Multiplier Ideal Sheaves and the Kähler-Ricci Flow

0.18 MB·English

by  D. H. Phong

MULTIPLIER IDEAL SHEAVES AND THE
¨
KAHLER-RICCI FLOW
1
D.H. Phong , Natasa Sesum , and Jacob Sturm
∗ ∗ †
7
Abstract
0
0
Multiplier ideal sheaves are constructed as obstructions to the convergence of the
2
Ka¨hler-Ricci flow on Fano manifolds, following earlier constructions of Kohn, Siu,

n
and Nadel, and using the recent estimates of Kolodziej and Perelman.
a
J

0
1

1 Introduction
]
G
D
The global obstruction to the existence of a Hermitian-Einstein metric on a holomorphic
.
h vector bundle is well-known to be encoded in a destabilizing sheaf, thanks to the works of
t
a Donaldson [9, 10] and Uhlenbeck-Yau [27]. It is expected that this should also be the case
m
for general canonical metrics in K¨ahler geometry. For K¨ahler-Einstein metrics on Fano
[
manifolds, obstructing sheaves have been constructed by Nadel [15] as multiplier ideal

2
sheaves, following ideas of Kohn [11] and Siu [23]. This formulation in terms of multiplier
v
4 idealsheaves opensupmanypossibilitiesforrelationswithcomplexandalgebraicgeometry
9
[24, 7, 28].
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1
The obstructing multiplier ideal sheaves are not expected to be unique. Nadel’s con-
1
6 struction is based on the method of continuity for solving a specific Monge-Amp`ere equa-
0
tionforK¨ahler-Einsteinmetrics. Ithasalwaysbeendesirabletoconstructalsoanobstruct-
/
h
ing multiplier ideal sheaf from the K¨ahler-Ricci flow. The purpose of this note is to show
t
a
that this can be easily done, using the recent estimates of Kolodziej [12, 13] and Perelman
m
[18]. In effect, Kolodziej’s estimates provide a Harnack estimate for the Monge-Amp`ere
:
v
equation, which is elliptic, and Perelman’s estimate reduces the K¨ahler-Ricci flow, which
i
X is parabolic, to the Monge-Amp`ere equation. Similar ideas were exploited by Tian-Zhu
r
[26] in their proof of an inequality of Harnack type for the Ka¨hler-Ricci flow.
a
2 The multiplier ideal sheaf
Let X be an n-dimensional compact K¨ahler manifold, equipped with a K¨ahler form ω
0
with µω c (X), where µ is a constant. The K¨ahler-Ricci flow is the flow defined by
0 1
∈
g˙ = (R µg ), (2.1)
k¯j k¯j k¯j
− −
1
Research supported in part by National Science Foundation grants DMS-02-45371, DMS-06-04657
and DMS-05-14003
where g = g (t) is a metric evolving in time t with initial value g (0) = gˆ , and
k¯j k¯j k¯j k¯j
R = ∂ ∂ log detg is its Ricci curvature. Since the K¨ahler-Ricci flow preserves the
k¯j j k¯ q¯p
−
K¨ahler class of the metric, we may set g = gˆ +∂ ∂ φ, and the K¨ahler-Ricci flow can be
k¯j k¯j j k¯
reformulated as
ωn
˙ φ ˆ
φ = log +µφ f, φ(0) = c , (2.2)
ωn − 0
0
where we have set ω = ig dzj dz¯k, and fˆ is the Ricci potential for the metric gˆ ,
φ 2 k¯j ∧ k¯j
that is, the C∞ function defined by the equation Rˆ µgˆ = ∂ ∂ fˆ, normalized by the
k¯j k¯j j k¯
−
condition that
efˆωn = ωn V. (2.3)
ZX 0 ZX 0 ≡
Here and henceforth, Rˆ denotes the Ricci curvature of gˆ , with similar conventions for
k¯j k¯j
all the other curvatures of gˆ . The initial potential c is a constant, so that the initial
k¯j 0
metric coincides with gˆ . The K¨ahler-Ricci flow exists for all time t > 0 [3], and the
k¯j
main issue is its convergence. Henceforth, we shall restrict to the case c (X) > 0 of Fano
1
manifolds unless indicated explicitly otherwise, and set µ = 1.
Theorem 1 Let X be an n-dimensional compact K¨ahler manifold with c (X) > 0.
1
(i) Consider the K¨ahler-Ricci flow (2.2) for potentials φ, with the initial value c spec-
0
ified by (2.10) below. If there exists some p > 1 with
sup e−pφωn < , (2.4)
t≥0ZX 0 ∞
then there exists a sequence of times t + with g (t ) converging in C∞ to a K¨ahler-
i k¯j i
→ ∞
Einstein metric. If in addition X admits no non-trivial holomorphic vector field, then the
whole flows (2.1) and (2.2) converge in C∞.
(ii) If X does not admit a K¨ahler-Einstein metric, then for each p > 1, there exists
a function ψ which is a L1 limit point of the K¨ahler-Ricci flow (2.2), with the following
property. Let the multiplier ideal sheaf (pψ) be the sheaf with stalk at z defined by
J
(pψ) = f; U z, f (U), f 2e−pψωn < , (2.5)
Jz { ∃ ∋ ∈ O ZU | | 0 ∞}
where U X is open, and (U) denotes the space of holomorphic functions on U. Then
⊂ O
(pψ) defines a proper coherent analytic sheaf on X, with acyclic cohomology, i.e.,
J
Hq(X,K−[p] (pψ)) = 0, q 1. (2.6)
X ⊗J ≥
If X admits a compact group G of holomorphic automorphisms, and gˆ is G-invariant,
k¯j
then (pψ) and the corresponding subscheme are also G-invariant.
J
2
In Part (i), once the convergence of a subsequence g (t ) has been established and
k¯j i
X is known to admit a K¨ahler-Einstein metric, it follows from an unpublished result of
Perelman that the full K¨ahler-Ricci flow must then converge. An extension of Perelman’s
result to K¨ahler-Ricci solitons is given in Tian-Zhu [26]. For the sake of completeness, we
have provided a short self-contained proof of the K¨ahler-Einstein case, in our context and
under the simplifying assumption of no non-trivial holomorphic vector fields.
Part (ii) is of course exactly the same as in the method of continuity for the Monge-
Amp`ere equation used by Nadel [15], in the formulation of Demailly-Koll´ar [7]. We divide
the proof of Theorem 1 into several lemmas.
First, we needtorecallthefundamental recent result ofPerelman. Let theK¨ahler-Ricci
flow be defined by (2.1), and for each time t, define the Ricci potential f by
1
R g = ∂ ∂ f, efωn = 1. (2.7)
k¯j − k¯j j k¯ V ZX φ
ˆ
(In particular, at time t = 0, the Ricci potential coincides with the function f defined
earlier). Then Perelman [18] (see also [22]) has shown that
sup ( f + f + ∆f ) < + , (2.8)
t≥0 || ||C0 ||∇ ||C0 || ||C0 ∞
with Laplacians and norms taken with respect to the metric g .
k¯j
Next, we specify the value of the initial potential c in (2.2), following Chen and Tian
0
[5] (see also [14]). The underlying observation is that φ˙ 2 , and hence the integral
||∇ ||L2
∞
e−t φ˙ 2 dt (2.9)
Z ||∇ ||L2
0
does not depend on the choice of initial value c for the flow (2.2). Indeed, given a flow φ
0
withinitialvaluec , thefunctionφ˜= φ+(c˜ c )et satisfies thesame flowwithinitialvalue
0 0 0
−
c˜ , and hence, by uniqueness, must coincide with the flow with initial value c˜ . Clearly,
0 0
˜
φ = φ, hence the assertion. Note also that the integral in (2.9) is always finite, in view
∇ ∇
of Perelman’s estimate (2.8). Following [5], we use this common value to choose the initial
value c in (2.2),
0
∞ 1
c = e−t φ˙ 2 dt+ fˆωn. (2.10)
0 Z0 ||∇ ||L2 V ZX 0
Aspecificchoiceofinitialdataisclearlynecessary todiscuss theconvergence oftheK¨ahler-
Ricci flow (2.2) for potentials, in view of the fact that different initial data for φ lead to
flows differing by terms blowing up in time. We will see below that the choice (2.10) is
the right choice.
The first indication is that, with the choice (2.10) for the initial data (2.2), Perelman’s
estimate for h is equivalent to
˙
sup φ < . (2.11)
t≥0|| ||C0 ∞
3
˙ ¯ ˙
To see this, we note that f + φ is a constant, since ∂∂(f + φ) = 0. It suffices to show
then that the average α(t) 1 φ˙ωn of φ˙ is uniformly bounded in absolute value, since
≡ V X φ
f already is, by Perelman’s estRimate (2.8). Now differentiating the equation (2.2) gives
| |
˙ ˙ ˙
∂ φ = ∆φ+φ, and hence
t
1 1 1 1
∂ ( φ˙ωn) = (∆φ˙ +φ˙)ωn + φ˙∆φ˙ωn = φ˙ωn φ˙ 2 . (2.12)
t V ZX φ V ZX φ V ZX φ V ZX φ −||∇ ||L2
This is a differential equation for α(t) which can be integrated, giving
t ∞
e−tα(t) = α(0) e−s φ˙(s) 2 ds = e−s φ˙(s) 2 ds, (2.13)
−Z0 ||∇ ||L2 Zt ||∇ ||L2
˙ ˆ
in view of (2.10) and the fact that at time t = 0, we have φ = c f. It follows that
0
−
∞ ∞
0 α(t) = e−(s−t) φ˙(s) 2 ds C e−(s−t)ds C, (2.14)
≤ Zt ||∇ ||L2 ≤ Zt ≤
˙
where we have applied Perelman’s uniform bound for φ . This proves (2.11).
C0
||∇ ||
More systematically, uniform bounds for φ and g are now equivalent:
k¯j
Lemma 1 Let X be a compact K¨ahler manifold, with K¨ahler form ω c (X), and
0 1
∈
consider the K¨ahler-Ricci flows (2.1) and (2.2) for g and φ respectively. Let the initial
k¯j
value c for φ be given by (2.10). Then φ is uniformly bounded for all m if and only
0 Cm
|| ||
if g is bounded for all m (here norms are taken with respect to a fixed reference
k¯j Cm
|| ||
metric, say gˆ ). The flow for g converges in C∞ if and only if the flow for φ converges
k¯j k¯j
in C∞.
Proof of Lemma 1: Clearly, the convergence/boundedness of the potentials φ’s implies the
convergence/boundedness of the metrics g . Conversely, the convergence/boundedness
k¯j
¯
of the metrics implies the convergence/boundedness of ∂∂φ, so it suffices to establish the
convergence/boundedness of the averages of φ with respect to the volume forms ωn. Since
φ
the flow implies
1 1 1 ωn 1
φωn = φ˙ωn log φωn + fˆωn, (2.15)
V ZX φ V ZX φ − V ZX ω0n φ V ZX φ
and α(t) is bounded in view of (2.14), it follows that 1 φωn is bounded in either case.
| | |V X φ|
Assume now that g converges. We wish to show the coRnvergence of 1 φωn, and thus
k¯j V X φ
of α(t). R
The convergence of g implies that X admits a K¨ahler-Einstein metric. By a theorem
k¯j
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 0 as t + (see e.g. [21] eq.
L2
||∇ || → → ∞
(2.10) and subsequent paragraph). But with the choice (2.10) for initial data for (2.2), we
have the estimate (2.14), which implies now that α(t) 0. Q.E.D.
→
4
Lemma 2 Let X be a compact K¨ahler manifold, and consider the K¨ahler-Ricci flow as
defined by (2.1) and (2.2) with ω c (X), and the initial value c for φ specified by
0 1 0
∈
(2.10). Then for any p > 1, we have
sup e−pφωn < sup φ < . (2.16)
t≥0ZX 0 ∞ ⇔ t≥0|| ||C0 ∞
Proof of Lemma 2. This lemma is a direct consequence of the above results of Perelman
combined with results of Kolodziej. Clearly, the uniform boundedness of the C0 norm of
φ implies the uniform boundedness of e−φ . To show the converse, we consider the
Lp(X)
|| ||
following Monge-Amp`ere equation
det(gˆ +∂ ∂ φ) = Φdetgˆ . (2.17)
k¯j j k¯ k¯j
where Φ is a smooth strictly positive function. Then Kolodziej [12, 13] has shown that,
for any p > 1, the solution φ must satisfy the a priori bound
osc φ sup φ inf φ C , (2.18)
X ≡ X − X ≤ p
for some constant C which is bounded if Φ is bounded. Now the K¨ahler-Ricci
p Lp(X)
|| ||
ˆ ˙
flow (2.2) can be rewritten in the form (2.17) with Φ = exp(f φ+ φ). By Perelman’s
−
estimate (2.11), Φ is uniformly bounded if and only if e−φ is uniformly
Lp(X) Lp(X)
|| || || ||
bounded. Combined with Kolodziej’s result, we see that the uniform boundedness of
e−φ implies the uniform boundedness of osc φ.
Lp(X) X
|| ||
To obtain a bound for φ from oscφ, it suffices to produce a lower bound for sup φ
C0 X
|| ||
and an upper bound for inf φ. Now, from Perelman’s estimate, we have
X
C efˆ−φ+φ˙ωn e−φωn C efˆ−φ+φ˙ωn, (2.19)
1 0 ≤ 0 ≤ 2 0
and hence, integrating and recalling that efˆ−φ+φ˙ωn = ωn has the same volume as ωn,
0 φ 0
1
C e−φωn C . (2.20)
1 ≤ V ZX 0 ≤ 2
This implies at once that
sup φ log C , inf φ log C . (2.21)
X ≥ − 2 X ≤ − 1
The proof of Lemma 2 is complete. Q.E.D.
Lemma 3 Let X be a compact K¨ahler manifold with K¨ahler form ω satifying µω
0 0
∈
c (X), where µ is any constant. Let the K¨ahler-Ricci flow be defined by (2.2). Then we
1
have the a priori estimates
sup φ A < sup φ A < , k N. (2.22)
t≥0|| ||C0 ≤ 0 ∞ ⇔ t≥0|| ||Ck ≤ k ∞ ∀ ∈
5
Proof of Lemma 3 This is the parabolic analogue of Yau’s and Aubin’s well-known result
[29, 1], namely, that the same statement holds for the solution φ of the elliptic Monge-
Amp`ere equation (2.17), with the corresponding constants A depending on the C∞ norms
k
of the right hand side Φ. Now the K¨ahler-Ricci flow can be rewritten in the form (2.17),
ˆ ˙
with Φ = exp(f φ+φ). The hypothesis φ A implies control of Φ , in view
C0 0 C0
− || || ≤ || ||
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 present a more streamlined version. The parabolic analogues of several key identities
are also made more explicit. They turn out to be quite simple, and may be more flexible
for future work.
Let , ∆ = p¯ , R l , etc. and ˆ, ∆ˆ, Rˆ l , etc. be the connections, laplacians,
p¯ q¯p m q¯p m
∇ ∇ ∇ ∇
and curvatures with respect to the metrics g and gˆ respectively. It is most convenient
k¯j k¯j
to formulate all the identities we need in terms of the endomorphism h = hα defined by
β
hα = gˆαλ¯g (2.23)
β λ¯β
For example, the difference between the connections and curvatures with respect to g
k¯j
and gˆ can be expressed as
k¯j
V ˆ V = V ( hh−1)α , Vl ˆ Vl = ( hh−1)l Vα
m l m l α m l m m m α
∇ −∇ − ∇ ∇ −∇ ∇
Rˆ α R α = ∂ ( hh−1)α (2.24)
k¯j β k¯j β k¯ j β
− ∇
In particular, taking V ∂ ∂ φ, we find
l k¯ l
→
φ ˆ ∂ ∂ φ = g ( hh−1)α . (2.25)
jk¯m m k¯ j k¯α m j
≡ ∇ − ∇
Henceforth, all indices are raised and lowered with respect to the metric g , unless indi-
k¯j
cated explicitly otherwise. We also set
ωn
φ
G = log . (2.26)
ωn
0
Proof of the C2 estimates: The basic identity for this step is the following,
1 1
(∆ ∂ )log Trh = ∆ˆ(G φ˙) Rˆ gpq¯g gˆrm¯Rˆ j
t m¯j q¯p r
− Trh{ − − }− Trh
+ gˆδk¯φγk¯pφγδp gδk¯∂k¯Trh∂δTrh (2.27)
{ Trh − (Trh)2 }
6
This identity follows from another well-known identity [29], which will also be of later use,
∆Trh = ∆ˆG Rˆ +gˆδk¯φ φγ p gpq¯g gˆrm¯Rˆ j , (2.28)
γk¯p δ m¯j q¯p r
− −
and can be seen as follows: ∆Trh = ∆¯Trh = gpq¯ Tr ( hh−1)h , and thus
q¯ p
∇ { ∇ }
∆Trh = gpq¯ Tr( hh−1)+gpq¯Tr ( hh−1) h (2.29)
q¯ p p q¯
∇ ∇ { ∇ ∇ }
The second term on the right hand side can be recognized as gˆδk¯φ φγ p using (2.25),
γk¯p δ
while, using (2.24), the first term can be rewritten as
gpq¯ Tr( hh−1) = gpq¯Rˆ α hβ Rα hβ = gpq¯g Rˆ α gˆβλ¯ Rα hβ . (2.30)
q¯ p q¯p β α β α λ¯α q¯p β β α
∇ ∇ − −
But the Ricci curvature R and be expressed in terms of G, R = Rˆ ∂ ∂ G. Sub-
γ¯β γ¯β γ¯β β γ¯
−
stituting in gives (2.28). Taking the log and subtracting the simple identity ∂ log Trh =
t
(∆ˆφ˙)(Trh)−1 gives (2.27).
So far the discussion has been general. Let now φ evolve by the K¨ahler-Ricci flow,
˙ ˆ
φ G = µφ f, (2.31)
− −
so that the term ∆ˆ(φ˙ G) in (2.27) can be replaced by the more tractable term µ∆ˆφ ∆ˆfˆ.
− −
In [29], it was shown that the expression in brackets in (2.27) was always non-negative,
while the curvature tensor term was bounded by
n 1+φ 1
gpq¯g Rˆj gˆαm¯ = ¯iiRˆ C(Trh) , (2.32)
m¯j pq¯α ¯ii¯jj
− − 1+φ ≥ − 1+φ
iX,j=1 ¯jj Xj ¯jj
in a system of local holomorphic coordinates where both g and gˆ were diagonal, and
k¯j k¯j
gˆ was the identity matrix at a given point. Thus we have
k¯j
1 n 1 n 1
(∆ ∂ ) log Trh µ C C µ C . (2.33)
t 1 2 3
− ≥ − − Trh − 1+φ ≥ − − 1+φ
jX=1 ¯jj jX=1 ¯jj
Let A be any constant. Since
n φ n 1
¯jj
∆φ = = n , (2.34)
1+φ − 1+φ
jX=1 ¯jj jX=1 ¯jj
we can write
n 1
˙
(∆ ∂ )(logTrh Aφ) C φ C +C ,
t 4 5 6
− − ≥ − 1+φ
jX=1 ¯jj
7
with A = C , C = µ+An, and C = A C > 0 forA largeenough. In view of Perelman’s
4 5 6 3
−
estimate (2.11), we conclude
n 1
(∆ ∂ )(log Trh Aφ) C +C . (2.35)
t 7 6
− − ≥ − 1+φ
jX=1 ¯jj
Let now [0,T] be any time interval, and (z ,t ) a point in X [0,T] where the function
0 0
×
log Trh Aφ attains its maximum. If this point is not at time t = 0, then the left hand
−
side of the above equation is 0, and we obtain the estimate
≤
1
C , 1 j n. (2.36)
8
1+φ ≤ ≤ ≤
¯jj
But then, at the point (z ,t ),
0 0
detgˆ n n 1
Trh = Trh( k¯j)efˆ−φ+φ˙ = efˆ−φ+φ˙ (1+φ )
¯ii
detg 1+φ
k¯j Xi=1 jY=1 ¯jj
n 1
= efˆ−φ+φ˙ C , (2.37)
9
1+φ ≤
Xi=1jY6=i ¯jj
using the boundedness of φ and again Perelman’s estimate. But now we have
C0
|| ||
supX×[0,T]Trh ≤ eA||φ||C0exp(log Trh−Aφ)(z0,t0) ≤ C10. (2.38)
Since T is arbitrary, this establishes the boundedness of the trace of gˆ +∂ ∂ φ, and since
k¯j j k¯
the matrix is positive, of all its entries. The proof of the C2 estimate is complete.
Proof of the C3 estimates: this step was established in [3] when c (X) = 0 or c (X) < 0,
1 1
and in [18] for c (X) > 0. We shall give below a simpler proof for all cases with completely
1
explicit formulas.
The main ingredient is a parabolic analogue of the Yau, Aubin, and Calabi identities
for the third derivatives of the Monge-Amp`ere equation. In their case, the Ricci curvature
is pre-assigned and hence all its derivatives can be controlled. In the present case, we
cannot control as yet the derivatives of the Ricci curvature, and it is crucial that they
cancel out in the desired identity. We show this by a completely explicit formula, the
main technical innovation being the use of the endomorphism hα instead of the potential
β
φ itself. The squared terms in the Calabi identity arise naturally as the familiar squared
terms in a formula of Bochner-Kodaira type.
Let S be defined as in [29] by
S = gjr¯gsk¯gmt¯φ φ (2.39)
jk¯m r¯st¯
In terms of hα , S is just the square of the L2 norm of the g connection,
β k¯j
S = gmγ¯g glα¯( hh−1)β ( hh−1)µ = hh−1 2 (2.40)
µ¯β m l γ α
∇ ∇ |∇ |
8
and its Laplacian leads immediately to a formula of Bochner-Kodaira type,
∆S = gmγ¯g glα¯( ∆( hh−1)β ( hh−1)µ +( hh−1)β ∆¯( hh−1)µ )
µ¯β m l γ α m l γ α
∇ ∇ ∇ ∇
+ ¯( hh−1) 2 + ( hh−1) 2 (2.41)
|∇ ∇ | |∇ ∇ |
where, more explicitly, ¯( hh−1) 2 = gqp¯gjm¯gβδ¯g ( hh−1)α ( hh−1)γ , etc.
γ¯α p¯ j β q¯ m δ
|∇ ∇ | ∇ ∇ ∇ ∇
The relation between ∆¯ and ∆ follows from commuting the and the derivatives,
q p¯
∇ ∇
(∆¯( hh−1))γ = (∆( hh−1))γ Rγ ( hh−1)µ +Rµ ( hh−1)γ
j α j α µ γ α α j µ
∇ ∇ − ∇ ∇
+Rµ ( hh−1)γ (2.42)
j µ α
∇
Thus we have
∆S = gmγ¯g glα¯( ∆( hh−1)β ( hh−1)µ +( hh−1)β ∆( hh−1)µ )
µ¯β m l γ α m l γ α
∇ } ∇ ∇ ∇
+ ¯( hh−1) 2 + ( hh−1) 2
|∇ ∇ | |∇ ∇ |
+( hh−1)β ( gmγ¯g Rlρ¯( hh−1)µ gmγ¯R glα¯( hh−1)ρ
m l µ¯β γ ρ ρ¯β γ α
∇ ∇ − ∇
+Rmρ¯g glα¯( hh−1)µ ) (2.43)
µ¯β ρ α
∇
In the case of the elliptic Monge-Amp`ere equation, this equation suffices already to es-
tablish the desired inequality ∆S C S C . This is because the Ricci tensor R is
1 2 α¯β
≥ − −
known in that case, and the Laplacian of hh−1 can be readily reduced to R , using
α¯β
∇ ∇
(2.24) and the Bianchi identity,
∆( hh−1)l = p¯∂ ( hh−1) = p¯R l p¯Rˆ l = Rl + p¯Rˆ l
j m p¯ j p¯j m k¯pj m j m p¯j m
∇ ∇ ∇ −∇ −∇ −∇ ∇
Since the connection p¯ is manifestly O( hh−1) = O(√S), the desired lower bound
∇ ∇
follows at once. The full expression for ∆S in terms of R may also be of interest,
α¯β
∆S = gmγ¯g glα¯( Rβ ( hh−1)µ +( hh−1)β Rµ )
µ¯β m l γ α m l γ α
− ∇ ∇ ∇ ∇
+ ¯( hh−1) 2 + ( hh−1) 2
|∇ ∇ | |∇ ∇ |
+( hh−1)β ( gmγ¯g Rlρ¯( hh−1)µ gmg¯R glα¯( hh−1)ρ
m l µ¯β γ ρ ρ¯β γ α
∇ ∇ − ∇
+Rmρ¯g glα¯( hh−1)µ )
µ¯β ρ α
∇
+gmγ¯g glα¯( p¯Rˆ β ( hh−1)µ +( hh−1)β p¯Rˆ µ ). (2.44)
µ¯β p¯m l γ α m l p¯γ α
∇ ∇ ∇ ∇
In the parabolic case, we do not have control of R and its derivatives, and need to
α¯β
eliminate these terms using the time derivative S˙ of S. We begin by giving a general
formula for S˙ in terms of h−1h˙, so that it is valid for all evolutions. First, note that the
derivatives of the connection are given by
( h)˙= (h−1h˙)h+( h)(h−1h˙), ( hh−1)˙= (h−1h˙). (2.45)
j j j j j
∇ ∇ ∇ ∇ ∇
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Indeed, note that g˙ = g h˙µ = g (h−1h˙)ν = (gh−1h˙) , and (gβ¯b)˙ = (g˙)β¯b =
α¯β 0α¯µ β α¯ν β α¯β
−
(h−1h˙)β gν¯b = (h−1h˙g−1)β¯b. Writing
ν
− −
( h)˙ p = g gpa¯∂ (gβ¯bg hα ) = g−1∂ (ghg−1)g p (2.46)
j q ¯bq j a¯α β j q
{ ∇ } { }
differentiating with respect to time, and substituting in the preceding formulas for g˙
α¯β
and (g¯α¯β)˙ gives at once ( h)˙= h−1h˙ h+( h)(h−1h˙)+ (h−1h˙h), from which the
j j j j
∇ − ∇ ∇ ∇
desired formula for ( h)˙ follows. The formula for ( hh−1)˙ is a simple consequence of
j j
∇ ∇
the one for ( h)˙. Next, differentiating S gives
j
∇
S˙ = +gmγ¯g glα¯( ∂ (( hh−1)β )( hh−1)µ +( hh−1)β )∂ ( hh−1)µ )
µ¯β t m l γ α m l t γ α
∇ ∇ ∇ ∇
( hh−1)β ( (h−1h˙)mγ¯g glα¯( hh−1)µ +gmγ¯(h−1h˙) glα¯( hh−1)µ
m l µ¯β γ α µ¯β γ α
− ∇ ∇ ∇
gmγ¯g (h−1h˙)lα¯( hh−1)µ ) (2.47)
µ¯β γ α
− ∇
Combining with (2.43), we obtain the following general heat equation,
(∆ ∂ )S = ¯( hh−1) 2 + ( hh−1) 2
t
− |∇ ∇ | |∇ ∇ |
+gmγ¯g glα¯ (∆ ∂ )( hh−1)β ( hh−1)µ
µ¯β t m l γ α
{ − ∇ ∇
+gmγ¯g glα¯ ( hh−1)β (∆ ∂ )( hh−1)µ
µ¯β m l t γ α
{ ∇ − ∇
+ (h−1h˙ +R)mγ¯g glγ¯ gmγ¯(h−1h˙ +R) glα¯ +gmγ¯g (h−1h˙ +R)lα¯
µ¯β µ¯β µ¯β
(cid:26) − (cid:27)
( hh−1)β ( hh−1)µ (2.48)
m l γ α
× ∇ ∇
We can now specialize to the K¨ahler-Ricci flow, where
(h−1h˙)β = (Rβ µδβ ), (2.49)
l l l
− −
and hence, using again (2.24),
(h−1h˙ +R)β = µδβ
λ λ
(∆ ∂ )( hh−1)l = q¯Rˆ l . (2.50)
t j m q¯j m
− ∇ ∇
Substituting this in the previous formula for (∆ ∂ )S, we obtain the following simple
t
−
and completely explicit parabolic analogue of the C3 identity of Yau, Aubin, and Calabi,
(∆ ∂ )S = ¯( hh−1) 2 + ( hh−1) 2 +µ hh−1 2
t
− |∇ ∇ | |∇ ∇ | |∇ |
+gmγ¯ q¯Rˆ β ( hh−1) ¯l +gmγ¯( hh−1) α¯ q¯Rˆ µ (2.51)
q¯m l γ β¯ m µ¯ q¯γ α
∇ ∇ ∇ ∇
Note that the terms in R and its derivatives have cancelled out. Since Rˆ l is a
k¯j q¯m β
fixed tensor, we obtain immediately the estimate
(∆ ∂ )S ¯( hh−1) 2 + ( hh−1) 2 C S C . (2.52)
t 1 2
− ≥ |∇ ∇ | |∇ ∇ | − −
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