The Kähler-Ricci flow on singular Calabi-Yau varieties 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-0847524 and a Sloan Foundation Fellowship. 1

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 1.2. 2 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

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

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)

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 2 0. 3.2 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

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

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

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

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.

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 0 + 1 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

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

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 0 + 1 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 3.8. 2. 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

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

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

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