The Kähler-Ricci flow on singular Calabi-Yau varieties 1

Transcription

1 The Kähler-Ricci flow on singular Calabi-Yau varieties 1 Dedicated to Professor Shing-Tung Yau Jian Song and Yuan Yuan Abstract In this note, we study the Kähler-Ricci flow on projective Calabi-Yau varieties with log terminal singularities. We prove that the flow has long time existence and it converges to the unique singular Ricci-flat Kähler metric in the initial Kähler class. 1 Introduction The study of Kähler-Einstein metrics has been an important subject in complex geometry and analysis, following Yau s celebrated solution to the Calabi Conjecture [Y1]. The existence of Ricci-flat Kähler metrics on Kähler manifolds of vanishing first Chern class is deeply related to mirror symmetry and such Calabi-Yau manifolds have been thoroughly studied in many fields of mathematics and physics [SYZ]. The degeneration of Calabi-Yau manifolds is also a central subject as one changes complex or Kähler structures. In many cases, the degenerated model is again a Calabi-Yau variety with mild singularities and lots of progress has been made in this direction. The existence and uniqueness of singular Ricci-flat Kähler metrics have been proved in [EGZ] on projective Calabi-Yau varieties with log terminal singularities. The behavior of Ricci-flat Kähler metrics is studied in [GW, To1, To2, RZ] on a projective Calabi-Yau manifold when the Kähler classes degenerate to the boundary of the Kähler cone. The Ricci flow [H] provides a canonical deformation of Kähler metrics toward canonical metrics of Einstein type. Cao [Ca] gives an alternative proof of the existence of Ricci-flat Kähler metrics on a compact Kähler manifold with vanishing first Chern class by the Kähler-Ricci flow. A projective Calabi-Yau variety is a Q-factorial projective variety with numerically trivial canonical divisor K. The weak Kähler-Ricci flow is defined in [ST3] on Q-factorial projective varieties with log terminal singularities. The short time existence and uniqueness is also obtained in [ST3] for the weak Kähler-Ricci flow with appropriate initial data. In this paper, we study the limiting behavior of the Kähler-Ricci flow on projective Calabi-Yau varieties with log terminal singularities. The following theorem is our main result. Theorem 1.1 Let be a projective Calabi-Yau variety with log terminal singularities and ι : CP N be a projective embedding of. Let ω 0 be a real smooth semi-positive closed (1, 1)-form on equivalent to the pull-back of the Fubini-Study metric by ι. Then the weak Kähler-Ricci flow t ω = Ric(ω), ω(0, ) = ω 0 on (1.1) 1 The first named author is supported in part by National Science Foundation grant DMS and a Sloan Foundation Fellowship. 1

2 has a unique weak solution ω(t, ) on [0, ), where ω(t, ) C ([0, ) reg ) and ω(t, ) admits bounded local potentials in L. Furthermore, ω(t, ) converges to the unique singular Ricci-flat Kähler metric ω CY [ω 0 ] in the sense of currents on and in C -topology on reg = \ sing. The assumption on the equivalence of ω 0 and ω F S, the pull-back of the Fubini-Study metric by ι, means that there exists κ > 0 such that κ 1 ω F S ω 0 κω F S. Theorem 1.1 generalizes the classical result of Cao [Ca]. The long time existence of the weak solution is already proved in [ST3]. We refer the readers to the precise definition of the weak Kähler-Ricci flow in Definition 2.4. The existence and uniqueness of such Ricci flat metrics is proved in [EGZ]. General Calabi-Yau equations are also studied on symplectic manifolds in [We, TW, TWY]. The Kähler-Ricci flow is expected to smooth out initial singular data. In particular, if the Calabi-Yau variety has only crepant singularities, the evolving singular Kähler metrics immediately have bounded scalar curvature. Theorem 1.2 Let be a projective Calabi-Yau variety of complex dimension n with crepant resolution. Let ω(t, ) be the unique solution of the Kähler-Ricci flow (1.1) starting with a real smooth semi-positive closed (1, 1)-form ω 0 on, equivalent to the pull-back of the Fubini-Study metric by the embedding ι. Then for any δ > 0, there exists C > 0 such that n t S(ω(t, )) C t (1.2) on reg for t > δ, where S(ω) is the scalar curvature of ω. In particular, the scalar curvature S(ω) converges to 0 uniformly in L () as t. We now give the outline of the current paper. In section 2, we give some background in birational geometry and reduce the Kähler-Ricci flow to the Monge-Ampère flow on singular varieties. In section 3, after deriving the uniform estimates, we prove the existence and convergence of the Monge-Ampère flow on smooth projective manifolds with degenerate data. In section 4, we give the proof of Theorem 1.1 and Theorem Preliminaries 2.1 Singular Calabi-Yau varieties In this section, we will give the definition of projective Calabi-Yau varieties with log terminal singularities. We always assume that the underlying projective varieties are normal. The analysis of holomorphic and plurisubharmonic functions is well studied on normal varieties (cf. [D]). The following definitions are standard in algebraic geometry (see [L]). Definition 2.1 A projective variety is Q-factorial if any Q-Weil divisor on is Q-Cartier. Definition 2.2 Let be a normal Q-factorial projective variety. Let π : ˆ be a resolution of singularities and let {E i } p i=1 be the irreducible components of the exceptional locus Exc(π) of π with simple normal crossings. Then there exists a unique collection a i Q such that 2

3 K ˆ = π K + p a i E i. 1. is said to have log terminal singularities if a i > 1 for all i. i=1 2. π : ˆ is called a crepant resolution if all ai = 0. Definition 2.3 A projective Calabi-Yau variety is a normal, Q-factorial projective variety with log terminal singularities and numerically trivial canonical divisor. Let be a Calabi-Yau variety with numerically trivial canonical divisor K. There exists τ H 0 (, mk ) for some integer m > 0 and it is nowhere vanishing. Ω := (τ τ) 1 m is called the Calabi-Yau volume form. Thus Ric(Ω) = 1 log Ω, is well defined and vanishes on reg. 2.2 Reduction of the Kähler-Ricci flow to the Monge-Ampère flow Let us recall the notion of the weak Kähler-Ricci flow on projective varieties with singularities introduced in [ST3]. Definition 2.4 Let be a Q-factorial projective Calabi-Yau variety with log terminal singularities. Let ω 0 be a real semi-positive closed (1, 1) form on equivalent to the pull-back of the Fubini-Study metric by a projective embedding of. A family of closed real positive (1, 1)- currents ω(t, ) [ω 0 ] on for t [0, ) is called a solution of the weak Kähler-Ricci flow if following conditions hold: 1. ω(t, ) is smooth on reg for t > 0. Moreover, ω = ω 0 + 1ϕ for some potential ϕ C ([0, ) reg ) and ϕ(t, ) P SH(, ω 0 ) for all t [0, ). Furthermore, for any T (0, ), ϕ L ([0, T ) ). 2. t ω = Ric(ω) ω(0, ) = ω 0 on. on (0, ) reg, (2.1) The existence and uniqueness of the weak Kähler-Ricci flow is shown in [ST3] to be equivalent to the existence of ϕ which satisfies the first condition in Definition 2.4 and solves the following parabolic Monge-Ampère equation: t ϕ = log (ω 0+ 1ϕ) n Ω on (0, ) reg, ϕ(0, ) = 0 on, where Ω is the Calabi-Yau volume form defined as in Section 2.1 and so Ric(Ω) = 0 on reg. (2.2) 3

4 3 Degenerate Monge-Ampère flow 3.1 Set-up Let be a projective manifold of complex dimension n. introduced in [ST3]. The following two conditions are Condition A. Let L be a big and semi-ample line bundle over. Let ϖ c 1 (L) be a real smooth semi-positive closed (1, 1)-form on. We assume that ϖ at worst vanishes along a projective subvariety of to a finite order, i.e., there exists an effective divisor E 0 on such that for any fixed Kähler metric ϑ, ϖ C ϑ S E0 2 h E0 ϑ, where C ϑ is a positive constant, S E0 is a defining section of E 0 and h E0 is a smooth hermitian metric on the line bundle associated to E 0. Condition B. Let Θ be a smooth volume form on. Let E = p i=1 a ie i and F = q j=1 b jf j be effective divisors on, where E i and F j are irreducible components with simple normal crossings. In addition, we assume a i 0 and 0 < b j < 1. Let Ω be a semi-positive (n, n)-form on such that Ω = S E 2 h E S F 2 h F Θ, where S E and S F are the defining sections of E and F, h E and h F are smooth hermitian metrics on the line bundles associated to E and F. We will always abuse the notation of Q-line bundles and Q-divisors without confusion. Since L is semi-ample and big, by Kodaira s lemma, there exists an effective Q-divisor D such that L ɛd is ample for sufficiently small positive ɛ Q. We can further assume that the support of E 0 E F is contained in that of D. Let h D be a fixed hermitian metric on the line bundle associated to [D] such that ϖ D,ɛ := ϖ ɛric(h D ) = ϖ + ɛ 1 log h D > 0. Let S D be the defining section of the divisor D, and let S D 2 h D = S D 2 h D be the norm of the section S D with respect to the metric h D. Then one can rewrite ϖ D,ɛ = ϖ + ɛ 1 log S D 2 h D on \ D. On the other hand, it follows from the simple fact in calculus that any (n, n)-form Ω satisfying Condition B is integrable. Let ω 0 be a real smooth closed (1, 1)-form on such that ω 0 Cϖ for some constant C > 0. Apparently, ω 0 is semi-positive and big. Moreover, ω D,ɛ := ω 0 + ɛ 1 log h D > 0 for sufficiently small ɛ > 0. It also holds that ω D,ɛ = ω 0 + ɛ 1 log S D 2 h D on \ D. Let Ω be the volume form satisfying Condition B and let s also assume that Ω = [ω 0 ] n =: V 0 after normalization. Let ϑ be a fixed Kähler metric on. Let ω s = ω 0 + sϑ for s (0, 1]. Let Ω r1,r 2 = S E 2 h E +r 1 S F 2 h F +r 2 Θ be the perturbed smooth positive volume form on. We consider the following family of Monge- Ampère flows: t ϕ s,r 1,r 2 = log (ωs+ 1ϕs,r1,r 2 ) n Ω r1,,r 2 ϕ s,r1,r 2 (0, ) = 0. 4 (3.1)

5 Equation (3.1) is exactly the parabolic Monge-Ampère equation studied by Cao [Ca] for s, r 1, r 2 > 0 and it is well-known that there exists a unique smooth solution ϕ s,r1,r 2 on [0, ). In order to get the C 0 estimates, we need to make the following normalization. Let V s = [ω s ] n, V r1,r 2 = Ω r 1,r 2 and c s,r1,r 2 = log Vs V r1. Let ϕ s,r1,r,r 2 = ϕ s,r1,r 2 tc s,r1,r 2. Then equation (3.1) 2 is equivalent to the following family of Monge-Ampère flows: t ϕ s,r 1,r 2 = log (ωs+ 1 ϕs,r1,r 2 ) n Ω r1 c s,r1,r,r 2, 2 (3.2) ϕ s,r1,r 2 (0, ) = 0. Obviously, the constants c s,r1,r 2 are uniformly bounded for s, r 1, r 2 (0, 1] and they approach 0 as s, r 1, r Estimates Let ω s,r1,r 2 = ω s + 1ϕ s,r1,r 2 and s,r1,r 2, s,r1,r 2 be the gradient and Laplacian operators with respect to the metric ω s,r1,r 2. Lemma 3.1 There exist p > 1 and C > 0, such that for all r 1, r 2 (0, 1], ( Ω r 1,r 2 Θ )p Θ C. (3.3) Proof Note that ( Ω r 1,r 2 Θ )p Θ = ( S E 2 h E + r 1 S F 2 ) p Θ C h F + r 2 S F 2p h F Θ for some constant C > 0. By choosing p > 1 such that p b j < 1 for all j, the lemma follows from the simple calculus fact for q < 1: x q dx <. (0,1] Corollary 3.1 Let φ s,r1,r 2 be the unique smooth solution satisfying the following Monge-Ampère equation (ω s + 1φ s,r1,r 2 ) n = Vs V r1 Ω r1,r,r 2, 2 (3.4) max φ s,r1,r 2 = 0 for s, r 1, r 2 > 0. Then there exists C > 0 such that for all s, r 1, r 2 (0, 1], V s V r1,r 2 Ω r1,r 2 φ s,r1,r 2 L () C. (3.5) Proof Since is uniformly bounded in L p (Θ) for some p > 1 for s, r 1, r 2 (0, 1], the corollary follows from the uniform estimates for degenerate complex Monge-Ampère equations in [Kol1], [EGZ] and [Zh1]. 5

6 Proposition 3.1 There exists C > 0, such that for all s, r 1, r 2 (0, 1], ϕ s,r1,r 2 L ([0, ) ) C. (3.6) Proof Let Φ = ϕ s,r1,r 2 φ s,r1,r 2. Then t Φ = log (ω s + 1 ϕs,r1,r 2 ) n (ω s + 1φ s,r1,r 2 ) = log (ω s + 1φs,r1,r 2 + 1Φ) n n (ω s +. 1φ s,r1,r 2 ) n Applying the maximum principle, t min Φ 0, so Φ(t, ) min Φ(0, ) = 0. Therefore, ϕ s,r1,r 2 φ s,r1,r 2 C by Corollary 3.1. The uniform upper bound for ϕ s,r1,r 2 can be derived in a similar manner. Proposition 3.2 There exist A, C > 0 such that for t [0, ) and s, r 1, r 2 (0, 1], t ϕ s,r 1,r 2 C A log S D 2 h D. Proof Let H + = ϕ s,r1,r 2 + A 1 ( ϕ s,r1,r 2 log S D 2ɛ h D ), H = ϕ s,r1,r 2 A 2 ( ϕ s,r1,r 2 log S D 2ɛ h D ). We fix ɛ > 0 sufficiently small. By choosing A 1 sufficiently large, we can assume H + (0, ) > C 1 by Condition B. Away from D, we calculate ( t s,r1,r 2 )H + = A 1 [ ϕ s,r1,r 2 c s,r1,r 2 s,r1,r 2 ( ϕ s,r1,r 2 log S D 2ɛ h D )] A 1 [ ϕ s,r1,r 2 c s,r1,r 2 tr ωs,r1,r 2 ( ω s,r1,r 2 ω D,ɛ )] A 1 ( ϕ s,r1,r 2 c s,r1,r 2 n). Suppose that H + achieves its minimum at (t 0, p 0 ) with p 0 \ D, t 0 > 0. Then it holds by the maximum principle that ϕ s,r1,r 2 (t 0, p 0 ) c s,r1,r 2 + n C 2. Hence, ϕ s,r1,r 2 + A 1 ( ϕ s,r1,r 2 log S D 2ɛ h D ) H + (t 0, p 0 ) C 3. If the minimum of H + is attained at t = 0, then ϕ s,r1,r 2 + A 1 ( ϕ s,r1,r 2 log S D 2ɛ h D ) H + (0, ) C 1. Therefore, one direction in the proposition is obtained with the uniform bound of ϕ s,r1,r 2 and the other direction follows from the similar argument applied to H. Lemma 3.2 There exist C, α > 0 such that for t [0, ) and s, r 1, r 2 (0, 1], tr ϑ ( ω s,r1,r 2 ) C S D 2α h D. 6

7 Proof The proof is a consequence by applying the maximum principle to ( t s,r1,r 2 )(log tr ϑ ( ω s,r1,r 2 ) A 2 ϕ s,r1,r 2 + A S D 2 h D ) for sufficiently large A and the detailed proof can be found in Lemma 3.10 in [ST3]. Although the argument in [ST3] is for short time, it still can be applied here as ϕ s,r1,r 2 is uniformly bounded. and Let (ϕ s,r1,r 2 ) i lp = ( s,r1,r 2 ) p i lϕ s,r1,r 2 T = ( ω s,r1,r 2 ) i j ( ω s,r1,r 2 ) k l( ω s,r1,r 2 ) p q (ϕ s,r1,r 2 ) i lp (ϕ s,r1,r 2 ) jk q. The following third order estimate follows from Lemma 3.4 in [ST3]. Lemma 3.3 There exist constants C, λ > 0, such that for t [0, ) and s, r 1, r 2 (0, 1], T C S D 2λ h D. The following proposition gives us the uniform estimates of the metric ω s,r1,r 2 along (3.2). Proposition 3.3 For any compact set K \ D and k > 0, there exists C K,k > 0, such that ϕ s,r1,r 2 C k ([0, ) K) C K,k. Proof The proof follows from standard Schauder s estimates. Lemma 3.4 The following monotonicity conditions hold for ϕ s,r1,r 2 on [0, ). 1. ϕ s,r1,r 2 ϕ s,r 1,r 2 for any 0 < s < s ; 2. ϕ s,r1,r 2 ϕ s,r 1,r 2 for any 0 < r 1 < r 1 ; 3. ϕ s,r1,r 2 ϕ s,r1,r 2 for any 0 < r 2 < r 2. Proof We give a proof for part 1 as below and proofs for part 2 and 3 are similar. Let ψ = ϕ s,r 1,r 2 ϕ s,r1,r 2. Then ψ(0, ) = 0 and t ψ = log ( ω s,r 1,r 2 + (s s)ϑ + 1ψ) n ω s,r n log ( ω s,r 2,r 2 + 1ψ) n 1,r 2 ω s,r n. 1,r 2 Hence t min ψ 0 by the maximum principle, yielding ψ 0. Let ϕ = lim s 0 lim r2 0 ( lim r 1 0 ϕ s,r 1,r 2 ), where (lim r1 0 ϕ s,r1,r 2 ) denotes the upper semi-continuous envelope of the increasing sequence ϕ s,r1,r 2 as r 1 0. Therefore, it is an ω s -plurisubharmonic function and is decreasing as r 2 0. Furthermore, lim r2 0(lim r1 0 ϕ s,r1,r 2 ) is also an ω s -plurisubharmonic function and decreasing as s 0. Hence, ϕ is an ω 0 -plurisubharmonic function. 7

8 Proposition 3.4 ϕ is the unique solution to the Monge-Ampère flow t ϕ = log (ω 0+ 1ϕ) n Ω on [0, ) ( \ D), ϕ(0, ) = 0 on such that ϕ L ([0, ) ) C ([0, ) ( \ D)) and ϕ(t, ) P SH(, ω 0 ) for each t [0, ). Proof We first notice that ϕ = lim s 0 lim r2 0 lim r1 0 ϕ s,r1,r 2 away from D by Proposition 3.3. Then the existence can be obtained by passing (3.2) to the limit as s, r 1, r 2 0 once we have the uniform estimates (Proposition 3.3) and the uniqueness follows from [ST3]. 3.3 Convergence In this section, we will prove the convergence of (3.7). We begin by formally defining the generalized Mabuchi K-energy with degenerate and singular data. Definition 3.1 Let ω = ω 0 + 1ϕ for ϕ P SH(, ω 0 ) L (). Suppose that log ωn Ω ωn is well-defined and absolutely integrable. Then the generalized Mabuchi K-energy is defined as µ ω0 (ϕ) = 1 log ωn V 0 Ω ωn. (3.8) Note that this generalizes the Mabuchi K-energy for a smooth Kähler metric ω 0 and a smooth volume form Ω. By perturbing the initial metric ω 0 and the volume form Ω as in the previous section, we have the standard Mabuchi K-energy [Ma] along the flow (3.1): µ ωs (ϕ s,r1,r 2 ) = 1 V s It is well known that along (3.1), we have t µ ω s (ϕ s,r1,r 2 ) = 1 V s for fixed s, r 1, r 2 > 0, by straightforward calculations. log ωn s,r 1,r 2 Ω r1,r 2 ω n s,r 1,r 2. s,r1,r 2 ϕ s,r1,r 2 2 ω s,r1,r 2 ω n s,r 1,r 2 0 Lemma 3.5 There exists C > 0 such that for t > 0, s, r 1, r 2 (0, 1], t ϕ s,r 1,r 2 (t, ) C + C t. Proof Let H(t, ) = t ϕ s,r1,r 2 ϕ s,r1,r 2 nt tc s,r1,r 2. Then H(0, ) C 1 for some C 1 > 0. ( t s,r1,r 2 )H(t, ) = s,r1,r 2 ϕ s,r1,r 2 n = tr ωs,r1,r 2 ( ω s,r1,r 2 ω s ) n 0. (3.7) It follows that Hence, The lemma is proved as ϕ s,r1,r 2 t max H(t, ) 0. t( ϕ s,r1,r 2 n c s,r1,r 2 ) C 1 + ϕ s,r1,r 2. is uniformly bounded. 8

9 Proposition 3.5 Let ϕ be the unique solution to (3.7) and ω = ω 0 + 1ϕ. Then the generalized Mabuchi K-energy (3.8) is well-defined along (3.7) for t [0, ). Moreover, µ ω0 (ϕ) 0 for t [0, ). Proof Notice that µ ω0 (ϕ(0, )) = 1 V 0 log ωn 0 Ω ωn 0. Then it is well defined at t = 0, by Condition A and Condition B as there is at worst log pole singularities in the integral. When t > 0, µ ω0 (ϕ(t, )) is bounded from above for each t by Lemma 3.5. Furthermore, by Jensen s inequality µ ω0 (ϕ) = 1 log Ω V 0 ω n ωn log( 1 Ω) = 0. V 0 Lemma 3.6 Fix t [0, ). We have lim µ ω s (ϕ s,r1,r 2 (t, )) = µ ω0 (ϕ(t, )). s,r 1,r 2 0 Proof Since there are at worse log poles in each integral (Proposition 3.2 and Lemma 3.5), for any δ > 0, there exists a tubular neighborhood D δ of D such that 1 ωn log Ω ωn δ 1 and 4 log ωn s,r 1,r 2 ωn s,r 1,r 2 δ 4. V 0 D δ V s D δ Ω r1,r 2 Since ϕ s,r1,r 2 converges uniformly to ϕ in C ( \ D δ ) by the uniform estimates (Proposition 3.3), there exist s δ, r δ, such that for any s (0, s δ ), r 1, r 2 (0, r δ ), the following inequality holds: 1 log ωn s,r1,r 2 ω s,r n V s \D δ Ω 1,r 2 1 log r1,r 2 V 0 \Dδ ωn Ω ωn δ 2, Therefore, we have The lemma follows by letting δ 0. δ µ ω0 (ϕ(t, )) µ ωs (ϕ s,r1,r 2 (t, )) δ. Let and be the gradient and Laplacian operators with respect to ω. Proposition 3.6 Let ϕ be the unique solution of (3.7). Then for 0 t 1 t 2, µ ω0 (ϕ(t 1, )) µ ω0 (ϕ(t 2, )) 1 t2 V 0 In particular, µ ω0 (ϕ(t, )) is decreasing for t [0, ). Proof Notice that on any compact set K \ D, µ ωs (ϕ s,r1,r 2 (t 1, )) µ ωs (ϕ s,r1,r 2 (t 2, )) = 1 t2 V s t 1 1 t2 V s t 1 9 t 1 \D K ϕ 2 ωω n dt. (3.9) s,r1,r 2 ϕ s,r1,r 2 2 ω s,r1 ω n,r 2 s,r 1,r 2 dt s,r1,r 2 ϕ s,r1,r 2 2 ω s,r1 ω n,r 2 s,r 1,r 2 dt.

10 Due to the uniform estimate of ϕ s,r1,r 2, by letting s, r 1, r 2 0, we get: µ ω0 (ϕ(t 1, )) µ ω0 (ϕ(t 2, )) 1 t2 V 0 t 1 K ϕ 2 ωω n dt. The proposition follows by letting K \ D. Corollary 3.2 We have 0 \D ϕ 2 ωω n dt <. Proof The corollary is a direct consequence of Proposition 3.5 and Proposition 3.6. Proposition 3.7 Let 0 be the gradient operator with respect to the metric ω 0. Then in any compact set K \ D, sup 0 ϕ(t, ) 2 ω 0 0 as t. K Proof Otherwise suppose that there exist δ > 0, z j K and t j such that 0 ϕ(t j, z j ) 2 ω 0 > δ. We can assume that t j+1 t j > 1 for all j 1 by passing to a subsequence. By the uniform C k estimates for ϕ s,r1,r 2 in [0, ) K where K K \ D, we have the uniformly bounded geometry of ω s,r1,r 2 on K \D along the Monge-Ampère flow (3.1), namely C 1 ω 0 ω 1 C 1 ω 0 on K for C 1 > 0. Therefore there exist ɛ > 0, r > 0 such that ϕ(t, z) 2 ω > C 2 δ for t (t j ɛ, t j + ɛ), z B(z j, r) K and C 2 > 0. This leads to the contradiction against Corollary 3.2. Now, we are in the position to prove the convergence. Proposition 3.8 Let v(t, ) = ϕ(t, ) max ϕ(t, ). Then v(t, ) converges in L ()-norm to ϕ P SH(, ω 0 ) L () C ( \ D), which solves the following degenerate Monge-Ampère equation (ω 0 + 1ϕ ) n = Ω (3.10) max ϕ = 0. Proof Notice that v is still a uniformly bounded solution of the equation (ω v) n = e ϕ Ω. First of all, we argue that v converges to a solution of the degenerate Monge-Ampère equation (3.10) along a subsequence. By the uniform estimates (Proposition 3.3), after passing to a convergent subsequence, we can always assume v(t j, ) ϕ P SH(, ω 0 ) L () C ( \ D). 10

11 Furthermore, on any connected K \ D, v(t j, ) ϕ in C -topology. Then on K, as t. Hence on K, 0 ϕ(t, ) ω0 = 0 log ωn Ω 0 = 0 log (ω 0 + 1ϕ ) n ω0 Ω log (ω 0 + 1ϕ ) n = C. Ω By letting K \ D, ϕ solves the equation (3.10) on \ D. ϕ extends to a bounded ω 0 -psh function and therefore C = 0. Secondly, we will show that v converges to ϕ in L () by using the stability theorem derived in [Kol2, DiZh, DP]. Notice that ϕ is uniformly bounded from above away from t = 0 by Lemma 3.5. Then for any δ > 0, there exists a compact set K δ \ D, such that for all t > 0, (e ϕ + 1)Ω δ \K δ 2. Note that ϕ tends to 0 uniformly on any compact subset of \ D. Therefore there exists T δ > 0 such that for t > T δ, K δ e ϕ 1 Ω < δ 2. Now for t > T δ, e ϕ 1 Ω e K ϕ 1 Ω + (e ϕ + 1)Ω δ. δ \K δ Then v ϕ L () Cδ 1 n+γ holds for some fixed C > 0, γ > 0 by applying the stability theorem in [DiZh]. The proposition is thus proved by letting δ 0 as t. 4 Proof of theorems 4.1 Convergence of the Kähler-Ricci flow on singular Calabi-Yau varieties Let be a projective Calabi-Yau variety of complex dimension n. Let ι : CP N be a projective embedding and π : ˆ be the resolution of singularities with exceptional divisor E i and F j. Let H be the pullback of the hyperplane divisor of CP N to by ι, and let ϖ [H] be the pull-back of the Fubini-Study metric by ι. Thus ϖ is a real semi-positive closed (1, 1)-form. Define ˆϖ := π ϖ in the semi-ample and big class [π H]. Therefore, ˆϖ satisfies Condition A in section 3: ˆϖ C ϑ S E 2 h E ϑ, where ϑ is a fixed Kähler metric on ˆ, C ϑ is a positive constant, S E is a defining section of a effective divisor E ˆ and h E is a smooth hermitian metric on the line bundle associated to E. Let initial metric ω 0 be a real smooth semi-positive closed (1, 1)-form on, equivalent to ϖ and define ˆω 0 := π ω 0. Hence ˆω 0 also satisfies Condition A. Let E = i a ie i, F = j b jf j be the linear combination of exceptional divisors with coefficient a i 0 and 0 < b j < 1. Let K ˆ/ = K ˆ/π K be the relative canonical line bundle and its local defining function can be written as the Jacobian of π, i.e. locally it could be written as S E 2 h E S F 2 h F near exceptional divisors, where S E, S F is the canonical defining section of E, F and h E, h F is the smooth hermitian metric on the line bundle associated to E, F respectively. Define ˆΩ := π Ω be the pull back of the Calabi-Yau volume form Ω. Then ˆΩ is a smooth, 11 ω0

12 semi-positive Calabi-Yau volume form on ˆ with zeros along E i of order a i and poles along F j of order b j. Thus one can choose Θ to be a non-vanishing smooth volume form on ˆ such that ˆΩ = S E 2 h E S F 2 h F Θ and so ˆΩ satisfies Condition B in section 3. Apparently, as the order of poles is less than 2. ˆ ˆΩ = ˆ S E 2 h E S F 2 h F Θ < The weak Kähler-Ricci flow (the Monge-Ampère flow) is studied in [ST3] on the projective variety by lifting the equation to the resolution ˆ. Following the idea of [ST3], we can prove the following proposition by the study of the Monge-Ampère flow in section 3. Proposition 4.1 The following statements hold. 1. There exists a unique solution ˆϕ C ((0, ) ( ˆ \ Exc(π))) L ([0, ) ˆ) and for all t [0, ), ˆϕ(t, ) L ( ˆ) P SH( ˆ, ˆω 0 ) satisfying the Monge-Ampère flow t ˆϕ = log (ˆω 0+ 1 ˆϕ) n ˆΩ ˆϕ(0, ) = 0 on, on (0, ) ( ˆ \ Exc(π)), (4.1) where Exc(π) is the exceptional locus of the resolution π. Moreover, ˆϕ(t, ) sup ˆ ˆϕ(t, ) converges in L ( ˆ) to a solution ˆϕ C ( ˆ \ Exc(π)) L ( ˆ) P SH( ˆ, ˆω 0 ) of the degenerate Monge-Ampère equation (ˆω ˆϕ ) n = ˆΩ. 2. ˆϕ(t, ) descends to ϕ(t, ) P SH(, ω 0 ). Then ϕ(t, ) solves (2.2). Moreover ϕ(t, ) sup ϕ(t, ) converges in L () to a solution ϕ CY C ( reg ) L () P SH(, ω 0 ) of the degenerate Monge-Ampère equation (ω 0 + 1ϕ CY ) n = Ω. Proof 1. Since ˆω 0, ˆΩ satisfies Condition A, B respectively, the existence and uniqueness follows from Proposition 3.4 and the convergence follows from Proposition Note that ˆϕ(t, ) is constant on each connected component in the fibre of π, as ˆω 0 0 restricted on each connected component. Then it is a direct consequence of part 1. Now we are ready to prove our main theorem. Proof of Theorem 1.1 The uniqueness of ω follows from Theorem 4.4 in [ST3] and the existence and the convergence follows from Proposition 4.1. Part 2. 12

13 4.2 Scalar curvature estimates for the Kähler-Ricci flow on Calabi-Yau varieties with crepant resolutions We are going to derive the scalar curvature estimates for the Kähler-Ricci flow on projective Calabi-Yau manifolds with degenerate initial metrics. Thus the corresponding curvature estimates on the projective Calabi-Yau variety can be obtained by lifting the Monge-Ampére flow to its crepant resolution. Let be a projective Calabi-Yau manifold. Let ω 0 be a big and real smooth semi-positive closed (1, 1)-form satisfying Condition A and Ω be a smooth Calabi-Yau volume form on. Thus, the existence of the Kähler-Ricci flow, and of the corresponding Monge-Ampère flow on follows from the argument in the previous section. And in this situation, (3.1) will be written in the following easier form: t ϕ s = log (ωs+ 1ϕ s) n Ω, ϕ s (0, ) = 0. We now prove the following smoothing lemma. Lemma 4.1 There exists C > 0 such that along (4.2) for t (0, ), we have C C t t ϕ s C. Proof The upper bound follows from the maximum principle directly, so it suffices to derive the lower bound. Let φ s be smooth solutions to the family of Monge-Ampère equations (3.4) with r 1 = r 2 = 0. Then we have C 1 φ s 0 for some constant C 1 > 0. Let H s = t ϕ s + A ϕ s Aφ s for some A > 0 with ϕ s := ϕ s,0,0 in (3.2). Hence H s (0) 0. Then (4.2) Hence, ( t s )H s = (A + 1) ϕ s A(n + c s ) + Atr ωs ( ω s 1ϕ s + 1φ s ) and it follows that (A + 1) ϕ s + An( ecs Ω ω s n ) 1 n A(n + cs ) (A + 1) ϕ s + C 2 e 1 n ϕs C 3 C 4 t min H s ( t s ) min H s C 4. (4.3) H s (t, ) min H s(0, ) C 4 t C 4 t. (4.4) The lower bound of t ϕ s is derived as ϕ s, φ s are uniformly bounded. We are ready to show that the scalar curvature decays at the rate of 1 t along (4.2). Proposition 4.2 Let ω s := ω s + 1ϕ s. For any δ > 0, there exists C > 0 such that along (4.2) the scalar curvature satisfies: n t S( ω s(t)) C t when t > δ. 13

14 Proof In the proof, we use <, > s, s to denote the inner product and norm with respect to the metric ω s. As computed in [ST3], one would have: ( t s )[ts( ω s (t))] S( ω s (t)) + t n S( ω s(t)) 2. By the maximum principle, one can show that S( ω s (t)) n t. For the upper bound, one needs to apply the gradient and Laplacian estimates as in [SeT] [ST3]. The proposition is for the first time proved in [ST3] for general projective varieties. In our situation, the proof is easier and we include the proof in the following. Let H = s ϕ s 2 s C ϕ s and K = s ϕ s C ϕ s + BH for sufficiently large B. Then the straightforward calculation gives the evolution: ( t s )H = s s ϕ s 2 s + s s ϕ s 2 s 2Re < s ϕ s, s H > s ɛ s ϕ s 4 s ( ) 3 (1 2ɛ) s s ϕ s 2 s + s s ϕ s 2 s (1 ɛ) 2Re < s ϕ s, s H > s and ( t s )K = (B 1) s s ϕ s 2 s + B s s ϕ s 2 s Similar calculation shows that 2Re < s ϕ s, s K > s. (4.5) ( t s )[(t δ)h] (t δ)ɛ s ϕ s 4 s ( ) 3 + s ϕ s 2 s (1 ɛ) 2Re < s ϕ s, s [(t δ)h] > s. Suppose that (t δ)h achieves maximum at (t 0, z 0 ) when t δ. maximum principle and Lemma 4.1 that: Then it follows from the (t 0 δ)h(t 0, z 0 ) C 1. Hence, when t > δ, we have It follows from (4.5) that: H(t, ) C 1 t δ. ( t s )[(t δ)k] C 2 (t δ) ( s ϕ s ) 2 s ϕ s + BH(t, ) 2Re < s ϕ s, s [(t δ)k] > s. Suppose that (t δ)k achieves maximum for t δ at (t, z ). maximum principle that Then it follows from the 0 C 2 [(t δ) s ϕ s (t, z )] 2 + [(t δ) s ϕ s (t, z )] + B(t δ)h(t, z ) C 2 [(t δ) s ϕ s (t, z )] 2 + [(t δ) s ϕ s (t, z )] + C 3. 14

15 Then we obtain: and hence Therefore when t > δ, (t δ) s ϕ s (t, z ) C 4, (t δ)k(t, z ) C 5. s ϕ s C 5 t δ () C 6 t δ as H is nonnegative and the upper bound of S( ω(t)) follows. Now Theorem 1.2 can be deduced from the above proposition. Proof of Theorem 1.2 Let π : ˆ be the crepant resolution with π K = K ˆ and so ˆ is a Calabi-Yau manifold. Let Ω be the Calabi-Yau volume form on, and ˆω 0 = π ω 0, ˆΩ = π Ω be the pull back of the initial metric and the Calabi-Yau volume form respectively. Then ˆω satisfies Condition A and ˆΩ is a smooth, strictly positive Calabi-Yau volume form in the sense that Ric(ˆΩ) = 0. Hence the theorem follows from Proposition 4.2 by letting s 0. Acknowledgements: Both authors would like to thank Professor D. H. Phong for his support, encouragement as well as many valuable suggestions. They are indebted to V. Tosatti and Z. Zhang for many useful comments. The second named author is also grateful to Professor. Huang for the constant support. References [A] Aubin, T., Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull. Sci. Math. (2) 102 (1978), no. 1, [BT] Bedford, E., Taylor, B. A., The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math. 37 (1976), no. 1, [C] Calabi, E., Extremal Kähler metrics, in Seminar on Differential Geometry, pp , Ann. of Math. Stud., 102, Princeton Univ. Press, Princeton, N.J., 1982 [Ca] Cao, H., Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985), no. 2, [D] D ly, J.P., Complex Analytic and Algebraic Geometry, online book available on the webpage of J. P. D ly. [DP] D ly, J.P. and Pali, N., Degenerate complex Monge-Ampère equations over compact Kähler manifolds, Internat. J. Math. 21 (2010), no. 3, [DiZh] Dinew, S. and Zhang, Z., Stability of Bounded Solutions for Degenerate Complex Monge-Ampère equations, ariv: [EGZ] Eyssidieux, P., Guedj, V. and Zeriahi, A., Singular Kähler-Einstein metrics, J. Amer. Math. Soc. 22 (2009), no. 3, [FN] Fornæss, J. E. and Narasimhan, R., The Levi problem on complex spaces with singularities, Math. Ann. 248 (1980), no. 1, [GW] Gross, M. and Wilson P.M.H., Large complex structure limits of K3 surfaces, J. Differential Geom. 55 (2000), no. 3, [H] Hamilton, R. Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982), no. 2, [Kol1] Kolodziej, S., The complex Monge-Ampère equation, Acta Math. 180 (1998), no. 1,

16 [Kol2] Kolodziej, S., The Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), no. 3, [L] Lazarsfeld, R., Positivity in algebraic geometry, volumn I. & II., Springer-Verlag, Berlin, [Ma] Mabuchi, T., K-energy maps integrating Futaki invariants, Tohoku Math. J. (2) 38 (1986), no. 4, [Pe1] Perelman, G., The entropy formula for the Ricci flow and its geometric applications, preprint math.dg/ [Pe2] Perelman, G., unpublished work on the Kähler-Ricci flow. [PS] Phong, D. H. and Sturm, J., On stability and the convergence of the Kähler-Ricci flow, J. Differential Geom. 72 (2006), no. 1, [PSSW] Phong, D. H., Song, J., Sturm, J. and Weinkove, B., The Kähler-Ricci flow and the operator on vector fields, J. Differential Geom. 81 (2009), no. 3, [RZ] Ruan, W.-D. and Zhang, Y., Convergence of Calabi-Yau manifolds, ariv: [SeT] Sesum, N. and Tian, G., Bounding scalar curvature and diameter along the Kähler-Ricci flow(after Perelman), J. Inst. Math. Jussieu 7 (2008), no. 3, [Si] Siu, Y.-T., Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, DMV Seminar, 8. Birkhäuser Verlag, Basel, pp. [So] Song, J., Finite time extinction of the Kähler-Ricci flow, ariv: [ST1] Song, J. and Tian, G., The Kähler-Ricci flow on minimal surfaces of positive Kodaira dimension, Invent. Math. 170 (2007), no. 3, [ST2] Song, J. and Tian, G., Canonical measures and Kahler-Ricci flow, ariv: [ST3] Song, J. and Tian, G., The Kähler-Ricci flow through singularities, ariv: [SW1] Song, J. and Weinkove, B. The Kähler-Ricci flow on Hirzebruch surfaces, ariv: [SW2] Song, J. and Weinkove, B. Contracting exceptional divisors by the Kḧler-Ricci flow, ariv: [SYZ] Strominger, A., Yau, S.T. and Zaslow, E., Mirror symmetry is T -duality, Nuclear Phys. B 479 (1996), no. 1-2, [T1] Tian, G., On Kähler-Einstein metrics on certain Kähler manifolds with C 1(M) > 0, Invent. Math. 89 (1987), no. 2, [T2] Tian, G., New results and problems on Kähler-Ricci flow, Géométrie différentielle, physique mathématique, mathématiques et société. II. Astérisque No. 322 (2008), [TiZha] Tian, G. and Zhang, Z., A note on the Kähler-Ricci flow on projective manifolds of general type, Chinese Ann. Math. Ser. B 27 (2006), no. 2, [To1] Tosatti, V., Limits of Calabi-Yau metrics when the Kähler class degenerates, J. Eur. Math. Soc. 11 (2009), no.4, [To2] Tosatti, V., Adiabatic limits of Ricci-flat Kahler metrics, J. Differential Geom. 84 (2010), no.2, [TW] Tosatti, V. and Weinkove, B., The Calabi-Yau equation, symplectic forms and almost complex structures, Geometry and Analysis, Vol. I, , Advanced Lectures in Math. 17, International Press, [TWY] Tosatti, V., Weinkove, B. and Yau, S.T. Taming symplectic forms and the Calabi-Yau equation, Proc. Lond. Math. Soc. (3) 97 (2008), no. 2, [Ts] Tsuji, H., Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type, Math. Ann. 281 (1988), [We] Weinkove, B., The Calabi-Yau equation on almost-kähler four-manifolds, J. Differential Geom. 76 (2007), no. 2, [Y1] Yau, S.-T., On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm. Pure Appl. Math. 31 (1978), [Y2] Yau, S.-T., Open problems in geometry, Proc. Symposia Pure Math. 54 (1993), 1 28 (problem 65). [Yu] Yuan, Y., On the convergence of a modified KählerCRicci flow, to appear in Math. Z., DOI: /s [Zh1] Zhang, Z., On degenerate Monge-Ampère equations over closed Kähler manifolds, Int. Math. Res. Not. 2006, Art. ID 63640, 18 pp. 16

17 [Zh2] Zhang, Z., Scalar curvature bound for Kähler-Ricci flows over minimal manifolds of general type, Int. Math. Res. Not. 2009, no. 20, [Zh3] Zhang, Z., Kähler-Ricci flow with degenerate initial class, ariv: Department of Mathematics, Rutgers University, New Brunswick, NJ Department of Mathematics, Johns Hopkins University, Baltimore, MD