Lecture Notes in Mathematics 2166 CIME Foundation Subseries Michel Boileau Gerard Besson Carlo Sinestrari Gang Tian Ricci Flow and Geometric Applications Cetraro, Italy 2010 Riccardo Benedetti Carlo Mantegazza Editors Lecture Notes in Mathematics Editors-in-Chief: J.-M Morel, Cachan B Teissier, Paris Advisory Board: Camillo De Lellis, Zurich Mario di Bernardo, Bristol Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, Paris and New York Catharina Stroppel, Bonn Anna Wienhard, Heidelberg More information about this series at http://www.springer.com/series/304 2166 Michel Boileau • Gerard Besson • Carlo Sinestrari • Gang Tian Ricci Flow and Geometric Applications Cetraro, Italy 2010 Riccardo Benedetti, Carlo Mantegazza Editors 123 Authors Michel Boileau Aix-Marseille Université, CNRS, Central Marseille Institut de Mathematiques de Marseille Marseille, France Carlo Sinestrari Dip di Ingegneria Civile e Ingegneria Informatica Università di Roma “Tor Vergata” Rome, Italy Editors Riccardo Benedetti Department of Mathematics University of Pisa Pisa, Italy ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-42350-0 DOI 10.1007/978-3-319-42351-7 Gerard Besson Institut Fourier Université Grenoble Alpes Grenoble, France Gang Tian Princeton University Princeton, NJ USA Carlo Mantegazza Department of Mathematics University of Naples Naples, Italy ISSN 1617-9692 (electronic) ISBN 978-3-319-42351-7 (eBook) Library of Congress Control Number: 2016951889 Mathematics Subject Classification (2010): 53C44, 57M50, 57M40 © Springer International Publishing Switzerland 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Preface Our aim in organizing this CIME course was to present to young students and researchers the impressive recent achievements in differential geometry and topology obtained by means of techniques based on the Ricci flow We then invited some of the leading researchers in the field of geometric analysis and low-dimensional geometry/topology to introduce some of the central ideas in their work Here is the list of speakers together with the titles of their lectures: • Gérard Besson (Grenoble) – The differentiable sphere theorem (after S Brendle and R Schoen) • Michel Boileau (Toulouse) – Thick/thin decomposition of three-manifolds and the geometrization conjecture • Carlo Sinestrari (Roma “Tor Vergata”) – Singularities of three-dimensional Ricci flows • Gang Tian (Princeton) – Kähler–Ricci flow and geometric applications The summer school had around 50 international attendees (mostly PhD students and postdocs) Even though they were sometimes technically heavy, the lectures were followed by all the students with interest The participants were very satisfied by the high quality of the courses The not-so-intense scheduling of the lectures gave the students many opportunities to interact with the speakers, who were always very friendly and available for discussion It should be mentioned that the wonderful location and the careful CIME organization were also greatly appreciated We think that the fast-growing field of geometric flows and more generally of geometric analysis, which has always received great attention in the international community, but which is still relatively “young” in Italy, will benefit from its diffusion by this CIME course We briefly describe the contents of the lectures collected in this volume Gérard Besson presented the impact of the Ricci flow technique on the theory of positively curved manifolds, the central result being the differentiable 1/4-pinched sphere theorem, proved by Brendle and Schoen It says that a complete, simply v vi Preface connected Riemannian manifold whose sectional curvature varies in 1=4; 1 is diffeomorphic to the standard sphere The problem was first proposed by H Hopf, and then in 1951, H.E Rauch showed that a complete Riemannian manifold whose sectional curvature is positive and varies between two numbers whose ratio is close to has a universal cover homeomorphic to a sphere In the 1960s, M Berger and W Klingenberg obtained the optimal result: a simply connected Riemannian manifold which is strictly 1=4pinched is homeomorphic to the sphere The analogous diffeomorphic conclusion remained open until S Brendle and R Schoen proved the following: Theorem (S Brendle and R Schoen, 2008) Let M be a pointwise strictly 1=4pinched Riemannian manifold of positive sectional curvature Then M carries a metric of constant sectional curvature Hence, it is diffeomorphic to the quotient of a sphere by a finite subgroup of O.n/ The proof relies on the use of the Ricci flow introduced by R Hamilton and culminating in the work of G Perelman The idea is to construct a deformation of the Riemannian metric, evolving it by means of the Ricci flow toward a constant curvature metric We recall that this was the method that R Hamilton used in his seminal paper, proving the following theorem: Theorem (R Hamilton, 1982) Let M be a closed 3-dimensional Riemannian manifold which carries a metric of positive Ricci curvature; then it also carries a metric of positive constant curvature The lectures also focus on the extension to higher dimensions of the following result, due to C Böhm and B Wilking Recall that a curvature operator is 2-positive if the sum of its two smallest eigenvalues is positive Theorem (C Böhm and B Wilking, 2008) Let M be a closed Riemannian manifold whose curvature operator is 2-positive; then M carries a constant curvature metric In the lectures, the connection between this method and the algebraic properties of the Riemann curvature operator is stressed, the main focus being the identification of those properties of the curvature operator which are preserved under the Ricci flow In his lectures, Michel Boileau gave an introduction to the geometrization of 3-manifolds Sections 2.1 and 2.2 cover Thurston’s classification of the eight 3dimensional geometries and the characterization of geometric (and Seifert) closed 3-manifolds in terms of basic topological properties This follows by combining Thurston’s hyperbolization theorems (in particular the characterization of hyperbolic 3-manifolds that are fibered over S1 ), Perelman’s general geometrization theorem, and Agol’s recent (2013) proof of a deep conjecture of Thurston that closed hyperbolic 3-manifolds are “virtually fibered.” Section 2.3 discusses the following: (1) A central result of classical 3dimensional geometric topology, that is, the canonical decomposition of a Preface vii 3-manifold by splitting it along spheres and tori (2) Thurston’s geometrization conjecture This roughly says that every piece of a canonical decomposition is geometric together with a prediction on the geometry carried by the piece in terms of basic topological properties It includes as a particular case the celebrated Poincaré conjecture (3) Thurston’s fundamental hyperbolization theorem for Haken manifolds Perelman’s proof of the general geometrization theorem deals with all of these topics and also allows us to recover, as a by-product, the canonical decomposition itself This is done by completing the program based on the Ricci flow with surgeries, first proposed by R Hamilton This is the subject of Boileau’s notes from Sect 2.4 Since the appearance of Perelman’s three celebrated preprints, several simplifications and variants of the original proofs have been developed by various authors At the end of the day, we can say that the Poincaré conjecture (i.e., the case when the Ricci flow with surgery becomes extinct in finite time) is in a sense the “simplest” case The general case (when the Ricci flow with surgery exists at all times, which includes the complete hyperbolization theorem) requires nontrivial extra arguments, in particular, to obtain a key non-collapsing theorem In Perelman’s original work, these come from the theory of Alexandrov spaces Bessières, Besson, Boileau, Maillot, and Porti developed instead an alternative approach where the basic tools are Thurston’s hyperbolization theorem for Haken manifolds and some well-established properties of Gromov’s simplicial volume, allowing one to bypass the need for the (somewhat more exotic) theory of Alexandrov spaces Boileau’s notes are largely based on the monograph by L Bessières, G Besson, M Boileau, S Maillot, and J Porti, Geometrisation of 3-Manifolds, EMS Tracts in Mathematics 13, 2010 In this tract, the authors developed a slightly different notion of surgery by defining the so-called Ricci flow with bubbling-off Actually, one might roughly say that the Ricci flow with bubbling-off reduces the general hyperbolization theorem to Thurston’s hyperbolization theorem for Haken manifolds Carlo Sinestrari provided an extensive introduction to the Ricci flow by first giving a survey of the basic results and examples, then concentrating on the analysis of the singularities of the flow in the three-dimensional case, which is needed in Hamilton and Perelman’s surgery construction After reviewing the properties of the Ricci flow and the fundamental estimates of the theory, such as Hamilton’s Harnack differential inequality, the Hamilton–Ivey pinching estimate, and Perelman’s no collapsing result, he presented Perelman’s analysis of kappa-solutions and the canonical neighborhood property which gives a full description of the singular behavior of the solutions in dimension All these results are central to the proof of the Poincaré and geometrization conjectures The exposition is quite accessible to nonexperts Indeed, the presentation is often informal, and the proofs are omitted except in some simple and significant cases, focusing more on the description of the results and their applications and consequences A final detailed bibliographical section gives to the interested reader all the references needed for an advanced study of these topics viii Preface Gang Tian’s expository notes, based on his lectures, discuss some aspects of the Analytic Minimal Model Program through the Kähler–Ricci flow, developed in collaboration with other authors, particularly, J Song and Z Zhang Very stimulating open problems and conjectures are also presented Section 4.2 contains a detailed account of the sharp version of the Hamilton– DeTurck local existence theorem, which holds in the Kähler case: the maximal time Tmax 0; C1 such that the flow exists on the interval Œ0; Tmax / is precisely determined in terms of a cohomological property of the initial Kähler metric As a corollary, one deduces that Tmax D C1 for every initial metric on a compact Kähler manifold with a numerically positive canonical bundle In Sect 4.3, the limit singularities that can arise when t ! Tmax < C1 are analyzed After having established a general convergence theorem (Theorem 4.3.1), one faces questions concerning regularity and the geometric properties of the limit A combination of partial results (particularly in the case of projective varieties and when one can apply deep results of algebraic geometry) and well-motivated conjectures outlines a pregnant scenario Sections 4.4 and 4.5 discuss the construction of a Kähler–Ricci flow with surgery (assuming the truth of a conjecture stated in Sect 4.3) and its asymptotic behavior Numerous conjectures arise throughout this discussion such as: the characterization (up to birational isomorphism) of “Fano-like” manifolds as those whose flow becomes extinct at a finite time; the characterization of uniruled manifolds (up to birational isomorphism) as those whose flow collapses in finite time; and the existence of a flow with surgery globally defined in time and with only finitely many surgery times In Sect 4.6, algebraic surfaces are considered, showing how most of the program is carried out in this case We are pleased to express our thanks to the speakers for their excellent lectures and to the participants for contributing with their enthusiasm to the success of the Summer School The speakers, the participants, and the CIME organizers collectively created a stimulating, rich, pleasant, and friendly atmosphere at Cetraro For this reason, we would finally like to thank the Scientific Committee of CIME and, in particular, Pietro Zecca and Elvira Mascolo Pisa, Italy Naples, Italy Riccardo Benedetti Carlo Mantegazza Acknowledgements CIME activity is carried out with the collaboration and financial support of: INdAM (Istituto Nazionale di Alta Matematica) and MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca) ix 124 G Tian where !N T is the pull-forward Q T !Q T on Y and f W Y 7! M is the developing map (possibly with singularities6 which maps generic point of Y to the corresponding fiber in the moduli space M of Kähler-Ricci solitons on generic fibers, and !WP is the L2 -metric, i.e., the Weil-Peterson metric, on the moduli M Obviously, it remains to prove that (4.28) is the limit of the Ricci flow !Q t (t < T) in a suitable sense One can use this modified Ricci flow to extend !Q t across T One of advantages for using this flow instead of the Kähler-Ricci flow is that it encodes more information of the original X through the developing map For simplicity, we will just use the Ricci flow to extend solution across the singular time T in our program It should not be difficult to prove a local existence theorem on the stationary solutions of (4.28) This is amount to solving a complex Monge-Ampere equation and analogous to the generalized Kähler-Einstein metrics studied in [15, 16] In fact, our discussions are inspired by those joint works 4.4 Extending Kähler-Ricci Flow Across Singular Time In this section, we discuss how to extend the Kähler-Ricci flow !Q t across the singular time T, assuming that we have solved Conjecture 4.3.7 proposed in last section Then we have a Kähler variety XT and a Gromov-Hausdorff limit !N T on XT which is smooth outside a subvariety B XT A natural question is how to continue the Kähler-Ricci flow on XT starting at !N T There are two difficulties: XT may not be smooth, so we need to study the Kähler-Ricci flow on a singular space It is not even clear whether or not we can make sense of the Kähler-Ricci flow in a useful way Even if XT is smooth, !N T may not be smooth, so we need to solve the KählerRicci flow with weak initial value !N T In short, we need a local existence theorem for (4.1) when the underlying space may be singular or initial Kähler metric is not smooth First we assume that XT is smooth Then we can choose the Ricci representative in setting up (4.2) such that !T is the pull-back of a smooth Kähler metric on XT , still denoted by !T for simplicity As we did in Sect p 4.2, we can reduce (4.1) N uT for some Kähler on XT to a scalar flow: First we write !N T D !T0 C @@N 0 metric !T Choose a (1,1)-form D Ric.˝ / for a volume form ˝ on XT Define 0 !t D !T t T/ for t Ä T, then (4.1) is reduced to a variant of (4.2): p N n @u !tCT C @@u D log ; lim u.t; / D uN T : t!0 @t ˝0 (4.29) The singularities are caused by those singular fibers, but the developing map should be meromorphic Notes on Kähler-Ricci Flow 125 If (4.29) has a solution u ; t/ for t 0; T /, then !Q TCt extends previous KählerRicci flow !Q t (t Œ0; T/) to T; T C T /, where !Q TCt D !TCt C p N ; t/: @@u Moreover, we have limt!T !Q t D !Q T and limt!TC !Q t D !N T in the sense of distribution It follows p from the potential theory that for any bounded function ' such that N !' D !00 C @@' as currents, one can have an associated volume form n !' In particular, we have a volume form !N Tn A direct computation shows that the function !N n =˝ is Lp -integrable for some p > Thus, the solvability of (4.29) for some T > is provided by the following Theorem 4.4.1 ([17]) Let X be a compact Kähler manifold and p !t be given as N above Assume that u0 is a bounded function such that !0 C @@u as n p currents and !u0 =˝ lies in L for some p > Then there is a unique smooth solution u ; t/ of (4.29) on XT 0; T / such that limt!0 u ; t/ D u0 This extends Theorem 4.2.1 to the case of weak initial metrics Previously, in [4], a weak version of theorem was proved under the condition that p > If the Kodaira dimension of X is non-negative, then L0 C aKX is nef and big on XT and dimC XT D n According to Conjecture 4.3.7, if XT is smooth, then !N T extends to be a Kähler class on XT Since @Q@tut is uniformly bounded from above for t 0; T/, we can show that the assumptions in the above theorem are satisfied Then one can extend (4.1) across T and continue the flow on XT until T2 > T when Œ!N T  t T/c1 XT / fails to be a Kähler class If T2 is finite, one can proceed as we did for !Q t at T However, in general, the resulting variety XT from the surgery at T may not be smooth.7 Nevertheless, we expect Conjecture 4.4.2 There is a Q-factorial variety8 X with a holomorphic map X 7! XT , referred as a flip of X, satisfying: X has only log-terminal singularities, that is, there is a smooth resolution XQ 7! X such that KXQ D KX C k X a i Ei ; T W W (4.30) iD1 where Ei are exceptional divisors of the resolution numbers For > sufficiently small, Œ !N T  C KX > and > are rational It will be interesting to construct an explicit example of such a singular XT , even though no one doubts its existence A projective variety is Q-factorial if it is normal and any Q-Weil divisor is Q-Cartier 126 G Tian Clearly, our description of flip X does not require that X is smooth If we can affirm Conjecture 4.4.2, then we need to extend Kähler-Ricci flow to X above with initial metric !N T First we carry out such an extension by establishing a scalar flow on potential functions in the way as we did from (4.1) to (4.2) Since X is Q-factorial, there is an m > such that mKX is a line bundle, that is, locally, it is of the form C ˛ for some local holomorphic section ˛ We can further assume that X is covered by finitely many open subsets fUi g satisfying: a mKX jUi D C ˛i for a local holomorphic section over Ui b For each i, there is a holomorphic embedding X \ Ui 7! CN for some large N.9 Then we can choose a smooth volume form ˝ on the regular part Reg.X / of X such that on each Reg.X 0/ \ Ui , it is of the form ˝ jReg.X 0/\Ui D fi ˛i ^ ˛N i / m ; (4.31) where fi is a smooth function on a neighborhood of Ui in CN Put D p N @@ log ˝ and !t0 D !N T C t T/ Though not really needed, we can even choose ˝ such that !t0 > on X for t > T and t T sufficiently small Now we introduce a scalar flow on X : p !t0 C @@N u/n ˝ @u D log on Reg.X / and u L1 X /: @t (4.32) The following theorem was proved in [17] and extends Theorem 4.4.1 to Qfactorial varieties which satisfy and Theorem 4.4.3 ([17]) Let X be a variety which arises from Conjecture 4.4.2 Define T1 D supft j Œ !N T  C t KX 0g: Then (4.29) has a unique solution u.t/ on X Œ0; T1 / such that (1) u is smooth on p 0 Reg.X / 0; T1 /; (2) !t C @@N u > on Reg.X /; (3) u is continuous on X This theorem provides a solution !Q t to Kähler-Ricci flow on X with initial data !N T in the sense of distribution, where T Ä t < T C T1 Thus we extend KählerRicci flow !Q t on X Œ0; T/ across T to a Kähler-Ricci flow on X ŒT; T C T1 / So we get a solution !Q t with surgery for (4.1) for t Œ0; T C T1 / satisfying: As usual, we call T a surgery time Assuming that Conjectures 4.3.7 and 4.4.2 are confirmed, we can repeat the above process to continue the flow beyond T C T1 and so on Thus, we can construct a Kähler-Ricci flow with surgery on X for all t By a Kähler-Ricci flow with Without loss of generality, we may assume that N is independent of i Notes on Kähler-Ricci Flow 127 surgery on X with initial metric !0 , we mean a sequence of Kähler-Ricci flows !i t/ on Xi ŒTi ; Ti /, where i D 0; 1; (possibly only finitely many), satisfying: and for any T < 1, there are only finitely many KR1 T D < T1 < Ti ’s which are less than T; KR2 X0 D X and each Xi for i is either an empty set or a Q-factorial Kähler variety with at most log-terminal singularities and which is obtained from Xi as described in Conjecture 4.4.2, that is, Xi is a flip of Xi ; KR3 For each i, !.t/ is a Kähler-Ricci flow on Xi Ti ; TiC1 / or Xi Ti ; 1/ if TiC1 does not exist, defined by the corresponding scalar flow either (4.2) or (4.29); KR4 For each i 0, as t ! Ti , Xi ; !i t// converge to XN i ; !N i / as described in Conjecture 4.3.7 Let iC1 W XiC1 7! Xi be the projection T0 given by Conjecture 4.4.2, then limt!Ti C !iC1 t/ D iC1 !N i We denote by f.Xi ; !i t//g such a Kähler-Ricci flow with surgery From the definition, a Kähler-Ricci flow with surgery may have infinite surgery times However, we expect that there should be only finitely many surgery times, that is, < TN < 1, we Conjecture 4.4.4 After finitely many surgeries at T1 < T2 < arrive at XN which is either an empty set or a Q-factorial Kähler variety Xmin with nef canonical bundle KXmin If XN D ;, we say that (4.1) becomes extinct at T D TN At each Ti (i D 1; ; N 1), we surgery along some “rational” components along which KXi integrates negatively In particular, Xi is birational to X for all i This leads us to expect Conjecture 4.4.5 The Kähler-Ricci flow (4.1) becomes extinct at finite time if and only if X is birational to a Fano-like manifold.10 Partial progress on this conjecture has been made by Jian Song In particular, he proved that if X is Fano manifold, then (4.1) becomes extinct at finite time, If X is an uni-rule manifold, then one can show the flow (4.1) collapses at some finite time We expect the converse holds, too Conjecture 4.4.6 The Kähler-Ricci flow (4.1) collapsed at finite time if and only if X is birational to an uni-ruled manifold There are strong supporting evidences for validity of both Conjectures 4.3.7 and 4.4.2 In [17], using deep results in algebraic geometry, we have established these two conjectures for X being a projective manifold of general type and !0 having rational Kähler class 10 It is likely that such a Fano-like manifold is actually Fano This is indeed the case if the dimension is not greater than 128 G Tian 4.5 Asymptotic Behavior of Kähler-Ricci Flow In last two sections, we have discussed results and speculations on singularity formation of the Kähler-Ricci flow at finite time We also conjectured that there is always a global solution Xt ; !Q t / with surgery of (4.1) with only finitely many surgery times This generalized solution with surgery becomes an usual solution !Q t of (4.1) on a variety with nef canonical bundle when t is sufficiently large In this section, we study the asymptotic behavior of !Q t as t goes to For simplicity, we assume that X is a compact Kähler manifold with KX nef The general case can be dealt with in a similar approach We refer the readers to [17] for a more detailed discussion in case of possible singular varieties It is known that (4.1) has a global solution !Q t for any given initial metric Set t D es and !.s/ Q D e s !Q t , then !.s/ Q is a solution of the following normalized Kähler-Ricci flow: @!.s/ Q D @s Ric.!.s// Q !.s/; Q !.0/ Q D !0 : (4.33) The advantage of doing this is that Œ!.s/ Q D e s Œ!0  e s /c1 X/, which converges to c1 X/ as s ! We also assume that there is a (1,1)-form representing c1 X/ This is of course the case if KX is semi-positive or equivalently, for m sufficiently large, H X; KXm / is free of base points The Abundance conjecture in algebraic geometry claims that it is true for any X with KX nef Since H X; KXm / is base-point free, any basis of it induces a holomorphic map W X 7! CPN for some N > so that OCPN 1/ D KXm The dimension of ’s image is just the Kodaira dimension Ä D Ä.X/ of X If Ä.X/ D 0, then c1 X/ D and by the result in [1], the global solution !Q t of (4.1) converges to a Calabi-Yau metric on X This is still true even if X is a Qfactorial variety with only log-terminal singularities If Ä.X/ D dim X D n, then X is minimal and of general type It follows from [27, 29] that !.s/ Q converges to the unique (possibly singular along a subvariety) Kähler-Einstein metric with scalar curvature n on X as s tends to The more tricky cases are for those X with Ä Ä.X/ Ä n If X is such a manifold, one can not expect the existence of any Kähler-Einstein metrics (even with possibly singular along a subvariety) on X since Ä 6D and KXn D Hence, the first problem is to find what limiting metrics for !.s/ Q one supposes to have as s tends to To solve this problem, we introduced a class of new canonical metrics which we call generalized Kähler-Einstein metrics in [15]11 and [16] Let us briefly describe them 11 Reference [15] is mainly for complex surfaces, but the part on limiting metrics works for any dimensions Notes on Kähler-Ricci Flow 129 Since we assume that KX is semi-ample, the canonical ring R.X/ D ˚m H X; KXm / is finitely generated, so there is a canonical model Xcan of X (possibly singular) Let W X 7! Xcan be the natural map from X onto its canonical model Xcan Then generic fibers of are Calabi-Yau manifolds of dimension n Ä, and consequently, there is 0 a holomorphic map f W Xcan 7! MCY which assigns p Xcan to the fiber p/ in the moduli MCY , where Xcan consists of all p such that p/ is smooth The moduli MCY admits a canonical metric, the Weil-Petersson metric Let us recall its definition Let X ! MCY be a universal family of Calabi-Yau manifolds Let UI t1 ; : : : ; t` / be a local holomorphic coordinate chart of MCY , where ` D dim M Then each @t@i corresponds to an element à @t@i / H Xt ; TXt / through the Kodaira-Spencer map à The Weil-Petersson metric is defined by the L2 -inner product of harmonic forms representing classes in H Xt ; TXt / In the case of Calabi-Yau manifolds, as shown in [20], it has the following simple expression: Let « be a nonzero holomorphic n Ä; 0/-form on the fiber Xt and « yà @t@i / be the contraction of « and @t@i Then the Weil-Petersson metric is given by  @ @ ; @ti @tNj R à D !WP Xt « yà @t@i / ^ « yà @t@i / R : Xt « ^ « (4.34) Now we can introduce the generalized Kähler-Einstein metrics Definition Let X, Xcan etc be as above A closed positive 1; 1/-current ! on Xcan is called a generalized Kähler-Einstein metric if it satisfies the following f ! c1 X/; 12 ! is smooth p on Xcan ; Ric.!/ D @@ log ! Ä lifts to a well-defined current on X and on Xcan Ric.!/ D ! C f !WP : (4.35) If Ä D n, then it is just the equation for Kähler-Einstein metrics with negative scalar curvature In fact, one can prove that !/Ä ^ extends to a continuous function on X, where is the (n-Ä, n-Ä)-form which restricts to polarized flat volume form on each smooth fiber (see [16, p 15]) 12 130 G Tian Remark 4.5.1 More generally, one can consider the generalized Kähler-Einstein equation: Ric.!/ D where ! C f !WP ; is a constant In [16], the following theorem was proved Theorem 4.5.2 Let X be an n-dimensional projective manifold with semi-ample canonical bundle KX Suppose that < Ä.X/ Ä n There exists a unique generalized Kähler-Einstein metric on Xcan To prove this theorem, we reduce (4.35) to a complex Monge-Ampere equation as in the proof of the Aubin-Yau theorem First we introduce a function which will appear in such a complex MongeAmpere equation Since KX is semi-ample, there is a semi-ample form representing c1 X/, where is defined in the following way: Xcan can be embedded into some projective space CPN by using any basis of H X; KXm / for a sufficiently large m, then D !FS jXcan : m Let ˝ be a volume form on X satisfying: p @@ log ˝ D : We push forward ˝ to get a current ˝, where For any continuous function on Xcan W X ! Xcan as above, as follows: Z Z ˝D Xcan / ˝: X It is easy to see that for any x Xcan , we have Z ˝.x/ D x/ ˝: Definition We define a function F on Xcan by F Ä D ˝: (4.36) There is another way of defining F: Choose any Kähler class ˇ on X, by using the Hodge theory, one can find a flat relative volume form on X D Xcan / n Ä n Ä in the cohomology class ˇ , this means a n Ä; n Ä/-form in ˇ whose Notes on Kähler-Ricci Flow restriction to each fiber 131 x/ for x Xcan is Ricci-flat, that is, @@ log j D 0: x/ This is possible because c1 X/ vanishes along each smooth fiber One can show  FD c à ˝ ^ ; Ä (4.37) where c is a constant determined by Z c x/ ˇn Ä D 1; For simplicity, assume that c D In particular, it where x is any point in Xcan Ä follows that ^ can be extended to X as a current Furthermore, one can show (see [20]) f !WP D p 1@@ log ^ Ä p / 1@@ log Ä : The function F may not extend smoothly to Xcan , but we have some controls on it along the subvariety Xcan nXcan Lemma 4.5.3 F is smooth on Xcan and is in L1C Xcan / for some p L -norm is defined by using the metric corresponding to > 0, where the To prove it, we notice Z F 1C Xcan Ä Z F 1C D Ä Z ^ D X F ˝: X Furthermore, one can show that if à W Y ! Xcan is any resolution of Xcan , then à F has at worst pole singularities on Y The proof is a bit technical and we refer the readers to [16] for details Consequently, F is integrable for sufficiently small > (see [16], Proposition 3.2) Consider C p 1@@'/Ä D Fe' Ä : If ' is a bounded solution for (4.38), then ! D C Kähler-Einstein metric To see this, we first observe that Œ Next we observe Ric.!/ D p @@ log ! Ä D p 1@@ log Ä p (4.38) p 1@@' is a generalized ! D Œ  D c1 X/ 1@@ log F p 1@@' 132 G Tian is a well-defined current on Xcan A direct computation shows p 1@@ log Ä C p D 1@@ log D!C D! p p p 1@@ log F C 1@@' à  p ˝ Ä C! C 1@@ log ^ Ä Á @@ log ^ Ä / C @@ log Ä f !WP : Therefore Ric.!/ D ! C f !WP : Thus, in order to prove Theorem 4.5.2, we only need to prove the following Theoremp 4.5.4 There exists a unique solution ' C0 Xcan / \ C1 Xcan / for (4.38) 1@@' with C This is proved by using the continuity method and establishing an a priori C3 estimate for solutions of (4.38) We refer the readers to [16] for its proof We would like to point out that !Ä ^ D ˝ ef ' is continuous since both ' and ˝ are continuous on X There are still unknown about the generalized Kähler-Einstein metric !can constructed above, for instance, we not know if it has finite diameter in general Conjecture 4.5.5 Let Xcan and !can be as above, then the metric completion of Xcan ; !can / coincides with Xcan In particular, the diameter of !can is finite Recently, J Song made some progress on this problem, especially, in the case of projective manifolds of general type Now we can discuss the limit of !.s/ Q in (4.33) as s tends to The following theorem was proved in [16] (also see [15] for complex surfaces) Theorem 4.5.6 Let X be a projective manifold with semi-ample canonical bundle KX So X admits an algebraic fibration W X ! Xcan over its canonical model Xcan Suppose < dim Xcan D Ä < dim X D n Then for any initial Kähler metric !0 , the solution !.s/ Q for (4.33) converges to !can as currents, where !can is the unique generalized Kähler-Einstein metric on Xcan Moreover, for any compact subset K Xcan , there is a constant CK such that jjR.!.s//jj Q L1 K// C e.n Ä/s sup jj!.s/ Q n Äj x2K x/ where R.!.s// Q denotes the scalar curvature of !.s/ Q jjL1 x// Ä CK ; (4.39) Notes on Kähler-Ricci Flow 133 If n D 2, then the above implies the convergence in the C1;˛ -topology for any ˛ 0; 1/ on any compact subset in Xcan We believe that the same can be proved in any dimensions Moreover, we also expect !GKE in the Conjecture 4.5.7 The solution !.s/ Q converges to the unique limit Gromov-Hausdorff topology and the convergence is in the smooth topology in Xcan / This is even open for complex surfaces In the above, we assume that X has semi-ample KX This is indeed true if the Abundance conjecture holds If KX is nef, (4.33) still has a global solution !.s/ Q Clearly, it will be extremely interesting to study the asymptotic behavior of !.s/ Q without assuming the Abundance Conjecture, namely, give a differential geometric proof of the convergence of !.s/ Q The success of such a direct approach will yield many deep applications to studying the structures of Kähler manifolds To solve the above conjecture or succeed in the above direct approach, we may need to develop a theory of compactness for Kähler metrics with bounded scalar curvature For Kähler surfaces, a compactness theorem of this sort was proved in [26] Also note that the scalar curvature is uniformly bounded along (4.33) on any compact projective manifold with semi-positive canonical bundle (see [18]) We also refer the readers to [13, 14] for corresponding scalar curvature estimate for Kähler-Ricci flow on Fano manifolds 4.6 The Case of Algebraic Surfaces In this section, we will carry out the program described above for complex surfaces Almost all the results in this section are taken from [27] (for surfaces of general type) and [15] (for elliptic surfaces) We just make a few simple observations in order to deduce the program from those previous works For simplicity, we assume that X is an algebraic surface The general case for Kähler surfaces can be done in a similar way As before, let !Q t be a maximal solution of (4.1) on X Œ0; T If T < 1, then Œ!0  Tc1 X/ is nef There are three possibilities: Q D Tt / !Q t , If Œ!0  Tc1 X/ D 0, then X is a Del-Pezzo surface and !.s/ t where s D T log.1 T /, converges to a Kähler-Ricci soliton as s ! or equivalently, t ! T (cf [21, 28, 30]) If Œ!0  Tc1 X/ 6D but Œ!0  Tc1 X//2 D 0, then there is a fibration W X 7! ˙ with rational curves as fibers (possibly with finitely many singular fibers) such that Œ!0  Tc1 X/ D Œ!˙  for some Kähler metric !˙ on ˙ It follows that p as t ! T, !Q t converges to a positive current of the form !˙ C 1@@uT / for some bounded function uT on ˙ To extend (4.1) across T, one needs to 134 G Tian solve (4.2) on ˙ with uT as the initial value This is the same as solving the following for t T, !˙ @u D log @t t T/ ˙ C ˝˙ p 1@@u ! ; u.T; / D uT ; (4.40) where ˝˙ is a volume form on ˙ with Ric.˝˙ / D ˙ One can solve this flow by using the standard potential theory in complex dimension Let !Q t be the resulting maximal solution of (4.40) (t T) If the genus g.˙/ of ˙ is zero, then !Q t becomes extinct at some finite time T2 > T or after appropriate scaling, these metrics converge to the standard round metric on ˙ D S2 as t ! T2 Hence, it verifies Conjecture 4.4.5 in case of algebraic surfaces If g.˙/ D 1, then !Q t exists for all t T and converges to a flat metric as t ! If g.˙/ > 1, then !Q t exists for all t T and after scaling, converges to a hyperbolic metric as t ! If Œ!0  Tc1 X//2 > 0, then Œ!0  Tc1 X/ is semi-ample, so it can vanish only along a divisor It is easy to see that for each irreducible component D of this divisor, KX D < Moreover, D2 < By the Adjunction Formula, D is a rational curve of self-intersection 1, so the divisor is made of finite disjoint (-1) rational curves and consequently, we can blow down them to get a new algebraic surface XT Moreover, the limit !Q T descends to a positive current with continuous potential and well-defined bounded volume form By Theorem 4.4.1, one can extend (4.1) across T Notice that the extension !Q t for t > T is smooth Either KXT is nef and there is a global solution on XT ,or !Q t develops finite-time singularity at some T2 > T In the later case, one can repeat the above steps 1, and Since H2 X; Z/ is finite, after finitely many surgeries, we will arrive at a minimal algebraic surface XN , that is, KXN is nef Then (4.1) has a global solution, denoted again by !Q t , on XN Let us study its asymptotic behavior There are three possibilities according to the Kodaira dimension Ä.X/ of X: If Ä.X/ D 0, then c1 X/R D or a finite cover of X is either a K3 surface or an Abelian surface In this case, the solution !Q t on XN converges to a Ricci flat Kähler metric In other two cases, we better use the normalized Kähler-Ricci flow (4.33) on XN : @!.s/ Q D Ric.!.s// Q @s !.s/; Q !.0/ Q D !0 ; where t D es and !.s/ Q D e s !Q t If Ä.X/ D 1, then XN is a minimal elliptic surface: W XN 7! ˙ It was proved in [15] that as s ! 1, !.s/ Q converges to a positive current of the form !Q / and the convergence is in the C1;1 -topology on any compact subset Notes on Kähler-Ricci Flow 135 outside singular fibers Fp1 ; ; Fpk , where p1 ; ; pk ˙ Furthermore, !Q satisfies the generalized Kähler-Einstein equation: Ric.!Q / D !Q C f !WP ; on ˙nfp1 ; :pk g; :pk g into the moduli of where f is the induced holomorphic map from ˙nfp1 ; elliptic curves If Ä.X/ D 2, then XN is a surface of general type and 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