The Hessian of quantized Ding functionals and its asymptotic behavior

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by  Ryosuke Takahashi

THE HESSIAN OF QUANTIZED DING FUNCTIONALS
AND ITS ASYMPTOTIC BEHAVIOR
7
1
RYOSUKETAKAHASHI
0
2

n Abstract. WecomputetheHessianofquantizedDingfunctionalsand
a giveanelementaryprooffortheconvexityofquantizedDingfunctionals
J alongBergmangeodesicsfromtheviewpointofprojectivegeometry. We

4 study also the asymptotic behavior of the Hessian using the Berezin-
2 Toeplitz quantization.

]
G
D
. 1. Introduction
h
t
a Let X be an n-dimensional Fano manifold and k a large integer such that
m
kK is very ample. Let ( K ) be the space of smooth fiber metrics φ
X X
[ o−n K with positive curvHat−ure ω := (√ 1/2π)∂∂¯φ and the space of
2 herm−itiXan forms on H0(X, kKX).φFor a m−etric φ ( KBXk), we denote
− ∈ H −
v the Monge-Amp`ere volume form by
7
3 ωn
φ
0 MA(φ) := ,
n!
6
0 and the canonical volume form by
.
1
2
0 ∂ ∂
e φ := (√ 1)n2 dz dz dz¯ dz¯ .
7 − 1 n 1 n
− ∂z ∧···∧ ∂z ∧···∧ ∧ ∧···∧
1 (cid:12) 1 n(cid:12)φ
(cid:12) (cid:12)
:
v This expression is re(cid:12)adily verified to(cid:12)be independent of the local holomor-
(cid:12) (cid:12)
i
X phic coordinates (z1,...,zn) and hence defines a volume form on X. We
normalize e φ to be a probability measure
r −
a
e φ
−
µ := .
φ e φ
X −
The space ( K ) admits a naturaRl Riemannian metric, called Mabuchi
X
metric, defiHned−by the L2-norm of a tangent vector f C (X,R) at φ:
∞
∈
f 2 := f2MA(φ).
| |φ X
Recall that Ding functionals on the space of infinite dimensional Rie-
R D
mannian manifold ( K ) is uniquely characterized (modulo an additive
X
H −
2010 Mathematics Subject Classification. 53C25.
Key words and phrases. Dingfunctional, quantization, K¨ahler-Einstein metric.
This work was supported by Grant-in-Aidfor JSPS Fellows Number16J01211.
1
2
constant) by the property
µ n!
φ
= ,
∇D|φ MA(φ) − ( KX)n
−
where denotes the gradient and ( K )n is the top intersection number.
X
∇ −
TheDing functionals are important in the study of Ka¨hler-Einstein metrics.
For instance, Donaldson [5] recently gave a “moment map” interpretation
of the Ding functional. More precisely, he showed that the ratio of volumes
µ /MA(φ) n!/( K )n arises as the moment map for a suitable infinite
φ X
− −
dimensional symplectic manifold and the Ding functional can be viewed as
the Kempf-Ness function. This interpretation provides us the direct link
between the existence problem of Ka¨hler-Einstein metrics and stability in
Geometric Invariant Theory.
On the other hand, quantization of the Ding functionals is also studied:
given φ ( K ), we define a hermitian form Hilb (φ) by
X k k
∈ H − ∈ B
s 2 := s 2e kφMA(φ).
k kHilbk(φ) | | −
ZX
Conversely, for a given H , we define a metric FS (H) ( K ) by
k k X
∈ B ∈H −
1 s 2
FSk(H) := log k−n sup | | .
k H(s,s)
s H0( kKX) 0 !
∈ − \{ }
In what follows, we fix some H and a reference H -ONB s :=
0 ∈ Bk 0 0
(s1,...,sNk), which defines an embedding ιs0: X ֒→ CPNk−1, where Nk :=
dimH0( kK ). For g GL(N ,C), let H be a hermitian form such
X k g k
− ∈ ∈ B
that s g is an H -orthonormal basis. Then the map g H gives an
isomor0ph·ism GL(Ng,C)/U(N ) , and the tangent spac7→e of g can be
k k k k
≃ B B
identified with √ 1u(N ). Thus the space admits a natural Riemannian
k k
− B
structure defined by the Killing form tr(AB) (A,B √ 1u(N )) at each
k
∈ −
tangent space.
Write M: CPNk−1 √ 1u(Nk) for the map:
→ −
Z Z
α β
M([Z ; ;Z ]) := .
1 ··· Nk Z 2
(cid:18) i| i| (cid:19)αβ
We define the center of mass M(g) by thePformula
M(g) := (M g)µ ,
◦ FSk(Hg)
ZX
where we identify X with ι (X) and the measure µ with its push
s0 FSk(Hg)
forward (ι ) µ . Quantized Ding functionals (k) on the space of in-
s0 ∗ FSk(Hg) D
finite dimensional Riemannian manifold ( K ) is uniquely characterized
X
H −
(modulo an additive constant) by the property
Id
(k) = k 1 M(g) .
∇D |Hg − (cid:18) − Nk(cid:19)
3
There is also a finite dimensional moment-map picture for this setting: the
space of all bases (k) GL(N ) is equipped with a Ka¨hler structure in-
k
Z ≃
duced from the Berndtsson metric and U(N ) acts on (k) isometrically
k
Z
with a moment map which is essentially M(g) Id/N . Critical points of
k
−
the quantized Ding functionals M(g) = Id/N are called anti-canonically
k
balanced metrics. There is a strong connection between the existence prob-
lemofanti-canonically balancedmetricsandKa¨hler-Einsteinmetrics(cf. [1,
Theorem 7.1]).
In this paper, we study the Hessian of quantized Ding functionals and its
asymptotic behavior as raising exponent k . We first give a formula
→ ∞
for the Hessian of the quantized Ding functional 2 (k):
∇ D
Theorem 1.1. The Hessian of the quantized Ding functional is computed
by
2 (k) (A,B) = k 1 Re(ξ ,ξ ) µ k 2 H(A)H(B)µ
∇ D |H0 − ZX A B FS FSk(H0)− − ZX FSk(H0)
+ k 2 H(A)µ H(B)µ ,
− FSk(H0)· FSk(H0)
ZX ZX
where ξA denotes the holomorphic vector filed on CPNk−1 corresponding to
R
A, and H(A) is the Hamiltonian for the Killing vector Jξ , (, ) is the
A · · FS
Fubini-Study inner product on tangent vectors.
As a corollary, we will show the following:
Corollary 1.1. We have 2 (k) (A,A) 0 for any A √ 1u(N ), and
∇ D |H0 ≥ ∈ − k
theequalityholdsifandonlyifA Lie(Aut(X, kK )), where Aut(X, kK )
X X
∈ − −
denotes the group of holomorphic automorphisms of the pair (X, kK ),
X
embedded into GL(N ,C) by means of the reference basis s . −
k 0
Although Corollary 1.1 is a direct consequence of Berndtsson’s convexity
theorem [2, Theorem 2.4] (see also [1, Lemma 7.2]), our proof is completely
independent, based on the viewpoint of projective geometry, and somewhat
elementary.
Next, we fix a reference metric φ ( K ) and set H := Hilb (φ ).
0 X 0 k 0
For f C (X,R), we associate with∈thHe −derivative of the Hilbert map in
∞
∈
the direction f:
d
Q := Hilb (φ tf) = (kf ∆ f)(s ,s )e kφ0MA(φ ) ,
f,k dt k 0− |t=0 (cid:18)ZX − φ0 α β − 0 (cid:19)αβ
where∆ denotes the(negative) ∂¯-Laplacian with respectto φ , andin the
φ0 0
last inequality, we identified Q with a hermitian matrix by means of the
f,k
reference basis s . With this operation, we can connect 2 (k) to 2 as
0 ∇ D ∇ D
follows:
Theorem1.2. For any functionsf,g C (X,R), we have the convergence
∞
∈
of the Hessian
(1.1) 2 (k) (Q ,Q ) 2 (f,g)
∇ D |H0 f,k g,k → ∇ D|φ0
4
as k . In particular, lim 2 (k) (Q ,Q ) = 0 implies that
the co→ndi∞tion characterizing degk→en∞er∇acyD 2|H0 (ff,,kf) =f,k0 follows. Finally,
the above convergence is uniform when ∇f,gDv|φa0ry in a subset of C (X,R)
∞
which is compact for the C -topology. It is also uniform for φ as long as
∞ 0
φ stays a compact set in the C -toplogy.
0 ∞
This is an analogue of Berndtsson’s result [2, Theorem 4.1], but quan-
tization schemes are different. Moreover, his argument is based on Hodge
theory, whereas an important technical tool we use in our proofs is Berezin-
Toeplitz quantization provided by Ma-Marinescu [8]. We should mention as
well that J. Fine [7] studied the quantization of the Lichnerowicz operator
on general polarized manifolds. Our method follows a strategy discovered
by him.
Acknowledgements. The author would like to express his gratitude to
his advisor Professor Shigetoshi Bando for useful discussions on this arti-
cle. This research is supported by Grant-in-Aid for JSPS Fellows Number
16J01211.
2. Foundations
2.1. Functionals on the space of metrics. We have a quick review on
several functionals over the space of metrics ( K ) or which play a
X k
H − B
central role in the study of Ka¨hler-Einstein metrics. The standard reference
for this section is [1]. We fix a reference metric φ ( K ). We define
0 X
∈ H −
the Monge-Amp`ere energy by
n
1
(φ) := (φ φ )ωn i ωi ,
E (n+1)( K )n − 0 φ− ∧ φ0
− X i=1ZX
X
and the Ding functional by
D
(φ):= (φ)+ (φ), (φ) := log e φ.
−
D −E L L −
ZX
Let h be the Ricci potential of ω :
φ φ
µ
φ
h := log .
φ
MA(φ)
A direct computation implies that the derivative of along a smooth curve
D
φ in ( K ) is
t X
H −
d2 n!
dt2D(φt) = − (φ¨t −|∂φ˙t|2φt) ( K )n −ehφt MA(φ)
ZX (cid:18) − X (cid:19)
2
+ ∂φ˙ 2 µ φ˙2µ + φ˙ µ .
| t|φt φt − t φt t φt
ZX ZX (cid:18)ZX (cid:19)
Hence the Hessian of is
D
(2.1) 2 (f,g) = Re(∂f,∂g) µ fgµ + fµ gµ .
φ φ φ φ φ φ
∇ D| ZX −ZX ZX ·ZX
5
Remark 2.1. We find that the Hessian 2 is non-negative by the modified
∇ D
Poincar´e inequality on Fano manifolds (for instance, see [10, Corollary 2.1]).
Set H := Hilb (φ ). we also define the quantized Monge-Amp`ere energy
0 k 0
by
1
(k)(H):= logdet(H H 1),
E −kN · 0−
k
and the quantized Ding functional by
(k)(H) := (k)(H)+( FS )(H).
k
D −E L◦
2.2. Berezin-Toeplitz quantization. The key technical result that we
use in the proof of Theorem 1.2 is the asymptotic expansion of Bergman
functionandtheirgeneralizations. Forφ ( K ),theBergmanfunction
0 X
ρ (φ ): X R is defined by ∈ H −
k 0
→
ρ (φ ) := s 2e kφ0,
k 0 i −
| |
i
X
where(s )isaHilb (φ )-ONBofH0( kK ). ThecentralresultforBergman
i k 0 X
−
function concerns the large k asymptotic of ρ (φ ), obtained by Bouche [3],
k 0
Catlin [4], Tian [9] and Zelditch [11]:
Theorem 2.1. We have the following asymptotic expansion of Bergman
function:
ρ (φ ) = b kn+b kn 1+b kn 2+ ,
k 0 0 1 − 2 −
···
where each coefficient b can be written as a polynomial in the Riemannian
i
curvature Riem(ω ), their derivatives and contractions with respect to ω .
φ0 φ0
In particular,
1
b = 1, b = S ,
0 1 2 φ0
where S(ω ) is the scalar curvature of ω . The above expansion is uniform
φ0 φ0
as long as φ stays in a compact set in the C -topology. More precisely, for
0 ∞
any integer p and l, there exists a constant C such that
p,l
p
ρ (φ ) b kn i < C kn p 1.
k 0 i − p,l − −
(cid:13) − (cid:13) ·
(cid:13) Xi=0 (cid:13)Cl
(cid:13) (cid:13)
We can take the c(cid:13)onstant C indepen(cid:13)dently of φ as long as φ stays in a
(cid:13) p,l (cid:13) 0 0
compact set in the C -topology.
∞
AnotherimportanttechnicaltoolinourproofsisprovidedbytheBerezin-
Toeplitz quantization [8]. For f C (X,R), the Berezin-Toeplitz operator
∞
∈
T is a sequence of linear operators
f,k
T : H0( kK ) H0( kK )
f,k X X
− → −
6
defined as two steps: first multiply a given section by f, then project to the
space of holomorphic sections using the L2-inner product Hilb (φ ). Using
k 0
the Hilb (φ )-ONB (s ), we obtain the explicit description of the kernel:
k 0 i
K (x,y) = f(z)(s ,s ) (z)s (y) s (x)MA(φ )(z).
f,k α β kφ0 β ⊗ ∗α 0
α,β ZX
X
e
If we restrict K to the diagonal, we have
f,k
Kf,k(x):= Kfe,k(x,x) = f(z)(sα,sβ)kφ0(z)(sβ,sα)kφ0(x)MA(φ0)(z).
α,β ZX
X
e
Theorem 2.2 ([8]). We have the following asymptotic expansion:
ρ (φ ) = b kn +b kn 1+
k 0 f,0 f,1 −
···
for smooth functions b . Moreover, there are the following formula for
f,j
coefficients:
b = f,
f,0
1
b = S f +∆ f.
f,1 2 φ0 φ0
The expansion is uniform in f varying in a subset of C (X,R) which is
∞
compact for the C -topology. It is also uniform for φ as long as φ stays
∞ 0 0
a compact set in the C -toplogy.
∞
For f,g C (X,R), we also use the kernel of the composition T T :
∞ f,k g,k
∈ ◦
K (x,y) = K (x,z)K (z,y)MA(φ )(z).
f,g,k f,k g,k 0
ZX
Restricting the diagonal, we obtain a function
e e e
K (x) := K (x,x)
f,g,k f,g,k
= f(y)g(z)(s ,s ) (y)(s ,s ) (z)(s ,s ) (x)
e α β kφ0 β γ kφ0 γ α kφ0
αX,β,γZX×X
MA(φ )(y) MA(φ )(z).
0 0
∧
Theorem 2.3 ([8]). We have the following asymptotic expansion:
ρ (φ )= b kn +b kn 1+
k 0 f,g,0 f,g,1 −
···
for smooth functions b . Moreover, there are the following formula for
f,g,j
coefficients:
b = fg,
f,g,0
1 1
b = S f2+2f∆ f + df 2 .
f,f,1 2 φ0 φ0 2| |φ0
The expansion is uniform in f,g varying in a subset of C (X,R) which is
∞
compact for the C -topology. It is also uniform for φ as long as φ stays
∞ 0 0
a compact set in the C -toplogy.
∞
7
3. Proof of the main theorem
3.1. The second variation formula for (k). Before going to the proof,
D
we define some notations that we will use later. For a hermitian matrix
A = (A ) √ 1u(N ), we write ξ for the corresponding holomorphic
i,j k A
∈ −
vector field on CPNk−1, i.e., the push forward of i,jAi,jZi∂∂Zj via the
standard projection CNk 0 CPNk−1. We set P
\{ } →
H(A) := tr(AM),
then H(A) is a real-valued smooth function satisfying
√ 1
i ω = − ∂¯H(A),
ξA FS 2π
where ω c ( (1)) denotes the Fubini-Study metric. Moreover, if we
FS 1
decompose ξ∈ = OξR √ 1JξR, we find that H(A) is the Hamiltonian for
R A A − − A
Jξ :
A
1
iJξARωFS = −4πdH(A).
For A∈ √−1u(Nk), we set g(t) := e12tA and s(t) := (s(1t),...,s(Nt)k) = s0·g(t).
Lemma 3.1.
d 1
(k)(H )= tr(A).
g(t)
dtE kN
k
In particular, (k) is affine along Bergman geodesics.
E
Proof. Since (H (s ,s )) = tg(t) 2, the direct computation shows that
g(t) i j −
d 1 d
(k)(H ) = tr (H (s ,s )) 1 (H (s ,s ))
g(t) g(t) i j − g(t) i j
dtE −kN dt
k (cid:18) (cid:19)
1 dg(t) 1 dg(t) 1
= tr g(t)2 − g(t) 1 +g(t) 1 −
− −
−kN · dt · · dt
k (cid:18) (cid:18) (cid:19)(cid:19)
1 1 1
= tr g(t) A g(t) 1+ A
−
kN · 2 · 2
k (cid:18) (cid:19)
1
= tr(A).
kN
k
(cid:3)
Lemma 3.2. (1) we have
d
(FS (H )) = k 1M(g(t)).
k g(t) −
dtL
(2) The second variation formula is
d2
(FS (H )) = k 1 ξ 2 µ k 2 H(A)2µ
dt2L k g(t) |t=0 − ZX| A|FS FSk(H0)− − ZX FSk(H0)
2
+ k 2 H(A)µ .
− FSk(H0)
(cid:18)ZX (cid:19)
8
Proof. (1) Direct computation shows that
1
d (FSk(Hg(t))) = e−FSk(Hg(t)) − d FSk(Hg(t)) e−FSk(Hg(t))
dtL − · −dt ·
(cid:18)ZX (cid:19) ZX(cid:18) (cid:19)
d
= FS (H )µ ,
dt k g(t) FSk(Hg(t))
ZX
and
d 1 d
FS (H ) = k 1 s(t) 2
dt k g(t) − · s(t) 2 · dt | i |
i| i | Xi
(3.1) = k 1tr((M g(t)) A).
− P
◦ ·
Thus we have
d
(FS (H )) = k 1 tr((M g(t)) A)µ
dtL k g(t) − ◦ · FSk(Hg(t))
ZX
= k 1M(g(t)).
−
(2) We compute
d2 d
(FS (H )) = k 1 tr((M g(t)) A) µ
dt2L k g(t) |t=0 − ZX dt ◦ · |t=0 FSk(H0)
d
+ k 1 H(A) µ .
− ZX · dt FSk(H0)|t=0
Since
d
R
tr((M g(t)) A) = tr(dM(ξ ) A)
dt ◦ · |t=0 A ·
R R
= 4πω (Jξ ,ξ )
− FS A A
= ξ 2 ,
| A|FS
the first term is
(3.2) k 1 ξ 2 µ .
− | A|FS FSk(H0)
ZX
On the other hand, using H(A) = tr(AM) and (3.1), we have
d d
µ = FS (H ) µ µ
dt FSk(H0)|t=0 (cid:18)ZX dt k g(t) |t=0 FSk(H0)(cid:19)· FSk(H0)
d
FS (H ) µ
− dt k g(t) |t=0· FSk(H0)
= k 1 H(A)µ H(A) µ .
− FSk(H0)− FSk(H0)
(cid:18)ZX (cid:19)
Hence the second term is
2
(3.3) k 2 H(A)2µ +k 2 H(A)µ .
− − FSk(H0) − FSk(H0)
ZX (cid:18)ZX (cid:19)
Combining (3.2) and (3.3) gives our conclusion. (cid:3)
9
When we take into account that the Hessian 2 (k) is a symmetric bi-
∇ D
linear form, we can easily get Theorem 1.1 from Lemma 3.1 and Lemma
3.2.
Proof of Corollary 1.1. For A √ 1u(N ), let ξ be the component of
∈ − k A⊤
ξ which is tangent to X and ξ the component which is perpendicular.
A|X A⊥
Then we have
√ 1
iξA⊤ωFSk(H0) = 2−π ∂¯ k−1H(A)
on X. It follows that k 1 ξ 2 = ξ 2 (cid:0) = k 2(cid:1)∂H(A)2 . Com-
− | A⊤|FS | A⊤|FSk(H0) − | |FSk(H0)
bining with the formula ξ 2 = ξ 2 + ξ 2 , we have
| A|FS | A⊤|FS | A⊥|FS
2 (k) (A,A) = k 2 2 (H(A),H(A))+k 1 ξ 2 µ
∇ D |H0 − ∇ D|FSk(H0) − ZX| A⊥|FS FSk(H0)
k 1 ξ 2 µ (by Remark 2.1)
≥ − | A⊥|FS FSk(H0)
ZX
0.
≥
Now we assume that 2 (k) (A,A) = 0, then we have ξ = 0, and
∇ D |H0 A⊥
hence A Lie(Aut(X, kK )) as desired. Conversely, the conditiion A
X
∈ − ∈
Lie(Aut(X, kK )) implies that ξ = 0 and ξ is holomorphic. Differenti-
− X A⊥ A⊤
ating the equation Ric(ω ) ω = √ 1∂∂¯h with respect
FSk(H0) − FSk(H0) 2−π FSk(H0)
to ξ , we obtain
A⊤
∆ H(A) (∂h ,∂H(A)) H(A)+ H(A)µ = 0.
− FSk(H0) − FSk(H0) FSk(H0)− FSk(H0)
ZX
MultiplingH(A)andintegratingbyparts,weobtain 2 (H(A),H(A)) =
0. Therefore, we have 2 (k) (A,A) = 0. ∇ D|FSk(H0) (cid:3)
∇ D |H0
3.2. Asymptotic of the Hessian 2 (k). Our starting point is the fol-
∇ D
lowing:
Lemma3.3([6],Lemma18). ForanyHermitianmatricesA,B √ 1u(N ),
k
∈ −
we have
H(A)H(B)+(ξ ,ξ ) = tr(ABM).
A B FS
By Lemma 3.3, we have
2 (k)(A,B) = k 1 Re(tr(ABM))µ k 1(1+k 1) H(A)H(B)µ
∇ DH0 − FSk(H0)− − − FSk(H0)
| ZX ZX
+ k 2 H(A)µ H(B)µ .
− FSk(H0)· FSk(H0)
ZX ZX
For given functions f,g C (X,R), we set A := Q , B := Q and
∞ f,k g,k
∈
compute the asymptotic of 2 (k)(Q ,Q ) as k . However, in
∇ DH0 f,k g,k → ∞
|
the course of the proof, we find that 2 (k)(Q ,Q ) has an asymptotic
∇ DH0 f,k g,k
expansion whosecoefficients are also symm|etric and bilinear with respect to
10
f and g. Hence it follows that we may assume f = g to prove Theorem 1.2.
We set
A := k 1 tr(Q2 M)µ ,
1 − ZX f,k Tk(φ0)
A := k 1(1+k 1) H(Q )2µ ,
2 − − − ZX f,k Tk(φ0)
2
A := k 2 H(Q )µ ,
3 − (cid:18)ZX f,k Tk(φ0)(cid:19)
and we will compute these terms separately, where := FS Hilb . The
k k k
T ◦
following arguments are based on [7, Section 2].
Lemma 3.4. The volume form e−Tk(φ0) has the asymptotic expansion
e−Tk(φ0) = (1+O(k−2))e−φ0.
Proof. Since (φ ) = φ +k 1log(k nρ (φ )), we have
k 0 0 − − k 0
T
e−Tk(φ0) = (k−nρk(φ0))−1/ke−φ
= (1+O(k 2))e φ,
− −
where we used the asymptotic expansion of ρ (φ ) in the last equality (cf.
k 0
Theorem 2.1). (cid:3)
Lemma 3.5. The term A has an asymptotic expansion
1
A = f2µ k+ ∂f 2 µ +O(k 1).
1 φ0 · | |φ0 φ0 −
ZX ZX
Proof. We can write M as
X
|
(s ,s )
M = α β kφ0 .
|X (cid:18) ρk(φ0) (cid:19)αβ
It follows that
A = k 1 Q Q M ,
1 − αβ βγ γα
α,β,γ
X
where
Q = (kf ∆ f)(s ,s ) MA(φ ),
αβ − φ0 α β kφ0 0
ZX
(s ,s )
M = γ α kφ0µ .
γα ZX ρk(φ0) Tk(φ0)
By Theorem 2.1 and Lemma 3.4, we have
e−Tk(φ0) = 1+O(k−2) e φ0
−
ρ (φ ) b kn+b kn 1+O(kn 2)
k 0 0 1 − −
= (k n b k n 1+O(k n 2))e φ0.
− 1 − − − − −
−