Gromov-Hausdorff limit of Kähler manifolds and the finite generation conjecture
aa r X i v : . [ m a t h . DG ] M a y GROMOV-HAUSDORFF LIMIT OF K ¨AHLER MANIFOLDS AND THE FINITEGENERATION CONJECTURE
GANG LIUA bstract . We study the uniformization conjecture of Yau by using the Gromov-Haudor ff convergence. As a consequence, we confirm Yau’s finite generation conjecture. Moreprecisely, on a complete noncompact K¨ahler manifold with nonnegative bisectional curva-ture, the ring of polynomial growth holomorphic functions is finitely generated. Duringthe course of the proof, we prove if M n is a complete noncompact K¨ahler manifold withnonnegative bisectional curvature and maximal volume growth, then M is biholomorphicto an a ffi ne algebraic variety. We also confirm a conjecture of Ni on the existence of poly-nomial growth holomorphic functions on K¨ahler manifolds with nonnegative bisectionalcurvature. Introduction
In [30], Yau proposed to study the uniformization of complete K¨ahler manifolds withnonnegative curvature. In particular, one wishes to determine whether or not a completenoncompact K¨ahler manifold with positive bisectional curvature is biholomorphic to acomplex Euclidean space. For this sake, Yau further asked in [30](see also page 117 in[31]) whether or not the ring of polynomial growth holomorphic functions is finitely gen-erated, and whether or not dimension of the spaces of holomorphic functions of polynomialgrowth is bounded from above by the dimension of the corresponding spaces of polynomi-als on C n . Let us summarize Yau’s questions in the three conjectures below: Conjecture 1.
Let M n be a complete noncompact K¨ahler manifold with positive bisec-tional curvature. Then M is biholomorphic to C n . Conjecture 2.
Let M n be a complete noncompact K¨ahler manifold with nonnegative bi-sectional curvature. Then the ring O P ( M ) is finitely generated. Conjecture 3.
Let M n be a complete noncompact K¨ahler manifold with nonnegative bi-sectional curvature. Then given any d > , dim ( O d ( M )) ≤ dim ( O d ( C n )) . On a complete K¨ahler manifold M , we say a holomorphic function f ∈ O d ( M ), if thereexists some C > | f ( x ) | ≤ C (1 + d ( x , x )) d for all x ∈ M . Here x is a fixed point on M . Let O P ( M ) = ∪ d > O d ( M ).Conjecture 1 is open so far. However, there have been many important progresses due tovarious authors. In earlier works, Mok-Siu-Yau [21] and Mok [20] considered embeddingby using holomorphic functions of polynomial growth. Later, with K¨ahler-Ricci flow,results were improved significantly [27][28][9][24][7].Conjecture 3 was confirmed by Ni [23] with the assumption that M has maximal volumegrowth. Later, by using Ni’s method, Chen-Fu-Le-Zhu [6] removed the extra condition.See also [17] for a di ff erent proof. The key of Ni’s method is a monotonicity formula forheat flow on K¨ahler manifold with nonnegative bisectional curvature. The author was partially supported by NSF grant DMS 1406593.
Despite great progresses of conjecture 1 and conjecture 3, not much is known aboutconjecture 2. In [20], Mok proved the following:
Theorem 1 (Mok) . Let M n be a complete noncompact K¨ahler manifold with positive bi-sectional curvature such that for some fixed point p ∈ M, • Scalar curvature ≤ C d ( p , x ) for some C > ; • Vol ( B ( p , r )) ≥ C r n for some C > .Then M n is biholomorphic to an a ffi ne algebraic variety. In Mok’s proof, the biholomorphism was given by holomorphic functions of polynomialgrowth. Therefore, O P ( M ) is finitely generated. In the general case, it was proved by Ni[23] that the transcendental dimension of O P ( M ) over C is at most n . However, this doesnot imply the finite generation of O P ( M ). The main result in this paper is the confirmationof conjecture 2 in the general case: Theorem 2.
Let M n be a complete noncompact K¨ahler manifold with nonnegative bisec-tional curvature. Then the ring O P ( M ) is finitely generated. During the course of the proof, we obtain a partial result for conjecture 1:
Theorem 3.
Let M n be a complete noncompact K¨ahler manifold with nonnegative bisec-tional curvature. Assume M is of maximal volume growth, then M is biholomorphic to ana ffi ne algebraic variety. Here maximal volume growth means
Vol ( B ( p , r )) ≥ Cr n for some C >
0. This seemsto be the first uniformization type result without assuming the curvature upper bound.If one wishes to prove conjecture 1 by considering O P ( M ), it is important to know when O P ( M ) , C . In [23], Ni proposed the following interesting conjecture: Conjecture 4.
Let M n be a complete noncompact K¨ahler manifold with nonnegative bi-sectional curvature. Assume M has positive bisectional curvature at one point p. Then thefollowing three conditions are equivalent:(1) O P ( M ) , C ;(2)M has maximal volume growth;(3) there exists a constant C independent of r so that − R B ( p , r ) S ≤ Cr . Here S is the scalarcurvature. − R means the average. In complex one dimensional case, the conjecture is known to hold, e.g., [15]. For higherdimensions, Ni proved (1) implies (3) in [23]. The proof used the heat flow method. Thenin [26], Ni and Tam proved that (3) also implies (1). Their proof employs the Poincare-Lelong equation and the heat flow method. Thus, it remains to prove (1) and (2) areequivalent. Under some extra conditions, Ni [24] and Ni-Tam [25] were able to prove theequivalence of (1) and (2). In [18], the author proved that (1) implies (2). In fact, thecondition that M has positive bisectional curvature at one point could be relaxed to that theuniversal cover of M is not a product of two K¨ahler manifolds.In this paper, we prove that (2) also implies (1). Thus conjecture 4 is solved in fullgenerality. More precisely, we prove Theorem 4.
Let ( M n , g ) be a complete K¨ahler manifold with nonnegative bisectional cur-vature and maximal volume growth. Then there exists a nonconstant holomorphic functionwith polynomial growth on M. This theorem might be compared with the following open question in Riemannian ge-ometry:
INITE GENERATION CONJECTURE 3
Conjecture 5.
Let M n be a complete noncompact Riemannian manifold with nonnegativeRicci curvature and maximal volume growth. Then there exists a nonconstant polynomialgrowth harmonic function. The strategy of the proofs in this paper is very di ff erent from earlier works. Here wemake use of several di ff erent techniques: the Gromov-Hausdor ff convergence theory de-veloped by Cheeger-Colding [1][2][3][4], Cheeger-Colding-Tian [5]; the heat flow methodby Ni [23] and Ni-Tam [25][26]; the Hormander L -technique [14][10] ; the three circletheorem [17].The first key point is to prove theorem 4. By Hormander’s L -technique, to produceholomorphic functions of polynomial growth, it su ffi ces to construct strictly plurisubhar-monic function of logarithmic growth. However, it is not obvious how to construct suchfunction by only assuming the maximal volume growth condition. In [21][20], Mok-Siu-Yau and Mok considered the Poincare-Lelong equation √− ∂∂ u = Ric . When the curva-ture has pointwise quadratic decay, they were able to prove the existence of a solution withlogarithmic growth. Later, Ni and Tam [25][26] were able to relax the condition to thatthe curvature has average quadratic decay. Then it su ffi ces to prove that maximal volumegrowth implies the average curvature decay.We prove theorem 4 by a di ff erent strategy. We first blow down the manifold. Thenby using the Cheeger-Colding theory, heat flow technique and Hormander L theory, weconstruct holomorphic functions with controlled growth in a sequence of exhaustion do-mains on M . Then three circle theorem ensures that we can take subsequence to obtain anonconstant holomorphic function with polynomial growth.Once theorem 4 is proved, Hormander’s L technique produces a lot of holomorphicfunctions of polynomial growth. It turns out O P ( M ) separates points and tangent spaces on M . However, since the manifold is not compact, it does not follow directly that M is a ffi nealgebraic. In Mok’s paper [20], the proof of this part took more than 35 pages, even withthe additional assumption that curvature has pointwise quadratic decay.In our case, there is a serious di ffi culty to prove that the map given by O d ( M ) is proper.We overcome this di ffi culty in theorem 11. Again, the idea is new. We will apply theinduction on the dimension of splitting factor for a tangent cone. All techniques above areemployed.Once we prove the properness of the holomorphic map, it is straightforward to prove M is a ffi ne algebraic by using techniques from complex analytic geometry. Here the ar-gument resembles some part in [11]. Then we conclude conjecture 2 when the manifoldhas maximal volume growth. For the general case, we apply the main result in [18]. Itsu ffi ces to handle the case when the universal cover of the manifold splits. Then we needto consider group actions. The final result follows from an algebraic result of Nagata [22].This paper is organized as follows. In section 2, we collect some preliminary resultsnecessary for this paper. In section 3, we prove a result which controls the size of a holo-morpic chart when the manifold is Gromov-Hausdor ff close to an Euclidean ball. As thefirst application, we prove in section 4 a gap theorem for the complex structure of C n . Sec-tion 5 deals with the proof of theorem 4. The proof of theorem 11 is contained in section6. Finally, the proof of theorem 2 is given section 7.There are two appendices. For appendix A , we present a result of Ni-Tam in [25] whichwas not stated explicitly (here we are not claiming any credits). In appendix B , we intro-duce some results of Nagata [22] to conclude the proof of the main theorem. Acknowledgment
GANG LIU
The author would like to express his deep gratitude to Professors John Lott, Lei Ni, JiapingWang for many valuable discussions during the work. He thanks Professor Xinyi Yuanfor some discussions on algebra. He also thanks Professors Peter Li, Luen-Fai Tam andShing-Tung Yau for the interest in this work.2.
Preliminary results
First recall some convergence results for manifolds with Ricci curvature lower bound.Let ( M ni , y i , ρ i ) be a sequence of pointed complete Riemannian manifolds, where y i ∈ M ni and ρ i is the metric on M ni . By Gromov’s compactness theorem, if ( M ni , y i , ρ i ) havea uniform lower bound of the Ricci curvature, then a subsequence converges to some( M ∞ , y ∞ , ρ ∞ ) in the Gromov-Hausdor ff topology. See [12] for the definition and basicproperties of Gromov-Hausdor ff convergence. Definition.
Let K i ⊂ M ni → K ∞ ⊂ M ∞ in the Gromov-Hausdor ff topology. Assume { f i } ∞ i = are functions on M ni , f ∞ is a function on M ∞ . Φ i are ǫ i -Gromov-Hausdor ff approximations, lim i →∞ ǫ i = . If f i ◦ Φ i converges to f ∞ uniformly, we say f i → f ∞ uniformly over K i → K ∞ . In many applications, f i are equicontinuous. The Arzela-Ascoli theorem applies to thecase when the spaces are di ff erent. When ( M ni , y i , ρ i ) → ( M ∞ , y ∞ , ρ ∞ ) in the Gromov-Hausdor ff topology, any bounded, equicontinuous sequence of functions f i has a subse-quence converging uniformly to some f ∞ on M ∞ .Let the complete pointed metric space ( M m ∞ , y ) be the Gromov-Hausdor ff limit of asequence of connected pointed Riemannian manifolds, { ( M ni , p i ) } , with Ric ( M i ) ≥
0. Here M m ∞ has Haudor ff dimension m with m ≤ n . A tangent cone at y ∈ M m ∞ is a completepointed Gromov-Hausdor ff limit (( M ∞ ) y , d ∞ , y ∞ ) of { ( M ∞ , r − i d , y ) } , where d , d ∞ are themetrics of M ∞ , ( M ∞ ) y respectively, { r i } is a positive sequence converging to 0. Definition.
A point y ∈ M ∞ is called regular, if there exists some k so that every tangentcone at y is isometric to R k . A point is called singular, if it is not regular. In [2], the following theorem was proved:
Theorem 5.
Regular points are dense in the Gromov-Haudor ff limit of manifolds withRicci curvature lower bound. Results of heat flow on K¨ahler manifolds by Ni-Tam [25]:
Theorem 6.
Let M n be a complete noncompact K¨ahler manifold with nonnegative bisec-tional curvature. Let u be a smooth function on M with compact support. Letv ( x , t ) = Z M H ( x , y , t ) u ( y ) dy . Here H ( x , y , t ) is the heat kernel of M. Let η ( x , t ) αβ = v αβ and λ ( x ) be the minimumeigenvalue for η ( x , . Let λ ( x , t ) = Z M H ( x , y , t ) λ ( y ) dy . Then η ( x , t ) − λ ( x , t ) g αβ is a nonnegative (1 , tensor for t ∈ [0 , T ] for T > . A detailed proof of this theorem is presented in appendix A .Hormander L theory: INITE GENERATION CONJECTURE 5
Theorem 7.
Let ( X , ω ) be a connected but not necessarily complete K¨ahler manifold withRic ≥ . Assume X is Stein. Let ϕ be a C ∞ function on X with √− ∂∂ϕ ≥ c ω for some posi-tive function c on X. Let g be a smooth (0 , form satisfying ∂ g = and R X | g | c e − ϕ ω n < + ∞ ,then there exists a smooth function f on X with ∂ f = g and R X | f | e − ϕ ω n ≤ R X | g | c e − ϕ ω n . The proof can be found in [10], page 38-39. Also compare lemma 4.4.1 in [14].Three circle theorem in [17]:
Theorem 8.
Let M be a complete noncompact K¨ahler manifold with nonnegative holo-morphic sectional curvature, p ∈ M. Let f be a holomorphic function on M. Let M ( r ) = max B ( p , r ) | f ( x ) | . Then log M ( r ) is a convex function of log r. Therefore, given any k > , M ( kr ) M ( r ) is monotonic increasing. This theorem has the following consequences:
Corollary 1.
Given the same condition as in theorem 8. If f ∈ O d ( M ) , then M ( r ) r d is non-increasing. Corollary 2.
Given the same condition as in theorem 8. If f ( p ) = , then M ( r ) r is non-decreasing. Remark.
The three circle theorem is still true for holomorphic sections on nonpositivebundles. See page of [17] for a proof. A multiplicity estimate by Ni [23](see also [6]):
Theorem 9.
Let M n be a complete noncompact K¨ahler manifold with nonnegative bisec-tional curvature. Then dim ( O d ( M )) ≤ dim ( O d ( C n )) . Note this result also follows from corollary 1.In this paper, we will denote by Φ ( u , ..., u k | .... ) any nonnegative functions dependingon u , ..., u k and some additional parameters such that when these parameters are fixed,lim u k → · · · lim u → Φ ( u , ..., u k | ... ) = . Let C ( n ) , C ( n , v ) be large positive constants depending only on n or n , v ; c ( n ) , c ( n , v ) besmall positive constants depending only on n or n , v . The values might change from line toline. 3. Construct holomorphic charts with uniform size
In this section, we introduce the following proposition which is crucial for the construc-tion of holomorphic functions.
Proposition 1.
Let M n be a complete K¨ahler manifold with nonnegative bisectional cur-vature, x ∈ M. There exist ǫ ( n ) > , δ = δ ( n ) > so that the following holds: For ǫ < ǫ ( n ) ,if d GH ( B ( x , ǫ r ) , B C n (0 , ǫ r )) < ǫ r, there exists a holomorphic chart ( w , ...., w n ) containingB ( x , δ r ) so that • w s ( x ) = ≤ s ≤ n ) . • | n P s = | w s | ( y ) − r ( y ) | ≤ Φ ( ǫ | n ) r in B ( x , δ r ) . Here r ( y ) = d ( x , y ) . • | dw s ( y ) | ≤ C ( n ) in B ( x , δ r ) . GANG LIU
Proof.
By scaling, we may assume r >> R = r >> .
23) in[3]), there exist real harmonic functions b , ..., b n in B ( x , r ) so that(1) − Z B ( x , r ) X j |∇ ( ∇ b j ) | + X j , l |h∇ b j , ∇ b l i − δ jl | ≤ Φ ( ǫ | n , r )and(2) b j ( x ) = ≤ j ≤ n ); |∇ b j | ≤ C ( n )in B ( x , r ). Moreover, the map F ( y ) = ( b ( y ) , ..., b n ( y )) is a Φ ( ǫ | n ) r Gromov-Hausdor ff approximation from B ( x , r ) to B R n (0 , r ). According to the argument above lemma 9 . − Z B ( x , r ) | J ∇ b s − − ∇ b s | ≤ Φ ( ǫ | n , r )for 1 ≤ s ≤ n . Set w ′ s = b s − + √− b s . Then(4) − Z B ( x , r ) | ∂ w ′ s | ≤ Φ ( ǫ | n , r ) . The idea is to perturb w ′ s so that they become a holomorphic chart. We would like to applythe Hormander L -estimate. First, we construct the weight function. Consider the function h ( y ) = n X j = b j ( y ) . Then in B ( x , r ),(5) | h ( y ) − r ( y ) | ≤ Φ ( ǫ | n ) r . By (2),(6) |∇ h ( y ) | ≤ C ( n ) r ( y )in B ( x , r ). The real Hessian of h satisfies(7) Z B ( x , R ) X u , v | h uv ( y ) − g uv | ≤ Φ ( ǫ | n , R ) . Now consider a smooth function ϕ : R + → R + with ϕ ( t ) = t for 0 ≤ t ≤ ϕ ( t ) = t ≥ | ϕ | , | ϕ ′ | , | ϕ ′′ | ≤ C ( n ). Let H ( x , y , t ) be the heat kernel on M and set(8) u ( y ) = R ϕ ( h ( y )5 R ) , u t ( z ) = Z M H ( z , y , t ) u ( y ) dy . Claim 1. u ( z ) satisfies that ( u ) αβ ( z ) ≥ c ( n ) g αβ > in B ( x , R ) .Proof. Let λ ( y ) be the lowest eigenvalue of the complex hessian u αβ . By (7), − Z B ( x , R ) | h αβ − g αβ | ≤ Φ ( ǫ | n , R ) . Then there exists E ⊂ B ( x , R ) with(9) vol ( B ( x , R ) \ E ) ≤ Φ ( ǫ | n , R ); h αβ ≥ g αβ INITE GENERATION CONJECTURE 7 on E . By (5), we may assume h ( y ) ≤ R in B ( x , R ). Then u = h in B ( x , R ). We have(10)( Z B ( x , R ) \ E | λ ( y ) | dy ) ≤ ( Z B ( x , R ) \ E X α,β | h αβ | ) ≤ Z B ( x , R ) \ E X α,β | h αβ − g αβ | dy ) + Z B ( x , R ) \ E X α,β | g αβ | dy ) ≤ Φ ( ǫ | n , R ) . (11) | λ | ≤ | u αβ | = | ϕ ′ h αβ + ϕ ′′ R h α h β |≤ | ϕ ′ ( h αβ − g αβ ) | + | ϕ ′ g αβ + ϕ ′′ R h α h β |≤ C ( n )( | h αβ − g αβ | + . Therefore,(12) Z B ( x , R ) | λ ( y ) | dy ≤ C ( n ) R n . Let λ ( z , = R H ( z , y , λ ( y ) dy . Note by definition (8), u is supported in B ( x , R ). By (9), λ ≥ in E . For z ∈ B ( x , R ),(13) Z H ( z , y , λ ( y ) dy = Z B ( x , R ) H ( z , y , λ ( y ) dy ≥ Z B ( x , R ) \ E H ( z , y , λ ( y ) dy + Z B ( x , R ) \ B ( x , R ) H ( z , y , λ ( y ) dy + Z E ∩ B ( z , H ( z , y , λ ( y ) dy . By heat kernel estimate of Li-Yau [19], H ( z , y , ≥ c ( n ) > y ∈ B ( z , H ( z , y , ≤ C ( n ) for y , z ∈ B ( x , R ).(14) Z B ( x , R ) \ E | H ( z , y , λ ( y ) | dy ≤ C ( n ) Z B ( x , R ) \ E | λ ( y ) | dy ≤ C ( n )( Z B ( x , R ) \ E | λ ( y ) | dy ) ( vol ( B ( x , R ) \ E )) ≤ Φ ( ǫ | n , R )(15) Z E ∩ B ( z , H ( z , y , λ ( y ) dy ≥ Z E ∩ B ( z , H ( z , y , dy ≥ c ( n ) > . Note d ( y , z ) ≥ R for y ∈ B ( x , R ) \ B ( x , R ). Heat kernel estimate says H ( y , z , ≤ C ( n ) e − R . Therefore, by (12),(16) Z B ( x , R ) \ B ( x , R ) | H ( z , y , λ ( y ) | dy ≤ C ( n ) e − R R n < Φ ( 1 R ) . Putting (14), (15), (16) in (13), we find(17) λ ( z , = Z H ( z , y , λ ( y ) dy ≥ c ( n ) − Φ ( 1 R | n ) − Φ ( ǫ | n , R ) GANG LIU for z ∈ B ( x , R ). We first let R be large, then ǫ be very small. Then λ ( z , > c ( n ). Weconclude the proof of the claim from theorem 6. (cid:3) Recall u t is defined in (8). We claim that there exists ǫ = ǫ ( n ) > R ,(18) min y ∈ ∂ B ( x , R ) u ( y ) > y ∈ B ( x ,ǫ R ) u ( y ) . This is a simple exercise by using the heat kernel estimate. One can also apply proposition5 to conclude the proof. From now on, we freeze the value of R . That is to say, R = R ( n ) > R ǫ > Ω be the connected component of { y ∈ B ( x , R ) | u ( y ) < y ∈ B ( x ,ǫ R ) u ( y ) } containing B ( x , ǫ R ). Then Ω is relatively compact in B ( x , R ) and Ω is a Stein manifold by claim 1.Now we apply theorem 7 to Ω , with the K¨ahler metric induced from M . Take smooth(0 ,
1) forms g s = ∂ w ′ s defined in (4); the weight function ψ = u . We find smooth functions f s in Ω with ∂ f s = g s and(19) Z Ω | f s | e − ψ ω n ≤ Z Ω | g s | c e − ψ ω n ≤ R Ω | ∂ w ′ s | ω n c ( n ) ≤ Φ ( ǫ | n ) . Here we used the fact that r = R = R ( n ). By proposition 5, we find ψ = u ≤ C ( R , n ) = C ( n ) in B ( x , R ). Therefore(20) Z B ( x , | f s | ω n ≤ Z Ω | f s | ω n ≤ Φ ( ǫ | n ) . Note w s = w ′ s − f s is holomorphic, as ∂ w s = ∂ w ′ s − g s =
0. Since w ′ s is harmonic(complex), f s is also harmonic. By the mean value inequality [16] and Cheng-Yau’s gradient estimate[8], we find that in B ( x , | f s | ≤ Φ ( ǫ | n ); |∇ f s | ≤ Φ ( ǫ | n ) . Therefore, (1) implies(22) Z B ( x , | ( w s ) i ( w t ) j g i j − δ st | ≤ Φ ( ǫ | n ) . Claim 2. w s ( s = , ...., n ) is a holomorphic chart in B ( x , .Proof. Recall that ( b , ..., b n ) is an Φ ( ǫ | n ) Gromov Hausdor ff approximation to the imagein R n . According to (21), on B ( x , w = ( w , ...., w n ) is also a Φ ( ǫ | n ) Gromov Hausdor ff approximation to B C n (0 , w − ( B C n (0 , B ( x , + Φ ( ǫ | n )). First weprove the degree d of the map w is 1. By (22) and that holomorphic maps preserves theorientation, d ≥
1. We also have(23) d · vol ( B C n (0 , = − √− n Z w − ( B (0 , dw ∧ dw ∧ · · · ∧ dw n ∧ dw n ≤ (1 + Φ ( ǫ | n )) Vol ( B ( x , (1 + Φ ( ǫ | n ))) + Φ ( ǫ | n )by (21) and (22). This means that if ǫ is su ffi ciently small, d =
1. That is to say ( w , ..., w n )is generically one to one in B ( x , w , ...., w n ) must be a finite map: thepreimage of a point must be a subvariety which is compact in the Stein manifold Ω , thusfinitely many points. According to Remmert’s theorem in complex analytic geometry, thisis an isomorphism. (cid:3) We can make a small perturbation so that w s ( x ) = ≤ s ≤ n . This completes theproof of proposition 1. (cid:3) INITE GENERATION CONJECTURE 9 A gap theorem for complex structure of C n As the first application of proposition 1, we prove a gap theorem for the complex struc-ture of C n . The conditions are rather restrictive. However, we shall expand some of theideas in later sections. Theorem 10.
Let M n be a complete noncompact K¨ahler manifold with nonnegative bisec-tional curvature and p ∈ M. There exists ǫ ( n ) > so that if ǫ < ǫ ( n ) and (24) vol ( B ( p , r )) r n ≥ ω n − ǫ for all r > , then M is biholomorphic to C n . Here ω n is the volume of the unit ball in C n .Furthermore, the ring O P ( M ) is finitely generated. In fact, it is generated by n functionswhich form a coordinate in C n .Proof. By scaling if necessary, we may assume B ( p ,
1) is C close to the Euclidean ball B (0 ,
1) in R n . In particular, ∂ B ( p ,
1) is C close to the sphere S n − . Consider the blowdown sequence ( M i , p i , g i ) = ( M , p , s i g ) for s i → ∞ . According to proposition 1 andthe Cheeger-Colding theory [1], if ǫ is su ffi ciently small, there exists a holomorphic chart( w i , ...., w in ) on B ( p i , w i , ...., w in ) is a Φ ( ǫ | n ) Gromov-Hausdor ff approximation to B C n (0 , w is ( p i ) = s = , ..., n . We can also regard w is as holomorphic functions on B ( p , s i ) ⊂ M . For each i , we can find a new basis v is for span { w is } so that(26) Z B ( p , v is v it = δ st . Set(27) M is ( r ) = max x ∈ B ( p , r ) | v is ( x ) | . Claim 3. M is ( s i ) M is ( s i ) ≤ + Φ ( ǫ | n ) for ≤ s ≤ n.Proof. It su ffi ces to prove this for s =
1. Let v i = n P j = c ij w ij . Without loss of generality,assume | c i | = max ≤ j ≤ n | c ij | >
0. Then v i c i = w i + n P j = α i j w ij and | α i j | ≤
1. Since on M i ,( w i , ..., w in ) is a Φ ( ǫ | n ) Gromov Hausdor ff approximation to B C n (0 , M is ( s i ) M is ( s i ) = max x ∈ B ( p , s i ) | w i ( x ) + n P j = α i j w ij ( x ) | max x ∈ B ( p , si ) | w i ( x ) + n P j = α i j w ij ( x ) | = max x ∈ B ( p i , | w i ( x ) + n P j = α i j w ij ( x ) | max x ∈ B ( p i , ) | w i ( x ) + n P j = α i j w ij ( x ) |≤ + Φ ( ǫ | n ) . This concludes the proof. (cid:3)
According to the three circle theorem 8, M is (2 r ) M is ( r ) is monotonic increasing for 0 < r < s i .Then claim 3 implies(29) M is (2 r ) M is ( r ) ≤ + Φ ( ǫ | n )for 0 < r < s i . From (26), we find M is ( ) ≤ C ( n ). (29) implies(30) M is ( r ) ≤ C ( n )( r α + α = + Φ ( ǫ | n ). As s i → ∞ , by taking subsequence, we can assume v is → v s uniformlyon each compact set of M . Set(31) M s ( r ) = max x ∈ B ( p , r ) | v s ( x ) | . Then(32) M s ( r ) ≤ C ( n )( r α + α = + Φ ( ǫ | n ) and r ≥
0. We may assume v s ∈ O ( M ). Note that v s also satisfies(33) v s ( p ) = ≤ s ≤ n ); Z B ( p , v s v t = δ st . Our goal is to prove ( v , ..., v n ) is a biholomorphism from M to C n . Claim 4.
Let ǫ in (24) be su ffi ciently small(depending only on n). If we rescale each v s sothat max B ( p , | v s | = , then in B ( p , , ( v , ...., v n ) is a n -Gromov-Haudor ff approximation toB C n (0 , .Proof. We argue by contradiction. Assume a positive sequence ǫ i → M ′ i , q i ) isa sequence of n -dimensional complete noncompact K¨ahler manifolds with nonnegativebisectional curvature and(34) vol ( B ( q i , r )) r n ≥ ω n − ǫ i for all r >
0. Assume there exist holomorphic functions u is ( s = , ..., n ) on M ′ i so that(35) u is ( q i ) = u is ∈ O ( M ′ i ); Z B ( q i , u is u it = c ist δ st ; max B ( q i , | u is | = . Here c ist are constants. Assume in B ( q i , u i , ..., u in ) is not a n -Gromov-Haudor ff ap-proximation to B C n (0 , M ′ i , q i ) con-verges to ( R n ,
0) in the pointed Gromov-Hausdor ff sense. By the three circle theoremand (35), we have uniform bound for u is in B ( q i , r ) for any r >
0. Let i → ∞ , there is asubsequence so that u is → u s uniformly on each compact set. Moreover, by remark 9 . R n . Thus we canidentify the limit space with C n . By lemma 4 in [18], the limit of holomorphic functionsare still holomorphic. Moreover, { u s } satisfy (35), according to the three circle theorem.Thus u s are all linear functions which form a standard complex coordinate in C n . There-fore in B C n (0 , u , ..., u n ) is an isometry to B C n (0 , u i , ..., u in ) is not a n -Gromov-Haudor ff approximation to B C n (0 , (cid:3) Recall that B ( p ,
1) is C close to the Euclidean ball B (0 ,
1) and ∂ B ( p ,
1) is C close tothe sphere S n − . By the degree theory and claim 4, we find that the degree of the map( v , ...., v n ) in B ( p ,
1) is 1. This means dv ∧ · · · ∧ dv n is not identically zero. By (32)and Cheng-Yau’s gradient estimate, | dv i | ≤ C ( n )( r Φ ( ǫ | n ) + | dv ∧ · · · ∧ dv n | ≤ INITE GENERATION CONJECTURE 11 C ( n )( r Φ ( ǫ | n ) + K M has nonpositive curvature. Note by theremark following corollary 2, three circle theorem also holds for holomorphic sections ofnonpositive bundles. Therefore, if the holomorphic n -form dv ∧ · · · ∧ dv n vanishes at somepoint in M , then | dv ∧···∧ dv n | must be of at least linear growth, by corollary 2. Therefore, dv ∧ · · · ∧ dv n is vanishing identically on M . This is a contradiction.Next we prove the map ( v , ...., v n ): M → C n is proper. Given any R >
1, we can definea norm | · | R for the span of v , ..., v n induced by R B ( p , R ) v s v t . There exists a basis v R , ...., v Rn for the span of v , ..., v n so that(36) Z B ( p , v Rs v Rt = δ st ; Z B ( p , R ) v Rs v Rt = c ( R ) st δ st . Here c ( R ) st are constants. That is, we diagonalize the two norms |·| and |·| R simultaneously.Obviously we have(37) n X s = | v s ( x ) | = n X s = | v Rs ( x ) | for any x ∈ M . To prove ( v , ..., v n ) is proper, it su ffi ces to prove n P s = | v Rs ( x ) | is large for x ∈ ∂ B ( p , R ) and large R . Define(38) w Rs ( x ) = v Rs ( x ) c Rs where c Rs are positive constants so that(39) max x ∈ B ( p , R ) w Rs ( x ) = s = , ...., n . Note R B ( p , | v Rs | = v Rs ( p ) =
0. According to corollary 2 and (36),(40) c Rs ≥ cR , where c = c ( n ) > R >
1. We can apply claim 4 to v Rs in B ( p , R ). Here we have torescale the radius to 1. Then we obtain that ( Rw R , ...., Rw Rn ) is a R n -Gromov-Hausdor ff approximation from B ( p , R ) to B C n (0 , R ). In particular, for any x ∈ ∂ B ( p , R ), there existssome s with | w Rs ( x ) | ≥ n . Then(41) | v Rs ( x ) | = c Rs | w Rs ( x ) | ≥ n cR ; n X s = | v s ( x ) | = n X s = | v Rs ( x ) | ≥ c ( n ) R . The properness is proved.As dv ∧ · · · ∧ dv n is not vanishing at any point on M and ( v , ...., v n ) is a proper map to C n , we conclude that ( v , ...., v n ) is a biholomorphism from M to C n .Next we prove O P ( M ) is generated by ( v , ...., v n ). We can regard ( v , ..., v n ) as a holo-morphic coordinate system on M . If f ∈ O d ( M ), we can think f = f ( v , ...., v n ). It su ffi cesto prove the right hand side is a polynomial. Indeed, | f ( x ) | ≤ C (1 + d ( x , p ) d ). Note by (41), | f ( v , ..., v n ) | ≤ C (( n P s = | v s | ) d + f is a polynomial of v , ...., v n . (cid:3) Proof of theorem 4
Proof.
We only consider the case for n ≥
2. Otherwise, the result is known. Pick p ∈ M .Let(42) α = lim r →∞ vol ( B ( p , r )) r n > . Consider the blow down sequence ( M i , p i , g i ) = ( M , p , s i g ) for s i → ∞ . By Cheeger-Colding theory [1], a subsequence converges to a metric cone ( X , p ∞ , d ∞ ). Define(43) r ( x ) = d ∞ ( x , p ∞ ) , x ∈ X ; r i ( x ) = d g i ( x , p i ) , x ∈ M i . Now pick two regular points y , z ∈ X with(44) r ( y ) = r ( z ) = d ∞ ( y , z ) ≥ c ( n , α ) > . Note the latter inequality is guaranteed by theorem 5. There exists δ > B ( y , δ ) ∩ B ( z , δ ) = Φ ; δ <
110 ;(46) d GH ( B ( y , ǫ δ ) , B R n (0 , ǫ δ )) ≤ ǫδ ; d GH ( B ( z , ǫ δ ) , B R n (0 , ǫ δ )) ≤ ǫδ . Here ǫ = ǫ ( n ), which is given by proposition 1. Therefore, if i is su ffi ciently large, wecan find points y i , z i ∈ M i with r i ( y i ) = r i ( z i ) = d GH ( B ( y i , ǫ δ ) , B R n (0 , ǫ δ )) ≤ ǫδ ; d GH ( B ( z i , ǫ δ ) , B R n (0 , ǫ δ )) ≤ ǫδ . Let w is and v is be the local holomorphic charts around y i and z i constructed in proposition1. Note that they have uniform size (independent of i ). By changing the value of δ , wemay assume w is , v is are holomorphic charts in B ( y i , δ ) and B ( z i , δ ). Moreover,(48) | dw is | , | dv is | ≤ C ( n ); w is ( y i ) = , v is ( z i ) = | n X s = | w is ( y ) | − d g i ( y , y i ) | ≤ Φ ( ǫ | n ) δ ;(50) | n X s = | v is ( z ) | − d g i ( z , z i ) | ≤ Φ ( ǫ | n ) δ for y ∈ B ( y i , δ ) , z ∈ B ( z i , δ ). We need to construct a weight function on B ( p i , R ) for somelarge R to be determined later. The construction is similar to proposition 1. Set(51) A i = B ( p i , R ) \ B ( p i , R ) . By Cheeger-Colding theory [1]((4.43) and (4.82)), there exists a smooth function ρ i on M i so that(52) Z A i |∇ ρ i − ∇ r i | + |∇ ρ i − g i | < Φ ( 1 i | R );(53) | ρ i − r i | < Φ ( 1 i | R )in A i . According to (4 . .
23) in [1],(54) ρ i =
12 ( G i ) − n ; ∆ G i ( x ) = , x ∈ B ( p i , R ) \ B ( p i , R ); INITE GENERATION CONJECTURE 13 (55) G i = r − ni on ∂ ( B ( p i , R ) \ B ( p i , R )) . Now(56) |∇ ρ i ( y ) | = C ( n ) |G i | n − n |∇G i ( y ) | . By (53)-(55) and Cheng-Yau’s gradient estimate,(57) |∇ ρ i ( y ) | ≤ C ( n ) r i ( y )for y ∈ A i and su ffi ciently large i . Now consider a smooth function ϕ : R + → R + given by ϕ ( t ) = t for t ≥ ϕ ( t ) = ≤ t ≤ | ϕ | , | ϕ ′ | , | ϕ ′′ | ≤ C ( n ). Let(58) u i ( x ) = R ϕ ( R ρ i ( x )) . We set u i ( x ) = x ∈ B ( p i , R ). Then u i is smooth in B ( p i , R ). Claim 5.
For su ffi ciently large i, R B ( p i , R ) |∇ u i −∇ r i | + |∇ u i − g i | < Φ ( R ); | u i − r i | < Φ ( R ) and |∇ u i | ≤ C ( n ) r i in B ( p i , R ) .Proof. We have(59) ∇ u i ( x ) = ϕ ′ ( R ρ i ( x )) ∇ ρ i ( x );(60) ∇ u i ( x ) = R ϕ ′′ ( R ρ i ( x )) ∇ ρ i ⊗ ∇ ρ i + ϕ ′ ( R ρ i ( x )) ∇ ρ i . The proof follows from a routine calculation, by (53), (54), (57). (cid:3)
Similar as in proposition 1, consider a smooth function ϕ : R + → R + with ϕ ( t ) = t for0 ≤ t ≤ ϕ ( t ) = t ≥ | ϕ | , | ϕ ′ | , | ϕ ′′ | ≤ C ( n ). Set(61) v i ( z ) = R ϕ ( u i ( z )3 R ) , v i , t ( z ) = Z M H i ( z , y , t ) v i ( y ) dy . Here H i ( x , y , t ) is the heat kernel on M i . Then v i is supported in B ( p i , R ). By similararguments as in claim 1, we arrive at the following: Proposition 2. v i , ( z ) satisfies that ( v ) αβ ( z ) ≥ c ( n , α ) g αβ > for z ∈ B ( p i , R ) . Here α > is given by (42). Now define(62) q i ( x ) = n (log( n X s = | w is | ) λ (4 n P s = | w is | δ ) + log( n X s = | v is | ) λ (4 n P s = | v is | δ )) . Here λ is a standard cut-o ff function R + → R + with λ ( t ) = ≤ t ≤ λ ( t ) = t ≥
2. Note by (49) and (50), q i ( x ) has compact support in B ( y i , δ ) ∪ B ( z i , δ ) ⊂ B ( p i , Lemma 1. √− ∂∂ q i ≥ − C ( n , δ ) ω i . Moreover, e − q i ( x ) is not locally integrable at y i and z i .Proof. (63) | √− ∂∂ | w is | | = | ∂ w is ∧ ∂ w is | ≤ | dw is | ≤ C ( n )in B ( y i , δ ). When λ ′ (4 n P s = | w is | δ ) , δ ≥ n X s = | w is | ≥ δ . Also note(65) √− ∂∂ log( n X s = | w is | ) ≥ x ∈ B ( y i , δ ), e − q i ( x ) = n P s = | w is | ) n . As w is ( y i ) = s , asimple calculation shows e − q i ( x ) is not locally integrable at y i . The same argument worksfor z i . (cid:3) Putting proposition 2 and lemma 1 together, we find C ( n , α, δ ) > √− ∂∂ ( q i ( x ) + C ( n , α, δ ) v i , ( x )) ≥ ω i in B ( p i , R ). Set(67) ψ i ( x ) = q i ( x ) + C ( n , α, δ ) v i , ( x ) . By the same argument as in proposition 1, we find ǫ = ǫ ( α, n ) > ffi -ciently large R ,(68) min y ∈ ∂ B ( p i , R ) v i , ( y ) > y ∈ B ( p i ,ǫ R ) v i , ( y ) . Of course, we can assume(69) ǫ R > . From now on, we freeze the value of R . That is,(70) R = R ( n , α ) > Ω i be the connected component of { y ∈ B ( p i , R ) | v i , ( y ) < y ∈ B ( p i ,ǫ R ) v i , ( y ) } containing B ( p i , ǫ R ). Then Ω i is relatively compact in B ( p i , R ) and Ω i is a Stein manifold, by proposition 2. Also B ( p i , ⊂ Ω i .Now consider a function f i ( x ) = x ∈ B ( y i , δ ); f i has compact support in B ( y i , δ ) ⊂ B ( p i , |∇ f i | ≤ C ( n , α, δ ). We solve the equation ∂ h i = ∂ f i in Ω i with(71) Z Ω i | h i | e − ψ i ≤ Z Ω i | ∂ f i | e − ψ i ≤ C ( n , α, δ ) . By lemma 1, h i ( y i ) = h i ( z i ) =
0. Therefore, the holomorphic function µ i = f i − h i is notconstant in Ω i . It is easy to see that ψ i ( x ) ≤ C ( n , α, δ ) in B ( p i , C ( n , α, δ ) Z B ( p i , | h i | ≤ Z Ω i | h i | e − ψ i ≤ C ( n , α, δ ) . Thus(73) Z B ( p i , | µ i | ≤ Z B ( p i , ( | h i | + | f i | ) ≤ C ( n , α, δ ) . Mean value inequality implies that(74) | µ i ( x ) | ≤ C ( n , α, δ )for x ∈ B ( p i , ν ∗ i ( x ) = µ i ( x ) − µ i ( p i ) INITE GENERATION CONJECTURE 15 is uniformly bounded in B ( p i , M ′ i ( r ) = max x ∈ B ( p i , r ) | ν i ( x ) | . Then(77) M ′ i (2) ≤ C ( n , α, δ ) . On the other hand, as µ i ( y i ) = f i ( y i ) − h i ( y i ) = µ i ( z i ) = f i ( z i ) − h i ( z i ) =
0, we find(78) M ′ i (1) ≥ . Therefore(79) M ′ i (2) M ′ i (1) ≤ C ( n , α, δ ) . Now we are ready to apply the three circle theorem. More precisely, we consider therescale functions ν ∗ i = β i ν ∗ i in B ( p , s i ) ⊂ M . Here β i are constants so that(80) Z B ( p , | ν ∗ i | = . This implies(81) | ν ∗ i | ≤ C ( n , α )in B ( p , M i ( r ) = max x ∈ B ( p , r ) | ν ∗ i | . The three circle theorem says M i (2 r ) M i ( r ) is monotonicincreasing for 0 < r ≤ s i . By (79) and similar arguments as in (32), we obtain that(82) M i ( r ) ≤ C ( n , α, δ )( r C ( n ,α,δ ) + i and s i ≥ r . Let i → ∞ , a subsequence of ν ∗ i converges uniformly on each compactset to a holomorphic function v of polynomial growth. v cannot be constant, as v satisfies v ( p ) = R B ( p , | v | =
1. Moreover, the degree at infinity is bounded by C ( n , α, δ ). (cid:3) Remark.
By Gromov compactness theorem, we can find δ = δ ( n , α ) , y , z satisfying(44), (45) and (46). Therefore, the degree of the holomorphic function at infinity is boundedby C ( n , α ) . The dependence on α is obvious necessary if we look at the complex onedimensional case. Corollary 3.
Let M n be a complete K¨ahler manifold with nonnegative bisectional cur-vature and maximal volume growth. Then the transcendental dimension of polynomialgrowth holomorphic functions is n. Moreover, O P ( M ) separates points and tangents on M.Proof. From theorem 4, there exists a nonconstant holomorphic function f of polynomialgrowth. First we assume the universal cover of M does not split as products. Then bytheorem 3 . | f | + L estimate(forexample, theorem 5 . O P ( M ) separates points and tangents on M . Together with the multiplicity estimate theorem 9, we proved that the transcendentaldimension of holomorphic functions of polynomial growth over C is n . If the universalcovering splits, we work on the universal covering space. Each factor must be of maxi-mal volume growth. Then we can find nonconstant holomorphic functions of polynomialgrowth. Then we run the heat flow for each factor to obtain strictly plurisubharmonic func-tions of logarithmic growth. Then we add these function together, which is still strictlyplurisubharmonic. Finally, to put these functions back to M , just observe that π ( M ) is finite, then we can symmetrize the function. Then it projects to M , still with logarithmicgrowth. Then the argument is the same for the nonsplitting case. (cid:3) Remark.
In this case, one can actually prove M n is biholomorphic to a quasi-a ffi ne va-riety. This follows from Mok’s deep work in [20] . However, with the aid of the theorembelow, we shall give a direct proof that M n is biholomorphic to an a ffi ne algebraic variety. A properness theoremTheorem 11.
Let M n be a complete noncompact K¨ahler manifold with nonnegative bisec-tional curvature and maximal volume growth. Then there exist finitely many holomorphicfunctions of polynomial growth f , ...., f N so that N P i = | f i ( x ) | ≥ cd ( x , p ) . Here p is a fixedpoint on M and c > is a constant independent of x.Proof. Put(83) v = lim r →∞ vol ( B ( p , r )) r n > . Proposition 3.
Let ( Y n , q ) be a complete K¨ahler manifold with nonnegative bisectionalcurvature. Assume vol ( B ( q , r )) r n ≥ v > for all r > . There exists ǫ ( n , v ) > so that if ǫ < ǫ ( n , v ) and if there exists a metric cone ( X , o ) (o is the vertex) with (84) d GH ( B ( q , ǫ R ) , B X ( o , ǫ R )) ≤ ǫ R , then there exist N = N ( v , n ) ∈ N , > δ > δ > δ ( v , n ) > and holomorphic functionsg , ..., g N in B ( q , δ R ) with g j ( q ) = and (85) min x ∈ ∂ B ( q , δ R ) N X j = | g j ( x ) | > x ∈ B ( q ,δ R ) N X j = | g j ( x ) | . Furthermore, for all j, (86) max x ∈ B ( q , δ R ) | g j ( x ) | max x ∈ B ( q , δ R ) | g j ( x ) | ≤ C ( n , v ) . Proof.
It is clear the proposition is independent of the value of R . Then, by scaling, wemay assume R is su ffi ciently large, to be determined. Assume(87) X = R k × Z . We will do induction on k . For the case k = n , the proposition reduces to proposition 1.Assume the proposition holds for k = s and fails for k = s −
2. Then there exist completeK¨ahler manifolds ( Y ni , q i )( i ∈ N ) with nonnegative bisectional curvature and vol ( B ( q i , r )) r n ≥ v > r >
0; metric cones ( X i , o i ) with(88) ( X i , o i ) = ( R s − , × ( Z i , z ∗ i ); d GH ( B ( q i , iR ) , B X i ( o i , iR )) ≤ Φ ( 1 i | R ) . But the proposition fails to hold uniformly for any subsequence of Y i . By passing to asubsequence, we may assume ( X i , o i ) converges in the pointed Gromov-Hausdor ff sense toa metric cone ( X , o ). Of course, there exists a sequence s i → ∞ with(89) ( X , o ) = ( R s − , × ( Z , z ∗ ); d GH ( B ( q i , s i R ) , B X ( o , s i R )) < Φ ( 1 i | R ) . INITE GENERATION CONJECTURE 17 Z does not split o ff a factor R , by induction hypothesis. Similar to the construction in(61), we have a function v i , so that in B ( q i , R ),(90) √− ∂∂ v i , ≥ c ( n , v ) ω i > y ∈ B ( q i , R ) \ B ( q i , R ) v i , ( y ) > y ∈ B ( q i ,δ R ) v i , ( y );(92) min y ∈ B ( q i , δ R ) \ B ( q i , δ R ) v i , ( y ) > y ∈ B ( q i ,δ R ) v i , ( y ) . Here δ s = δ s ( n , v ) > s = , y ∈ B ( q i ,δ R ) v i , ( y ) >
12 ; δ R > . Now we freeze the value of R . That is to say,(94) R = R ( n , v ) > . Then(95) | v i , ( y ) | ≤ C ( R , n , v ) = C ( n , v ) , y ∈ B ( q i , R ) . Let Ω i be the connected component of { z | v i , ( z ) < max B ( q i ,δ R ) v i , } containing B ( q i , δ R ). Asbefore, we see Ω i is Stein.According to (89) and (2 . .
11) in [5], there exist harmonic functions b il (1 ≤ l ≤ s −
2) in B ( q i , R ) with(96) Z B ( q i , R ) X ≤ l , l ≤ s − |h∇ b il , ∇ b il i − δ l l | + X l |∇ b il | < Φ ( 1 i | n );(97) b il ( q i ) = |∇ b il | ≤ C ( n )in B ( q i , R ). Moreover, in B ( q i , R ), ( b i , ..., b i s − ) approximates the ( y , ..., y s − ) with error Φ ( i | n ). Here ( y , ..., y s − ) is the Euclidean coordinate in ( X , o ) = ( R s − , × ( Z , z ∗ ).By similar arguments as before, we may assume that(98) Z B ( q i , R ) | J ∇ b i m − − ∇ b i m | ≤ Φ ( 1 i | n )for 1 ≤ m ≤ s −
1. Set ˜ w im = b i m − + √− b i m . Then(99) Z B ( q i , R ) | ∂ ˜ w im | ≤ Φ ( 1 i | n ) . Then by solving ∂ problem as before, we find holomorphic functions w im (1 ≤ m ≤ s − w im ( q i ) = | w im − ˜ w im | ≤ Φ ( 1 i | n )in B ( q i , R ). Recall δ appeared in (92). For su ffi ciently large i , define(101) E i = { x | x ∈ ∂ B ( q i , δ R , s − X m = | w im | ≤ ( δ R ) } ;(102) E = { x | x ∈ ∂ B R s − × Z ((0 , z ∗ ) , δ R , s − X k = | y k | ≤ ( δ R ) } . Then the limit of E i is contained in E under the Gromov-Hausdor ff approximation. Ob-serve from the definition and (93), if x ∈ ∂ B ( q i , δ R ) \ E i ,(103) s − X m = | w im ( x ) | > ( δ R ) > . For x ∈ E , let C x be a tangent cone. Then C x must split o ff a factor R s − . Since C x is theGromov-Hausdor ff limit of K¨ahler manifolds with noncollapsed volume and nonnegativeRicci curvature, C x splits o ff a factor R s , by [5]. Thus there exists(104) δ R > r x > d GH ( B ( x , ǫ r x ) , B W ( w , ǫ r x )) < ǫ r x ; ( W , w ) = ( R s , × ( H , h ∗ ) . Here ǫ < ǫ ( n , v ) satisfies proposition 3 for the case X splits o ff R s ; ( H , h ∗ ) is a metric conewith vertex at h ∗ . By compactness, there is a uniform positive lower bound of r x , say(106) r x ≥ R ( X ) > r x > R = R ( n , v ) > . Then for su ffi ciently large i and any point x i ∈ E i ,(108) d GH ( B ( x i , ǫ r x i ) , B W i ( w i , ǫ r x i )) < ǫ r x i . (109) δ R > r x i ≥ R >
0; ( W i , w i ) = ( R s , × ( H i , h ∗ i ) . Here ǫ is the same as in (105); ( H i , h ∗ i ) is a metric cone with vertex at h ∗ i . We apply theinduction to B ( x i , ǫ r x i ). By induction hypothesis, there exist(110) 1 > δ > δ > δ ( n , v ); N = N ( v , n ) ∈ N and holomorphic functions g ji (1 ≤ j ≤ N ) in B ( x i , δ r x i ) with(111) g ji ( x i ) =
0; min x ∈ ∂ B ( x i , δ rxi ) N X i = | g ji ( x ) | > x ∈ B ( x i ,δ r xi ) N X i = | g ji ( x ) | ;(112) max x ∈ B ( x i , δ r xi ) | g ji ( x ) | max x ∈ B ( x i , δ r xi ) | g ji ( x ) | ≤ C ( n , v ) . By normalization, we can also assume(113) max j max y ∈ B ( x i ,δ r xi ) | g ji ( y ) | = . Note by three circle theorem,(114) max y ∈ B ( x i , rxi δ ) | g ji ( y ) | ≤ C ( n , v ) . INITE GENERATION CONJECTURE 19
Set(115) F i ( x ) = N X j = | g ji | . Let λ be a standard cut-o ff function: R + → R + given by λ ( t ) = ≤ t ≤ λ ( t ) = t ≥ | λ ′ | , | λ ′′ | ≤ C ( n ). Consider(116) h i ( x ) = n log F i ( x ) λ ( F i ( x )) . By (111) and (113), h i ( x ) is supported in B ( x i , δ r xi ). Similar to lemma 1, It is easy to checkthat(117) √− ∂∂ h i ( x ) ≥ − C ( n , v ) ω i . Therefore, there exists ξ = ξ ( n , v ) > √− ∂∂ ( ξ v i , + h i ) ≥ ω i in Ω i . We will assume such ξ is large, which will be determined later. Set(119) φ ( x ) = ξ v i , ( x ) + h i ( x ) . Now consider a function(120) µ i ( x ) = ϕ ( d ( x , x i ) δ r x i ) . Here ϕ ( t ) = t ≤ ; ϕ ( t ) = t ≥ | ϕ ′ | ≤ C ( n ). Then it is clear µ i is supportedin B ( x i , δ r x i ). Also, by (109),(121) |∇ µ i | ≤ C ( n , v ) . We solve the ∂ problem ∂ s i = ∂µ i in Ω i satisfying(122) Z Ω i e − φ | s i | ≤ Z Ω i e − φ | ∂µ i | = Z B ( x i ,δ r xi ) \ B ( x i , δ rxi ) e − φ | ∂µ i | ≤ exp( − ξ min y ∈ B ( q i , δ R ) \ B ( q i , δ R ) v i , ( y )) C ( n , v ) . Here we used that h i is supported in B ( x i , δ r xi ). We also used that(123) B ( x i , δ r x i ) ⊂ B ( q i , δ R \ B ( q i , δ R , by (109). Observe by (92), µ i vanishes in B ( q i , δ R ). Hence s i is holomorphic in B ( q i , δ R ).Mean value inequality implies for x ∈ B ( q i , δ R ),(124) | s i ( x ) | ≤ R B ( q i ,δ R ) | s i | c ( n , v )( δ R ) n ≤ exp( ξ max y ∈ B ( q i ,δ R ) v i , ( y )) R Ω i e − φ | s i | c ( n , v )( δ R ) n ≤ exp( − ξ ( min y ∈ B ( q i , δ R ) \ B ( q i , δ R ) v i , ( y ) − max y ∈ B ( q i ,δ R ) v i , ( y ))) 1 c ( n , v )( δ R ) n ≤ e − ξ c ( n , v )( δ R ) n . Here we used (92) and (93). If ξ is large (depending only on n , v ), then we can make(125) | s i ( x ) | ≤ x ∈ B ( q i , δ R ). Now we freeze the value of ξ = ξ ( n , v ). Note that the local integrabilityof s i forces s i ( x i ) =
0. Set(126) w i ( x ) = µ i ( x ) − s i ( x ) . Then(127) w i ( x i ) = | w i ( x ) | ≤ B ( q i , δ R ). Set(128) f i ( x ) = w i ( x ) − w i ( q i ) . Then(129) f i ( q i ) = | f i ( x i ) | ≥ . By (122), we find(130) | f i ( x ) | ≤ C ( n , v ) , |∇ f i ( x ) | ≤ C ( n , v ) , x ∈ B ( q i , δ R . Therefore, there exists δ ( n , v ) > | f i ( x ) | ≥ B ( x i , δ R ). We can take x j ∈ E with j = , , ..., K , K = K ( v , n ). Also(132) ∪ j B ( x j , δ R ⊃ E ; δ R > r x j ≥ R ( n , v ) > d GH ( B ( x j , ǫ r x j ) , B W j ( w j , ǫ r x j )) < ǫ r x j ; ( W j , w j ) = ( R s , × ( H j , ( h j ) ∗ ) . Here ( H j , ( h j ) ∗ ) is a metric cone with vertex at ( h j ) ∗ . Then for su ffi ciently large i , we canfind x ji ∈ E i , j = , ..., K with(134) d GH (( B ( x ji , ǫ r x j ) , B W j ( w j , ǫ r x j )) < ǫ r x j ;(135) ∪ j B ( x ji , δ R ⊃ E i . Now we can apply the induction argument above for each geodesic ball B ( x ji , ǫ r x j ). Weobtain holomorphic functions f ji in B ( q i , δ R ) satisfying(136) | f ji ( x ) | ≥ x ∈ B ( x ji , δ R );(137) | f ji ( x ) | ≤ C ( n , v ) , x ∈ B ( q i , δ R f ji ( q i ) = . Put G i ( x ) = K P j = | f ji | + s − P m = | w im | . Then by (100), (103) and (137),(138) |∇ G i ( x ) | ≤ C ( n , v ) , x ∈ B ( q i , δ R G i ( q i ) = INITE GENERATION CONJECTURE 21 (139) | G i ( x ) | ≥ , x ∈ ∂ B ( q i , δ R . Therefore, there exists δ = δ ( n , v ) > x ∈ B ( q i ,δ R ) | G i ( x ) | ≤ . This contradicts the assumption that the proposition does not hold uniformly for ( Y ni , q i ).The proof of proposition 3 is complete. (cid:3) We continue the proof of theorem 11. For any sequence r i → ∞ , Set ( M i , p i ) = ( M , p , r − i g )(we shall make r i explicit in proposition 4 below). Then there exist R ′′ i → ∞ and metric cones ( X i , x ∗ i )( x ∗ i is the vertex) with(141) d GH ( B ( p i , R ′′ i ) , B X i ( x ∗ i , R ′′ i )) < R ′′ i . Let d i ( x ) = d i ( x , p i ) for x ∈ M i . Following the construction in (52) and claim 5, we find asequence R ′ i → ∞ , functions ρ i in M i satisfying(142) Z B ( p i , R ′ i ) |∇ ρ i − ∇ d i | + |∇ ρ i − g i | < Φ ( 1 i ) . Also, in B ( p i , R ′ i ),(143) | ρ i − d i | < Φ ( 1 i ); |∇ ρ i | ≤ C ( n ) d i . As before, consider a smooth function ϕ : R + → R + with ϕ ( t ) = t for 0 ≤ t ≤ ϕ ( t ) = t ≥ | ϕ | , | ϕ ′ | , | ϕ ′′ | ≤ C ( n ). Set(144) v i ( z ) = R ′ i ) ϕ ( ρ i ( z )3( R ′ i ) )Then v i is supported in B ( p i , R ′ i ). Let H i ( x , y , t ) be the heat kernel of M i . Consider thefunction τ i ( x ) = log(1 + v i ( x )) and define(145) τ i , t ( z ) = Z M i H i ( z , y , t ) τ i ( y ) dy . Let δ = δ ( n , v ) be given by proposition 3. By (143), we have(146) min B ( p i , ) \ B ( p i , ) τ i − max B ( p i , ) τ i ≥ c ( n , v ) > .τ i is of logarithmic growth uniform for all i . By heat kernel estimates, there exists t = t ( n , v ) > B ( p i , ) \ B ( p i , ) τ i , t − max B ( p i , ) τ i , t ≥ c ( n , v ) > . On a smooth K¨ahler metric cone, let r be the distance function to the vertex. Then √− ∂∂ log(1 + r ) is positive (1 ,
1) form away from the vertex. Since τ i resembles log(1 + r ), by similar arguments as in proposition 2, we find that in B ( p i , √− ∂∂τ i , t ≥ c ( n , v ) > . By proposition 5, for any fixed R and su ffi ciently large i , in B ( p i , R ),(149) c ( n , v ) log( d i ( x ) + − C ( n , v ) ≤ τ i , t ( x ) ≤ C ( n , v ) log( d i ( x ) + √− ∂∂τ i , t ( x ) > , x ∈ B ( p i , R ) . Therefore, there exist sequences ˜ R i → ∞ , R i → ∞ , c i → ∞ so that τ − i , t ( { c | c ≤ c i } ) ∩ B ( p i , ˜ R i )is relatively compact on B ( p i , ˜ R i ). Also(151) τ − i , t ( { c | c ≤ c i } ) ∩ B ( p i , ˜ R i ) ⊃ B ( p i , R i );(152) √− ∂∂τ i , t > τ − i , t ( { c | c ≤ c i } ) ∩ B ( p i , ˜ R i ). Let Ω i be the connected component of τ − i , t ( { c | c < c i } ) con-taining B ( p i , R i ). Then Ω i is a Stein manifold.According to proposition 3, there exist holomorphic functions w ij (1 ≤ j ≤ K = K ( n , v ))in B ( p i , R = δ in proposition 3) so that(153) w ij ( p i ) =
0; max j max B ( p i , | w ij | = x ∈ ∂ B ( p i , K X j = | w ij ( x ) | > x ∈ B ( p i , δ δ ) K X i = | w ij ( x ) | ;(155) max x ∈ B ( p i , ) | w ij ( x ) | max x ∈ B ( p i , | w ij ( x ) | ≤ C ( n , v ) . Then of course, in B ( p i , ),(156) | w ij ( x ) | ≤ C ( n , v ) . Also, by three circle theorem, we have(157) max j max B ( p i , δ δ ) | w ij | ≥ c ( n , v ) > . Thus(158) min x ∈ ∂ B ( p i , K X j = | w ij ( x ) | ≥ c ( n , v ) > . Now consider a cut o ff function λ i ( x ) = λ ( d i ( x )) with λ i = B ( p i , ); λ i has com-pact support in B ( p i , ); |∇ λ i | ≤ C ( n , v ). Let ˜ w ij = λ i w ij . Then ∂ ˜ w ij is supported in B ( p i , ) \ B ( p i , ). We solve the ∂ -problem ∂ ˜ f ij = ∂ ˜ w ij in Ω i with the weight function ψ i = ητ i , t . Here η = η ( n , v ) is a very large number to be determined. Then by (148), wehave(159) Z Ω i | ˜ f ij | e − ψ i ≤ R Ω i | ∂ ˜ w ij | e − ψ i c ( n , v ) . This implies that(160) Z B ( p i , ) | ˜ f ij | e − ψ i ≤ R B ( p i , ) \ B ( p i , ) | ∂ ˜ w ij | e − ψ i c ( n , v ) . Let(161) f ij ( x ) = ˜ w ij ( x ) − ˜ f ij ( x ) − ( ˜ w ij ( p i ) − ˜ f ij ( p i )) . INITE GENERATION CONJECTURE 23
By (147), (153), (154), (156), (158) and similar arguments as in (124), if η = η ( n , v ) islarge enough, we can make | ˜ f ij | so small in B ( p i ,
1) that(162) C ( n , v ) ≥ min x ∈ ∂ B ( p i , K X j = | f ij ( x ) | >
32 max x ∈ B ( p i , δ δ ) K X i = | f ij ( x ) | ≥ c ( n , v )Now we freeze the value η = η ( n , v ). (149) says ψ i is of logarithmic growth uniform for all i . By (159) and the mean value inequality, we find C = C ( n , v ) > R > i is su ffi ciently large,(163) | f ij ( x ) | ≤ C ( d i ( x ) C + x ∈ B ( p i , R ). By passing to subsequence, we can assume ( M i , p i ) → ( M ∞ , p ∞ ) in theGromov-Hausdor ff sense. Also, f ij converges to f ∞ j which is of polynomial growth of order C on M ∞ .For C in (163), let V = span { g ∈ O C ( M ) | g ( p ) = } and let k = dim ( V ). Take a basis g s of V satisfying(164) Z B ( p , g s g t = δ st . Proposition 4.
There exist constants R > and c > with P s | g s ( x ) | ≥ cr ( x , p ) forr ( x , p ) ≥ R.Proof.
Assume the proposition is not true. There exist r i → ∞ and points x i with(165) d ( p , x i ) = r i , X s | g s ( x i ) | ≤ r i i . We follow the notations from page 21 to page 23. For each i , There exists a basis g is of V with(166) Z B ( p , g is g it = δ st ; Z B ( p i , g is g it = λ ist δ st . Here λ ist are constants. Then (164) and (166) imply(167) X s | g s | = X s | g is | . Note by three circle theorem and mean value inequality,(168) λ iss ≥ cr i for some c = c ( n , v ) >
0. Then h is = g is √ λ iss satisfies(169) Z B ( p i , h is h it = δ st . Three circle theorem and mean value inequality imply(170) 0 < c ( n , v ) ≤ max B ( p i , | h is ( x ) | ≤ C ( n , v ) . After passing to subsequence, we may assume M i → M ∞ and h is , f ij all converge. Say h is → h ∞ s ; f ij → f ∞ j uniformly on each compact set. Clearly h ∞ s ( s = , .., k ) are linearlyindependent on M ∞ . Claim 6. span { f ∞ j } ⊂ span { h ∞ s } on M ∞ . Proof.
Assume the claim is not true. Set V ′ = span { f ∞ j , h ∞ s } . Then dim ( V ′ ) > k . Bythree circle theorem, f ∞ j , h ∞ s are of polynomial growth of order 2 C . Take a basis u , ..., u m of V ′ , m ≥ k +
1. Therefore, u l (1 ≤ l ≤ m ) are of polynomial growth of order 2 C . Forany f ∈ V ′ , f satisfies three circle theorem. That is, if M ( f , r ) = max B ( p ∞ , r ) | f ( x ) | , log M ( f , r )is convex in terms of log r . The reason is that f is a limit of holomorphic functions ofpolynomial growth on M i . Write u l = k P s = a sl h ∞ s + K P j = b jl f ∞ j . Here a sl , b jl are constants.Define u il = k P s = a sl h is + K P j = b jl f ij . Then u il → u l uniformly on each compact set. As u l is abasis for V ′ , for su ffi ciently large i , u il are linearly independent on B ( p i , u il as holomorphic functions on B ( p , r i ) on M . Let v il be a basis of span { u il } with R B ( p , v il v it = δ lt . Let us write v il = m P t = C ilt u it . Here C ilt are constants. We are interested in(171) F i , l = max B ( p i , | v il | max B ( p i , | v il | = max B ( p i , | m P t = C ilt u it | max B ( p i , | m P t = C ilt u it | . In the quotient, we can normalize the coe ffi cients C ilt so that max ≤ t ≤ m | C ilt | =
1. As u l arelinearly independent on M ∞ , by a simple compactness argument and three circle theoremfor V ′ on M ∞ , we see that for i su ffi ciently large, 1 ≤ l ≤ m ,(172) F i , l ≤ (2 + ǫ ) C for any given ǫ >
0. As before, we can apply the three circle theorem to find a subsequenceof v il converging to linearly independent holomorphic functions v l on M , satisfying v l ( p ) = deg ( v l ) ≤ C . As l is from 1 to m and m > k , this contradicts that dim ( V ) = k . (cid:3) Given claim 6, we find f ij is almost in the span { h is } . More precisely,(173) lim i →∞ max B ( p i , | f ij ( x ) − X s c ijs h is | = c ijs = R B ( p i , f ij h is . In particular, | c ijs | ≤ C ( n , v ). By (162),(174) min ∂ B ( p i , K X j = | f ij ( x ) | >
32 max B ( p i , δ δ ) K X j = | f ij ( x ) | ≥ c ( n , v ) > . Hence(175) C ( n , v ) min ∂ B ( p i , X s | h is | ≥ c ( n , v ) > . Finally by (168),(176) | h is | = | g is | λ iss ≤ | g is | cr i . Then from (167),(177) min ∂ B ( p , r i ) X s | g s | = min ∂ B ( p i , X s | g s | = min ∂ B ( p i , X s | g is | ≥ c ( n , v ) r i > . This contradicts (165). (cid:3)
INITE GENERATION CONJECTURE 25
To conclude the proof of theorem 11, we just need to add the constant function 1 to V . (cid:3) Completion of the proof of theorem 2Theorem 12.
Let M be a complete noncompact K¨ahler manifold with nonnegative bi-sectional curvature and maximal volume growth. Then M is biholomorphic to an a ffi nealgebraic variety. Also the ring of holomorphic functions of polynomial growth is finitelygenerated.Proof. Given any k ∈ N , let n k = dim C ( O k ( M )). Define a holomorphic map from M to C n k by F k ( x ) = ( g ( x ) , ...., g n k ( x )). Here g , ...., g n k is a basis for O k ( M ). When k is gettinglarger, we only add new functions to the basis (that is, we do not change the previousfunctions). Our goal is to prove that for su ffi ciently large k , F k is a biholomorphism to ana ffi ne algebraic variety.Below the value k might change from line to line, basically we shall increase its valuein finite steps. First assume k is large so that the functions f , ..., f N constructed in theorem11 are in O k ( M ) and they separate the tangent space at a point p ∈ M . Let α be the ideal ofpolynomial relations of functions g , ..., g n k . That is to say,(178) α = { p ( g , ..., g n k ) | p ( g ( x ) , ..., g n k ( x )) = , ∀ x ∈ M } . Here p is a polynomial. Then α is a prime ideal. Let Σ k be the a ffi ne algebraic varietydefined by α . Then dim ( Σ k ) = n , as the transcendental dimension of ( g , ..., g n k ) over C is n . Moreover, dim ( F k ( M )) = n , as the tangent space at p is separated. By theorem 11, F k is a proper holomorphic map from M to C n k . Hence the image of F k is closed. Byproper mapping theorem, the image of F k is an analytic subvariety of dimension n . As Σ k is irreducible, F k ( M ) = Σ k .Our argument below is very similar to some parts of [11]. Given any point in Σ k , thepreimage of F k is a compact subvariety of M , as F k is proper. As M is a Stein manifold( M is exhausted by Ω i which are Stein), The preimages contain only finitely many points.Given a generic point y ∈ Σ k , we can find polynomial growth holomorphic functions sep-arating F − k ( y ). Therefore, by increasing k , we may assume F k is generically one to one.Note that if x ∈ Σ k and the preimage of x contain more than one point, then x is in thesingular set of Σ k , say S ( Σ k ). Write S ( Σ k ) as a finite union of irreducible algebraic sub-varieties Σ ′ s (1 ≤ s ≤ t k ). Set h = dim ( S ( Σ k )). Let us assume dim ( Σ ′ s ) = dim ( S ( Σ k )) for1 ≤ s ≤ r k ≤ t k . For a generic point x ∈ Σ ′ s , the preimages under F k contain finitely manypoints. Therefore, we can increase the value of k so that the preimages of x and their tan-gent spaces are separated. In this way, the dimension of S ( Σ k ) is decreased. After finitelymany steps, F k becomes a biholomorphism from M to Σ k which is a ffi ne algebraic. Claim 7.
We can identify polynomial growth holomorphic functions on M with regularfunctions on Σ k via F k . Thus O P ( M ) is finitely generated.Proof. First, by theorem 3 . Σ k are identified with the a ffi necoordinate ring of Σ k . Thus, any regular function is of polynomial growth. Since the tran-scendental dimension of O P ( M ) is n over C , we may assume the a ffi ne coordinate functionsgenerates the field of O P ( M ). Then every polynomial growth holomorphic function is ra-tional, hence a regular function on M . (cid:3) The proof of theorem 12 is complete. (cid:3)
Corollary 4.
Let M be a complete noncompact K¨ahler manifold with nonnegative bisec-tional curvature. Then the ring of holomorphic functions of polynomial growth is finitelygenerated.Proof.
We first consider the case when the universal cover does not split. By theorem 2 in[18], if there exists a nonconstant holomorphic function of polynomial growth on M , then M is of maximal volume growth. Then then the result follows from the theorem above.In the general case, let ˜ M be the universal cover. Let G be the fundamental group of M .Let E be the set of G -invariant holomorphic functions of polynomial growth on ˜ M . We canidentify E with O P ( M ). Given any f ∈ E , consider(179) u ft ( x ) = Z ˜ M H ˜ M ( x , y , t ) log( | f ( y ) | + dy , where H ˜ M ( x , y , t ) is the heat kernel of ˜ M . By theorem 3 . √− ∂∂ u ft ≥ t >
0. Let D tf be the null space of √− ∂∂ u ft . Theorem 3 . D tf is a paralleldistribution. Claim 8. D tf is invariant for t > . Then we define D f = D tf , t > .Proof. By theorem 2 .
1, part ( ii ) in [25](see also the second sentence in the proof of corol-lary 2 . t > t > dim ( D t f ) ≤ dim ( D t f ) . De Rham theorem says we can write ˜ M = N × N where D t f is the tangent space of N . u ft is of logarithmic growth by proposition 5. Moreover, u ft is pluriharmonic on each slice N ,hence harmonic on N . As N has nonnegative bisectional curvature, the Ricci curvature of N is nonnegative. By a theorem of Cheng-Yau [8], u ft is constant on each slice N . That isto say, u ft is a function on N . By uniqueness of the heat flow, u ft is also constant on eachslice of N . Combining this with (180), we obtain that D t f = D t f . (cid:3) Hence, u ft is constant on N for t ≥
0. This implies f is constant on the factor N . Nowdefine the parallel distribution(181) D = ∩ f ∈ E D f . By De Rham theorem, we can assume ˜ M = M × M where D is the tangent space of M .Then, for any f ∈ E , f is constant on the factor M . Note D is invariant under G -action.Fix an inclusion i of a slice: M ֒ → M × M . Now for any g ∈ G , g ( i ( M )) must be anotherslice of M . Let π be the projection from M × M to M . For x ∈ M and g ∈ G , Definea holomorphic isometry u g of M by u g ( x ) = π ( g ( i ( x ))). Of course, u g is a subgroup of theholomorphic isometry group of M . Let G ′ be the closure of u g . Then we can identify E with polynomial growth holomorphic functions on M invariant under G ′ . Claim 9. G ′ is a compact group.Proof. It su ffi ces to prove that for x ∈ M , u g ( x ) is bounded for g ∈ G . Assume this is nottrue, then there exists a sequence g i ∈ G ′ with x i = g i ( x ) → ∞ on M . Let ( U , z , .., z m )be a holomorphic chart on M around x with z ( x ) =
0. Let ( U i = g i ( U ) , z is = z s ◦ g − i )be the holomorphic chart in U i . By taking subsequence of x i , we may assume U i aredisjoint. We will use some construction in [23]. First, pick finitely many f j ∈ E sothat √− ∂∂ P j u f j > M . Let u = P j u f j . Then u is a strictly plurisubharmonic INITE GENERATION CONJECTURE 27 function on M with logarithmic growth. Moreover, u is invariant under G ′ action. Let U ⊂⊂ U ⊂⊂ U be open sets containing x . Consider a smooth cut-o ff function ϕ with ϕ = U ; ϕ = M \ U . Define ϕ i = ϕ ◦ g − i . Then ϕ i is supported in U i . Let(182) ψ ( x ) = m X i ϕ i ( x ) log( m X s = | z is ( x ) | ) + Cu ( x ) . Here C is a positive constant so that √− ∂∂ψ ≥ ω on U i ; ω is the K¨ahler form on M .Then √− ∂∂ψ > M . Now we solve the ∂ -problem ∂ h i = ∂ϕ i with(183) Z M | h i | e − ψ ≤ Z M | ∂ϕ i | e − ψ . One sees that λ i = h i − ϕ i are holomorphic functions of polynomial growth. The growthorders are uniformly bounded. Moreover, h i ( x k ) = k ∈ N . Thus λ i are linearlyindependent, as ( h i − ϕ i )( x j ) = − δ i j . This contradicts theorem 9. (cid:3) Claim 10. M is of maximal volume growth. In particular, the ring of polynomial growthholomorphic functions is finitely generated.Proof. As M is simply connected, write M as a product of K¨ahler manifolds which arenot products anymore. For each factor, there exists polynomial growth holomorphic func-tion on M which is not constant on that factor. Then each factor must be of maximalvolume growth by theorem 2 in [18]. (cid:3) By claim 10 and theorem 12, O P ( M ) is finitely generated. O P ( M ) is just the subring of O P ( M ) invariant under G ′ . Since G ′ is compact, the finite generation of O P ( M ) followsfrom a theorem of Nagata [22] (the detailed argument is in the appendix B ). (cid:3) A ppendix A. P roof of theorem Proof.
This part basically follows from [25]. For any a > η ( x , t ) satisfies (184), (185)below.(184) ( ∂∂ t − ∆ ) η γδ = R βαγδ η αβ −
12 ( R γ p η p δ + R p δ η γ p )(185) Z M || η ( x , || exp( − ar ( x )) dx < ∞ We will assume the equation below at this moment. The proof is given at the end of thissection.(186) lim r →∞ inf Z T Z B ( p , r ) || η || ( x , t ) exp( − ar ( x )) dxdt < ∞ Recall corollary 1 . Proposition 5.
Let ( M n , p ) be a complete noncompact K¨ahler manifold with nonnegativebisectional curvature. r ( x ) = d ( x , p ) . Let u be a nonnegative function on M satisfying (187) − Z B ( p , r ) u ( y ) dy ≤ exp( a + br ) for some constants a , b > . Let (188) v ( x , t ) = Z M H ( x , y , t ) u ( y ) dy . H is the heat kernel on M. Then given any ǫ > , T > , there exists C ( n , ǫ, a , b ) > suchthat for any x satisfying r = r ( x ) ≥ √ T , (189) − C ( n , ǫ, a , b ) + C ( n , ǫ ) inf B ( x ,ǫ r ) u ≤ v ( x , t ) ≤ C ( n , ǫ, a , b ) + sup B ( x ,ǫ r ) ufor ≤ t ≤ T . Here C ( n , ǫ ) > . Fix a point p ∈ M . Let r ( x ) = d ( x , p ). Let φ ( x ) = exp( r ( x )). Define(190) φ ( x , t ) = e t Z M H ( x , y , t ) φ ( y ) dy . Then(191) ( ∂∂ t − ∆ ) φ = φ and(192) φ ( x , t ) ≥ ce c r for 0 ≤ t ≤ T , by proposition 5. Here c , c are positive constants. Let(193) h ( x , t ) = Z M H ( x , y , t ) || η || ( y , dy . The proposition below is just lemma 2 . Proposition 6.
There exists a positive function τ ( R ) so that for ≤ t ≤ T , h ( x , t ) ≤ τ ( R ) for x ∈ B ( p , R ) \ B ( p , R ) . Moreover, lim R →∞ τ ( R ) = . The next proposition is lemma 2.1 in [25]. Note (184), (185) and (186) are used.
Proposition 7. || η || ( x , t ) is a subsolution of the heat equation. Moreover, || η || ( x , t ) ≤ h ( x , t ) . Given ǫ >
0, define(194) ( ˜ η ) αβ = η αβ + ( ǫφ − λ ( x , t )) g αβ . At t =
0, ˜ η >
0. Also, for 0 ≤ t ≤ T , if R is su ffi ciently large, by proposition 6, we have˜ η > ∂ B ( p , R ). Suppose at some t ∈ [0 , T ], ˜ η ( x , t ) < x ∈ B ( p , R ). Then thereexists 0 ≤ t < T with ˜ η ( x , t ) ≥ x ∈ B ( p , R ) and 0 ≤ t ≤ t . Moreover, the minimumeigenvalue of ˜ η ( x , t ) is zero for some x ∈ B ( p , R )(note x cannot be on the boundary).Now we apply the maximal principle. Let us assume(195) ˜ η ( x , t ) γγ = γ ∈ T , x M , | γ | =
1. We may diagonize ˜ η at ( x , t ). Of course, we can assume γ is oneof the basis of the holomorphic tangent space. Then at ( x , t ),(196) ( ∂∂ t − ∆ ) ˜ η γγ ≤ . On the other hand, by (184),(197) ( ∂∂ t − ∆ ) η γγ = X α R γγαα η αα − X α R γγαα η γγ = X α R γγαα ( ˜ η αα − ˜ η γγ ) ≥ . INITE GENERATION CONJECTURE 29
Note(198) ( ∂∂ t − ∆ )(( ǫφ − λ ( x , t )) g γγ ) = ǫφ g γγ > . Hence at ( x , t ),(199) ( ∂∂ t − ∆ ) ˜ η γγ > . This is a contradiction. Now let R → ∞ and then ǫ →
0, we proved that η − λ ( x , t ) g αβ ≥ ≤ t ≤ T .Next we verify (186). Basically we follow page 487 −
488 on [25]. Note our conditionis more special. First, we have that | v ( x , t ) | ≤ C for all x , t , as u has compact support. Note(200) ( ∆ − ∂∂ t ) v = |∇ v | . Multiplying (200) by suitable cuto ff functions, using integration by parts, we find(201) Z T − Z B ( p , r ) |∇ v | ≤ C ( r − Z T − Z B ( p , r ) v + − Z B ( p , r ) u ) ≤ C ( T + r ≥
1. Bochner formula gives(202) ( ∆ − ∂∂ t ) |∇ v | ≥ |∇ v | . Multiplying (202) by suitable cuto ff functions, using integration by parts, we find(203) Z T − Z B ( p , r ) |∇ v | ≤ C ( r − Z T − Z B ( p , r ) |∇ v | + − Z B ( p , r ) |∇ u | ) ≤ C ( T + r ≥
1. From this, (186) follows easily. (cid:3) A ppendix B. S ome algebraic results of N agata We continue the proof of theorem 2. The ring R = O P ( M ) is finitely generated. Wemay assume the generators are in F = O d ( M ) for some d >
0. Let g , ..., g m be a basis for F . Obviously F is an invariant space of G ′ . Then we may think O P ( M ) is C [ g , ..., g m ] /α . Here α is an ideal. Then the G ′ action on R is induced by the representation G ′ → GL ( m , C ). Let I G ′ ( R ) be the subring of R fixed by G ′ . In [22], page 370, the followingdefinition was made: Definition.
A group G is reductive if every rational representation is completely reducible.
It was pointed out on page 370 of [22] that all rational representations of G in [22] aregiven by some specific finite dimensional representations of G . In our case, as G ′ is com-pact, every finite dimensional representation(complex) is completely reducible. Therefore,according to the definition above, G ′ is reductive. In [22], the following was proved: Theorem 13 (Nagata) . I G ( R ) is finitely generated if G is semi-reductive. It was pointed out in the first sentence of part 5, page 373 of [22] that a reductive group isobviously semi-reductive. Putting all these things together, we proved the finite generationof I G ′ ( R ) = O P ( M ). R eferences [1] J. Cheeger and T. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products ,Ann. of Math. (2) 144 (1996), no. 1, 189-237.[2] J. Cheeger and T. Colding,
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INITE GENERATION CONJECTURE 31 D epartment of M athematics , U niversity of C alifornia , B erkeley , B erkeley , CA 94720 E-mail address ::