Almost scalar-flat Kähler metrics on affine algebraic manifolds
aa r X i v : . [ m a t h . DG ] J u l Almost scalar-flat K¨ahler metrics on affine algebraicmanifolds
Takahiro Aoi
Abstract
Let (
X, L X ) be an n -dimensional polarized manifold. Let D be a smooth hyper-surface defined by a holomorphic section of L X . In this paper, we show the existenceof a complete K¨ahler metric on X \ D whose scalar curvature is flat away from somedivisor if there are positive integers l ( > n ) , m such that the line bundle K − lX ⊗ L mX is very ample and the ratio m/l is sufficiently small. Contents D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 K¨ahler potential near F . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Ricci-flat K¨ahler metric away from D ∪ F . . . . . . . . . . . . . . . . . . 62.4 Gluing plurisubharmonic functions . . . . . . . . . . . . . . . . . . . . . . 7 C -estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 The C ,ǫ -estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 The third and the forth order estimates . . . . . . . . . . . . . . . . . . . 16 Let (
X, L X ) be a polarized manifold of dimension n , i.e., X is an n -dimensional compactcomplex manifold and L X is an ample line bundle over X . Assume that there is a smoothhypersurface D ⊂ X with D ∈ | L X | . Mathematics Subject Classification . Primary 53C25; Secondary 32Q15, 53C21.
Key words and phrases . constant scalar curvature K¨ahler metrics, complex Monge-Amp`ere equations,plurisubharmonic functions, K¨ahler manifolds. lmost scalar-flat K¨ahler metrics on affine algebraic manifolds L D := O ( D ) | D = L X | D over D . Since L X is ample, there existsa Hermitian metric h X on L X which defines a K¨ahler metric θ X on X , i.e., the curvatureform of h X multiplied by √− h X to L D defines also a K¨ahler metric θ D on D . Let ˆ S D be the average of the scalar curvature S ( θ D )of θ D defined by ˆ S D := Z D S ( θ D ) θ n − D Z D θ n − D = ( n − c ( K − D ) ∪ c ( L D ) n − c ( L D ) n − , where K − D is the anti-canonical line bundle of D . Note that ˆ S D is a topological invariantin the sense that it is representable in terms of Chern classes of the line bundles K − D and L D . In this paper, we treat the following case :ˆ S D > . (1.1)Let σ D ∈ H ( X, L X ) be a defining section of D and set t := log || σ D || − h X . Following [3],we can define a complete K¨ahler metric ω by ω := n ( n − S D √− ∂∂ exp ˆ S D n ( n − t ! on the noncompact complex manifold X \ D . This K¨ahler metric ω is of asymptoticallyconical geometry (see [1]).In [1], we show that there exists a complete scalar-flat K¨ahler metric which is ofasymptotically conical geometry if the following conditions hold : (1) n ≥ X vanishing on D , (2) θ D is a cscK metric and0 < ˆ S D < n ( n − ω is sufficiently small in the weightedBanach space (see Condition 1.2 and Condition 1.3 in [1]). In this paper, we construct acomplete K¨ahler metric on X \ D whose scalar curvature can be made small arbitrarilyby gluing plurisubharmonic functions.To show this, we consider a degenerate (meromorphic) complex Monge-Amp`ere equa-tion. Take positive integers l > n and m such that the line bundle K − lX ⊗ L mX is veryample. Let F ∈ | K − lX ⊗ L mX | be a smooth hypersurface defined by a holomorphic section σ F ∈ H ( X, K − lX ⊗ L mX ) such that the divisor D + F is simple normal crossing. For adefining section σ D ∈ H ( X, L X ) of D , set ξ := σ F ⊗ σ − mD . From the result due to Yau [14, Theorem 7], we can solve the following degenerate complexMonge-Amp`ere equation: ( θ X + √− ∂∂ϕ ) n = ξ − /l ∧ ξ − /l . Moreover, it follows from a priori estimate due to Ko lodziej [11] that the solution ϕ isbounded on X . Thus, we can glue plurisubharmonic functions by using the regularizedmaximum function. To compute the scalar curvature of the glued K¨ahler metric, we needto study behaviors of higher order derivatives of the solution ϕ . So, we give explicitestimates of them near the intersection D ∩ F : lmost scalar-flat K¨ahler metrics on affine algebraic manifolds Theorem 1.1.
Let ( z i ) ni =1 = ( z , z , ..., z n − , w F , w D ) be local holomorphic coordinatessuch that { w F = 0 } = F and { w D = 0 } = D . Then, there exists a positive integer a ( n ) depending only on the dimension n such that (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂z i ∂z j ∂ α ϕ (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) | w D | − a ( n ) m/l | w F | − a ( n ) /l (cid:1) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂w F ∂w F ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) | w D | − a ( n ) m/l | w F | − − a ( n ) /l (cid:1) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂w D ∂w D ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) | w D | − − a ( n ) m/l | w F | − a ( n ) /l (cid:1) , as | w F | , | w D | → , for any ≤ i, j ≤ n − and multi-index α = ( α , ..., α n ) with ≤ P i α i ≤ . By applying Theorem 1.1 and gluing plurisubharmonic functions, we have the followingresult :
Theorem 1.2.
Assume that there exist positive integers l > n and m such that a ( n ) m l < ˆ S D n ( n −
1) (1.2) and the line bundle K − lX ⊗ L mX is very ample. Here, a ( n ) is the positive integer in Theorem1.1. Take a smooth hypersuface F ∈ | K − lX ⊗ L mX | such that D + F is simple normal crossing.Then, for any relatively compact domain Y ⋐ X \ ( D ∪ F ) , there exists a complete K¨ahlermetric ω F on X \ D whose scalar curvature S ( ω F ) = 0 on Y and is arbitrarily small onthe complement of Y . In addition, ω F = ω on some neighborhood of D \ ( D ∩ F ) . For example, if the anti-canonical line bundle K − X of the compact complex manifold X is nef (in particular, X is Fano), the assumption (1.2) in Theorem 1.2 holds, i.e., we canalways find such integers l, m . In this article, we treat the case that K − X has positivity inthe senses of (1.1) and (1.2). From [1], if there exists a complete K¨ahler metric which isof asymptotically conical geometry and satisfies Condition 1.2 and Condition 1.3, X \ D admits a complete scalar-flat K¨ahler metric. In fact, Theorem 1.2 gives a K¨ahler metricwhose scalar curvature is under control. However, the K¨ahler metric ω F in Theorem 1.2 isnot of asymptotically conical geometry (near the intersection of D and F ). This problemwill be solved in [2].This paper is organized as follows. In Section 2, we construct K¨ahler potentials, i.e.,strictly plurisubharmonic functions, whose scalar curvature is under control. In addition,we glue these plurisubharmonic functions by using the regularized maximum function. InSection 3, we prove Theorem 1.1. To show this, we recall the C ,ǫ -estimate of a solutionof the degenerate complex Monge-Amp`ere equation. In Section 4, we prove Theorem 1.2. Acknowledgment.
The author would like to thank Professor Ryoichi Kobayashi whofirst brought the problem in this article to his attention, for many helpful comments. lmost scalar-flat K¨ahler metrics on affine algebraic manifolds In this section, we prepare K¨ahler potentials, i.e., strictly plurisubharmonic functions,whose scalar curvature is under control. D In this subsection, we consider a K¨ahler potential near D and study the scalar curvatureof it. Recall that t = log || σ D || − (2.1)and θ X = √− ∂∂t = √− ∂∂ log || σ D || − on X \ D . SetΘ( t ) = n ( n − S D exp ˆ S D n ( n − t ! . (2.2)Following [3], we can define a complete K¨ahler metric by ω := √− ∂∂ Θ( t ) = n ( n − S D √− ∂∂ exp ˆ S D n ( n − t ! on X \ D . Following [1], recall the asymptotic behavior of the scalar curvature of ω . Lemma 2.1.
The scalar curvature S ( ω ) can be estimated as follows : S ( ω ) = O (cid:16) || σ D || S D /n ( n − (cid:17) as σ D → . Remark 2.2.
Moreover, from Theorem 1.1 in [1], if θ D is cscK, we have the followingstrong result : S ( ω ) = O (cid:16) || σ D || S D /n ( n − (cid:17) as σ D → F In this subsection, we construct a K¨ahler metric on X whose scalar curvature is smallnear the smooth hypersurface F ∈ | K − lX ⊗ L mX | . Here, l, m are positive integers such thatthe line bundle K − lX ⊗ L mX is very ample. For a fixed Hermitian metric on K − lX ⊗ L mX , set b := log || σ F || − . Since the holomorphic line bundle K − lX ⊗ L mX is very ample, we mayassume that √− ∂∂b is a K¨ahler metric on X . For parameters v > β ∈ Z > , definea function by G βv ( b ) := Z bb (cid:18) e − y + v (cid:19) /β dy (2.3)for some fixed b ∈ R . Note that G βv ( b ) is defined smoothly outside F and lim b →∞ G βv ( b ) =+ ∞ for any v > lmost scalar-flat K¨ahler metrics on affine algebraic manifolds Lemma 2.3.
For Z ∋ β ≥ , γ βv := √− ∂∂G βv ( βb ) defines a K¨ahler metric on X .Proof. In fact, √− ∂∂G βv ( βb ) = β √− ∂ "(cid:18) e − βb + v (cid:19) /β ∂b = (cid:18) e − βb + v (cid:19) /β (cid:18) β √− ∂∂b + e − βb e − βb + v √− ∂b ∧ ∂b (cid:19) . Note that the last term e − βb e − βb + v √− ∂b ∧ ∂b is defined smoothly on X from the assumption that Z ∋ β ≥
1. Since √− ∂∂b is a K¨ahlermetric on X , we finish the proof.Next, the scalar curvature of γ βv is given by Lemma 2.4.
For β ≥ , we obtain S ( γ βv ) = S ( √− ∂∂G βv ( βb )) = O (( || σ F || β + v ) /β ) as || σ F || → .Proof. This lemma follows from the similar way in the computation of the scalar curvatureof ω . In fact, since (cid:0) ( √− ∂∂G βv ( βb ) (cid:1) n = β n (cid:18) e − βb + v (cid:19) n/β (cid:18) e − βb β ( e − βb + v ) || ∂b || (cid:19) ( √− ∂∂b ) n , we haveRic( √− ∂∂G βv ( βb )) = Ric( √− ∂∂b ) − √− ∂∂ log (cid:18) e − βb β ( e − βb + v ) || ∂b || (cid:19) + nβ (cid:18) e − βb + v √− ∂∂e − βb + β ( e − βb + v ) √− ∂e − βb ∧ ∂e − βb (cid:19) . Note that second and last terms above are zero on F . Thus, when we consider the scalarcurvature S ( γ βv ), it is enough to see the term 1 / ( e − βb + v ) /β √− ∂∂b and the Ricci formRic( √− ∂∂b ). Therefore the desired result is obtained. Remark 2.5.
If the value of the function e − βb = || σ F || β is compatible with v , i.e., || σ F || β ≈ v , we have the following estimate of S ( √− ∂∂G βv ( βb )) : S ( √− ∂∂G βv ( βb )) = O (1) . However, we will consider the case that || σ F || β ≈ v k for sufficiently large k ∈ N whichwill be specified in [2]. Namely, it suffices to consider a sufficiently small neighborhood of F defined by the inequality || σ F || β ≤ v k and Lemma 2.4 holds on this region. lmost scalar-flat K¨ahler metrics on affine algebraic manifolds D ∪ F In this subsection, we study an incomplete Ricci-flat K¨ahler metric away from the supportof the divisor D + F . Recall the setting in Theorem 1.2. Let l > n and m be positiveintegers such that there exists a holomorphic section σ F ∈ H ( K − lX ⊗ L mX ) which defines asmooth hypersurface F ⊂ X , i.e., ( σ F ) = F . It follows from the hypothesis of the averagevalue ˆ S D of the scalar curvature that divisors D and F intersect to each other. Set ξ := σ F ⊗ σ − mD . Note that ξ is a meromorphic section of K − lX . Then, define a singular and degeneratevolume form V by V := ξ − /l ∧ ξ − /l . From the construction above, V has finite volume on X and its curvature form, i.e., theRicci form, is zero on the complement of D ∪ F . For the K¨ahler metric θ X on X , write V = f θ nX for some non-negative function f on X with the normalized condition Z X V = Z X f θ nX = Z X θ nX . We know that f is smooth away from D ∪ F . From the result due to Yau [14, Theorem7], recall the solvability of a meromorphic complex Monge-Amp`ere equation : Theorem 2.6.
Let L and L be holomorphic line bundles over a compact K¨ahler manifold ( X, θ X ) . Let s and s be nonzero holomorphic sections of L and L , respectively. Let F bea smooth function on X such that R X | s | k | s | − k exp( F ) θ nX = Vol( X ) , where k ≥ and k ≥ . Suppose that R X | s | − nk < ∞ for n = dim X . Then, we can solve the followingequation ( θ X + √− ∂∂ϕ ) n = | s | k | s | − k exp( F ) θ nX so that ϕ is smooth outside divisors of s and s with sup X ϕ < + ∞ . Then, we can solve the following complex Monge-Amp`ere equation( θ X + √− ∂∂ϕ ) n = f θ nX = ξ − /l ∧ ξ − /l . (2.4)with ϕ ∈ C ∞ ( X \ D ∪ F ). Thus, we obtain a Ricci-flat K¨ahler metric θ X + √− ∂∂ϕ onthe complement of D ∪ F . For this solution ϕ , we obtain the following a priori estimatedue to Ko lodziej [11] (see also [9]): Theorem 2.7. If f is in L p ( θ nX ) for some p > , we have Osc X ϕ ≤ C for some C > depending only on θ X and || f || L p . lmost scalar-flat K¨ahler metrics on affine algebraic manifolds In this subsection, following [6, Chapter I], we consider gluing K¨ahler potentials, i.e.,plurisubharmonic functions, obtained in previous subsections. Let ρ ∈ C ∞ ( R , R ) be anonnegative function with support in [ − ,
1] such that R R ρ ( h ) dh = 1 and R R hρ ( h ) dh = 0. Lemma 2.8 (the regularized maximum) . For arbitrary η = ( η , ..., η p ) ∈ (0 , + ∞ ) p , thefunction M η ( t , ..., t p ) = Z R p max { t + h , ..., t p + h p } Y ≤ j ≤ p η − j ρ ( h j /η j ) dh ...dh p called the regularized maximum possesses the following properties : a) M η ( t , ..., t p ) is non decreasing in all variables, smooth and convex on R p ; b) max { t , ..., t p } ≤ M η ( t , ..., t p ) ≤ max { t + η , ..., t p + η p } ; c) M η ( t , ..., t p ) = M ( η ,..., ˆ η j ,...,η p ) ( t , ..., ˆ t j , ..., t p ) if t j + η j ≤ max k = j { t k − η k } ; d) M η ( t + a, ..., t p + a ) = M η ( t , ..., t p ) + a ; e) if u , ..., u p are plurisubharmonic and satisfy H ( u j ) z ( ξ ) ≥ γ z ( ξ ) where z γ z is acontinuous hermitian form on T M , then u = M η ( u , ..., u p ) is a plurisubharmonicand satisfies Hu z ( ξ ) ≥ γ z ( ξ ) . Remark 2.9.
Lemma 2.8 is a key in the proof of Richberg theorem (see [6, p.43]). Inour case, we have already prepared three plurisubharmonic functions and must computethe Ricci form of the glued K¨ahler metric later. Therefore, we need the explicit formulaof the glued function.In addition, we obtain
Lemma 2.10.
There exists a constant
C > such that (cid:12)(cid:12)(cid:12)(cid:12) ∂ | α | M η ∂t α ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C min { η j | α j = 0 } Y α i =0 η − α i i for any multi index α = ( α i ) i with ≤ | α | ≤ . Recall that the K¨ahler potential of ω is given byΘ( t ) = n ( n − S D exp ˆ S D n ( n − t ! . For κ ∈ (0 , G βv ( b ) := G βv ( βb ) + κ Θ( t ) . (2.5)This constant κ will be specified later. For this K¨ahler potential, we have lmost scalar-flat K¨ahler metrics on affine algebraic manifolds Lemma 2.11.
For the complete K¨ahler metric √− ∂∂ ˜ G βv ( b ) on X \ D , we have S ( √− ∂∂ ( ˜ G βv ( b ))) = (cid:26) O ( || σ D || S D /n ( n − ) near D , O (( || σ F || β + v ) /β ) near F . (2.6) Proof.
First, we study the behavior of the scalar curvature near D . Since || σ D || S/ ( n − (cid:16) √− ∂∂ ( ˜ G βv ( b )) (cid:17) n is a smooth volume form on X , the Ricci form of √− ∂∂ ( ˜ G βv ( b )) given byRic( √− ∂∂ ( ˜ G βv ( b )) = − ˆ Sn − ! θ X − √− ∂∂ log || σ D || S/ ( n − (cid:16) √− ∂∂ ( ˜ G βv ( b )) (cid:17) n is defined smoothly on X. Recall that √− ∂∂ ( ˜ G βv ( b )) = κω + γ βv . As ω is of asymptotically conical geometry, we have the desired result near D . Similarly,the volume form ( || σ F || β + v ) n/β (cid:16) √− ∂∂ ( ˜ G βv ( b )) (cid:17) n is smooth near F \ ( D ∩ F ). Then, the following identityRic( √− ∂∂ ( ˜ G βv ( b )) = nβ (cid:18) e − βb + v √− ∂∂e − βb + β ( e − βb + v ) √− ∂e − βb ∧ ∂e − βb (cid:19) − √− ∂∂ log( || σ F || β + v ) n/β (cid:16) √− ∂∂ ( ˜ G βv ( b )) (cid:17) n implies the desired result near F .In summary, we have prepared the three strictly plurisubharmonic functions Θ( t ) =( n ( n − / ˆ S D ) exp(( ˆ S D /n ( n − t ) , ˜ G βv ( b ) = G βv ( βb ) + κ Θ( t ) , t + ϕ = log || σ D || − + ϕ whosescalar curvature is under control. From Lemma 2.8, we immediately have Proposition 2.12.
For parameters c, v, η and κ ∈ (0 , , a function defined by M c,v,η := M η (cid:16) Θ( t ) , ˜ G βv ( b ) , t + ϕ + c (cid:17) is a strictly plurisubharmonic function on X \ ( D ∪ F ) . Here, the functions above aredefined in (2.1), (2.2), (2.3), (2.4) and (2.5). Remark 2.13.
From a priori estimate due to Ko lodziej [11], the solution ϕ is boundedon X . Thus, by taking c > ϕ can be ignored when we consider thevalue of M c,v,η . lmost scalar-flat K¨ahler metrics on affine algebraic manifolds c >
0, we have M c,v,η = Θ( t ) near D and away from F ,˜ G βv ( b ) near F and away from D , t + ϕ + c away from F and D . (2.7)Set ω c,v,η := √− ∂∂M (cid:16) Θ( t ) , ˜ G βv ( b ) , t + ϕ + c (cid:17) . The reason why we consider the second K¨ahler potential which contains the term κ Θ( t ) is that we want to make ω c,v,η complete on X \ D . The function M c,v,η is definedon X \ ( D ∪ F ). On the other hand, Lemma 2.3 implies that ω c,v,η is defined on X \ D since the K¨ahler metric γ βv is a smooth K¨ahler metric on X . From (2.7), we know thatthe scalar curvature of ω c,v,η is small on three regions above (in particular, away from D and F , S ( ω c,v,η ) = 0 since t + ϕ + c is a K¨ahler potential whose Ricci form is zero).The explicit formula of ω c,v,η is written as ω c,v,η = ∂M c,v,η ∂t ω + ∂M c,v,η ∂t ( γ βv + κω ) + ∂M c,v,η ∂t √− ∂∂ ( t + ϕ )+ (cid:2) ∂ Θ( t ) ∂ ˜ G βv ( b ) ∂ ( t + ϕ ) (cid:3) h ∂ M c,v,η ∂t i ∂t j i (cid:2) ∂ Θ( t ) ∂ ˜ G βv ( b ) ∂ ( t + ϕ ) (cid:3) t . Thus, when we compute the scalar curvature of ω c,v,η , higher order derivatives of ϕ arisein the components of the Ricci tensor of ω c,v,η . So, we must study the behavior of higherorder derivatives of ϕ near D ∪ F . In this section, we prove Theorem 1.1. Firstly, we use the C -estimate due to Pˇaun [12](see also [7], [9, p.366, Theorem 14.3]) for the solution φ of the complex Monge-Amp`ereequation (2.4) in the previous section to obtain the estimate of the ellipticity. i.e., themaximal ratio of the maximal eigenvalue to the minimal eigenvalue, of the K¨ahler metric θ X + √− ∂∂ϕ . Secondly, we study how the C ,ǫ -estimate of ϕ depends on the ellipticity of θ X + √− ∂∂ϕ on a fixed relatively compact domain in X \ ( D ∪ F ). Finally, we estimatethe higher order derivatives of ϕ by using the Schauder estimate. C -estimate To study the behavior of the higher order derivatives of ϕ , the elliptic operator definedby the K¨ahler metric θ X + √− ∂∂ϕ plays an important role. To obtain the ellipticity of θ X + √− ∂∂ϕ , we use the C -estimate due to Pˇaun [12] (see also [7], [9, p.366, Theorem14.3]). Theorem 3.1.
Let dV be a smooth volume form. Assume that ϕ ∈ PSH(
X, θ X ) satisfies ( θ X + √− ∂∂ϕ ) n = e ψ + − ψ − dV with Z X ϕθ nX = 0 . Here, ψ + , ψ − are quasi-plurisubharmonic functions on X . Assume thatwe are given C > and p > such that lmost scalar-flat K¨ahler metrics on affine algebraic manifolds (i) √− ∂∂ψ + ≥ − Cθ X and sup X ψ + ≤ C . (ii) √− ∂∂ψ − ≥ − Cθ X and || e − ψ − || L p ≤ C .Then there exists A > depending only on θ X , p and C such that ≤ θ X + √− ∂∂ϕ ≤ Ae − ψ − θ X . Set ψ + := log || σ D || m/l and ψ − := log || σ F || /l . Then, Theorem 3.1 implies the follow-ing inequality 0 ≤ θ X + √− ∂∂ϕ ≤ A || σ F || − /l θ X . (3.1)Recall that the singular and degenerate volume form( θ X + √− ∂∂ϕ ) n = ξ − /l ∧ ξ − /l , ξ = σ F ⊗ σ − mD (3.2)vanishes along D with order 2 m/l and has a pole along F of order 2 /l . So, we obtain thebehavior of the product of the eigenvalues of the K¨ahler metric θ X + √− ∂∂ϕ . From (3.1)and (3.2), we can estimate the eigenvalues of θ X + √− ∂∂ϕ . Namely, the maximal eigen-value Λ and the minimal eigenvalue λ of the K¨ahler metric θ X + √− ∂∂ϕ are estimatedas follows : Λ = O ( || σ F || − /l ) , λ − = O ( || σ D || − m/l ) . In the next subsection, to consider the third and the forth order derivatives, we recall the C ,ǫ -estimate of ϕ . C ,ǫ -estimate This subsection follows from [9, Chapter 14]. In this subsection, we study the relationbetween the ellipticity of θ X + √− ∂∂ϕ and the C ,ǫ -estimate of ϕ . This subsection is thecore of the proof of Theorem 1.1 because the estimate of the higher order derivatives ofthe solution ϕ are obtained by the C ,ǫ -estimate and the Schauder estimate.Let H be the set of all n × n Hermitian matrices and set H + := { A ∈ H| A > } . In addition, for 0 < λ < Λ < ∞ , let S ( λ, Λ) be the subset of H + whose eigenvalues liein the interval [ λ, Λ]. First, recall the following result from linear algebra (see [8, p.454,Lemma 17.13], [9, p.372, Lemma 14.10]):
Lemma 3.2.
One can find unit vectors ζ , ..., ζ N ∈ C n and < λ ∗ < Λ ∗ < ∞ , dependingonly on n, λ and Λ , such that every A ∈ S ( λ, Λ) can be written as A = N X k =1 β k ζ k ⊗ ζ k , i . e ., a i,j = X k β k ζ ki ζ kj , where β k ∈ [ λ ∗ , Λ ∗ ] . The vectors ζ , ..., ζ N ∈ C n can be chosen so that they contain a givenorthonormal basis of C n . lmost scalar-flat K¨ahler metrics on affine algebraic manifolds Remark 3.3.
In the proof of Lemma 3.2, they use the following covering U ( ζ , ..., ζ n ) = (X k β k ζ k ⊗ ζ k | < β k < ) of the compact subset S ( λ/ , Λ) (see [8, p.454, Lemma 17.13], [9, p.372, Lemma 14.10]).Here, ζ , ..., ζ n ∈ C n are unit vectors such that the matrices ζ k ⊗ ζ k span H over R . Thus,it follows from the form of the covering U ( ζ , ..., ζ n ) that the number N in Lemma 3.2 isdepending only on the dimension n . In particular, N is independent of the ellipticity of θ X + √− ∂∂ϕ .Take local holomoriphic coordinates ( z i ) ni =1 = ( z , z , ..., z n − , w F , w D ) such that { w F =0 } = F and { w D = 0 } = D . On this coordinate chart, we can write t = a + log | w D | − forsome smooth plurisubharmonic function a . Since θ X + √− ∂∂ϕ = √− ∂∂ ( a + ϕ ) on thiscoordinate chart, it is enough to consider the following complex Monge-Amp`ere equationdet( u i,j ) = f on an open subset Ω ⋐ C n \ ( D ∪ F ) by setting u = a + ϕ. (3.3)It follows from our construction that we may assume that the function f is a form of f = | w F | − /l | w D | m/l . Fix an unit vector ζ ∈ C n . Differentiating the following equation :log det( u i,j ) = log f, we have u i,j u ζ,ζ,i,j = (log f ) ζ,ζ + u i,l u k,j u ζ,i,j u ζ,k,l ≥ (log f ) ζ,ζ = 0 . Here we use the standard Einstein convention and the notation ( u i,j ) = (( u i,j ) t ) − . Set a i,j = f u i,j . Then, for any i , we have( a i,j ) j = f j u i,j − f u i,l u k,j u j,k,l = f u k,l u j,k,l u i,j − f u i,l u k,j u j,k,l = 0 . Thus, we obtain ( a i,j u ζ,ζ,i ) j = ( a i,j ) j u ζ,ζ,i + a i,j u ζ,ζ,i,j ≥ f (log f ) ζ,ζ = 0 . Note that u ζ,ζ is a subsolution of the equation Lv = 0, where Lv := P i,j ( a i,j v i ) j . Theassumption of u and the later lemma ensure that the operator L is uniformly elliptic (inthe real sense). Then, we have the following estimate (see [8, Theorem 8.18]). lmost scalar-flat K¨ahler metrics on affine algebraic manifolds Lemma 3.4.
The weak Harnack inequality r − n Z B r (sup B r u ζ,ζ − u ζ,ζ ) ≤ C H (sup B r u ζ,ζ − sup B r u ζ,ζ ) , holds. Here, B r := B ( z , r ) ⊂ Ω with d ( z , ∂ Ω) > r . Moreover, in our case, we havethe following estimate of the constant C H in Harnack inequality : C H = O (Λ /λ ) . Proof.
It suffices to show the estimate of the constant C H . In our case, we will onlyconsider the behavior of ϕ in the neighborhood of D ∪ F and the C -estimate of ϕ impliesthat u ζ,ζ = O ( || σ F || − /l ) = O (Λ) u − ζ,ζ = O ( || σ D || − m/l ) = O ( λ − )as || σ F || → || σ D || →
0. Thus, the weak Harnack inequality implies that the lemmafollows.
Remark 3.5.
From the proof of [8, Theorem 8.18], we know that the optimal Harnackconstant C H is estimated by C H = C √ Λ /λn , where C n depends only on n .Set U := ( u i,j ). For x, y ∈ B r , we obtain a i,j ( y ) u i,j ( x ) = f ( y ) u i,j ( y ) u i,j ( x ) = f ( y )tr( U ( y ) − U ( x )) . In particular, a i,j ( y ) u i,j ( y ) = nf ( y ). Since det( f ( y ) /n U ( y ) − ) = 1, we have a i,j ( y ) u i,j ( x ) = f ( y ) − /n tr( f ( y ) /n U ( y ) − U ( x )) ≥ nf ( y ) − /n det( U ( x )) /n = nf ( y ) − /n f ( x ) /n . Here, we have used the following lemma (see [9, Lemma 5.8]) :
Lemma 3.6.
For any A ∈ H + , we have (det A ) /n = 1 n inf { tr( AB ) | B ∈ H + , det B = 1 } . Therefore, for any x, y ∈ B r and ǫ ∈ (0 , a i,j ( y )( u i,j ( y ) − u i,j ( x )) ≤ nf ( y ) − nf ( y ) − /n f ( x ) /n = nf ( y ) − /n ( f ( y ) /n − f ( x ) /n ) ≤ C ( ǫ ) | x − y | ǫ , lmost scalar-flat K¨ahler metrics on affine algebraic manifolds C ( ǫ ) := n sup Ω ( f − /n ) H¨ol ǫ, Ω ( f /n )and H¨ol ǫ, Ω denotes an ǫ -H¨older constant. In this case, the following estimatesH¨ol ǫ, Ω ( f /n ) = O ( || σ F || − /nl − ǫ || σ D || m/nl − ǫ ) (3.4)sup Ω ( f − /n ) = O ( || σ F || − n − /nl || σ D || m ( n − /nl ) (3.5)implies that we have C ( ǫ ) = O ( || σ F || − /l − ǫ || σ D || m/l − ǫ ) . (3.6) Remark 3.7.
In [9, p.375], they used the Lipscitz constant of f . But in our case, it isenough to use the H¨older constant of f for sufficiently small ǫ .Set λ, Λ > a i,j ( y )) lie in the interval [ λ, Λ]. Then, Lemma3.2 implies that we can find unit vectors ζ , ..., ζ N ∈ C n such that for any x, y ∈ Ω, a i,j ( y )( u i,j ( y ) − u i,j ( x )) = N X k =1 β k ( y )( u ζ k ,ζ k ( y ) − u ζ k ,ζ k ( x )) , where β k ( y ) ∈ [ λ ∗ , Λ ∗ ] and λ ∗ , Λ ∗ > N X k =1 β k ( y )( u ζ k ,ζ k ( y ) − u ζ k ,ζ k ( x )) ≤ C ( ǫ ) | x − y | ǫ . Set M k,r := sup B r u ζ k ,ζ k , m k,r := inf B r u ζ k ,ζ k , and η ( r ) := N X k =1 ( M k,r − m k,r ) . To establish the H¨older condition η ( r ) ≤ Cr ˜ ǫ for some 0 < ˜ ǫ <
1, we need the following lemma from [8, p.201, Lemma 8.23] :
Lemma 3.8.
Let η and σ be non-decreasing functions defined on the interval (0 , R ] suchthat there exist τ, α ∈ (0 , satisfying η ( τ r ) ≤ αη ( r ) + σ ( r ) for all r ∈ (0 , R ] . Then, for any µ ∈ (0 , , we have η ( R ) < α (cid:18) RR (cid:19) (1 − µ )(log α/ log τ ) + σ ( R − µ R µ )1 − α . lmost scalar-flat K¨ahler metrics on affine algebraic manifolds η ( r ) ≤ δη (4 r ) + Cr ǫ , < r < r , where δ, ǫ ∈ (0 ,
1) and r > k , Harnack inequality implies that r − n Z B r X l = k ( M l, r − u ζ l ,ζ l ) = X l = k r − n Z B r ( M l, r − u ζ l ,ζ l ) ≤ X l = k C H ( M l, r − M l,r ) ≤ X l = k C H ( η (4 r ) − η ( r ))= ( N − C H ( η (4 r ) − η ( r )) . For x ∈ B r and y ∈ B r , we have β k ( y )( u ζ k ,ζ k ( y ) − u ζ k ,ζ k ( x )) ≤ C ( ǫ ) | x − y | ǫ + X l = k β l ( y )( u ζ l ,ζ l ( x ) − u ζ l ,ζ l ( y )) ≤ C ( ǫ ) r ǫ + Λ ∗ X l = k ( M l, r − u ζ l ,ζ l ( y )) . Thus, for all y ∈ B r , we have u ζ k ,ζ k ( y ) − m k, r ≤ λ ∗ C ( ǫ ) r ǫ + Λ ∗ X l = k ( M l, r − u ζ l ,ζ l ( y )) ! . Therefore, r − n Z B r ( u ζ k ,ζ k ( y ) − m k, r ) ≤ r − n Z B r λ ∗ C ( ǫ ) r ǫ + Λ ∗ X l = k ( M l, r − u ζ l ,ζ l ( y )) ! ≤ C ( ǫ ) λ ∗ r ǫ + Λ ∗ λ ∗ r − n Z B r X l = k ( M l, r − u ζ l ,ζ l ) ≤ C ( ǫ ) λ ∗ r ǫ + Λ ∗ λ ∗ ( N − C H ( η (4 r ) − η ( r )) . Using Harnack inequality again, we have M k, r − m k, r = r − n Z B r (sup B r u ζ k ,ζ k − u ζ k ,ζ k ) + r − n Z B r ( u ζ k ,ζ k ( y ) − m k, r ) ≤ C H ( M k, r − M k,r ) + 5 C ( ǫ ) λ ∗ r ǫ + Λ ∗ λ ∗ ( N − C H ( η (4 r ) − η ( r )) ≤ (cid:18) C H + Λ ∗ λ ∗ ( N − C H (cid:19) η (4 r ) − (cid:18) C H + Λ ∗ λ ∗ ( N − C H (cid:19) η ( r ) + 5 C ( ǫ ) λ ∗ r ǫ . lmost scalar-flat K¨ahler metrics on affine algebraic manifolds k , we have η (4 r ) ≤ N (cid:18) C H + Λ ∗ λ ∗ ( N − C H (cid:19) η (4 r ) − N (cid:18) C H + Λ ∗ λ ∗ ( N − C H (cid:19) η ( r ) + N C ( ǫ ) λ ∗ r ǫ . Thus, we obtain η ( r ) ≤ N (cid:16) C H + Λ ∗ λ ∗ ( N − C H (cid:17) − N (cid:16) C H + Λ ∗ λ ∗ ( N − C H (cid:17) η (4 r ) + C ( ǫ ) λ ∗ C H + Λ ∗ λ ∗ ( N − C H r ǫ . (3.7)Since we can take arbitrary λ ∗ N < λ and Λ ∗ > Λ, we may assume that λ ∗ N = λ andΛ ∗ = Λ. Thus, we have Lemma 3.9.
By taking ǫ ≤ /l , there exists < ˜ ǫ < ǫ with || u || C , ˜ ǫ = O (cid:18)(cid:18) Λ λ (cid:19) C H (cid:19) . Proof.
In order to show this lemma, we apply Lemma 3.8 to the inequality (3.7). Set α := N (cid:0) C H + Λ λ ( N − C H (cid:1) − N (cid:0) C H + Λ λ ( N − C H (cid:1) , where this is the coefficient of η (4 r ) in (3.7). Then, we have the following estimates :1 α = O (1) , − α = O ((Λ /λ ) C H ) . Here, we have used the fact that the number N depends only on the dimension n (Remark3.3). Define a non-decreasing function σ by σ ( r ) := C ( ǫ ) λ C H + Λ λ ( N − C H r ǫ . Here, this is the second term in the right hand side of the inequality (3.7). Recall theestimate (3.6) C ( ǫ ) = O ( || σ F || − /l − ǫ || σ D || m/l − ǫ )and Lemma 3.4. The assumption that ǫ ≤ /l implies that we have the following C ( ǫ ) λ C H + Λ λ ( N − C H = O (1) . Lemma 3.8 implies that we have η ( r ) < α (cid:18) rr (cid:19) (1 − µ )(log α/ log(1 / + σ ( r − µ r µ )1 − α . lmost scalar-flat K¨ahler metrics on affine algebraic manifolds µ ∈ (0 , µ ∈ (0 ,
1) so that(1 − µ )(log α/ log(1 / > µǫ. Thus, we have η ( r ) < O ((Λ /λ ) C H ) σ ( r − µ r µ )Set ˜ ǫ := ǫµ < ǫ . From the interior H¨older estimate for solutions of Poisson’s equation [8,Theorem 4.6], we finish the proof.Recall the relation (3.3) between u and ϕ . Lemma 3.4 implies Proposition 3.10.
For the domain Ω ⋐ X \ ( D ∪ F ) , we have || ϕ || C , ˜ ǫ (Ω) = O (cid:16)(cid:0) || σ D || − m/l || σ F || − /l (cid:1) (cid:17) as σ D , σ F → . In this subsection, we prove Theorem 1.1. This subsection also follows from [9, Chapter14]. To consider higher order estimates, we recall the Schauder estimate with respect tothe elliptic linear operator defined by the K¨ahler metric θ X + √− ∂∂ϕ . The complexMonge-Amp`ere operator F ( D u ) = det( u i,j )is elliptic if the 2 n × n real symmetric matrix A := ( ∂F/∂u p,q ) is positive (we denotehere by u p,q the element of the real Hessian D u ). The matrix A is determined by ddt F ( D u + tB ) | t =0 = tr( A t B ) . From [4] (see also [9, Exercise 14.8]), we have
Lemma 3.11.
One has λ min ( ∂F/∂u p,q ) = det( u i,j )4 λ max ( u i,j ) , λ max ( ∂F/∂u p,q ) = det( u i,j )4 λ min ( u i,j ) , where λ min ( ∂F/∂u p,q ) and λ max ( ∂F/∂u p,q ) denote minimal and maximal eigenvalue of thematrix ( ∂F/∂u p,q )) p.q respectively. Then, we can estimate the ellipticity in the real sense. We apply the standard elliptictheory to the equation F ( D u ) = f. For a fixed unit vector ζ and small h >
0, we consider u h ( x ) := u ( x + hζ ) − u ( x ) h lmost scalar-flat K¨ahler metrics on affine algebraic manifolds a p,qh ( x ) := Z ∂F∂u p,q ( tD u ( x + hζ ) + (1 − t ) D u ( x )) dt. Thus, we have a p,qh ( x ) u hp,q ( x ) = 1 h Z ddt F ( tD u ( x + hζ ) + (1 − t ) D u ( x )) dt = f h ( x ) . From the definition of a p,qh , we obtain || a p,qh || C , ˜ ǫ ≤ C || u || n − C , ˜ ǫ = O ((Λ /λ ) n − )for sufficiently small h > Proposition 3.12.
There exists C S > such that || u h || C , ˜ ǫ ≤ C S ( || f h || C , ˜ ǫ + || u h || C ) for any h > . Therefore, we can obtain the estimate of derivatives of the solution ϕ in the desireddirection by taking a suitable vector ζ and h →
0. The constant C S in Proposition 3.12is also depending on the maximal ratio of the eigenvalues Λ /λ and the dimension n . Byexamining the proof of [8, Lemma 6.1 and Theorem 6.2], there is a positive constant s ( n )depending only on the dimension n such that C S = O ((Λ /λ ) s ( n ) ) . As h →
0, we have the following third order estimates of ϕ : Proposition 3.13.
For any multi-index α = ( α , ..., α n ) satisfying P i α i = 2 , we have (cid:12)(cid:12)(cid:12)(cid:12) ∂∂z i ∂ α ϕ (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) C S | w D | − m/l | w F | − /l (cid:1) , (cid:12)(cid:12)(cid:12)(cid:12) ∂∂w F ∂ α ϕ (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) C S | w D | − m/l | w F | − − /l (cid:1) , (cid:12)(cid:12)(cid:12)(cid:12) ∂∂w D ∂ α ϕ (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) C S | w D | − − m/l | w F | − /l (cid:1) , as | w D | , | w F | → . From the discussion above, we can prove Theorem 1.1.
Proof of Theorem 1.1
Let ˙ a p,qh be a differential of a p,qh in some direction. From thedefinition of a p,qh , we know that || ˙ a p,qh || C , ˜ ǫ ≤ C || ˙ u || C , ˜ ǫ || u || n − C , ˜ ǫ Thus, by differentiating the equation a p,qh ( x ) u hp,q ( x ) = f h ( x ), Schauder estimate impliesagain the following inequality: || ˙ u h || C , ˜ ǫ ≤ C S ( || ˙ f h − ˙ a p,qh u hp,q || C , ˜ ǫ + || ˙ u h || C )Thus, we finish the proof pf Theorem 1.1 by taking a suitable vector ζ and h → (cid:3) lmost scalar-flat K¨ahler metrics on affine algebraic manifolds Remark 3.14.
By examining the proof of [8, Lemma 6.1 and Theorem 6.2] and thediscussion above, we can find that a = a ( n ) = O ( n ) . In this section, we prove Theorem 1.2. To compute the scalar curvature of the K¨ahlermetric ω c,v,η , we have to consider the inverse matrix (see Lemma 3.4 in [1]). Since weassume that the divisor D + F is simple normal crossing, we can choose block matricesin suitable directions in local holomorphic coordinates defining hypersurfaces D and F .To prove Theorem 1.2, we consider the case that the parameter η = ( η , η , η ) dependson c >
0. More precisely, we set η i := a i c for i = 1 , a i ∈ (0 ,
1) and η a fixed positivereal number. We use many parameters, i.e., c, v, β, κ, η, a i . When we want to make thescalar curvature S ( ω c,v,η ) small, we take sufficiently large c and sufficiently small v . Onthe other hand, we don’t make other parameters β, κ, a i close to ∞ , 0 or 1. Namely, theparameters β, κ, a i are bounded in this sense. Settings of these bounded parameters willbe given later. Proof of Theorem 1.2.
Take a relatively compact domain Y ⋐ X \ ( D ∪ F ). Recallthat the function G βv ( βb ) is defined by G βv ( βb ) := Z βbb (cid:18) e − y + v (cid:19) /β dy. Immediately, we have G βv ( βb ) < βe b and G βv ( βb ) → βe b as v →
0. So, we can find asufficiently large number c = c ( Y ) > Y ⋐ n t + ϕ + c > max { Θ( t ) , ˜ G βv ( b ) } o ⋐ X \ ( D ∪ F )for any v >
0. Here, ˜ G βv ( b ) = G βv ( βb ) + κ Θ( t ). For simplicity, we write ϕ + c by the samesymbol ϕ .Recall that the property d) of the regularized maximum in Lemma 2.8. If the followinginequality max j = k { t j + η j } < t k − η k holds for some k , we have M η ( t ) = t k . For instance, in our case, if we consider the regiondefined by the following inequalitymax { ˜ G βv ( βb ) + η , t + ϕ + c + η } < Θ( t ) − η , we have M c,v,η = Θ( t ). Note that this region is contained in a sufficiently small neigh-borhood of D . In this case, we don’t have to estimate the scalar curvature S ( ω c,v,η )since S ( ω c,v,η ) = S ( ω ) on this region and the estimate of S ( ω ) have been obtained inLemma 2.1 before. Similarly, if the value of M c,v,η corresponds to one of the other variables˜ G βv ( b ) , t + ϕ + c , Lemma 2.11 and the Ricci-flatness of the K¨ahler metric √− ∂∂ ( t + ϕ ) lmost scalar-flat K¨ahler metrics on affine algebraic manifolds S ( ω c,v,η ) is under control on such regions. Thus, it suffices for us to study the S ( ω c,v,η ) on the other regions defined by the inequalities t k + η k < max j = k { t j − η j } , | t i − t j | < η i + η j , for i, j = k and | t − t | < η + η , | t − t | < η + η , | t − t | < η + η . So we have to study S ( ω c,v,η ) on four regions defined by the inequalities above.Directly, we have ω c,v,η = √− g i,j dz i ∧ dz j = ∂M c,v,η ∂t ω + ∂M c,v,η ∂t ( γ βv + κω ) + ∂M c,v,η ∂t √− ∂∂ ( t + ϕ )+ (cid:2) ∂ Θ( t ) ∂ ˜ G βv ( b ) ∂ ( t + ϕ ) (cid:3) h ∂ M c,v,η ∂t i ∂t j i (cid:2) ∂ Θ( t ) ∂ ˜ G βv ( b ) ∂ ( t + ϕ ) (cid:3) t . It follows from the convexity of M η that the last term is semi-positive. When we com-pute the scalar curvature of ω c,v,η , the difficulty comes from terms ∂ Θ( t ) ∧ ∂ Θ( t ) and ∂ ˜ G βv ( b ) ∧ ∂ ˜ G βv ( b ). For these terms, since functions t and b are defined by Hermitian normsof holomorphic sections, it suffices to focus on derivatives in normal directions of smoothhypersurfaces D and F by taking suitable local trivializations of line bundles L X and K − lX ⊗ L mX respectively. The reason why scalar curvatures of two K¨ahler metrics ω , γ βv are under control near these hypersurfaces D, F is that Ricci curvatures are bounded andK¨ahler metrics grow asymptotically near these hypersurfaces. Thus, it suffices for us tofocus on derivatives of ϕ and M η arising in Ricci tensors. The higher order derivativesof ϕ are estimated in the previous section (Theorem 1.1). In addition, the definition of aparameter η = ( η i ) = ( a c, a c, η ) and Lemma 2.10 imply that the higher order deriva-tives in the first or the second variable of M η are estimated by some negative power of c >
0. To estimate S ( ω c,v,η ) on each region, we divide the proof of Theorem 1.2 into thefollowing four claims. Claim 1.
On the region defined by ( t + ϕ + c ) + η < max { Θ( t ) − η , ˜ G βv ( b ) − η } , | Θ( t ) − ˜ G βv ( b ) | < η + η , we can make the scalar curvature S ( ω c,v ) small arbitrarily by taking a sufficiently large c .Proof. On this region, we can write as ω c,v,η = ∂M c,v,η ∂t ω + ∂M c,v,η ∂t ( γ βv + κω )+ (cid:2) ∂ Θ( t ) ∂ ˜ G βv ( b ) (cid:3) h ∂ M c,v,η ∂t i ∂t j i (cid:2) ∂ Θ( t ) ∂ ˜ G βv ( b ) (cid:3) t . To prove this claim, we need the following lemma. lmost scalar-flat K¨ahler metrics on affine algebraic manifolds Lemma 4.1.
Take a point p ∈ D ∩ F and local holomorphic coordinates ( z , ..., z n − , w F , w D ) centered at p satisfying D = { w D = 0 } and F = { w F = 0 } . By taking suitable lo-cal trivializations of L X and K − lX ⊗ L mX , we may assume that if ( z , ..., z n − , w F , w D ) =(0 , ..., , w F , w D ) , we have ∂ Θ( t ) ∧ ∂ Θ( t ) = O ( | w F | | w D | − S D /n ( n − ) dw F ∧ dw F + O ( | w F || w D | − − S D /n ( n − )( dw D ∧ dw F + dw F ∧ dw D )+ O ( | w D | − − S D /n ( n − ) dw D ∧ dw D ,∂G βv ( βb ) ∧ ∂G βv ( βb ) = O (( | w F | β + v ) − /β | w F | − ) dw F ∧ dw F + O ( | w F | − | w D | ( | w F | β + v ) − /β )( dw D ∧ dw F + dw F ∧ dw D )+ O (( | w F | β + v ) − /β | w D | ) dw D ∧ dw D . From the definition of this region, we obtain ω c,v,η = g , · · · g ,n − g ,n − g ,n ... . . . ... ... ... g n − , · · · g n − ,n − g n − ,n − g n − ,n g n − , · · · g n − ,n − ( | w F | β + v ) − /β | w F | − | w F | − | w D | ( | w F | β + v ) − /β g n, · · · g n,n − | w F | − | w D | ( | w F | β + v ) − /β | w D | − − S D /n ( n − as w D , w F → g i,j for 1 ≤ i, j ≤ n − ω and γ βv . Thus, g , · · · g ,n − ... . . . ... g n − , · · · g n − ,n − = O ( | w D | − S D /n ( n − + ( | w F | β + v ) − /β ) . For other blocks, we similarly have g ,n − g ,n ... ... g n − ,n − g n − ,n = O ( | w D | − S D /n ( n − + ( | w F | β + v ) − /β ) . From Lemma 3.4 in [1], we have g i,j = g , · · · g ,n − g ,n − g ,n ... . . . ... ... ... g n − , · · · g n − ,n − g n − ,n − g n − ,n g n − , · · · g n − ,n − c ( | w F | β + v ) /β | w F | c | w D | S D /n ( n − | w F | g n, · · · g n,n − c | w D | S D /n ( n − | w F | c | w D | S D /n ( n − as w D , w F →
0. Since metric tensors g i,j with i, j = n − , n come from K¨ahler metrics ω and γ βv whose scalar curvature have been already known. Thus, it is enough to study the lmost scalar-flat K¨ahler metrics on affine algebraic manifolds i = n − , n and j = n − , n . Recall that the components of the Ricci tensor aredefined by R i,j := − g p,q ∂ g p,q /∂z i ∂z j + g k,q g p,l ( ∂g k,l /∂z i )( ∂g p,q /∂z j ). So, the Ricci formRic( ω c,v,η ) is written as R , · · · R ,n − R ,n − R ,n ... . . . ... ... ... R n − , · · · R n − ,n − R n − ,n − R n − ,n R n − , · · · R n − ,n − c − ( | w F | β + v ) − /β | w F | − c − | w F | − | w D | ( | w F | β + v ) − /β R n, · · · R n,n − c − | w F | − | w D | ( | w F | β + v ) − /β c − | w D | − − S D /n ( n − as w D , w F → R i,j for 1 ≤ i ≤ n − S ( ω c,v,η ) = O ( c − ) . Remark 4.2.
On the region in the previous claim, there are the terms ∂ Θ( t ) ∧ ∂ Θ( t )and ∂G βv ( βb ) ∧ ∂G βv ( βb ) in the complete K¨ahler metric ω c,v,η . Thus, ( X \ D, ω c,v,η ) is notof asymptotically conical geometry and we can’t use the analysis in Section 5 of [1] withrespect to this K¨ahler metric ω c,v,η . This problem will be solved in [2].We proceed to the estimate of S ( ω c,v,η ) on another region. Claim 2.
Consider the region defined by ˜ G βv ( b ) + η < max { Θ( t ) − η , ( t + ϕ + c ) − η } , | Θ( t ) − ( t + ϕ + c ) | < η + η . Take parameters η, κ so that (1 − κ ) c + κη − η = (1 − κ + κa − a ) c = 0 (4.1) for any c > . Then, we can make the scalar curvature S ( ω c,v ) small arbitrarily by takinga sufficiently large c .Proof. On this region, since M c,v,η = M η (Θ( t ) , t + ϕ + c )from Lemma 2.8, we have ω c,v,η = ∂M c,v,η ∂t ω + ∂M c,v,η ∂t √− ∂∂ ( t + ϕ )+ (cid:2) ∂ Θ( t ) ∂ ( t + ϕ ) (cid:3) h ∂ M c,v,η ∂t i ∂t j i (cid:2) ∂ Θ( t ) ∂ ( t + ϕ ) (cid:3) t . lmost scalar-flat K¨ahler metrics on affine algebraic manifolds G βv ( βb ) < ( t + ϕ + c ) + η − κ Θ( t ) − η < (1 − κ )( t + ϕ + c ) + κ ( η + η ) + η − η = (1 − κ )( t + ϕ ) + (1 + κ ) η . By taking a small v > b in the definition of the function G βv ( βb ), wemay assume that βb < G βv ( βb ) . From a priori estimate due to Ko lodziej [11] again, ϕ is bounded on X . So, on this region,we have the following inequality: || σ F || − β/ (1 − κ ) < C || σ D || − for some constant C > C -norm of ϕ . By taking κ close to 1which depends on m, l and a = a ( n ) in Theorem 1.1, we may assume that || σ F || − − a/l < C || σ D || − am/l . Thus, on this region, the growth of derivatives of ϕ can be controlled by the K¨ahlermetric ω . Take a point in D \ ( D ∩ F ) and local holomorphic coordinates ( z i ) ni =1 =( z , ..., z n − , w D ) satisfying D = { w D = 0 } . Then, we have (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂z i ∂z j ∂ α ϕ (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) | w D | − am/l (cid:1) , if 1 ≤ i, j ≤ n − (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂w D ∂w D ∂ α ϕ (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) | w D | − − am/l (cid:1) . Similarly, we have
Lemma 4.3.
By taking a suitable local holomorphic trivialization of L X , we may assumethat if ( z , ..., z n − , w D ) = (0 , ..., , w D ) , we have ∂ Θ( t ) ∧ ∂ Θ( t ) = O ( | w D | − − S D /n ( n − ) dw D ∧ dw D . Recall the hypothesis am l < ˆ S D n ( n − . So, Theorem 1.1 implies that the growth of the K¨ahler metric ω c,v,η is greater then thegrowth of the higher order derivatives of ϕ . Thus, Lemma 3.4 in [1] shows that higherorder derivatives including ∂ ϕ/∂w ∂w are controlled by taking the trace with respectto ω c,v,η . Therefore, we can ignore derivatives of ϕ arising in the components of the Riccitensor and we have S ( ω c,v,η ) = O ( c − ) . lmost scalar-flat K¨ahler metrics on affine algebraic manifolds S ( ω c,v,η ) the following region. Claim 3.
Consider the region defined by Θ( t ) + η < max { ˜ G βv ( b ) − η , ( t + ϕ + c ) − η } , | ˜ G βv ( b ) − ( t + ϕ + c ) | < η + η . By choosing sufficiently small number v > so that ( || σ F || β + v ) /β < || σ F || am/l holds on this region, we can make the scalar curvature S ( ω c,v ) small arbitrarily by takinga sufficiently large c .Proof. The reason why we can find a sufficiently small number v > {|| σ F ||} on this region increase as v → am/l < Lemma 4.4.
By taking a suitable local trivialization of K − lX ⊗ L mX , we may assume thatif ( z , ..., z n − , w F , z n ) = (0 , ..., , w F , , we have ∂G βv ( βb ) ∧ ∂G βv ( βb ) = O (( | w F | β + v ) − /β | w F | − ) dw F ∧ dw F . Thus, we can prove this claim by using the same way in the previous claim.The remained case is the following claim.
Claim 4.
On the region defined by | Θ( t ) − ˜ G βv ( b ) | < η + η , | ˜ G βv ( b ) − ( t + ϕ + c ) | < η + η , | Θ( t ) − ( t + ϕ + c ) | < η + η , we can make the scalar curvature S ( ω c,v,η ) small arbitrarily by taking a sufficiently large c .Proof. On this region, we can show that S ( ω c,v,η ) = O ( c − ) similarly. Thus, we havefinished proving Theorem 1.2. References [1] T. Aoi, Toward a construction of scalar-flat K¨ahler metrics on affine algebraic mani-folds, arXiv 1907.09780.[2] T. Aoi, Complete scalar-flat K¨ahler metrics on affine algebraic manifolds, arXiv1910.12317.[3] S. Bando and R. Kobayashi, Ricci-flat K¨ahler metrics on affine algebraic manifolds.II, Math. Ann. 287 (1990), 175–180. lmost scalar-flat K¨ahler metrics on affine algebraic manifolds ∼ demailly/manuscripts/agbook.pdf.[7] E. Di Nezza and H. C. Lu, Complex Monge-Amp`ere equations on quasi-projectivevarieties, J. Reine Angew. Math. (2014), DOI 10.1515/crelle-2014-0090.[8] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order(2nd edn.), Grundlehren Math. Wiss. 224. Springer, Berlin, 1983.[9] V. Guedj and A. Zeriahi, Degenerate Complex Monge-Amp`ere Equations, EMS TractsMath, 2017.[10] A. D. Hwang and M. A. Singer, A momentum construction for circle-invariant K¨ahlermetrics, Trans. Amer. Math. Soc. 354 (2002), no. 6, 2285–2325.[11] S. Ko lodziej, The complex Monge-Amp`ere equation, Acta Math. 180 (1998), 69–117.[12] M. Pˇaun, Regularity properties of the degenerate Monge-Amp`ere equations on com-pact K¨ahler manifolds, Chin. Ann. Math. Ser. B. 29 (2008), 623–630.[13] G. Tian and S. T. Yau, Complete K¨ahler manifolds with zero Ricci curvature. II,Invent. Math. 106 (1991), no. 1, 27–60.[14] S. T. Yau, On the Ricci curvature of a compact K¨ahler manifold and the complexMonge-Amp`ere equations, I, Comm. Pure Appl. Math. 31 (1978), 339–411. Department of MathematicsGraduate School of ScienceOsaka UniversityToyonaka 560-0043Japan