ≤ C(Trωt π ωΣ) · (T − t) . (3.3.7)

Combining Lemma 3.26 with (3.3.7), we see than (3.3.6) can be restated as

√ r r 1 ¯ 1 ω0,z + −1∂∂ϕ˜t −1 = Fz(ξ, t) ω0,z T π (z) T

−1 where Fz(ξ, t): π (z) × [0,T ) → R>0 is uniformly bounded. R r ∞ Since π−1(z) ϕ˜tω0,z = 0, by applying Yau’s L -estimate (Lemma 3.3) on (3.3.5), we then have

sup |ϕ˜t| ≤ Cz, π−1(z)×[0,T )

−1 where Cz depends on n, r, ω0, ωΣ,B, supπ−1(z)×[0,T ) Fz, Volω0,z (π (z)), the Sobolev and Poincar´e −1 2 constants of π (z) with respect to metric ω0,z, all of which can be bounded uniformly independent of z. It completes the proof of the lemma.

3.3.2 Collapsing rate of Fano ﬁbers

Using the estimates proved in the previous subsection, we can now give the proof of Theorem 3.23.

Proof of Theorem 3.23. We apply maximum principle to the following quantity

A Q := log((T − t)Tr ω ) − (ϕ − Φ ), ωt 0 T − t t t

2See Remark 3.10 for a discussion of the uniform boundedness of these quantities. CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 58

where A is a positive constant to be chosen. Denote = ∂t − ∆ωt , we have

1 1 i¯j k¯l log((T − t)Trωt ω0) ≤ − + g g Rm(g0)i¯jk¯l (3.3.8) T − t Trωt ω0 1 ≤ − + C˜Tr ω T − t ωt 0 where C˜ depends only on the curvature of g0.

A A ∂ϕ ∂Φ A (ϕ − Φ ) = t − t − (ϕ − Φ ) T − t t t T − t ∂t ∂t (T − t)2 t t A − (∆ϕ − ∆Φ ) T − t t t n Z ! A ωt ∂ϕt r CA ≥ log r − ω0,z − T − t (T − t) Ω π−1(z) ∂t T − t A − (n − Tr ωˆ − ∆Φ ) T − t ωt t t

Note that n 1/n r 1/n ωˆt (T − t) Ω Trωt ωˆt ≥ n n ≥ cn n . ωt ωt

1 1/n Since x 7→ − log x + 2 cnx is uniformly bounded from below, we have

n r r 1/n ωt 1 (T − t) Ω 1 (T − t) Ω log r + Trωt ωˆt ≥ − log n + cn n ≥ −C (T − t) Ω 2 ωt 2 ωt for some uniform constant C > 0. Hence, we have

A AC (ϕ − Φ ) ≥ − (3.3.9) T − t t t T − t Z ! A ∂ϕt r + ∆Φt − ω0,z T − t π−1(z) ∂t A + Tr ωˆ . 2(T − t) ωt t CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 59

Combining (3.3.8) and (3.3.9), we have

AC A T − t t Q ≤ + C˜Tr ω − Tr ω + π∗ω (3.3.10) T − t ωt 0 2(T − t) ωt T 0 T Σ Z ! A ∂ϕt r − ∆Φt − ω0,z T − t π−1(z) ∂t AC A ≤ + C˜ − Tr ω T − t 2T ωt 0 Z ! A ∂ϕt r − ∆Φt − ω0,z T − t π−1(z) ∂t

∂ϕt By Lemma 3.25, we have ∂t for some uniform constant C. It follows that

Z ∂ϕt r −1 0 ω0,z ≤ CVolω0,z (π (z)) ≤ C . π−1(z) ∂t

−1 Note that Volω0,z (π (z)) is independent of z. For the Laplacian term of Φt, we have

∗ R r ∗ r By the fact that Trωt π ωΣ ≤ C and π−1(z) ω0 ∧ ω0,z + π ωΣ ∧ ω0,z is a (1, 1)-form on Σ independent of t, we have Z r ∆ ϕtω0,z ≥ −C. π−1(z) for some uniform constant C. Back to (3.3.10), we have

AC A AC Q ≤ + C˜ − Tr ω ≤ − Tr ω T − t 2T ωt 0 T − t ωt 0

˜ A if we choose A suﬃciently large such that C − 2T < −1. Hence, for any ε > 0, at the point where Q achieves its maximum over M × [0,T − ε], we have

Trωt (T − t)ω0 ≤ C for some uniform constant C independent of ε. Together with Lemma 3.27, it CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 60

shows that for any t ∈ [0,T ) we have,

−1 C (T − t)ω0 ≤ ωt. (3.3.11)

−1 ∗ Combining this with the fact that ωt ≥ C π ωΣ (by Lemma 3.26), we have

−1 C ωˆt ≤ ωt. (3.3.12)

Together with (3.3.4) and (3.3.12), we also have ωt ≤ Cωˆt for any t ∈ [0,T ). It completes the proof of the theorem.

3.3.3 Blow-up rate of curvature

We would like to end this section by a discussion of the implications of Theorem 3.23 on the blow-up rate of curvatures. There are “folklore conjectures” concerning the blow-up of curvatures along the Kähler-Ricciflow (2.2.1) with finite-time singularity (see Section 7 in [80]). It is known by Hamilton [42], Sesum [65] and Zhang [87] that sup kRmkg(t), sup kRickg(t) and the scalar curvature sup R(g(t)) must blow-up to +∞ as t → T when T < ∞ is the singular time. However, it is still open whether they blow-up at the rate of O((T − t)−1) for the Kähler-Ricciflow (2.2.1).

In the case where [ω0] = c1(M) > 0, it was established by Perelman (see [66]) that R(ωt) = −1 O((T − t) ) along (2.2.1). For the normalized Kähler-Ricciflow ∂tωt = −Ric(ωt) − ωt with finite- time singularity, Zhang established in [87] that R(˜g(t)) = O((T − t)−2) under a cohomological assumption analogous to (3.3.1). Under the cohomological setting in this article, the implications of the boundedness of Trω0 Ric(ωt) on the curvature blow-up rates are as follows:

Proposition 3.28. Under the same assumptions as in Theorem 3.23, we have

Ric(ωt) ≤ Bω0 for some uniform constant B > 0

−1 −2 ⇒ R(ωt) = O((T − t) ), and kRmkωt = O((T − t) ).

−1 Proof. The O((T − t) ) blow-up rate of the scalar curvature R(ωt) = Trωt Ric(ωt) follows trivially from Ric(ωt) ≤ Bω0 and (3.3.11). For the blow-up of the Riemann curvature tensor, we use the result in the proof of Theorem 3.23 CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 61

that there exists a uniform constant C > 0 such that for any t ∈ [0,T ), we have

−1 C (T − t)ω0 ≤ ωt ≤ Cω0.

Using the estimates in [59, 67] one can establish that

−2 kRmkωt = O((T − t) ).

1 −4 Note that in [67] the authors assumed Nω0 ≤ ωt ≤ N ω0 and asserted that kRmkωt = O(N ). One can easily check this result can be extended without much diﬃculty so that N is any positive non- 1 increasing function N(t) deﬁned on [0,T ). Furthermore, if the metric upper bound N ω0 is replaced −2 by Cω0 for some uniform constant C > 0, the result can be strengthened to kRmkωt = O(N ) with an almost identical proof. For reader’s convenience, we have included the proof below in Theorem 3.30.

−2 Remark 3.29. It is not known whether the blow-up rate O((T − t) ) of kRmkωt can be achieved. The following theorem is a curvature estimate used in the discussion of Proposition 3.28. It is a slightly modiﬁed interior derivative estimate based on Sherman-Weinkove’s work [67].

Theorem 3.30 (c.f. Phong-Sesum-Sturm [59], Sherman-Weinkove [67]). Let (M, ωt) be a solution to the Kähler-Ricci flow ∂tωt = −Ric(ωt) on a closed manifold M which encounters finite time singularity at T < ∞. Suppose there exists a fixed (time independent) Kählermetric ωˆ on M such that there exist constants C and α independent of t such that

−1 α C (T − t) ωˆ ≤ ωt ≤ Cω,ˆ t ∈ [0,T ). (3.3.13)

−2α Then, the Riemann curvature kRmkωt up at most at the rate of O((T − t) ).

Proof. Throughout the proof, we denote gt to be the K¨ahlermetric associated to ωt andg ˆ to be the

K¨ahlermetric associated toω ˆ. All of the ∇, ∆ and norms are with respect to gt unless otherwise speciﬁed.

Choose a sequence {Ti} such that Ti → T as i → ∞. Let (xi, ti) be a point such that

(T − t )2αkRm(x )k = sup (T − t)2αkRmk . (3.3.14) i i gti gt M×[0,Ti]

After passing to a subsequence, we assume xi → x∞ ∈ M as i → ∞. We then choose a ball CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 62

(with respect to metricω ˆ) B3r = Bgˆ(x∞, 3r) centered at x∞ such that B3r is contained in a single holomorphic coordinate chart. In doing so, one can locally deﬁne an Euclidean metric gE such that −1 A gE ≤ gˆ ≤ AgE on B(x∞, 3r) where A depends onω, ˆ x∞ and r. We let

¯ S = k∂g k2 = gi¯jgklgqp¯∂ g ∂ g t gt i kp¯ j lq¯

where ∂ is the ﬂat connection on B(x∞, 3r). We claim

Lemma 3.31. There exists a constant C > 0 depending on ω0, ω,ˆ x∞ and r such that

C S ≤ , on B(x , 2r). (T − t)2α ∞

Proof of lemma. Choose a cut-oﬀ function φ with compact support in B(x∞, 3r) and identically equal to 1 on B(x∞, 2r). Assume further there is a constant C = C(ˆω, x∞, r) such that on B(x∞, 3r), we have

C k∂φk2 ≤ gE r2

X ∂2φ C ≤ ∂z2 r2 i i

By (3.3.13), we then have

C k∇φk2 ≤ gt (T − t)αr2 C |∆φ| ≤ , (T − t)αr2 where ∇ and ∆ are with respect to gt. Denote = ∂t − ∆. One can compute

2α 2 2α−1 2 2α 2 (T − t) φ S = −2α(T − t) φ S + (T − t) φ S (3.3.15)

j jk¯ Let h be an endomorphism locally deﬁned on B(x∞, 3r) by hi = (gE) gik¯. Direct compution (see [59]) shows, −1 2 ¯ −1 2 S = −(k∇(∇h · h )k + k∇(∇h · h )k ). (3.3.16) CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 63

Combining (3.3.15) and (3.3.16), we have

2α 2 (T − t) φ S 2α−1 2 2α 2 = − 2α(T − t) φ S + (T − t) φ S 2α 2 2 2 ≤(T − t) (φ S + 2h∇φ , ∇Si + ∆φ · S) ≤(T − t)2α{−φ2(k∇(∇h · h−1)k2 + k∇¯ (∇h · h−1)k2) CS + + 2h∇φ2, ∇Si} (3.3.17) (T − t)αr2

Note that S = k∇h · h−1k2 . Using holomorphic normal coordinate with respect to g , one can check gt t

2 X 2|h∇φ , ∇Si| ≤ 4 (|φ∂iφ · ∂¯iS| + |φ∂¯iφ · ∂iS|) (3.3.18) i

Note that

X X −1 j −1 j |φ∂iφ · ∂¯iS| = 2|φ∂iφ · ∂i(∇h · h )kl · (∇h · h )kl| i i,j,k,l φ2k∇(∇h · h−1)k2 ≤ + 8k∇φk2S 8 φ2k∇(∇h · h−1)k2 CS ≤ + . 8 (T − t)αr2

Similarly, we have X φ2k∇¯ (∇h · h−1)k2 CS |φ∂¯φ · ∂ S| ≤ + i i 8 (T − t)αr2 i Using (3.3.18), we have

1 CS 2|h∇φ2, ∇Si| ≤ φ2(k∇(∇h · h−1)k2 + k∇¯ (∇h · h−1)k2) + . 2 (T − t)αr2

Combining with (3.3.17), we have

(T − t)2α CS(T − t)α (T − t)2αφ2S ≤ − φ2(k∇(∇h · h−1)k2 + k∇¯ (∇h · h−1)k2) + . (3.3.19) 2 r2 CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 64

By Lemma 3.1 (by takingω ˆt ≡ ωE), we have

i¯j k¯l pq¯ Trh = −g (gE) g ∂igkq¯∂¯jgp¯l (3.3.20) ≤ −C−1(T − t)αS, using (3.3.13)

Therefore, one can ﬁnd a suﬃciently large constant C˜ depending only on C and r such that

2α 2 ˜ ((T − t) φ S + CTrh) ≤ 0 on B(x∞, 3r) (3.3.21)

Applying the maximum principle on (3.3.21), we know the supremum of

(T − t)2αφ2S + C˜Trh

is attained at (∂B(x∞, 3r) × [0,T )) ∪ (B(x∞, 3r) × {0}). Since φ ≡ 0 on ∂B(x∞, 3r) and Trh ≤ C

(by (3.3.13)), one can ﬁnd K depending on ω0, ω,ˆ x∞ and r such that

K (T − t)2αφ2S ≤ (T − t)2αφ2S + C˜Trh ≤ on B(x , 3r) 2 ∞

As φ ≡ 1 on B(x∞, 2r), it completes the proof of the lemma.

Back to the proof of Theorem 3.30, we denote

2α H = (T − t) S + C˜Trh on B(x∞, 2r), where C˜ is a constant chosen in the proof of the above lemma. K We proved in the lemma that H ≤ 2 on B(x∞, 2r) for some constant K, and hence we have K 2 ≤ K − H ≤ K on B(x∞, 2r).

Let Ψ be a time independent cut-oﬀ function with compact support on B(x∞, 2r) and that Ψ ≡ 1 on B(x∞, r). As before, we choose Ψ such that

C k∇Ψk2 ≤ gt (T − t)αr2 C |∆Ψ| ≤ gt (T − t)αr2 CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 65

We are going to apply maximum principle on the quantity

Ψ2(T − t)4αkRmk2 Q = . K − H

∂ Again we denote = ∂t − ∆t, note that

2 4α 4α−1 2 4α 2 (Ψ (T − t) ) = −4α(T − t) Ψ − (T − t) ∆Ψ (3.3.22) 2 2 ¯ 2 3 kRmk ≤ −k∇Rmk − k∇Rmk + CkRmk (3.3.23) 2α −1 2 ¯ −1 2 H ≤ −(T − t) (k∇(∇h · h )k + k∇(∇h · h )k ) (3.3.24)

−1 Since ∇h · h is the diﬀerence of Christoﬀel symbols of gt and gE, we have

k∇¯ (∇h · h−1)k = kRmk, and so (3.3.24) can be rewritten as

2α −1 2 2 H ≤ −(T − t) (k∇(∇h · h )k + kRmk ) (3.3.25)

Combining (3.3.22), (3.3.23) and (3.3.25), using the formula

fg g f fg 2fg = f + g − h − k∇hk2 h h h h2 h3 2 2g 2f −

(∆Ψ2)kRmk2 Ψ2 Q ≤ (T − t)4α − + (−k∇Rmk2 − k∇¯ Rmk2 + CkRmk3) K − H K − H Ψ2(T − t)2αkRmk2 2Ψ2kRmk2 − (k∇(∇h · h−1)k2 + kRmk2) − k∇Hk2 (K − H)2 (K − H)3 4 4kRmk2 −

We then apply Young’s and Cauchy-Schwarz’s inequalities to the last three terms.

4 1 −

(∆Ψ2)kRmk2 CΨ2kRmk3 Q ≤ (T − t)4α − + K − H K − H Ψ2(T − t)2α(kRmk4 + kRmk2k∇(∇h · h−1)k2) − (K − H)2 2Ψ2kRmk2 N k∇Ψk2kRmk2 N Ψ2k∇Hk2kRmk2 − k∇Hk2 + 1 + 2 (K − H)3 K − H (K − H)3 4kRmk2 Ψ2k∇Hk2 + (K − H)k∇Ψk2 (K − H)2 (K − H) CkRmk2 CΨ2kRmk3 CΨ2k∇Hk2kRmk2 ≤ (T − t)4α + + (T − t)αr2(K − H) K − H (K − H)3 Ψ2(T − t)2α(kRmk4 + kRmk2k∇(∇h · h−1)k2) − (K − H)2

2 We ﬁrst estimate the term k∇Hk . Applying (3.1.1) withω ˆ = gE, we have

2 i¯j k¯l pq¯ k∇Trhk ≤ (Trh)(g δ g · ∂igkq¯ · ∂¯jgp¯l).

Recall from our assumption (3.3.13) that ωt ≤ Cωˆ0, hence we can ﬁnd C depending onω ˆ0, x∞ and r such that k∇Trhk2 ≤ C(Trh)S (3.3.26) CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 67

To estimate k∇Hk2, we have

k∇Hk2 ≤ k∇((T − t)2αS + C˜Trhk2

≤ 2(T − t)4αk∇Sk2 + 2C˜2k∇Trhk2

≤ 2(T − t)4αS(k∇(∇h · h−1)k2 + k∇¯ (∇h · h−1)k2) + 2CS

C ¯ −1 Since we proved in the lemma that S ≤ (T −t)2α , and note also that k∇(∇h · h )k = kRmk, we have

C k∇Hk2 ≤ C(T − t)2α(k∇(∇h · h−1)k2 + kRmk2) + (T − t)2α

1 2 C Back to the estimate of Q, since K−H ≤ K , one can let K large enough such that (K−H)3 − 1 2(K−H)2 < 0 and so we have

CkRmk2 CΨ2kRmk3 Ψ2(T − t)2αkRmk4 Q ≤ (T − t)4α + − (T − t)2α K − H 2(K − H)2 CkRmk2 Ψ2kRmk3 ≤ (T − t)4α + [C(K − H) − (T − t)2αkRmk] (T − t)2α (K − H)2 Ψ2kRmk3 ≤ (T − t)2αCkRmk2 + (T − t)4α [CK − (T − t)2αkRmk] (K − H)2

Deﬁne Q˜ = Q + A(T − t)2αS − t, where A is a suﬃciently large constant, we have

˜ 2α Q ≤ Q + A(T − t) S − 1

From (3.3.16) and the fact that k∇¯ (∇h · h−1)k = kRmk, one can choose A large enough such that

Ψ2kRmk3 Q˜ ≤ (T − t)4α [CK − (T − t)2αkRmk] − 1 (3.3.27) (K − H)2

Using (3.3.27), one can easily argue Q˜ (and hence Q) is uniformly bounded: given any ε > 0, let

(xε, tε) ∈ B(x∞, 2r) × [0,T − ε] at which Q˜ attains it’s maximum in B(x∞, 2r) × [0,T − ε]. Since 2α Ψ ≡ 0 on ∂B(x∞, 2r) and (T − t) S is uniformly bounded by the lemma, one can assume without loss of generality that xε is in the interior of B(x∞, 2r). Also, we may also assume at (xε, tε) one has CK − (T − t)2αkRmk < 0 CHAPTER 3. COLLAPSING OF HOLOMORPHIC FIBRATIONS 68

otherwise Q˜(xε, tε) is again uniformly bounded. However, it will imply

˜ 0 ≤ Q(xε, tε) ≤ −1

which is absurd. Hence Q˜ attains it’s maximum at ∂B(x∞, 2r) which proves uniform boundedness of Q˜ and hence Q. It completes the proof of our theorem. Chapter 4

Singularity Model of Projective Bundles

It is of popular interest in understanding the singularity developments of the Ricci flow in both Riemannian and Kählersettings. Hamilton introduced in [42] a method of studying singularity formation of the Ricci flow by considering the Cheeger-Gromov limit of a sequence of rescaled dilated metrics. The singularity model obtained, which is often an ancient or eternal solution, captures the geometry of the singularity formation near the blow-up time of the flow. For closed 3-manifolds, the study of ancient κ-solutions formed by the dilated sequence limit in Hamilton-Perelman’s works (e.g. [42, 56]) leads to a solid understanding of singularity formation of closed 3-manifolds. Based on the collapsing results discussed in Chapter 3, this chapter is devoted to a specific type 1 of holomorphic fibrations, namely CP -bundles over Kähler-Einsteinmanifolds. We will adopt a cohomogeneity-1 symmetry, known as the Calabi’s ansatz, which is preserved under the Kähler- Ricci flow. We will classify the blow-up model of these bundles by rescaling procedures. 1 The organization of this chapter is as follows. In Section 4.1, we define the class of CP -bundles being studied and summarize the main result of this chapter. In Section 4.2, we discuss the Calabi’s ansatz and give a survey of literatures which adopted this ansatz. We also show that the Calabi’s ansatz is preserved by the Kähler-Ricciflow. Section 4.3 gives important estimates along the flow with Calabi’s ansatz which eventually lead to Gromov-Hausdorff’s convergence. Section 4.4 classifies the singularity models and rule out the possibility of Type II singularity under our setting.

69 CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 70

4.1 Hirzebruch surfaces and projective bundles

In this section, we will define and elaborate on the projective bundles under consideration in this n chapter. We first start with a compact Kähler-Einsteinmanifold Σ with dimC = n. A Kähler manifold is called Kähler-Einsteinif it admits a Kählerform ωΣ whose Ricci form is a real constant multiple of ωΣ, i.e. Ric(ωΣ) = νωΣ, ν ∈ R. Clearly a necessary condition for a compact Kähler manifold to be Kähler-Einsteinis that the first Chern class c1 has a definite sign. It is well-known by results of Aubin [2] and Yau [85] that when c1 < 0 or = 0, Kähler-Einsteinmetric always exists. However, if c1 > 0 (i.e. Fano manifolds), Kähler-Einsteinmetrics do not exist in general.

For compact Riemann surfaces, i.e. dimC = 1, K¨ahler-Einsteinmetric must exist according to the classical uniformization theorem. See also Cheng-Yau’s work [18] on pseudoconvex domains in the complete non-compact case.

We take this K¨ahler-Einsteinmanifold (Σ, ωΣ) to be our base manifold, and build a projective 1 CP -bundle upon it. Precisely, we construct our class of projective bundles as follows:

M = P(OΣ ⊕ L).

Here OΣ is the trivial line bundle, and L → Σ is a holomorphic line bundle which is equipped with a Hermitian-Einstein metric h such that

√ ¯ −1∂∂ log h = λωΣ, λ ∈ R.

Here P denotes the projectivization of the holomorphic rank-2 bundle OΣ ⊕ L over Σ. The local trivialization (z, u) of this rank-2 bundle has transition functions of the form (zα, uα) ≈ (zβ, ηαβuα) 1 ∗ for some ηαβ ∈ H (Σ, OΣ). Passing to the projectivization quotient, every element under this u trivialization can be expressed as either [1 : z ] for z 6= 0 or [0 : 1] and we may regard [0 : 1] as the infinity. One can check easily that the projectivization factors through the identification by the transition functions OΣ ⊕ L. Therefore, one can regard the projectivization of OΣ ⊕ L as compactifying each fiber by adding an infinity point (x, [0 : 1]) and hence M can be regarded as 1 a CP -bundle over Σ. We define Σ0 to be the zero section {x : [1 : 0]} and Σ∞ to be the infinity section {x : [0 : 1]}. It is easy to see that the zero section Σ0 and the infinity section Σ∞ are global over Σ. The class of holomorphic line bundles over Σ with tensor product as the operation form a group n which is known as the Picard group, denoted by Pic(Σ). For Σ = CP , it is well-known that CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 71

n n n Pic(CP ) = Z and the line bundles over CP are given by OCP (k), k ∈ Z. In particular if (Σ,L) = 1 ( , O 1 (−k)), k > 0, the projective bundles M = (O 1 ⊕ O 1 (−k)) are called the Hirzebruch CP CP P CP CP 2 2 2 surfaces. When k = 1, the projective bundles is CP #(−CP ), i.e. CP blown-up at a point. When Σ = C/Λ, i.e. an elliptic curve or a 2-torus, the class of line bundles are classified by a classical result by Appell-Humbert. In general for Riemann surface Σg of genus g, the Picard group Pic(Σg) g is isomorphic to J(Σg) × Z where J(Σg) is a compact complex manifold C /Λ of dimension g. The projective bundle M under our consideration is characterized by the pair (Σ,L) where Σ is a compact Kähler-Einsteinmanifold and L a holomorphic line bundle over Σ which is equipped with a Hermitian metric h such that the Chern cuvature is of the form λωΣ. In particular, the line ⊗k k ⊗k bundles KM , k > 0 all fall into this category since h = det(ωΣ) is a Hermitian-Einstein for KM √ ¯ and −1∂∂ log h = kνωΣ. Moreover, if Σ is a compact Riemann surface, then such h always exists given any holomorphic line bundle L. We will only focus on negative line bundles L with λ > 0, ∗ since the projective bundle P(OΣ ⊕ L) is biholomorphic to its dual cousin P(OΣ ⊕ L ). One can ∗ replace L by L in case c1(L) is positive. This Hermitian-Einstein metric h will be used to construct the Calabi’s ansatz defined in Section 4.2. The main results in this section are as follows. We study the (unnormalized) Kähler-Ricci

ﬂow ∂tωt = −Ric(ωt) starting with an initial K¨ahlermetric ω0 constructed by the Calabi’s ansatz.

Suppose that the initial K¨ahlerclass [ω0] satisﬁes

∗ [ω0] − T c1(M) = [π ωΣ] (c.f. (3.3.1) in p52), where T < ∞ is the singular time. Then, along the K¨ahler-Ricciﬂow, we have

• (M, g(t)) converges in Gromov-Hausdorﬀ sense to (Σ, ωΣ) (Theorem 4.6); 1 • the associated ancient κ-solution is Cn × CP (Theorem 4.13); • the Ricci ﬂow solution must have a Type I singularity (Theorem 4.14).

4.2 Calabi’s ansatz

We will discuss the construction of cohomogeneity-1 K¨ahlermetrics on these projective bundles in this section. Recall that ωΣ is the K¨ahler-Einsteinform on the manifold Σ and we have Ric(ωΣ) =

νωΣ for some ν ∈ R. Using the Hermitian-Einstein metric h described above, one can deﬁne a height CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 72

parameter ρ on M\(Σ0 ∪ Σ∞) given by

2 ρ = log k · kh.

Note that {ρ = −∞} corresponds to the zero section Σ0 and {ρ = ∞} corresponds to the infinity section Σ∞. Our next step is to define Kählermetrics on M whose Kählerpotential depends only on the height parameter ρ, i.e.

√ ¯ ω = −1∂∂u(ρ) on M0 = M\(Σ0 ∪ Σ∞).

For simplicity, we denote f = uρ and hence ω on M0 is characterized by an strictly increasing function f(ρ): R → R and we have

√ √ ¯ ω = f(ρ) −1∂∂ρ + −1fρ(ρ)∂ρ ∧ ∂ρ.¯

Denote [Σ0] and [Σ∞] to be the Poincaréduals (with respect to a fixed background volume form) R R of Σ0 and Σ∞ in H2(M, ) respectively, i.e. [Σ∞] = − [Σ0] = 1. Any Kählerclass [ω] is a R Σ∞ Σ0 linear combinations of [Σ0] and [Σ∞], i.e.

bλ[Σ∞] − aλ[Σ0] with b > a > 0.

√ ¯ ∗ Note also that c1(L) = [− −1∂∂ρ] = −λπ [ωΣ] = λ[Σ0] − λ[Σ∞]. One can extend the metric ω induced by f to the whole manifold M by requiring the following asymptotic conditions:

0 1. There exists a smooth function G0 : [0, ∞) → R with G0(0) > 0 and G0(0) > 0, so that ρ f(ρ) = G0(e ) as ρ → −∞.

0 2. There exists a smooth function G∞ : [0, ∞) → R with G∞(0) > 0 and G∞(0) > 0 so that −ρ f(ρ) = G∞(e ) as ρ → ∞.

Note that f has to be a strictly increasing function so that ω is positive deﬁnite. In order for such a

K¨ahlermetric ω to be in the K¨ahlerclass bλ[Σ∞]−aλ[Σ0], the function f has to satisfy the following CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 73

conditions:

a < f(ρ) < b for ρ ∈ R, lim f(ρ) = a, ρ→−∞ lim f(ρ) = b, and ρ→∞

lim fρ(ρ) = 0. ρ→±∞

To show that the K¨ahlerclass [ω] = bλ[Σ∞] − aλ[Σ0], we consider:

Z √ ¯ h[ω], Σ∞i = b[ −1∂∂ρ] Σ∞ Z = bλ[Σ∞] = bλ, Σ∞ Z √ ¯ h[ω], Σ0i = a[ −1∂∂ρ] Σ0 Z = −aλ[Σ0] = aλ. Σ0

Example 4.1 (Hirzebruch manifolds). Hirzebruch surfaces/manifolds are a special case of these n n projective bundles with. (Σ, L, ωΣ) = (CP , OCP (−k), ωFS), where ωFS is the Fubini-Study metric. One can take h to be the following: n !k X 2 h = |zi| i=1

n where (z1, . . . , zn) are inhomogeneous coordinates of the base manifold CP .

Next we derive the local expression of the K¨ahlermetric ω constructed by the Calabi’s ansatz as well as its Ricci curvature. Let (z1, . . . , zn, ξ) be local holomorphic coordinates of M where z = (z1, . . . , zn) are the base coordinates and ξ is the ﬁber coordinate. The height parameter ρ can be locally written in the following form:

ρ(z, ξ) = log(|ξ|2h(z)) = log |ξ|2 + log h(z), (4.2.1)

where ϕ(z) is a positive function. Using this, one can easily check ρξξ¯ = ρiξ¯ = ρξ¯i = 0 for any √ ¯ ∗ ∗ i j i = 1, 2, . . . , n. Moreover, −1∂∂ρ = λπ ωΣ, so we can let λπ ωΣ = ρi¯jdz ∧dz¯ . Hence, the K¨ahler CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 74

metric in (z, ξ) coordinates is given by

√ n √ n X i ¯j X i ¯ ω = −1 (fρi¯j + fρρiρ¯j)dz ∧ dz + −1fρ ρiρξ¯dz ∧ dξ i,j=1 i=1 √ n √ X ¯i 2 ¯ + −1fρ ρξρ¯idξ ∧ dz + −1fρ|ρξ| dξ ∧ dξ. i=1

Let g be the metric associated to the K¨ahlerform ω, and gΣ be that of ωΣ. The determinant of the metric g and its logarithm are given by

n n −2 det(g) = λ f fρ det(gΣ)|ξ| ,

2 log det(g) = n log λ + n log f + log fρ + log det(gΣ) − log |ξ| .

√ Using this, one can then compute the Ricci form − −1∂∂¯log det(g):

√ Ric(ω) = − −1∂∂¯log det(g)

−1 i ¯j = {(νλ − ∂ρ(n log f + log fρ))ρi¯j − ∂ρρ(n log f + log fρ)ρiρ¯j}dz ∧ dz i ¯ ¯i − ∂ρρ(n log f + log fρ)ρiρξ¯dz ∧ dξ − ∂ρρ(n log f + log fρ)ρξρ¯idξ ∧ dz 2 ¯ − ∂ρρ(n log f + log fρ)|ρξ| dξ ∧ dξ.

√ ¯ In the computation of the Ricci form, we used the fact that ωΣ is K¨ahler-Einsteinso that − −1∂∂ log det(gΣ) =

νωΣ. To summarize, the local components of g, g−1 and Ric(g) are given by:

 fρ ¯ + f ρ ρ¯ if (A, B) = (i, ¯j)  ij ρ i j  g = ¯ , AB fρρiρξ¯ if (A, B) = (i, ξ)   2 ¯ fρ|ρξ| if (A, B) = (ξ, ξ)

 ¯  1 ρij if (A, B) = (i, ¯j)  f  AB 1 n ik¯ g = − P ρ ρ¯ if (A, B) = (i, ξ¯) , fρξ¯ k=1 k  ¯  Pn ρklρ ρ  1 1 k,l=1 k l¯ ¯  2 + if (A, B) = (ξ, ξ)  |ρξ| fρ f CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 75

 −1 (νλ − F )ρ ¯ − F ρ ρ¯ if (A, B) = (i, ¯j)  ρ ij ρρ i j  Ric = ¯ AB −Fρρρiρξ¯ if (A, B) = (i, ξ)   2 ¯ −Fρρ|ρξ| if (A, B) = (ξ, ξ) where F = n log f + log fρ. Observing that the ω and Ric(ω) have similar linear-algebraic expressions as far as ω has Calabi’s ansatz, one can see easily that the Kähler-Ricci flow on M is equivalent to a parabolic equation that evolves f which is sometimes called the momentum profile. In other words, the Kähler-Ricciflow preserves Calabi’s ansatz. Precisely, we have

Proposition 4.2. Suppose ω0 is the initial Kählerform on M with momentum profile f0(ρ), then the solution ωt, t ∈ [0,T ) to the Kähler-Ricci flow ∂tωt = −Ric(ωt) also admits a momentum profile f(ρ, t) at each time t ∈ [0,T ) where f(ρ, t) evolves by

∂f ∂ ν = (n log f(ρ, t) + log f (ρ, t)) − , f(ρ, 0) = f (ρ); ∂t ∂ρ ρ λ 0 or equivalently, ∂f fρρ fρ ν = + n − , f(ρ, 0) = f0(ρ). (4.2.2) ∂t fρ f λ The Calabi’s ansatz was introduced by Calabi in [5] on the subject of extremal Kählermet- n rics. When the base manifold Σ is CP , gradient Kähler-Ricci solitons were constructed on both O(−k)-bundles and their projectivizations by Feldman-Ilmanen-Knopf [28], Cao [7] and Koiso [45]. Extremal Kähler metrics and Kähler-Riccisolitons for Σ being a general Kähler-Einsteinmanifold were also constructed and studied by Apostolov–Calderbank–Gauduchon–Tønnesen-Friedman [1], Hwang-Singer [43], Dancer-Wang [21], Futaki-Wang [33], Székelyhidi [81], etc. The Ricci flow behavior assuming Calabi’s ansatz was studied by Song-Weinkove in [77] (see Theorem 1.5 of Chapter 1) which characterized the convergence behavior in the Gromov-Hausdorff π n sense of the Hirzebruch manifolds P(O ⊕ O(−k)) −→ CP . In their paper, it was proved that the Kähler-Ricciflow exhibits three distinct behaviors:

1 1. collapsing the CP -ﬁbers; 2. contracting the zero section; or 3. shrinking to a point.

This trichotomy is determined explicitly by the triple (n, k, [ω0]) where [ω0] is the initial K¨ahler class. Later in [78, 79] by the same authors, case (2) is much generalized and the assumption on CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 76

the symmetry is removed. The Calabi’s ansatz assumption is removed in case (1) later by Song- Sz´ekelyhidi-Weinkove in [72].

4.3 Estimates under Calabi’s ansatz

From now on, we will consider the Kähler-Ricciflow ∂tωt = −Ric(ωt) on M whose the flow metric

ωt has the aforesaid Calabi’s ansatz. We say T is the blow-up time of the Ricci flow if [0,T ) is the maximal time interval for the Ricci flow to exist. As discussed and adopted in previous chapters, the blow-up time T is for the Kähler-Ricciflow (2.2.1) is uniquely determined by [ω0] and c1(M) (see Theorem 2.9), namely

T = sup{t :[ω0] − tc1(M) > 0}.

Recall that the Kählerclass [ωt] at any time t is given by [ωt] = [ω0] − tc1(M). In order to work out the evolving Kählerclasses and the blow-up time, one needs to understand the first Chern class of KM , which can be computed by the adjunction formula. Note that c1(KM ) = −c1(M). Given any smooth divisor D of a compact Kählermanifold M, the adjunction formula relates

KM and KD by

KD = (KM ⊗ NM/D) D , (4.3.1) where NM/D is the normal bundle of D in M.

Using (4.3.1), one can easily work out c1(KM ) by taking D = Σ0, Σ∞ in turn. For example, taking D = Σ∞, we have

K = (K ⊗ L∗)| , Σ∞ M Σ∞

hc1(KΣ∞ ), Σ∞i = hc1(KM ) − c1(L), Σ∞i.

Since Σ is K¨ahler-Einsteinsuch that Ric(ωΣ) = νωΣ, we then have

hc1(KΣ∞ ), Σ∞i = −ν.

Since c1(L) = λ[Σ0] − λ[Σ∞], we have hc1(L), Σ∞i = −λ, and hence

hc1(KM ), Σ∞i = −ν − λ. CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 77

Similarly, one can also show by taking D = Σ0 in (4.3.1) (now NM\D = L) to show

hc1(KM ), Σ0i = −ν + λ.

Therefore, the ﬁrst Chern class of the canonical line bundle KM is given by:

c1(KM ) = (−ν − λ)[Σ∞] − (−ν + λ)[Σ0].

Hence, under the Kähler-Ricciflow ∂tωt = −Ric(ωt) with initial class [ω0] = b0λ[Σ∞] − a0λ[Σ0], the Kählerclass evolves by

[ωt] = (b0λ − (ν + λ)t)[Σ∞] − (a0λ − (ν − λ)t)[Σ0].

We denote [ωt] = λbt[Σ∞] − λat[Σ0] where at, bt are deﬁned by

(ν − λ) a := a − t, (4.3.2) t 0 λ (ν + λ) b := b − t. (4.3.3) t 0 λ

∗ Note also that [π ωΣ] = [Σ∞] − [Σ0], therefore the K¨ahlerclass can also be expressed as

∗ [ωt] = λat[π ωΣ] + λ(bt − at)[Σ∞].

Hence, by Theorem 2.9, the maximal time is characterized by λ and ν in the following way:

• Case 1: ν ≤ λ

b0−a0 In this case, [ωt] ceases to be K¨ahlerwhen bt = at, namely, at T := 2 . The limiting K¨ahler class is given by ∗ [ωT ] = λaT [π ωΣ].

This holds true for any given b0 > a0 > 0. • Case 2: ν > λ We further divide it into three sub-cases

(i) (ν − λ)b0 < (ν + λ)a0: CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 78

[ωt] ceases to be K¨ahlerwhen bt = at. Likewise, the limiting K¨ahlerclass is given by

∗ [ωT ] = λaT [π ωΣ].

(ii) (ν − λ)b0 = (ν + λ)a0:

−1 a0λ [ωt] is then proportional to c1(KM ), i.e. canonical class. The ﬂow stops at T = ν−λ and

the limiting class [ωT ] = 0. It is well-known (see e.g. [66]) that in such case (M, g(t)) extincts and converges to a point in the Gromov-Hausdorﬀ sense as t → T .

(iii) (ν − λ)b0 > (ν + λ)a0:

[ωt] ceases to be K¨ahlerwhen at T = a0, and the limit class is given by [ωT ] = λbT [Σ∞].

This trichotomy resembles that in Song-Weinkove’s work [77] on Hirzebruch surfaces and Hirze- n n n bruch manifolds, i.e. (Σ,L) = (CP , OCP ⊕ OCP (−k)). In their work, from which our study was motivated, similar trichotomy of the blow-up time of the Kähler-Ricciflow with initial Kähler class [ω0] was also exhibited as it is characterized by the triple (n, k, [ω0]). It was shown in [77] assuming Calabi’s ansatz and in [72] assuming Σ is projective that in case of having limiting Kähler ∗ 1 class aT [π ωΣ], the Kähler-Ricciflow collapses the CP -fiber of the projective bundle, which hereof converges to some Kählermetric of Σ as metric spaces in Gromov-Hausdorff sense. Case 2(iii) is in reminiscence of Song-Weinkove’s recent works [78] and [79] of contracting exceptional divisors, in which O(−k)-blow-up of arbitrary compact Kählermanifold X are considered. In their works, a cohomological condition is given on the initial Kählerclass and the first Chern class, under which the blown-up manifold will converge in Gromov-Hausdorff sense back to X with orbifold singularity of type O(−k). There is no symmetry assumption in these works. 1 We will only focus on Case 1 and Case 2(i) which exhibit collapsing of CP -fiber. From now on we assume that the triple (ν, λ, [ω0]) satisfies either

ν ≤ λ, or (4.3.4)

ν > λ and (ν − λ)b0 < (ν + λ)a0.

Recall that ν is the Ricci curvature of the K¨ahler-Einsteinmanifold Σ and λ is the Chern curvature CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 79

of the Hermitian-Einstein line bundle L, i.e.

Ric(ωΣ) = νωΣ, √ ¯ ∗ −1∂∂ρ = λπ ωΣ.

Recall that the ﬁrst Chern class of KM and the evolving K¨ahlerclass are given by:

c1(KM ) = (−ν − λ)[Σ∞] − (−ν + λ)[Σ0]

∗ = (−ν + λ)[π ωΣ] − 2λ[Σ∞],

[ωt] = λbt[Σ∞] − λat[Σ0]

∗ = λat[π ωΣ] + 2λ(T − t)[Σ∞]

where at and bt defined in (4.3.2) and (4.3.3). As in Chapter 3, we rewrite the Kähler-Ricciflow as a complex Monge-Ampèreequation. We chooseω ˆt to be the cohomogeneity-1 Kählermetric induced by the following momentum profile:

(b − a )eρ 2λ(T − t)eρ fˆ(ρ, t) := a + t t = a + . t 1 + eρ t 1 + eρ

This momentum proﬁle gives the following K¨ahlermetric:

√ √ eρ eρ ωˆ = a −1∂∂ρ¯ + 2 −1λ(T − t) ∂∂ρ¯ + ∂ρ ∧ ∂ρ¯ . t t 1 + eρ (1 + eρ)2

ˆ Clearly, f satisfies the asymptotic conditions for extendingω ˆt to the whole M. Also, we have ˆ ˆ [ˆωt] = [ωt] because f → at as ρ → −∞ and f → bt as ρ → ∞. √ ∂ωˆt ¯ Take Ω be a fixed volume form of M such that ∂t = −1∂∂ log Ω, then the Kähler-Ricciflow

∂tωt = −Ric(ωt) is equivalent to the following complex Monge-Ampèreequation, c.f (3.3.2): √ ∂ϕ det(ˆω + −1∂∂ϕ¯ ) t = log t t , ϕ | = ϕ ∂t (T − t)Ω t t=0 0 √ ¯ in a sense that ωt =ω ˆt + −1∂∂ϕt, t ∈ [0,T ) is a solution to the Kähler-Ricciflow ∂tωt = −Ric(ωt) √ ¯ with initial data ω0 =ω ˆ0 + −1∂∂ϕ0 if and only if ϕ : M × [0,T ) is a solution to (3.3.2). Working similarly as in Chapter 3, we have CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 80

Lemma 4.3. There exists a constant C = C(n, ω0, ν, λ) > 1 such that the following holds

1. |ϕt| ≤ C, n+1 2. ωt ≤ CΩ, and ∗ 3. Trωt π ωΣ ≤ C.

Proof. It suﬃces to show (2) as both (1) and (3) were proved in Chapter 3.

∂ϕt −1 We consider Q := ∂t − |λ − ν|a ϕt + log(T − t) where a := inf[0,T ) at > 0. We rewriteω ˆt as the following form: √ √ ¯ ωˆt = at −1∂∂ρ + 2 −1λ(T − t)Θ

eρ ¯ eρ ¯ where Θ = 1+eρ ∂∂ρ + (1+eρ)2 ∂ρ ∧ ∂ρ > 0. By direct computation, we have

∂Q ∂ϕ = Tr ((λ − ν)π∗ω − 2λΘ) + ∆ t (4.3.5) ∂t ωt Σ ∂t 1 − |λ − ν|a−1(Q + |λ − ν|a−1ϕ − log(T − t)) − t T − t −1 ∗ = ∆Q + |λ − ν|a ∆ϕt + (λ − ν)Trωt π ωΣ − 2λTrωt Θ 1 − |λ − ν|a−1(Q + |λ − ν|a−1ϕ − log(T − t)) − . t T − t √ ∗ ¯ Since ωt = atπ ωΣ + 2λ(T − t)Θ + −1∂∂ϕt, taking trace with respect to ωt yields

∗ ∗ n + 1 = atTrωt π ωΣ + 2λ(T − t)Trωt Θ + ∆ϕt ≥ atTrωt π ωΣ + ∆ϕt.

Hence we have, −1 −1 ∗ |λ − ν|a ∆ϕt ≤ |λ − ν|a (n + 1) − |λ − ν|Trωt π ωΣ. (4.3.6)

Note that at ≥ a for any t ∈ [0,T ). Combining (4.3.5) and (4.3.6), we have

−1 −1 −1 −1 Q ≤ (n + 1)|λ − ν|a − |λ − ν|a (Q + |λ − ν|a ϕt) + |λ − ν|a log T. (4.3.7)

As ϕt is uniformly bounded from (1), (4.3.7) implies a uniformly upper bound for Q. Since Q =

det ωt −1 log Ω − |λ − ν|a ϕt, again together with the uniform bound for ϕt, we proved (2).

Next, we will derive estimates on the Kähler-Ricciflow by assuming the metric satisfies Calabi’s ansatz and admits a evolving momentum profile f(ρ, t). First note that because fρ(ρ, t) > 0 for any CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 81

t and also limρ→−∞ f(ρ, t) = at, limρ→∞ f(ρ, t) = bt, we have

at < f(ρ, t) < bt, for any (ρ, t) ∈ R × [0,T ).

Note that at and bt are both bounded away from zero as t → T , (2) in Lemma 4.3 implies fρ is also uniformly bounded. Using these, one is able to derive the following estimates.

Lemma 4.4. There exists a constant C = C(n, ω0, ν, λ) > 0 such that

1. C−1 ≤ f ≤ C,

2. fρρ ≤ C, fρ

3. fρ ≤ C(T − t) for any (ρ, t) ∈ R × [0,T ).

Proof. As discussed above, (1) clearly holds because at is bounded away from zero and bt is uniformly bounded above on [0,T ).

fρρ For (2), ﬁrst note that by the asymptotic conditions of the momentum proﬁle f(ρ, t), limρ→±∞ = fρ

1 for any t ∈ [0,T ), so sup fρρ exists for every > 0. We will derive the uniform lower R×[0,T −) fρ fρρ bound for on [0,T ) since the upper bound is similar. Given any > 0, let (ρ, t) ∈ × [0,T − ) fρ R be the point such that

fρρ fρρ = sup . fρ fρ (ρ,t) R×[0,T −)

2 ∂ fρρ ∂ fρρ ∂ fρρ Then at (ρ, t), one has ≥ 0, = 0, and 2 ≤ 0. ∂t fρ ∂ρ fρ ∂ρ fρ Recall that f satisﬁes heat-type equation (4.2.2), i.e. ∂f = fρρ +n fρ − ν . By direct computation, ∂t fρ f λ one has

2 2 3 ∂ fρρ 2nfρ 2nfρρ nfρρ 3fρρ nfρρρ 4fρρfρρρ fρρρρ = 3 − 2 − 2 + 4 + − 3 + 2 , ∂t fρ f f ffρ fρ ffρ fρ fρ 2 ∂ fρρ fρfρρρ − fρρ = 2 , ∂ρ fρ fρ 2 3 ∂ fρρ 2fρρ 3fρρfρρρ fρρρρ 2 = 3 − 2 + . ∂ρ fρ fρ fρ fρ

2 ∂ fρρ fρρ Evaluating at (ρ, t), the fact that = 0 implies fρρρ = at (ρ, t). By substituting ∂ρ fρ fρ 2 2 fρρ ∂ fρρ ∂ fρρ fρρρ = into the expressions of and 2 , one can check that after cancellation of fρ ∂t fρ ∂ρ fρ CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 82

terms, we have 2 2 ∂ 1 ∂ fρρ 2nfρ 2nfρρ 0 ≤ − 2 = 3 − 2 at (ρ, t). ∂t fρ ∂ρ fρ f f

fρρ fρρ fρ It shows sup = ≤ . Since fρ is uniformly bounded from above and R×[0,T −) fρ fρ f (ρ,t) (ρ,t) f > C−1, there exists C > 0 independent of such that sup fρρ ≤ C. Similar approach R×[0,T −) fρ proves inf fρρ ≥ −C˜ for some uniform constant C˜ > 0. It completes the proof of (2). R×[0,T ) fρ

Part (3) follows from part (2). Precisely, (2) implies |(log fρ)ρ| ≤ C. If we let ρt ∈ R such that f (ρ ) = sup f . Then by the mean-value theorem, ρ t ρ∈R ρ

| log fρ(ρ, t) − log fρ(ρt, t)| ≤ C|ρ − ρt|.

−1 −1 Thus for ρ ∈ [ρt − C , ρt + C ], we have

f (ρ, t) log ρ ≥ −1, fρ(ρt, t)

−1 or equivalently, fρ(ρ, t) ≥ e fρ(ρt, t). We then have

−1 Z Z ρt+C −1 −1 fρdρ ≥ fρdρ ≥ 2C e fρ(ρt, t). −1 R ρt−C

On the other hand, we have

Z fρdρ = f(∞) − f(−∞) = bt − at = 2λ(T − t). R

Hence sup f ≤ C(T − t) for some uniform constant C. ρ∈R ρ

1 Lemma 4.4 implies the CP -ﬁber of our manifold M is collapsing along the ﬂow. Precisely we have the following:

Lemma 4.5. Assume (Σ, L, [ω0]) satisﬁes the condition stated in Case 1 and Case 2(i) in p77. Let 1 1 Vx ∈ TxM be a tangent vector of M at x ∈ M\(Σ0 ∪ Σ∞) which lies TxCPx. Here we denote CPx 1 as the CP -ﬁber passing through x. Then we have kVxkg(t) → 0 as t → T .

Proof. It suﬃces to express kVxkg(t) in terms of f and fρ. Since the metric g(t) is given by

∗ ¯ g(t) = fλπ gΣ + fρ∂ρ ⊗ ∂ρ. CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 83

∗ ¯ 2 Since Vx is parallel to the ﬁber, we have π∗Vx = 0 and so π gΣ(Vx, Vx) = 0. Hence kVxkg(t) =

∂Vx ∂V¯x fρ ∂ρ ∂ρ → 0 as t → T . Here we have used part (3) of Lemma 4.4.

Furthermore, Lemmas 4.3 and 4.4 provide enough estimates in order to show (M, ωt) converges to (Σ, aT ωΣ) as metric spaces in Gromov-Hausdorﬀ sense.

Theorem 4.6 (F [30]; Theorem 1.9 of Chapter 1). Suppose (Σ, L, [ω0]) satisfies the condition stated in Case 1 and Case 2(i) in p77, then (M, g(t)) converges to the Kähler-Einsteinmanifold (Σ, aT ωΣ) in Gromov-Hausdorff sense as t → T .

Remark 4.7. Unlike Proposition 3.12 in Chapter 3 or in [72], this theorem does not require to take subsequences for the Gromov-Hausdorﬀ convergence, and also that the limit metric is explicitly given.

Proof. The proof goes almost the same as in Song-Weinkove’s paper [77] on Hirzebruch surfaces with Calabi ansatz. The main ingredients of the argument are as follows:

1. the metric g(t) is degenerating along the ﬁber direction on compact subsets of M\(Σ0 ∪ Σ∞), 2. g(t) is bounded above uniformly g(0), and ∗ α 3. for any 0 < α < 1, g(t) converges to aT π ωΣ in C -sense on compact subsets of M\(Σ0 ∪Σ∞).

We have proved (1) in Lemma 4.5. (2) can be proved by a uniform estimate on fρ which can n+1 be obtained easily by the bound on the volume form ωt in Lemma 4.3. For (3), note that √ √ ω = f(ρ, t) −1∂∂ρ¯ + −1f (ρ, t)∂ρ ∧ ∂ρ¯ . One can compute that k∇ g(t)k2 is a polynomial t ρ g0 g0 expression of f(ρ, t), fρ(ρ, t) and fρρ(ρ, t) where the coeﬃcients are time-independent. Lemma 4.4 then shows for any compact subset K ∈ M\(Σ ∪ Σ ), so we have sup k∇ g(t)k2 ≤ C 0 ∞ K×[0,T ) g0 g0 K for some time independent constant CK > 0. It proves (3).

To show the Gromov-Hausdorff convergence, first fix a leave of Σ in M\(Σ0 ∪ Σ∞). We denote it by σ(Σ). Using (2), one can choose a sufficiently small tubular neighborhood of Σ0 and Σ∞ such that their complement contains σ(Σ). Then given any two points x1, x2 ∈ M, we project them down to the base Σ via the bundle map π. Consider the length of the geodesic γ joining π(x1) and

π(x2), by lifting the geodesic up by σ, we know that the lifted γ has length arbitrarily close to the aT ωΣ-length by (3). Finally, using (1), one can show xi is arbitrarily close to σ ◦ π(xi) as t → T .

Using triangle inequality, one can then prove the g(t)-distance between x1 and x2 are is arbitrarily close to the aT ωΣ-distance as t → T . CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 84

4.4 Singularity analysis

In the singularity analysis of closed (real) 3-manifolds as in [42] and [56], one often consider a rescaled −1 dilation, which is a rescaled sequence of metrics gi(t) = Kig(ti +Ki t) where Ki are chosen such that

Ki = kRm(xi)kg(ti) → ∞ and kRmgi(t)kgi(t) ≤ C for some uniform constant C > 0 independent of i. By Hamilton-Cheeger-Gromov’s compactness [42] and Perelman’s local non-collapsing theorem [56], one can extract a subsequence, still call it gi(t), such that (M, gi(t), xi) → (M∞, g∞(t), x∞) on compact subsets in Cheeger-Gromov sense. The convergence is in C∞-topology because once the curvature tensor is uniformly bounded, Shi’s derivative estimate in [68] asserts all the higher order derivatives of Rm are uniformly bounded. The limit obtained is often called a singularity model. According to the curvature blow-up rate, a singularity model may be an ancient or eternal solution, and is κ-non-collapsed by Perelman’s result [56]. These singularity models encode crucial geometric data near the singularity region of the ﬂow.

In this section, we study the singularity formations on our projective bundle M = P(O ⊕ L) over Σ in the ﬁber-collapsing case. The ultimate goal is to show that the limit model obtained from the 1 above rescaling procedure is Cn × CP . Moreover, we want to rule out the possibility of Type II singularity using Perelman’s local non-collapsing theorem.

4.4.1 Splitting lemma

The ﬁrst preliminary result for classifying the limit model is a splitting lemma, which asserts that the n 1 limit model must split into a product of two manifolds N1 × N2 after passing to the universal cover. From the local expressions of g and g−1, one can easily derive local components of the Christoﬀel symbols which we will need for deriving our splitting result.

Lemma 4.8. The Christoffel symbols of the Kählermetric g on M constructed by momentum profile f are given by

i ξ fρρ ρξξ fρρ Γξξ = 0, Γξξ = ρξ + = − 1 ρξ, fρ ρξ fρ ξ fρρ fρ j fρ j Γiξ = − ρi, Γiξ = δi ρξ, fρ f f ξ fρρ 2fρ ρiρj 1 lk¯ k fρ k k k¯l Γij = − − ρ ρlρjki¯ + ρij , Γij = (ρiδj + ρjδi ) + ρ ρj¯li. fρ f ρξ ρξ f

i Remark 4.9. We will see that the vanishing of Γξξ is crucial when dealing with the curvature tensor in the blow-up analysis in the next section. CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 85

γ γδ¯ Proof. Using the formula Γαβ = g ∂αgβδ¯ for K¨ahlermanifolds, one can compute the Christoﬀel symbols directly:

i i¯j ∂ iξ¯ ∂ Γ = g g ¯ + g g ¯ ξξ ∂ξ ξj ∂ξ ξξ

1 i¯j ∂ 1 ik¯ ∂ = ρ (fρρξρ¯j) − ρ ρk¯ (fρρξρξ¯) f ∂ξ fρξ¯ ∂ξ

1 i¯j 1 ik¯ = ρ (fρρρξρξρ¯j + fρρξξρ¯j) − ρ ρk¯(fρρρξρξρξ¯ + fρρξξρξ¯) f fρξ¯ = 0. n ξ X ξ¯i ∂ ξξ¯ ∂ Γ = g g ¯ + g g ¯ ξξ ∂ξ ξi ∂ξ ξξ i=1 n n k¯l ! P ρ ρ ρ¯ 1 X k¯i ∂ 1 1 k,l=1 k l ∂ = − ρ ρk (fρρξρ¯i) + 2 + (fρρξρξ¯) fρξ ∂ξ |ρξ| fρ f ∂ξ k=1 n 1 X k¯i ∂ 2 = − ρ ρk (fρρρξρ¯i + fρρξξρ¯i) fρξ ∂ξ k=1 n k¯l ! P ρ ρ ρ¯ 1 1 k,l=1 k l 2 + 2 + (fρρρξρξ¯ + fρρξξρξ¯) |ρξ| fρ f fρρ ρξξ fρρ = ρξ + = − 1 ρξ. fρ ρξ fρ n ξ X ξ¯j ∂ ξξ¯ ∂ Γ = g g ¯ + g g ¯ iξ ∂z ξj ∂z ξξ j=1 i i n k¯l ! P ρ ρ ρ¯ 1 k¯j ∂ 1 1 k,l=1 k l ∂ = − ρ ρk (fρρξρ¯j) + 2 + (fρρξρξ¯) fρξ ∂zi |ρξ| fρ f ∂zi

1 k¯j = − ρ ρk(fρρρiρξρ¯j + fρρξρi¯j) fρξ Pn k¯l ! 1 1 k,l=1 ρ ρkρ¯l + 2 + (fρρρiρξρξ¯) |ρξ| fρ f fρρ fρ = − ρi. fρ f CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 86

n j X jk¯ ∂ jξ ∂ Γiξ = g gξk¯ + g gξξ¯ ∂zi ∂zi k=1

1 jk¯ ∂ 1 jk¯ ∂ = ρ (fρρξρk¯) − ρ ρk¯ (fρρξρξ¯) f ∂zi fρξ¯ ∂zi

1 jk¯ 1 jk¯ = ρ (fρρρiρξρk¯ + fρρξρik¯) − ρ ρk¯(fρρρiρξρξ¯) f fρξ¯ f = ρ δjρ . f i ξ n ξ X ξk¯ ∂ ξξ¯ ∂ Γij = g gjk¯ + g gjξ¯ ∂zi ∂zi k=1 k¯l ! 1 kl¯ ∂ 1 1 ρ ρkρ¯l ∂ = − ρ ρl (fρjk¯ + fρρjρk¯) + 2 + (fρρjρξ¯) fρξ ∂zi |ρξ| fρ f ∂zi

fρ lk¯ 1 lk¯ fρρ 1 = − ρ ρl(ρiρjk¯ + ρjρik¯) − ρ ρjki¯ ρl + ρiρ¯j + ρi¯j fρξ ρξ fρρξ ρξ fρρ 2fρ ρiρj 1 lk¯ = − − ρ ρlρjki¯ + ρij . fρ f ρξ ρξ n k X k¯l ∂ kξ¯ ∂ Γij = g gj¯l + g gjξ¯ ∂zi ∂zi k=1

1 k¯l = ρ (f ρ ρ ¯ + fρ ¯ + f ρ ρ ρ¯ + f ρ ρ¯ + f ρ ρ ¯) f ρ i jl jli ρρ i j l ρ ij l ρ j il

1 k¯l − ρ ρ¯l(fρρρiρjρξ¯ + fρρijρξ¯) fρξ¯

fρ k k k¯l = (ρ δ + ρ δ ) + ρ ρ ¯ . f i j j i jli

The splitting lemma is stated in Lemma 4.10 below. We will use the well-known de Rham’s holonomy splitting theorem, which asserts that if the tangent bundle TM∞ admits an irreducible Lk decomposition i=1 Ei under the holonomy group action, i.e. parallel translation, then the universal

Qk dim Ei dim Ei cover of M∞ splits isometrically as (M∞, g) = i=1 Ni with TNi = Ei. Note that in the

Kählercase where the holonomy group is a subgroup of the unitary group, each Ni is also Kähler. There are many fruitful applications of the de Rham’s splitting theorem on classification problems such as in [3, 40, 50] etc.

Lemma 4.10. Let M = P(OΣ ⊕ L) be the projective bundle such that the triple (ν, λ, ω0) satisfies the assumptions stated in p78 and ω0 is constructed by Calabi’s ansatz. Let (M, ωt), t ∈ [0,T ) be the Kähler-Ricci flow ∂tωt = −Ric(ωt) with initial Kählerclass [ω0]. Let (xi, ti) ∈ M × [0,T ) be a sequence such that ti → T and Ki := kRm(xi)kg(ti) → ∞ as i → ∞. Define gi(t) to be rescaled CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 87

dilated sequence by Ki and ti, i.e.

−1 gi(t) := Kig(ti + Ki t), t ∈ [−βi, αi]

where βi → ∞, αi ≥ 0 and αi → A ∈ [0, ∞]. Suppose the curvature tensor of gi(t), t ∈ [−βi, αi] is uniformly bounded independent of i, i.e. there exists C > 0 independent of i such that

sup kRmkgi(t) ≤ C. M×[−βi,αi]

n+1 Then, after passing to a subsequence, (M , gi(t), xi) converges smoothly in pointed Cheeger-

Gromov sense to a complete ancient Kähler-Ricci flow (M∞, g∞(t), x∞) whose universal cover is of the form n 1 (N1 × N2 , h1(t) ⊕ h2(t)), t ∈ (−∞,A] where (Ni, hi(t)), i = 1, 2, are Kähler-Ricci flow solutions.

Proof. By the uniform boundedness condition of kRmkgi(t) over M × [−βi, αi], the subsequential Cheeger-Gromov convergence can be done by Hamilton’s compactness theorem and Perelman’s local non-collapsing theorem. See [19, 42, 56], etc. From now on we suppress the time parameter t of gi(t) for simplicity. Furthermore, we may assume the complex structure of J of M converges after passing to a subsequence to a complex structure J∞ of M∞. That makes (M∞,J∞) K¨ahlerbecause

g∞ gi ∇ J∞ = limi→∞ ∇ J = 0.

Suppose (M∞, g∞(t), x∞) is the pointed Cheeger-Gromov limit obtained above. We would like to show it (precisely, the universal cover) splits isometrically into a product. According to the nature 1 of the collapsing of the CP -fiber, it is natural to guess that one factor of the split product should correspond to the base and the other should correspond to the fiber. Denote π : M → Σ to be the bundle map. We define the vertical and horizontal distributions as follows:

V = ker(π∗),

(i) ⊥ H = V gi ,

so that TM = V ⊕Hgi where the direct sum is with respect to gi. We consider the following operator CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 88

P (i) : V × V × H(i) deﬁned by

(i) (i) P (X,Y,Z) = h∇X Y,Zigi ,X,Y ∈ V and Z ∈ H

(i) where ∇ is the Levi-Civita connection with respect to gi. P measures how much Y deviates from Z when travelling along X. As Y ⊥ Z, it is easy to check that P (i) is tensorial in each slot. We will (i) ∞ show that asymptotically as i → ∞, P → 0 and so the limit vertical distribution V on (M∞, g∞) is invariant under parallel transport.

Let (z1, . . . , zn, ξ) be the holomorphic base-ﬁber local coordinates, the following unit vectors deﬁne a local basis of TM in the coordinate chart:

1 ∂ 1 ∂ ∂ˆ(i) = = ; ξ ∂ ∂ξ p 2 ∂ξ k ∂ξ kgi Kifρ|ρξ| ∂ ∂ ! 1 ∂ h , ¯igi ∂ ∂ˆ(i) = − ∂zj ∂ξ j h ∂ , ∂ i ∂ 2 ∂z ∂ξ¯ gi ∂zj k k ∂ξ ∂ j ∂ ∂ξ gi − ∂ ∂zj k k2 ∂ξ ∂ξ gi gi 1 ∂ ρj ∂ = p − Kifρj¯j ∂zj ρξ ∂ξ for j = 1, . . . , n. We have used the local expressions of g listed in p74. It is easily to see that ˆ(i) ˆ(i) (i) ˆ(i) ˆ(i) n V = span{∂ξ , ∂ξ } and H = span{∂j , ∂j }j=1.

(i) ˆ(i) ˆ(i) ˆ(i) D ˆ(i) ˆ(i)E P (∂ξ , ∂ξ , ∂j ) = ∇∂ˆ(i) ∂ξ , ∂j ξ gi 1 ∂ ˆ(i) = ∇ ∂ , ∂j K f |ρ |2 ∂ξ ∂ξ i ρ ξ gi 1 ∂ ∂ = Γξ + Γl , ∂ˆ(i) . K f |ρ |2 ξξ ∂ξ ξξ ∂z j i ρ ξ l gi

∂ ˆ(i) l As ∂ξ ⊥ ∂j with respect to gi and by Lemma 4.8 we have Γξξ = 0 for any l, we conclude

(i) ˆ(i) ˆ(i) ˆ(i) P (∂ξ , ∂ξ , ∂j ) = 0 for any j = 1, . . . , n.

l l Since Γξξ¯ = 0 and Γ¯jξ = 0, one can easily verify that

(i) ˆ(i) ˆ(i) ˆ(i) (i) ˆ(i) ˆ(i) ˆ(i) P (∂ξ , ∂ξ , ∂j ) = P (∂k , ∂ξ , ∂j ) = 0 for any k, j = 1, . . . , n. CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 89

(i) ˆ(i) ˆ(i) ˆ(i) Finally, we need to compute P (∂k , ∂ξ , ∂j ) for any k, j = 1, . . . , n:

(i) ˆ(i) ˆ(i) ˆ(i) D ˆ(i) ˆ(i)E P (∂k , ∂ξ , ∂j ) = ∇∂ˆ(i) ∂ξ , ∂j k gi 1 1 ∂ ˆ(i) = p · p ∇ ∂ − ρk ∂ , ∂j 2 ∂zk ρξ ∂ξ ∂ξ Kifρkk¯ Kifρ|ρξ| gi 1 1 ∂ ρ ∂ = · Γl − k Γl , ∂ˆ(i) . p p 2 kξ ∂z ρ ξξ ∂z j Kifρkk¯ Kifρ|ρξ| l ξ l gi

∂ ˆ(i) ∂ Here we have used the fact that ∂ξ ⊥ ∂j and so ∂ξ can be ignored when computing the connection l term. Note also that Γξξ = 0, we have

(i) ˆ(i) ˆ(i) ˆ(i) P (∂k , ∂ξ , ∂j ) 1 1 1 ∂ ∂ ρ¯ ∂ = · · Γl , − j p p 2 p kξ ¯ K fρ ¯ K f |ρ | Kifρρ ¯ ∂zl ∂z¯j ρ¯ ∂ξ i kk i ρ ξ jj ξ gi 1 1 1 fρ l ρ¯j = · · · δ ρξKi fρρlρ¯ + fρ ¯ − fρρlρ¯ p p 2 p k j lj ξ Kifρkk¯ Kifρ|ρξ| Kifρρj¯j f ρξ¯ 1 1 1 fρ = · · · ρξKifρ ¯ p p 2 p kj Kifρkk¯ Kifρ|ρξ| Kifρρj¯j f p p 1 ρξ fρ ρk¯j 1 |ξ| fρ ρk¯j = √ · · · √ √ = √ · · · √ √ . Ki |ρξ| f ρj¯j ρkk¯ Ki ξ f ρj¯j ρkk¯

√ fρ By Lemma 4.4, we have C−1 ≤ f ≤ C and f = O((T − t)), hence √ → 0 as i → ∞. ρ K f i −1 ti+Ki t ρ ¯ (i) (i) (i) Note that √ k√j is independent of i and ξ. Therefore, we have |P (i)(∂ˆ , ∂ˆ , ∂ˆ )| → 0 as i → ∞. ρj¯j ρkk¯ k ξ j

Furthermore, on Σ0 we have |ξ| = 0. By the asymptotic condition for f near Σ0 (see p72), ρ f = G(e ) for some smooth function G : [0, ∞) such that Gρ(0) > 0. Therefore, near Σ0 one has ρ ρ 2 ρ (i) ˆ(i) ˆ(i) ˆ(i) fρ = e Gρ(e ) = |ξ| h(z)Gρ(e ) and P (∂k , ∂ξ , ∂j ) can also be expressed as

2 p ρ ρ ¯ (i) ˆ(i) ˆ(i) ˆ(i) 1 |ξ| h(z)Gρ(e ) kj P (∂k , ∂ξ , ∂j ) = √ · · · √ √ Ki ξ f ρj¯j ρkk¯ ¯ p ρ ξ h(z)Gρ(e ) ρk¯j = √ · · √ √ Ki f ρj¯j ρkk¯

(i) ˆ(i) ˆ(i) ˆ(i) Letting |ξ| → 0 yields P (∂k , ∂ξ , ∂j ) = 0 on Σ0 for each i. At the same token, one can check (i) ˆ(i) ˆ(i) ˆ(i) P (∂k , ∂ξ , ∂j ) = 0 on Σ∞ for each i by the asymptotic condition of f near Σ∞. CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 90

We would like to remark that if we consider instead the holomorphic chart

1 z = z , . . . , z ; ζ := 1 n ξ

1 which covers the ∞ of the CP -fibers but not 0, then ρ = − log(|ζ|2h(z)) and hence all the above ∂ computations will only differ by a minus sign. The unit vectors along ∂ζ also span the same vertical distribution V. Moreover, since Σ is compact and one can cover it by a finite collection of bundle coordinate charts (z, ξ). One can check that the transitions between these charts over Σ do not affect the above computations. For example, suppose two bundle coordinate charts UZ :(z, ξ(z)) and Uw :(w, ξ(w)) are related by:

w = w(z)

ξ(w) = ψ(z) · ξ(z)

for some non-vanishing holomorphic functions ψ(z) on Uz ∩ Uw. By the chain rule, we have

∂ ∂w ∂ ∂ξ ∂ ∂ = · + (w) · = ψ(z) · ∂ξ(z) ∂ξ(z) ∂w ∂ξ(z) ∂ξ(w) ∂ξ(w) which gives the same unit vector ﬁelds on Uz ∩ Uw.

Finally, we will now show that when passing to the limit manifold M∞, the vertical distribution ∞ ∞ V is invariant under parallel transport with respect to g∞. To see this, pick V0 ∈ Vx , x ∈ M∞ and let V (s) ∈ TM be the parallel translation of V0 along a curve γ(s), i.e.

∇γ˙ V (s) = 0,V (0) = V0,

∞ where ∇ is the Levi-Civita connection of g∞. Write V (s) = VT (s) + V⊥(s) where VT (s) ∈ V and ∞ (i) (i) (i) (i) V⊥(s) ∈ H for any s. Take VT ∈ V ⊂ TM such that VT → VT and V⊥ ∈ H such that (i) (i) (i) V⊥ → V⊥ for any s. Also, we take a sequence of curves γ (s) ∈ M such that γ → γ. Since P (i) : V × V × H(i) is tensorial, the above computations and remarks show that

(i) (i) (i) (i) P (γ ˙ ,VT ,V⊥ ) → 0

and hence h∇γ˙ VT ,V⊥ig∞ = 0 for any s. CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 91

Since h∇γ˙ (VT + V⊥),V⊥ig∞ = h∇γ˙ V,V⊥ig∞ = 0, we must have h∇γ˙ V⊥,V⊥ig∞ = 0 and so

d kV k2 = 2h∇ V ,V i = 0 ds ⊥ γ˙ ⊥ ⊥ g∞

∞ which implies kV⊥(s)kg∞ = kV⊥(0)kg∞ = 0 for any s since V (0) ∈ V . In other words, V (s) ∈ V∞ for any s. Therefore, V∞ is invariant under parallel transport. By the de Rham’s decomposition theorem, our splitting lemma follows.

4.4.2 Analysis of kRmk

The splitting lemma in the previous subsection allows a dimension reduction for our singularity analysis. The ultimate goal of this subsection is to analyze the singularity formation of the Ricci 1 flow on our projective bundles M = P(OΣ ⊕ L) whose CP -fiber collapses near the singularity. We are going to prove that the Kähler-Ricci flow (M, g(t)) must be of Type I (see definition below) and n 1 the singularity model is C × CP , in a sense that one can choose a sequence (xi, ti) in space-time in the high curvature region such that the universal cover of the Cheeger-Gromov limit of the rescaled n 1 dilated sequence is isometric to (C × CP , δ ⊕ ωFS(t)). Here ωFS(t) is the shrinking Fubini-Study metric. The Riemann curvature kRmk must blow up under a Ricci flow with finite time singularity by Theorem 2.8. It is crucial to classifying singularity types according to the blow-up rate of curvatures. In the literature of the Ricci flow, it is accustomed to divide finite time singularities into two types stated below (see [42]).

Definition 4.11. Let (M, g(t)) be a Ricci flow solution ∂tg(t) = −Ric(g(t)) on a closed manifold M which becomes singular at a finite time T . We call the Ricci flow encounters

• Type I singularity if supM×[0,T )(T − t)kRmkg(t) < ∞

• Type II singularity if supM×[0,T )(T − t)kRmkg(t) = ∞.

We would like to remark that although this classification of finite-time singularity was proposed in the early 90’s, surprisingly the first compact Type II solution was constructed by Gu-Zhu in [38] only recently in 2007. In order to understand the singularity formation, we need to bring curvatures into the topic. Therefore, we will compute and analyze the Riemann curvature tensor of our projective bundle CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 92

M which is equipped with momentum proﬁle f. Recall that for K¨ahlermanifolds, the Riemann curvature (3, 1)-tensor can be computed using the formula

∂ RD = − ΓD ABC¯ ∂z¯B AC where A, B, C, D = 1, . . . , n or ξ. The non-zero components of the Riemann curvature tensor are given below.

l lp¯ Ri¯jk = −(log f)ρρρ¯j(ρiδkl + ρkδil) − (log f)ρ(δijδkl + δjkδil) − (ρ ρikp¯)¯j. l Riξk¯ = −(log f)ρρρξ¯(ρiδkl + ρkδil), l Ri¯jξ = −(log f)ρρρ¯jρξδil,

ξ 1 Ri¯jk = − (log fρ − 2 log f)ρρρ¯jρiρk, ρξ

1 1 lp¯ − (log fρ − 2 log f)ρ(ρi¯jρk + ρk¯jρi) + (ρ ρlρipk¯ + ρik)¯j, ρξ ρξ l Rξ¯jk = −(log f)ρρρ¯jρξδkl. ξ ρξ¯ Riξk¯ = −(log fρ − 2 log f)ρρ ρiρk, ρξ l 2 Riξξ¯ = −(log f)ρρ|ρξ| δik, l 2 Rξξk¯ = −(log f)ρρ|ρξ| δkl, l Rξ¯jξ = 0, ξ Rξ¯jk = −(log fρ − log f)ρρρ¯jρk − (log fρ − log f)ρρk¯j, ξ Rl¯jξ = −(log fρ − log f)ρρρ¯jρl − (log fρ − log f)ρρl¯j. ξ Rlξξ¯ = −(log fρ − log f)ρρρξ¯ρi, l Rξξξ¯ = 0, ξ Rξξk¯ = −(log fρ − log f)ρρρξ¯ρk, ξ Rξ¯jξ = −(log fρ)ρρρ¯jρξ. ξ 2 Rξξξ¯ = −(log fρ)ρρ|ρξ| .

Since the understanding of kRmk is crucial in analyzing the singularity according their type (I or II), we need an organized expression of kRmk that is written in terms of our momentum proﬁle f. For the purpose of our study of the singularity models, it suﬃces to understand the asymptotics of CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 93

kRmk2 in terms of f and its derivatives. 1 Recall from Lemma 4.4 that f = O(1), f = O(1), fρρ/fρ = O(1). Therefore we have the following asymptotics

f (log f) = ρ = O(f ), ρ f ρ f f 2 (log f) = ρρ − ρ = O(f ), ρρ f f 2 ρ fρρ (log fρ)ρ = = O(1). fρ

The asymptotic of (log fρ)ρρ is not yet known because it involves the third ρ-derivative of f which we have not derived. Also, the local expressions of g and g−1 have the following asymptotics

gi¯j = O(1),

giξ¯ = g¯iξ = gξξ¯ = O(fρ), ¯ gi¯j = giξ = g¯iξ = O(1),

ξξ¯ −1 g = O(fρ ).

We claim that the norm kRmk2 can be expressed in the following asymptotic form

Lemma 4.12.

2 −2 2 −1 kRmkg(t) = fρ (log fρ)ρρ + O(fρ (log fρ)ρρ) (4.4.1)

−1 2 2 + O(fρ (log fρ)ρρ) + O((log fρ)ρρ)

+ O((log fρ)ρρ) + O(1).

Proof. A generic term in kRmk2 can be expressed as

CD¯ EF¯ GH¯ A B gAB¯ g g g RCFG¯ RDEH¯ (**)

−1 where A, . . . , H ∈ {1, . . . , n, ξ}. From Lemma 4.4, we know fρ = O(T − t) and so fρ is a bad term −1 ξξ¯ as it diverges as t → T . The only factor in (**) which can contribute to a fρ is g , and there are at most three gξξ¯’s in (**). We are going to check that CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 94

−1 (1) whenever fρ appears in (**) exactly once, there must at least one factor of (log fρ)ρρ from the curvature components; −2 2 (2) whenever fρ appears in (**), there must be a (log fρ)ρρ factor from the curvature components; −3 (3) it is impossible for fρ to appear in (**).

Combining these, it is not diﬃcult to see kRmk2 satisﬁes the asymptotic form (4.4.1). −1 We start by arguing (1). Suppose there is exactly one fρ factor in (**), we can assume without loss of generality that either (C,D) = (ξ, ξ) or (E,F ) = (ξ, ξ). Suppose the former, we can check A from the table of Riemann curvatures in p92 that almost all RξFG¯ terms have either asymptotics −1 ξ O(fρ) (which cancels out fρ ) or a (log fρ)ρρ factor. There is only one exception: Rξ¯jk which has A B an O(1)-term from (log fρ)ρ. However, if both of RCFG¯ and RDEH¯ are taken to be in this form, then (**) becomes ξξ¯ p¯j kq¯ ξ ξ gξξ¯g g g Rξ¯jkRξpq¯ ,

−1 −1 where the gξξ¯ = O(fρ) cancels out the undesirable fρ factor, and end up with no fρ at all. Similar argument applies to the case (E,F ) = (ξ, ξ), and (1) is proved. ξξ¯ −1 For (2), since g is the only possible contribution to fρ , at least two of C,F,G (and their corresponding two of D,E,G) must be ξ. Check again the table of Riemann curvature components in p92, we see all the terms with two lower ξ-indexes must either of O(fρ)-type or has a (log fρ)ρρ factor. It proves (2). −3 For (3), the only possible case for fρ to appear is that all of (C,D), (E,F ) and (G, H) are (ξ, ξ). l ξ The only possible choice for the curvature components are Rξξξ¯ and Rξξξ¯ . However, the former is 0. For the latter case, all indexes will be ξ and (**) becomes

ξξ¯ ξξ¯ ξξ¯ ξ ξ gξξ¯g g g Rξξξ¯ Rξξξ¯

−2 2 which can be computed easily as fρ (log fρ)ρρ. ξξ¯ ξξ¯ ξξ¯ ξ ξ −2 2 Finally, we remark that gξξ¯g g g Rξξξ¯ Rξξξ¯ is the only term that fρ (log fρ)ρρ appears, thanks i −2 2 to the fact that Rξξξ¯ = 0. As a result, the leading term of (4.4.1) is fρ (log fρ)ρρ with coeﬃcient 1 which can be easily veriﬁed by computing

ξξ¯ ξξ¯ ξξ¯ ξ ξ gξξ¯g g g Rξξξ¯ Rξξξ¯ . CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 95

4.4.3 Type I singularity

Having understood the asymptotics of kRmk2, we are in a position to study the singularity models. We ﬁrst consider the Type I case:

Theorem 4.13 (F [30]; Theorem 1.10 of Chapter 1). Let M = P(OΣ ⊕ L) be the projective bundle with the triple (Σ, L, [ω0]) satisfying the conditions listed in p77 and with ω0 constructed by Calabi’s ansatz. Let (M, ωt) be the Kähler-Ricci flow ∂tωt = −Ric(ωt), t ∈ [0,T ) with initial Kählerclass

[ω0]. Suppose the ﬂow encounters Type I singularity, then choose (xi, ti) in space-time such that

Ki := kRm(xi, ti)kg(ti) = maxM kRmkg(ti) and ti → T . Consider the rescaled dilated sequence of metrics −1 gi(t) := Kig(ti + Ki t), t ∈ [−tiKi, (T − ti)Ki).

Then the pointed sequence (M, gi(t), xi) converges, after passing to a subsequence, smoothly in pointed Cheeger-Gromov sense to an ancient κ-solution (M∞, g∞(t), x∞), whose universal cover splits isometrically as n 1 n (C × CP , δC ⊕ (1 − t)ωFS),

n where δC is the Euclidean metric and ωFS denotes the Fubini-Study metric.

Proof. Suppose C = C(n) is a constant depending only on n such that |R| ≤ C(n)kRmk. Since the blow-up factor Ki is deﬁned by Ki = maxM kRmkg(ti) = kRm(xi)kg(ti), the scalar curvature at time ti satisﬁes |Rg(ti)| ≤ CKi on M. One can compute the scalar curvature explicitly:

Rg(t) = Trωt Ric(ωt) n(ν − F ) F = ρ − ρρ , f fρ where F = log fρ + n log f. Hence,

1 Rg(t) = − (log fρ)ρρ + O(1). fρ CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 96

Therefore, for any ρ ∈ [−∞, ∞] at ti, we have

1 − (log fρ)ρρ + O(1) ≤ CKi, fρ

−1 −1 (log fρ)ρρ + O(Ki ) ≤ C. Kifρ

Recall that Ki → ∞. Letting i → ∞ yields

lim sup K−1f −1(log f ) ≤ C. (4.4.2) i ρ ρ ρρ (ρ,t ) i→∞ i

By considering the asymptotic expression of kRmk2 given by (4.4.1), we have for any ρ ∈ [−∞, ∞] at time ti,

1 ≥ K−2kRmk2 = (K f )−2(log f )2 + O(K−2f −1(log f )2 ) i g(ti) i ρ ρ ρρ i ρ ρ ρρ −2 −1 −2 2 + O(Ki fρ (log fρ)ρρ) + O(Ki (log fρ)ρρ) −2 −2 + O(Ki (log fρ)ρρ) + O(Ki ),

where equality is achieved at xi.

Letting i → ∞ and using (4.4.2) and the fact that fρ = O(T − t) from Lemma 4.4, we can deduce:

−2 2 lim sup(Kifρ) (log fρ)ρρ ≤ 1, ρ ∈ [−∞, ∞], t = ti, i→∞ −2 2 lim (Kifρ) (log fρ)ρρ = 1. (4.4.3) i→∞ (xi,ti)

−1 Recall that gi(t) = Kig(ti + Ki t), we then have

1 −1 Rgi(t) = − (log fρ)ρρ + O(Ki ) . Kifρ −1 ti+Ki t

Letting i → ∞, we have

1 Rg∞(t) = − lim (log fρ)ρρ . (4.4.4) i→∞ Kifρ −1 ti+Ki t By strong maximum principle, the scalar curvature of every ancient solution must be either identically zero or everywhere positive. In our case, (4.4.3) and (4.4.4) together implies Rg∞(0) = 1 and hence Rg∞(t) > 0 on M × (−∞, 0]. By our splitting lemma 4.10, we know that the limit manifold CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 97

n 1 n ∞ 1 ∞ M∞ splits isometrically as a product N1 × N2 , such that TN1 = H and TN2 = V (see the deﬁnitions of V’s and H’s in the proof of Lemma 4.10). As a result, the curvature tensors also split as Rm = Rm n ⊕ Ric 1 . Next, we would like to compute the curvatures of each factor. M∞ N1 N2

First, for each bundle coordinate chart (z1, . . . , zn, ξ) on M, deﬁne unit vector ﬁelds

1 ∂ 1 ∂ Zˆ(i) = = , j ∂ p 2 k kg ∂zj Ki(fρ ¯ + fρ|ρj| ) ∂zj ∂zj i jj 1 ∂ 1 ∂ ∂ˆ(i) = = . ξ ∂ ∂ξ p 2 ∂ξ k ∂ξ kgi Kifρ|ρξ|

ˆ(i) ˆ(i) (i) Note that ∂ξ is the same as in Lemma 4.10 but Zj is no longer in H . However, one can show pf |ξ| hZˆ(i), ∂ˆ(i)i = ρ · ρ j ξ gi p 2 j fρj¯j + fρ|ρj| ξ ˆ(i) ˆ(i) and by Lemma 4.4 we have fρ = O((T − t)) and so |hZj , ∂ξ igi | → 0 as i → ∞. Moreover, using ˆ(i) ˆ(i) the asymptotic condition (p72) of f, one has fρ = 0 on Σ0 and Σ∞, and hence hZj , ∂ξ igi = 0 on ˆ(i) ˆ∞ ∞ Σ0 ∪ Σ∞. These imply when passing to a subsequence, we have Zj → Zj ∈ H as i → ∞. We compute

q ξ Rjkl¯ p¯ = Rjkl¯ gqp¯ + Rjkl¯ gξp¯ 1 = O(1)O(1) + O(1) · fρρξρp = O(1) ρξ and hence

ˆ(i) ˆ(i) ˆ(i) ˆ(i) |Rmgi(t)(Zj , Zk , Zl , Zp )| (4.4.5) (i) (i) (i) (i) = |KihRm(Zˆ , Zˆ )Zˆ , Zˆp i −1 | j k l g(ti+Ki t)

1 4 1 1 1 1 √ ≤ Ki p p p p Rjkl¯ p¯ Ki fρj¯j fρkk¯ fρl¯l fρpp¯ 1 = O(1) → 0 as i → ∞. Ki CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 98

Hence Rm n = 0. Similarly, we have N1

1 1 1 Ric (∂ˆ(i), ∂ˆ(i)) = (−(n log f + log f ) |ρ |2) (4.4.6) gi(t) ξ ξ p p 2 ρ ρρ ξ Kifρ Kifρ |ρξ|

1 −1 = − (log fρ)ρρ + O(Ki ). Kifρ

ˆ∞ ˆ∞ By (4.4.4) and the positivity of Rg∞(t), we know that Ricg∞(t)(∂ξ , ∂ξ ) > 0. Since the Kähler-Ricci flow g(t) is of Type I, the ancient solution obtained by the blow-up sequence is also of Type I, i.e. supM×(−∞,0] |t|kRmkg∞(t) < ∞, and is κ-non-collapsed. The limit n 1 n solution splits as a product (N1 , h1(t)) × (N2 , h2(t)) of complete manifolds which we know N1 is 1 flat and N2 has positive curvature. According to Hamilton’s classification of ancient κ-solution [42] 1 1 (see also [19]), (N2 , h2(t)) must be the shrinking round 2-sphere, which is equivalent to CP having a shrinking Fubini-Study metric. It completes the proof of the theorem.

4.4.4 Type II singularity

Next, we will rule out the possibility of Type II singularity on (M, g(t)). We will show that by a standard point-picking argument for Type II singularity, one can form a rescaled dilated sequence of metrics which converges, after passing to a subsequence, to a product of the cigar soliton and a ﬂat factor. By Perelman’s local non-collapsing result, such limit model is not possible.

Theorem 4.14 (F [30]; Theorem 1.11 of Chapter 1). Let M = P(OΣ ⊕ L) be the projective bundle with the triple (Σ, L, [ω0]) satisfying the conditions listed in p77 and ω0 is constructed by Calabi’s ansatz. Let (M, ωt) be the Kähler-Ricci flow ∂tωt = −Ric(ωt), t ∈ [0,T ) with initial Kählerclass

[ω0]. Then (M, g(t)) must be of Type I, i.e. Type II singularity is not possible.

Proof. First take an increasing sequence Ti → T . Let (xi, ti) ∈ M × [0,Ti] be such that

(Ti − ti)kRmk(xi, ti) = max (Ti − t)kRmkg(t) M×[0,Ti]

−1 = max (Ti − (ti + K t))kRmk −1 . i g(ti+Ki t) M×[−Kiti,Ki(Ti−ti)]

We denote Ki = kRmk(ρi, ti), then Ki(Ti − ti) → ∞ by the Type II condition.

As in the Type I case, we let C = C(n) be a constant depending only n such that |Rg(t)| ≤ CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 99

CkRmkg(t). Recall that scalar curvature has the following asymptotic expression:

1 Rg(t) = − (log fρ)ρρ + O(1). fρ

Hence for any ρ ∈ [−∞, ∞], t ∈ [0,Ti], we have

1 C(Ti − ti)Ki − (log fρ)ρρ + O(1) ≤ −1 , fρ Ti − (ti + Ki t)

−1 −1 C(Ti − ti) (log fρ)ρρ + O(Ki ) ≤ −1 , Kifρ Ti − (ti + Ki t)

−1 where we evaluate the left-hand side at ti + Ki t. Letting i → ∞, and using the fact that (T −t )−K−1t i i i = 1 − t → 1, one can show Ti−ti Ki(Ti−ti)

−1 lim sup | (Kifρ) (log fρ)ρρ (x,t +K−1t) ≤ 1 for any (x, t). (4.4.7) i→∞ i i

2 2 2 2 At (xi, ti) we have (Ti − ti) kRmk (xi, ti) = (Ti − ti) Ki . Consider the asymptotic expression of kRmk2 as in the Type I case, one can then show

1 2 lim (log fρ)ρρ = 1. (4.4.8) i→∞ K2f 2 i ρ (xi,ti)

As Ki → ∞, our splitting lemma 4.10 also implies the limit solution (M∞, g∞(t)) splits isomet- n 1 rically as a product (N × N , h (t) ⊕ h (t)). As in the Type I case, Rm n and Ric 1 can be found 1 2 1 2 N1 N2 by (4.4.5) and (4.4.6):

ˆ(i) ˆ(i) ˆ(i) ˆ(i) 1 Rmgi(t)(Zj , Zk , Zl , Zp ) = O(1), Ki ˆ∞ ˆ∞ 1 −1 Ricgi(t)(∂ξ , ∂ξ ) = − (log fρ)ρρ + O(Ki ). Kifρ

Letting i → ∞, we have Rm n (h (t)) = 0 and by (4.4.7) and (4.4.8), we have N1 1

1 ≥ Ric 1 (h (t)) > 0, (4.4.9) N2 2

Ric 1 (x , h (0)) = h (0). N2 ∞ 2 2

(M∞, g∞(t)) is an eternal solution to the K¨ahler-Ricciﬂow since we have (Ti − ti)Ki → ∞. 1 By our splitting lemma, so does (N2 , h2(t)). From (4.4.9), the space-time maximum of the scalar CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 100

1 curvature of (N2 , h2(t)) is achieved at (x∞, 0). Hence by Hamilton’s classiﬁcation of eternal solutions 1 (see the Main Theorem of [41]), (N2 , h2(t)) is a steady gradient soliton. In case of dimR = 2, it must be the cigar soliton (see Section 26.3 of [42]). However, by Perelman’s local non-collapsing [56], the

Cheeger-Gromov limit (M∞, g(t)) must be κ-non-collapsed at all scales, and so the product of cigar soliton and a ﬂat space is not a possible singularity model. It leads to a contradiction and hence completes our proof.

Remark 4.15. Throughout this chapter we have focused on Case 1 and Case 2(i) in p77. We would like to point out as a ﬁnal remark that for Case 2(iii) we expect one could mimic Section 5.2 in [77] and also [78, 79] to show the contraction of Σ0 near the singular time. For singularity models n obtained by rescaling analysis in Case 2(iii), it was conjectured in [28] that for (Σ, ωΣ) = (CP , ωFS) n the singularity should be modelled on K¨ahler-Riccisolitons on O(−k)-bundles over CP . It was recently shown by Song in [71] that this conjecture is true when k = 1. List of symbols

∧ Wedge product ⊕ Direct sum ⊗ Tensor product ∼ Equivalence relation ' Uniform equivalence ≡ Biholomorphic ∆ Laplacian ∇ Levi-Civita connection, gradient ∂ ∂z ∂z C Set of complex numbers Cn Complex Euclidean space of complex dimension n n CP Complex projective space of complex dimension n c1(M) First Chern class of M C∞(M, R) Smooth functions from M to R diam Diameter dimC Complex dimension dimR Real dimension idX Identity map on X inf Greatest lower bound

P Projectivization

OΣ Trivial line bundle over Σ N Set of natural numbers, excluding 0 π∗ Pull-back by the map π Rm Riemann curvature tensor

101 CHAPTER 4. SINGULARITY MODEL OF PROJECTIVE BUNDLES 102

kRmk Norm of Riemann curvature tensor Ric Ricci curvature tensor or Ricci form R Scalar curvature

R Set of real numbers Q Set of rational numbers sup Least upper bound −1 Trωη Trace of the endomorphism ω η Vol Volume Bibliography

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