Sano, Yuji

Institutional Repositories DataBase (IRDB)

Numerical algorithm for finding balanced metrics

Osaka University Knowledge Archive

Osaka City University Repository

Sano, Y.
Osaka J. Math.
43 (2006), 679–688
NUMERICAL ALGORITHM
FOR FINDING BALANCED METRICS
YUJI SANO
(Received December 24, 2004, revised November 10, 2005)
Abstract
The purpose of this paper is to give an explicit statement with respect to a nu-
merical algorithm for finding balanced metrics, which has already been pointed out
by Donaldson [3].
1. Introduction
First of all, we shall recall the definition of balanced metrics. Let (X; L) be a
compact complex manifold of complex dimension n with a very ample line bundle.
Let h be a Hermitian metric on L , which defines a Ka¨hler form !h := (
p
 1=(2))2(h)
on X , where 2(h) is the curvature form induced by an unitary connection with respect
to h. The metric h and the Ka¨hler form !h define an inner product h  ;  ih on the
(N + 1)-dim vector space E := H 0(X; L) by
hS; S0ih :=
Z
X
h(S; S0) dV
!h
where dV
!h := (1=(n!))!nh is the volume form with respect to h. Let fS0; : : : ; SN g be
an orthonormal basis with respect to the above inner product. The metric form !h is
called balanced when the Bergman kernel
B
!h (p) :=
N
X
k=0
jSk(p)j2h
is constant on X .
The existence of balanced metrics is equivalent to the existence of the specific
projective embedding of X . For a basis fSi g of E , Kodaira embedding theorem im-
plies a projective embedding

fSi g : X ,! CPN
2000 Mathematics Subject Classification. Primary 14D20; Secondary 14C05.
This research was partially supported by the Ministry of Education, Science, Sports and Culture,
Grant-in-Aid for JSPS Fellows, 03340, 2003.
680 Y. SANO
by 
fSi g(p) := [S0(p) :    : SN (p)]. The embedding fSi g is called balanced when
N + 1
V
Z
X
Si ¯S j
PN
k=0 jSk j2
dV


!FS = Æi j ;
where V and !FS denote the volume of X and the Fubini-Study metric form on CPN
respectively. At this moment, we find that !FS is balanced.
The existence of balanced metrics is closely related to stability of the projective
variety (X )  CPN in the sense of Geometric Invariant Theory (cf. [6], [9], [12]).
Using the asymptotics of the Bergman kernel, Donaldson [2] and Mabuchi [7] proved
that the existence of a constant scalar curvature metric in c1(L) implies asymptotic sta-
bility of (X; L).
The purpose of this paper is to carry out explicitly a numerical algorithm for find-
ing balanced metrics. This algorithm has already been obtained by Donaldson in Re-
mark at the end of [3]. However, in view of the importance of the problem, it would
be interesting to carry out the procedure very explicitly. For that purpose, we shall use
two maps defined by Donaldson as follows. Let K and M be the set of metrics h on
L such that !h is a positive (1; 1) form on X and the set of Hermitian metrics on
E respectively. We fix a base point h0 in K. We can identify K as the set of Ka¨hler
potentials, that is,
K =
(
 2 C1(X )





!h0  
p
 1
2

¯
 > 0
)
;
and M as the set of (N + 1)  (N + 1) positive Hermitian matrices with respect to a
fixed basis of E respectively. We define two maps on K and M .
• Define a map
Hilb : K! M
by
kSk2Hilb(h) :=
N + 1
V
Z
X
jSj2h dV!h :
• For a given H 2 M , let fSHi g be an orthonormal basis of E with respect to H ,
which is occasionally abbreviated as H -ONB. So we define a map
FS : M ! K
by
FS(H ) := log 1
PN
k=0

SHk


2
h0
:
NUMERICAL ALGORITHM FOR FINDING BALANCED METRICS 681
Of course, FS(H ) is independent of the choice of an H -ONB and
(1.1)
N
X
k=0

SHk


2
FS(H )  1:
We note that the induced form !FS(H ) equals to 
fSHi g
!FS.
DEFINITION 1.1. A point h 2 K (resp. H 2 M) are called balanced if and
only if
FS ÆHilb(h) = h (resp. Hilb ÆFS(H ) = H ).
By definitions, a balanced metric corresponds to a balanced pair (h; H ) up to scalar
multiplication. Donaldson pointed out that the iteration of the map Hilb ÆFS : M ! M
converges to a balanced point which is the minimiser of a functional ˜Z : M ! R de-
fined in [3]. We denote Aut(X; L) be the group of automorphisms of the pair (X; L)
and the trivial automorphisms C act by constant scalar multiplication on the fibre
of L . Our main result is the following.
Theorem 1.2. Fix a point H0 2 M . Let Hl := Hilb ÆFS(Hl 1). Suppose that
Aut(X; L)=C is discrete. If there is a balanced point H
1
2 M , then there is a con-
stant  > 0 such that fHlg converges to H1 as l !1. Moreover,
(1.2) ˜Z (H
1
) = min
H2M
˜Z (H ) = lim
l!1
˜Z (Hl ):
REMARK 1.3. The convergence stated above makes sense by virtue of the oper-
ator norm when we regard H 2 M as a positive Hermitian matrix with respect to an
H0-ONB.
2. A functional ˜Z on M
Donaldson [3] proved that a balanced point in M is characterized as a critical
point of a functional ˜Z on M . In this section, we shall run though the definition and
the property of ˜Z (see [3] for the full details).
Fix a basis fSi g of E . We regard any H 2 M as a positive Hermitian matrix with
respect to fSi g. Thus we have a map
log det : M ! R:
A different choice of a basis just changes this map by the addition of a constant. Cor-
respondingly, we introduce a functional on K. For a path ht = et h0 with 0 = 0 in
682 Y. SANO
K, we define
I : K! R
by
Ih0 (h1) :=
Z 1
0
dt
Z
X
˙
t dV!ht :
This functional is equivalent to the well-known functional (cf. [4])
F0
!h0
( 1) := J!h0 ( 1) +
1
V
Z
X
1!
n
h0
where
J
!h0
( 1) := 1V
n 1
X
k=0
k + 1
n + 1
Z
X
1
p
 1
2

¯
1 ^ !
k
h0 ^ !
n k 1
h1
=  
1
V
Z 1
0
dt
Z
X
˙
t
 
!
n
h0   !
n
ht

;
hence the functional I is independent of the choice of a path.
We now define maps on K and M respectively as follows:
L := log det ÆHilb : K! R; ˜L := L 
N + 1
V
I;
Z :=  I Æ FS : M ! R; ˜Z := Z +
V
N + 1
log det :
We note that ˜L and ˜Z are invariant by scalar multiplication on K and M respectively.
Donaldson [3] proved:
(1) A point h 2 K (resp. H 2 M) is balanced if and only if it is a critical point
of ˜L (resp. ˜Z ) on K (resp. M). Moreover, if there is a balanced point in K (resp. M),
then it is an absolute minimum of the ˜L (resp. ˜Z ) on K (resp. M).
(2) The map Hilb ÆF S : M ! M decreases ˜Z , i.e., for any H 2 M
(2.1) ˜Z (H )  ˜Z (Hilb ÆFS(H )):
Because of these two results, it seems reasonable that Hl in the main theorem would
converge to the minimiser of ˜Z if balanced points exist.
3. Proof of Theorem 1.2
Fix a point H0 in M and let Hl := Hilb ÆFS(Hl 1). With respect to an H0-ONB,
we identify M as the set of (N + 1)  (N + 1) positive Hermitian matrices. Then, we
can discuss the boundedness of a subset U = fHg  M .
NUMERICAL ALGORITHM FOR FINDING BALANCED METRICS 683
DEFINITION 3.1. A subset U  M is bounded if and only if there is a positive
constant R > 1 satisfying the following condition: For any H 2 U , there is a constant
H > 0 such that
(3.1) H
R
 min
 6= 0
jH ( )j
j j
 max
 6= 0
jH ( )j
j j
 H R:
Here max
 6= 0(jH ( )j=j j) is so-called the operator norm kHkop which is the largest
eigenvalue of H , and min
 6= 0(jH ( )j=j j) is the smallest one.
Note that when U is bounded with respect to a basis of E , it is bounded with
respect to a different one, too.
We shall give a sufficient condition for the existence of balanced points in M .
Proposition 3.2. If the sequence fHlg is bounded, fHlg has a subsequence which
converges to some point H
1
2 M such that
(3.2) ˜Z (H )  ˜Z (H
1
) = inf ˜Z (Hl )
for any H 2 M . In particular, H
1
is balanced.
Proof. From the boundedness of fHlg, we find that there is a subsequence of


 1
l Hl
	
which converges to some point in M . Since ˜Z (Hl ) = ˜Z
 

 1
l Hl

decreases
monotonously with respect to l, so ˜Z is bounded from below on fHlg.
Now we shall show that the map Hilb ÆF S : M ! M decreases Z and log det
respectively. (This can be also found from Lemma 4 and Lemma 5 in [3].) For any
H 2 M , let H 0 := Hilb ÆFS(H ).
n!
V
(Z (H )  Z (H 0))
=  F0
!0
( FS(H )) + F0
!0
( FS(H 0))
= F0
!FS(H ) (FS(H )  FS(H 0))
(because of the cocycle property of F0
!
)
= F0
!FS(H )

log

V
N + 1
B
!FS(H )

= J
!FS(H )

log

V
N + 1
B
!FS(H )

 
n!
V
Z
X
log

V
N + 1
B
!FS(H )

dV
!FS(H )
 J
!FS(H )

log

V
N + 1
B
!FS(H )

(since the logarithm is a concave function)
 0:
684 Y. SANO
The last inequality is one of the well-known properties of the functional J
!
([1], [11]).
From the definition of Hilb ÆF S, we get
Tr(H 0H 1) =
N
X
i=0
N + 1
V
Z
X

SHi


2
h0
PN
j=0

SHj


2
h0
dV
!FS(H )
= N + 1;
where

SHi
	
is an H -ONB. So we find that log det(H 0H 1)  0. We remark that
Tr(H 0H 1) and log det(H 0H 1) are independent of the choice of a basis of E .
Therefore, we find that Z (Hl ) and log det Hl decrease monotonously, respectively.
Since ˜Z (Hl ) is bounded from below, so we find that
(3.3) det  Hl+1 H 1l

! 1
as l !1 and that det Hl is bounded. So, flg is also bounded. Consequently, we get
that there is a subsequence of fHlg which converges to some point H1 2 M .
To prove (3.2), it suffices to show that for any H 2 M and " > 0
(3.4) ˜Z (H )  ˜Z (Hl )  "
for l sufficiently large. Since Hl+1 H 1l is a Hermitian matrix w.r.t. any Hl -ONB, we
can take an Hl-ONB

Sli
	
of E such that Hl+1 H 1l is regarded as a diagonal matrix by
N + 1
V
diag
0

Z
X

Sl0


2
h0
P
k

Slk


2
h0
dV
!FS(Hl ) ; : : : ;
Z
X

SlN


2
h0
P
k

Slk


2
h0
dV
!FS(Hl )
1
A
;
where h0 is a fixed base point in K. So (3.3) implies that
(3.5)






N + 1
V
Z
X

Sli


2
h0
P
k

Slk


2
h0
dV
!FS(Hl )   1






< "
for any l sufficiently large and i = 0; : : : ; N . The boundedness of fHlg implies that
there is a constant RH > 1 independent of l such that
(3.6) l
RH
 min
 6= 0
jHl( )j
j j
 max
 6= 0
jHl ( )j
j j
 l RH
where Hl is regarded as a matrix with respect to an H -ONB and l is some positive
constant. Then, we find that H can be written by a diagonal matrix
(3.7) diag(ea0 ; : : : ; eaN ); jak + log l j  log RH
NUMERICAL ALGORITHM FOR FINDING BALANCED METRICS 685
with respect to a suitable Hl-ONB

ˆSli
	 (which may be not the same as Sli
	
in (3.5)).
By the one-parameter subgroup of GL(N + 1;C)
(t) := diag  eta0=2; : : : ; etaN =2 ; 0  t  1;
we denote a path H (t) := T(t)(t) in M from Hl to H , so H (0) = Hl and H (1) = H .
Let f (t) := ˜Z (H (t)). We have
d
dt
f (t) = d
dt
( I Æ FS(H (t))) + V
N + 1
 N
X
k=0
ak
!
=  
Z
X

d
dt
FS(H (t))

dV
!FS(H (t)) +
V
N + 1
 N
X
k=0
ak
!
=  
Z
X
P
ake
 tak

 ˆSlk


2
h0
P
e tak

 ˆSlk


2
h0
dV
!FS(H (t)) +
V
N + 1
 N
X
k=0
ak
!
:
(3.8)
The equality (3.8) holds, since e tai =2 ˆSli
	
is an H (t)-ONB. Therefore we get
d
dt
f (t)




t=0
=  
Z
X
P
ak

 ˆSlk


2
h0
P

 ˆSlk


2
h0
dVl +
V
N + 1
 N
X
k=0
ak
!
=  
Z
X
P(ak + log l )

 ˆSlk


2
h0
P

 ˆSlk


2
h0
dVl +
V
N + 1
N
X
k=0
(ak + log l )
(3.9)
where dVl denotes dV!FS(Hl ) . From (3.5), (3.7) and (3.9), we have
(3.10) d
dt
f (t)




t=0
  "V log RH :
Donaldson proved that ˜Z is convex along geodesics in M (see Proposition 1 in [3],
and also [2], [9], and [12]), therefore we get
˜Z (H )  ˜Z (Hl )  "V log RH
for l large enough. Since (3.2) means that H
1
is a minimiser (in particular, a critical
point) of ˜Z on M , so H
1
is balanced. Hence, the proof is complete.
We shall prove the boundedness of fHlg if balanced points exist.
Proposition 3.3. Suppose that Aut(X; L)=C is discrete. If there is a balanced
point H
1
in M , fHlg is bounded in M .
686 Y. SANO
Proof. The existence of balanced points in M implies that ˜Z is bounded from
below on M . In the same manner as the proof of Proposition 3.2, we find that det Hl
is bounded. Now we normalize M (with respect to a fixed basis fSi g of E) to
MSL :=

HSL := (det H ) 1=(N+1) H

 H 2 M
	
 M:
We note that this normalization unchanges ˜Z , i.e., ˜Z (H ) = ˜Z (HSL), and that MSL can
be identified as SL(N + 1;C)=SU (N + 1).
The functional ˜Z (HSL) on MSL equals to Z (HSL) which also equals to
 (V =(n!))F0
!h0
( FS(HSL)). Let  2 SL(N + 1;C) with HSL = T ¯ and


 1 := log
T
¯S T¯ 1 1S
T
¯SS
;
where S(p) := T (' Æ S0(p); : : : ; ' Æ SN (p)) is the column vector of the values of holo-
morphic sections of L at p 2 X in a local trivialization '. It is clear that 

 1 is
well-defined and independent of the choice of local trivialization. Then, by using the
cocycle property of F0
!
, we get
(3.11)  F0
!h0
( FS(HSL)) =  F0


!FS
(

 1 ) + C
for some constant C , where  is the projective embedding of X to CPN induced by
fSi g. It is known by many authors that  F0


!FS
(

 1 ) is equivalent to a specific norm
k  kCH on the set of Chow points, that is to say,
(3.12)  V (n + 1)F0


!FS
(

 1 ) = log k
 1
 CH(X )k2CH
kCH(X )k2CH
where CH(X ) is the Chow point of (X ) (See Theorem 5 in [9] and [12]. See also
Section 3 in [10] for the detail of the norm mentioned above).
Since it is known that the assumption of Proposition 3.3 implies the uniqueness
of balanced metrics (see Theorem 1 in [2]), so we find that for any one parameter
subgroup ft g of SL(N + 1;C)
(3.13) ˜Z (Ht ) := ˜Z
 T
¯tt

!1 as t !1
due to the convexity of ˜Z along ft g. From (3.11), (3.12) and (3.13), the numerical
criterion of GIT (cf. [5]) implies that (X )  CPN is Chow stable. In other words, ˜Z
is proper when we regard it as the functional on SL(N +1;C). Since ˜Z (Hl ) is bounded,
so f(Hl )SLg is bounded in M . Therefore we find that fHlg is also bounded due to the
boundedness of det Hl . The proof is complete.
From Proposition 3.2 and the proof of Proposition 3.3, we obtain a necessary and
sufficient condition for Chow stability of (X )  CPN .
NUMERICAL ALGORITHM FOR FINDING BALANCED METRICS 687
Corollary 3.4. Let (X; L) be a complex manifold with a very ample line bundle.
Suppose that Aut(X; L)=C is discrete. The projective variety X  P(H 0(X; L)) is
Chow stable if and only if the sequence fHlg as above is bounded.
We shall prove Theorem 1.2.
Proof of Theorem 1.2. The assumption that Aut(X; L)=C is discrete, implies the
uniqueness of balanced metrics, so the sequence fHlg has a subsequence which con-
verges to H
1
for some positive constant  due to Proposition 3.2 and Proposition 3.3.
Since det Hl decreases monotonously with respect to l and is bounded from below, so
(3.14) det Hl ! (det H1)
as l ! 1. Because of the proof of Proposition 3.2, the uniqueness of balanced met-
rics and (3.14), we find that any subsequence of fHlg has a subsequence which con-
verges to H
1
. This means that fHlg converges to H1. The equality (1.2) follows
from (3.2). The proof is complete.
4. Remarks
Donaldson proved (Theorem 2 in [2]): Let (X; L) be a complex manifold with a
positive line bundlle. Suppose that Aut(X; L)=C is discrete. If (X; Lk) admits a bal-
anced metric !k 2 kc1(X ) for all sufficiently large k and (1=k)!k converges in C1, its
limit !
1
has constant scalar curvature.
Hence it may be natural that the algorithm of balanced metrics proved in this pa-
per would be considered as the algorithm of constant scalar curvature metrics (if it
exists). In particular, the author guesses that this algorithm would substitute for Con-
tinuity method in the case of Ka¨hler-Einstein metrics. He also hope that this prospect
would be one of the approaches for trying the extension of the relationship between
the existence of Ka¨hler-Einstein metrics and the non-existence of multiplier ideal
sheaves defined by Nadel [8] to the case of constant scalar curvature metrics.
References
[1] T. Aubin: Re´duction du cas positif de l’e´quation de Monge-Ampe`re sur les varie´te´s ka¨hle´riennes
compactes a´ la de´monstration d’une ine´qalite´, J. Funct. Anal. 57 (1984), 143–153.
[2] S.K. Donaldson: Scalar curvature and projective embeddings, I, J. Diff. Geom. 59 (2001),
479–522.
[3] S.K. Donaldson: Scalar curvature and projective embeddings, II, arXiv: math.DG/0407534.
[4] A. Futaki: On a character of the automorphism group of a compact manifold, Invent. Math. 87
(1987), 655–660.
[5] D. Gieseker: Geometric invariant theory and applications to moduli problems; in Invariant The-
ory (Montecatini, 1982), Lecture Notes in Math. 996, Springer, Berlin, 1983, 45–73.
688 Y. SANO
[6] H. Luo: Geometric criterion for Gieseker-Mumford stability of polarized manifolds, J. Diff.
Geom. 49 (1998), 577–599.
[7] T. Mabuchi: An energy-theoretic approach to the Hitchin-Kobayashi correspondence for mani-
folds, I, Invent. Math. 159 (2005), 225–243.
[8] A. Nadel: Multiplier ideal sheaves and Ka¨hler-Einstein metrics of positive scalar curvature,
Annals of Math. 132 (1990), 549–596.
[9] D.H. Phong and J. Sturm: Stability, energy functionals, and Ka¨hler-Einstein metrics, Comm. in
Analysis and Geometry 11 (2003) 565–597.
[10] Y. Sano: On stability criterion of complete intersections, J. Geom. Anal. 14 (2004), 533–544.
[11] G. Tian: On Ka¨hler-Einstein metrics on certain Ka¨hler manifolds with C1(M) > 0, Invent. Math.
89 (1987) 225–246.
[12] S. Zhang: Heights and reductions of semi-stable varieties, Compositio Math. 104 (1996),
77–105.
Department of Mathematics
Hokkaido University
Kita 10, Nishi 8, Kita-Ku
Sapporo, Hokkaido, 060-0810, Japan
e-mail: ysano@math.sci.hokudai.ac.jp


http://dlisv03.media.osaka-cu.ac.jp/infolib/user_contents/sugaku/111F0000002-04303-12.pdf

Numerical algorithm for finding balanced metrics

Abstract

Similar works

Full text

Available Versions

Institutional Repositories DataBase (IRDB)

Osaka University Knowledge Archive

Osaka City University Repository