a r X i v :m a t h /0611794v 2 [m a t h .D G ] 10 J a n 2007MULTIPLIER IDEAL SHEA VES AND THE K ¨AHLER-RICCI FLOW 1
D.H.Phong ∗,Natasa Sesum ∗,and Jacob Sturm †
Abstract Multiplier ideal sheaves are constructed as obstructions to the convergence of the K¨a hler-Ricci flow on Fano manifolds,following earlier constructions of Kohn,Siu,and Nadel,and using the recent estimates of Kolodziej and Perelman.1Introduction The global obstruction to the existence of a Hermitian-Einstein metric on a holomorphic vector bundle is well-known to be encoded in a destabilizing sheaf,thanks to the works of Donaldson [9,10]and Uhlenbeck-Yau [27].It is expected that this should also be the ca for general canonical metrics in K¨a hler geometry.For K¨a hler-Einstein metrics on Fano manifolds,obstructing sheaves have been constructed by Nadel [15]as multiplier ideal sheaves,following ideas of Kohn [11]and Siu [23].This formulation in terms of multiplier ideal sheaves opens up many possibilities for relations with complex and algebraic geometry [24,7,28].The obstructing multiplier ideal sheaves are not expected to be unique.Nadel’s con-struction is bad on the method of continuity for solving a specific Monge-Amp`e re equa-tion for K¨a hler-Einstein metrics.It has always been desirable to construct also an obstruct-ing multiplier ideal sheaf from the K¨a hler-Ricci flow.The purpo of this note is to show that this can be easily done,using the recent estimates of
Kolodziej [12,13]and Perelman [18].In effect,Kolodziej’s estimates provide a Harnack estimate for the Monge-Amp`e re equation,which is elliptic,and Perelman’s estimate reduces the K¨a hler-Ricci flow,which is parabolic,to the Monge-Amp`e re equation.Similar ideas were exploited by Tian-Zhu [26]in their proof of an inequality of Harnack type for the K¨a hler-Ricci flow.
2The multiplier ideal sheaf
Let X be an n -dimensional compact K¨a hler manifold,equipped with a K¨a hler form ω0with µω0∈c 1(X ),where µis a constant.The K¨a hler-Ricci flow is the flow defined by
˙g ¯kj =−(R ¯kj −µg ¯kj ),(2.1)
where g¯kj=g¯kj(t)is a metric evolving in time t with initial value g¯kj(0)=ˆg¯kj,and R¯kj=−∂j∂¯k log det g¯q p is its Ricci curvature.Since the K¨a hler-Ricciflow prerves the K¨a hler class of the metric,we may t g¯kj=ˆg¯kj+∂j∂¯kφ,and the K¨a hler-Ricciflow can be reformulated as
amai˙φ=logωn φ
2
g¯kj dz j∧d¯z k,andˆf is the Ricci potential for the metricˆg¯kj, that is,the C∞function defined by the equationˆR¯kj−µˆg¯kj=∂j∂¯kˆf,normalized by the condition that
X
eˆfωn0= Xωn0≡V.(2.3)
Here and henceforth,ˆR¯kj denotes the Ricci curvature ofˆg¯kj,with similar conventions for all the other curvatures ofˆg¯kj.The initial potential c0is a constant,so that the initial metric coincides withˆg¯kj.The K¨a hler-Ricciflow exists for all time t>0[3],and the main issue is its convergence.Henceforth,we shall restrict to the ca c1(X)>0of Fano manifolds unless indicated explicitly otherwi,and tµ=1.
Theorem1Let X be an n-dimensional compact K¨a hler manifold with c1(X)>0.
(i)Consider the K¨a hler-Ricciflow(2.2)for potentialsφ,with the initial value c0spec-ified by(2.10)below.If there exists some p>1with
sup t≥0 X e−pφωn0<∞,(2.4)
欧洲债务危机then there exists a quence of times t i→+∞with g¯kj(t i)converging in C∞to a K¨a hler-Einstein metric.If in addition X admits no non-trivial holomorphic vectorfield,then the wholeflows(2.1)and(2.2)converge in C∞.
(ii)If X does not admit a K¨a hler-Einstein metric,then for each p>1,there exists a functionψwhich is a L1limit point of the K¨a hler-Ricciflow(2.2),with the following property.Let the multiplier ideal sheaf J(pψ)be the sheaf with stalk at z defined by做鬼脸 英文
J z(pψ)={f;∃U∋z,f∈O(U),
U|
f|2e−pψωn0<∞},(2.5)
where U⊂X is open,and O(U)denotes the space of holomorphic functions on U.Then J(pψ)defines a proper coherent analytic sheaf on X,with acyclic ,
H q(X,K−[p]
X⊗J(pψ))=0,q≥1.(2.6) If X admits a compact group G of holomorphic automorphisms,andˆg¯kj is G-invariant, then J(pψ)and the corresponding subscheme are also G-invariant.
In Part(i),once the convergence of a subquence g¯kj(t i)has been established and X is known to admit a K¨a hler-Einstein metric,it follows from an unpublished result of Perelman that the full K¨a hler-Ricciflow must then converge.An extension of Perelman’s result to K¨a hler-Ricci solitons is given in Tian-Zhu[26].For the sake of completeness,we have provided a short lf-contained proof of the K¨a hler-Einstein ca,in our context and under the simplifying assumption of no non-trivial holomorphic vectorfields.
Part(ii)is of cour exactly the same as in the method of continuity for the Monge-Amp`e re equation ud by Nadel[15],in the formulation of Demailly-Koll´a r[7].We divide the proof of Theorem1into veral lemmas.
First,we need to recall the fundamental recent result of Perelman.Let the K¨a hler-Ricci flow be defined by(2.1),and for each time t,define the Ricci potential f by
R¯kj−g¯kj=∂j∂¯k f,1
V
X
ˆfωn
.(2.10)
A specific choice of initial data is clearly necessary to discuss the convergence of the K¨a hler-Ricciflow(2.2)for potentials,in view of the fact that different initial data forφlead to flows differing by terms blowing up in time.We will e below that the choice(2.10)is the right choice.
Thefirst indication is that,with the choice(2.10)for the initial data(2.2),Perelman’s estimate for h is equivalent to
sup t≥0||˙φ||C0<∞.(2.11)
To e this,we note that f+˙φis a constant,since∂¯∂(f+˙φ)=0.It suffices to show then that the averageα(t)≡1
V
scheduleX
˙φωn
φ
)=
1
V
mediumbuildX
˙φ∆˙φωn
φ
=
1
V
X
φωnφ=
anemometer1
V
X
awardinglogu的用法
ωnφ
V
X
ˆfωn
φ
,(2.15)
and|α(t)|is bounded in view of(2.14),it follows that|1
V Xφωnφ,and thus ofα(t).
The convergence of g¯kj implies that X admits a K¨a hler-Einstein metric.By a theorem of Bando-Mabuchi[2],the Mabuchi K-energy functional must be then bounded from below.This is well-known to imply in turn that||∇˙φ||L2→0as t→+∞([21]eq.
(2.10)and subquent paragraph).But with the choice(2.10)for initial data for(2.2),we have the estimate(2.14),which implies now thatα(t)→0.Q.E.D.
嗤之以鼻什么意思Lemma2Let X be a compact K¨a hler manifold,and consider the K¨a hler-Ricciflow as defined by(2.1)and(2.2)withω0∈c1(X),and the initial value c0forφspecified by (2.10).Then for any p>1,we have
sup t≥0 X e−pφωn0<∞⇔sup t≥0||φ||C0<∞.(2.16) Proof of Lemma2.This lemma is a direct conquence of the above results of Perelman combined with results of Kolodziej.Clearly,the uniform boundedness of the C0norm of φimplies the uniform boundedness of||e−φ||L p(X).To show the conver,we consider the following Monge-Amp`e re equation
det(ˆg¯kj+∂j∂¯kφ)=Φdetˆg¯kj.(2.17) whereΦis a smooth strictly positive function.Then Kolodziej[12,13]has shown that, for any p>1,the solutionφmust satisfy the a priori bound
osc Xφ≡sup Xφ−inf Xφ≤C p,(2.18) for some constant C p which is bounded if||Φ||L p(X)is bounded.Now the K¨a hler-Ricci flow(2.2)can be rewritten in the form(2.17)withΦ=exp(ˆf−φ+˙φ).By Perelman’s estimate(2.11),||Φ||L p(X)is uniformly bounded if and only if||e−φ||L p(X)is uniformly bounded.Combined with Kolodziej’s result,we e that the uniform boundedness of ||e−φ||L p(X)implies the uniform boundedness of osc Xφ.
To obtain a bound for||φ||C0from oscφ,it suffices to produce a lower bound for sup Xφand an upper bound for inf Xφ.Now,from Perelman’s estimate,we have
C1eˆf−φ+˙φωn0≤e−φωn0≤C2eˆf−φ+˙φωn0,(2.19) and hence,integrating and recalling that eˆf−φ+˙φωn0=ωnφhas the same volume asωn0,脾组词语
C1≤1
Proof of Lemma 3This is the parabolic analogue of Yau’s and Aubin’s well-known result
[29,1],namely,that the same statement holds for the solution φof the elliptic Monge-Amp`e re equatio
n (2.17),with the corresponding constants A k depending on the C ∞norms of the right hand side Φ.Now the K¨a hler-Ricci flow can be rewritten in the form (2.17),with Φ=exp(ˆf
−φ+˙φ).The hypothesis ||φ||C 0≤A 0implies control of ||Φ||C 0,in view of Perelman’s estimate.However,we do not have control of all the C ∞norms of Φ,and hence Yau’s a priori estimates cannot be quoted directly.
Thus we have to go through a full parabolic analogue of Yau’s arguments,and make sure that it goes through without any estimate on ˙φ
which is not provided by Perelman’s result.The arguments here are completely parallel to Yau’s,but we take this opportunity to prent a more streamlined version.The parabolic analogues of veral key identities are also made more explicit.They turn out to be quite simple,and may be more flexible for future work.
Let ∇,∆=∇¯p ∇¯p ,R ¯q p l m ,etc.and ˆ∇,ˆ∆,ˆR ¯q
p l m ,etc.be the connections,laplacians,and curvatures with respect to the metrics g ¯kj and ˆg ¯kj respectively.It is most convenient to formulate all the identities we need in terms of the endomorphism h =h αβdefined by
h αβ=ˆg α¯λg ¯λβ(2.23)
For example,the difference between the connections and curvatures with respect to g ¯kj and ˆg ¯kj can be expresd as
∇m V l −ˆ∇m V l =−V α(∇m h h −1)αl ,∇m V l −ˆ∇m V l =(∇m h h −1)l αV αˆR ¯kj αβ−R ¯kj
αβ=∂¯k (∇j h h −1)αβ(2.24)In particular,taking V l →∂¯k ∂l φ,we find
φj ¯km ≡ˆ∇m ∂¯k ∂j φ=−g ¯kα(∇m h h −1)αj .(2.25)
Henceforth,all indices are raid and lowered with respect to the metric g ¯kj ,unless indi-cated explicitly otherwi.We also t
G =log ωn φ
Tr h {ˆ∆(
G −˙φ)−ˆR }−1Tr h −g δ¯k ∂¯k Tr h ∂δTr h
