A remark on 'Some numerical results in complex differential geometry'

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A remark on ’Some numerical results in complex differential geometry’

Kefeng Liu, Xiaonan Ma

To cite this version:

Kefeng Liu, Xiaonan Ma. A remark on ’Some numerical results in complex differential geometry’.

2006. �hal-00016826v2�

ccsd-00016826, version 2 - 18 Jan 2006

DIFFERENTIAL GEOMETRY’

KEFENG LIU AND XIAONAN MA

Abstract. In this note we verify certain statement about the operator Q

_K

constructed by Donaldson in [3] by using the full asymptotic expansion of Bergman kernel obtained in [2] and [4].

In order to find explicit numerical approximation of K¨ahler-Einstein metric of projec- tive manifolds, Donaldson introduced in [3] various operators with good properties to approximate classical operators. See the discussions in Section 4.2 of [3] for more details related to our discussion. In this note we verify certain statement of Donaldson about the operator Q

K

in Section 4.2 by using the full asymptotic expansion of Bergman kernel derived in [2, Theorem 4.18] and [4, § 3.4]. Such statement is needed for the convergence of the approximation procedure.

Let (X, ω, J ) be a compact K¨ahler manifold of dim

C

X = n, and let (L, h

^L

) be a holo- morphic Hermitian line bundle on X. Let ∇

^L

be the holomorphic Hermitian connection on (L, h

^L

) with curvature R

^L

. We assume that

√ − 1

2π R

^L

= ω.

(1)

Let g

^{T X}

( · , · ) := ω( · , J · ) be the Riemannian metric on T X induced by ω, J . Let dv

X

be the Riemannian volume form of (T X, g

^{T X}

), then dv

X

= ω

ⁿ

/n!. Let dν be any volume form on X. Let η be the positive function on X defined by

dv

X

= η dν.

(2)

The L

²

–scalar product h i

^ν

on C

∞

(X, L

^p

), the space of smooth sections of L

^p

, is given by

h σ

1

, σ

2

i

^ν

:=

Z

X

h σ

1

(x), σ

2

(x) i

^Lp

dν(x) . (3)

Let P

ν,p

(x, x

^′

) (x, x

^′

∈ X) be the smooth kernel of the orthogonal projection from ( C

∞

(X, L

^p

), h i

^ν

) onto H

⁰

(X, L

^p

), the space of the holomorphic sections of L

^p

on X, with respect to dν(x

^′

). Note that P

ν,p

(x, x

^′

) ∈ L

^p_x

⊗ L

^p_x^∗′

. Following [3, § 4], set

K

p

(x, x

^′

) := | P

ν,p

(x, x

^′

) |

²_h^Lp_{x ⊗h}^Lp∗

x′

, R

p

:= (dim H

⁰

(X, L

^p

))/Vol(X, ν), (4)

here Vol(X, ν) := R

X

dν. Set Vol(X, dv

X

) := R

X

dv

X

.

Let Q

Kp

be the integral operator associated to K

p

which is defined for f ∈ C

∞

(X),

(5) Q

Kp

(f )(x) := 1

R

p

Z

X

K

p

(x, y)f(y)dν(y).

Date: January 18, 2006.

1

2 KEFENG LIU AND XIAONAN MA

Let ∆ be the (positive) Laplace operator on (X, g

^{T X}

) acting on the functions on X.

We denote by | |

L²

the L

²

-norm on the function on X with respect to dv

_X

. Theorem 1. There exists a constant C > 0 such that for any f ∈ C

^∞

(X), p ∈ N ,

Q

Kp

− Vol(X, ν)

Vol(X, dv

X

) η exp

− ∆ 4πp

f

L²

≤ C p | f |

L²

,

∆

p Q

Kp

− Vol(X, ν) Vol(X, dv

_X

)

∆

p η exp

− ∆ 4πp

f

L²

≤ C p | f |

L²

. (6)

Moreover, (6) is uniform in that there is an integer s such that if all data h

^L

, dν run over a set which are bounded in C

^s

and that g

^{T X}

, dv

X

are bounded from below, then the constant C is independent of h

^L

, dν.

Proof. We explain at first the full asymptotic expansion of P

ν,p

(x, x

^′

) from [2, Theorem 4.18

^′

] and [4, § 3.4]. For more details on our approach we also refer the readers to the recent book [5].

Let E = C be the trivial holomorphic line bundle on X. Let h

^E

the metric on E defined by | 1 |

²h^E

= 1, here 1 is the canonical unity element of E. We identify canonically L

^p

to L

^p

⊗ E by Section 1.

As in [4, § 3.4], let h

^E_ω

be the metric on E defined by | 1 |

²h^E_ω

= η

⁻¹

, here 1 is the canonical unity element of E. Let h i

ω

be the Hermitian product on C

^∞

(X, L

^p

⊗ E) = C

^∞

(X, L

^p

) induced by h

^L

, h

^E_ω

, dv

_X

as in (3). Then by (2),

(7) ( C

^∞

(X, L

^p

⊗ E), h i

ω

) = ( C

^∞

(X, L

^p

), h i

ν

).

Observe that H

⁰

(X, L

^p

⊗ E) does not depend on g

^{T X}

, h

^L

or h

^E

. If P

ω,p

(x, x

^′

), (x, x

^′

∈ X) denotes the smooth kernel of the orthogonal projection P

ω,p

from ( C

^∞

(X, L

^p

⊗ E), h · , ·i

ω

) onto H

⁰

(X, L

^p

⊗ E) = H

⁰

(X, L

^p

) with respect to dv

X

(x), from (2), as in [4, (3.38)], we have

(8) P

ν,p

(x, x

^′

) = η(x

^′

) P

ω,p

(x, x

^′

).

For f ∈ C

^∞

(X ), set

K

ω,p

(x, x

^′

) = | P

ω,p

(x, x

^′

) |

²(h^Lp⊗h^E_ω)x⊗(h^Lp∗⊗h^E_ω^∗)_x′

, (K

ω,p

f)(x) =

Z

X

K

ω,p

(x, y)f(y)dv

X

(y).

(9)

By the definition of the metric h

^E

, h

^E_ω

, if we denote by 1

^∗

the dual of the section 1 of E, we know

(10) 1 = | 1 ⊗ 1

^∗

|

²h^E⊗h^E∗

(x, x

^′

) = | 1 ⊗ 1

^∗

|

²h^E_ω⊗h^E_ω^∗

(x, x

^′

)η(x)η

⁻¹

(x

^′

).

Recall that we identified (L

^p

, h

^Lp

) to (L

^p

⊗ E, h

^Lp

⊗ h

^E

) by Section 1. Thus from (4), (8) and (10), we get

(11) K

_p

(x, x

^′

) = | P

_ν,p

(x, x

^′

) |

²_(h^Lp_⊗h^E_)x⊗(h^Lp∗_⊗h^E∗_)x′

= η(x) η(x

^′

) K

_ω,p

(x, x

^′

), and from (2), (5) and (11),

(12) Q

Kp

(f)(x) = 1

R

_p

Z

X

K

ω,p

(x, y )η(x)f (y)dv

X

(y).

Now for the kernel P

ω,p

(x, x

^′

), we can apply the full asymptotic expansion [2, Theorem 4.18

^′

]. In fact let ∂

^L

p⊗E,∗^ω

be the formal adjoint of the Dolbeault operator ∂

^L

p⊗E

on the Dolbeault complex Ω

^0,•

(X, L

^p

⊗ E) with the scalar product induced by g

^{T X}

, h

^L

, h

^E_ω

, dv

_X

as in (3), and set

(13) D

p

= √

2(∂

^Lp⊗E

+ ∂

^Lp⊗E,∗ω

).

Then H

⁰

(X, L

^p

⊗ E) = Ker D

p

for p large enough, and D

p

is a Dirac operator, as g

^{T X}

( · , · ) = ω( · , J · ) is a K¨ahler metric on T X .

Let ∇

^E

be the holomorphic Hermitian connection on (E, h

^E_ω

). Let ∇

^{T X}

be the Levi- Civita connection on (T X, g

^{T X}

). Let R

^E

, R

^{T X}

be the corresponding curvatures.

Let a

^X

be the injectivity radius of (X, g

^{T X}

). We fix ε ∈ ]0, a

^X

/4[. We denote by B

^X

(x, ε) and B

^TxX

(0, ε) the open balls in X and T

x

X with center x and radius ε. We identify B

^TxX

(0, ε) with B

^X

(x, ε) by using the exponential map of (X, g

^{T X}

).

We fix x

0

∈ X. For Z ∈ B

^Tx0X

(0, ε) we identify (L

Z

, h

^L_Z

), (E

Z

, h

^E_Z

) and (L

^p

⊗ E)

Z

to (L

x⁰

, h

^L_x0

), (E

x⁰

, h

^E_x0

) and (L

^p

⊗ E)

x⁰

by parallel transport with respect to the con- nections ∇

^L

, ∇

^E

and ∇

^Lp⊗E

along the curve γ

Z

: [0, 1] ∋ u → exp

^X_x0

(uZ ). Then under our identification, P

ω,p

(Z, Z

^′

) is a function on Z, Z

^′

∈ T

x⁰

X, | Z | , | Z

^′

| ≤ ε, we denote it by P

ω,p,x⁰

(Z, Z

^′

). Let π : T X ×

^X

T X → X be the natural projection from the fiberwise product of T X on X. Then we can view P

ω,p,x⁰

(Z, Z

^′

) as a smooth func- tion on T X ×

^X

T X (which is defined for | Z | , | Z

^′

| ≤ ε) by identifying a section S ∈ C

^∞

(T X ×

X

T X, π

^∗

End(E)) with the family (S

_x

)

_x∈X

, where S

_x

= S |

π⁻¹(x)

, End(E) = C . We choose { w

i

}

ⁿi=1

an orthonormal basis of T

x^(1,0)0

X, then e

2j−1

=

^√1₂

(w

j

+ w

j

) and e

2j

=

^√√−₂¹

(w

j

− w

j

) , j = 1, . . . , n forms an orthonormal basis of T

x⁰

X. We use the coordinates on T

_x0

X ≃ R

²ⁿ

where the identification is given by

(14) (Z

1

, · · · , Z

2n

) ∈ R

²ⁿ

−→ X

i

Z

i

e

i

∈ T

x⁰

X.

In what follows we also introduce the complex coordinates z = (z

1

, · · · , z

n

) on C

ⁿ

≃ R

²ⁿ

. By [2, (4.114)] (cf. [4, (1.91)]), set

P

^N

(Z, Z

^′

) = exp

− π 2

X

i

| z

i

|

²

+ | z

_i^′

|

²

− 2z

i

z

^′_i

(15) .

Then P

^N

is the classical Bergman kernel on C

ⁿ

(cf. [4, Remark 1.14]) and (16) | P

^N

(Z, Z

^′

) |

²

= e

^{−π|Z−Z′|2}

.

By [2, Proposition 4.1], for any l, m ∈ N , ε > 0, there exists C

l,m,ε

> 0 such that for p ≥ 1, x, x

^′

∈ X,

| P

ω,p

(x, x

^′

) |

^Cm(X×X)

≤ C

l,m,ε

p

^−l

if d(x, x

^′

) ≥ ε.

(17)

Here the C

^m

-norm is induced by ∇

^L

, ∇

^E

, ∇

^{T X}

and h

^L

, h

^E

, g

^{T X}

.

By [2, Theorem 4.18

^′

], there exist J

r

(Z, Z

^′

) polynomials in Z, Z

^′

, such that for any

k, m, m

^′

∈ N , there exist N ∈ N , C > 0, C

0

> 0 such that for α, α

^′

∈ N

ⁿ

, | α | + | α

^′

| ≤ m,

4 KEFENG LIU AND XIAONAN MA

Z, Z

^′

∈ T

x⁰

X, | Z | , | Z

^′

| ≤ ε, x

0

∈ X, p ≥ 1, (18)

∂

^|α|+|α′|

∂Z

^α

∂Z

^′α′

1

p

ⁿ

P

_ω,p,x0

(Z, Z

^′

) −

k

X

r=0

(J

_r

P

^N

)( √ pZ, √ pZ

^′

)p

^−r/2

!

_Cm′(X)

≤ Cp

^{−(k+1−m)/2}

(1 + | √ pZ | + | √ pZ

^′

| )

^N

exp( − C

0

√ p | Z − Z

^′

| ) + O (p

^−∞

).

Here C

^m′

(X) is the C

^m′

norm for the parameter x

0

∈ X. The term O (p

^−∞

) means that for any l, l

1

∈ N , there exists C

l,l1

> 0 such that its C

^l1

-norm is dominated by C

_l,l1

p

^−l

. (In fact, by [2, Theorems 4.6 and 4.17, (4.117)] (cf. [4, Theorem 1.18, (1.31)]), the polynomials J

r

(Z, Z

^′

) have the same parity as r and deg J

r

(Z, Z

^′

) ≤ 3r, whose coefficients are polynomials in R

^{T X}

, R

^E

and their derivatives of order 6 r − 1).

Now we claim that in (18),

(19) J

₀

= 1, J

₁

(Z, Z

^′

) = 0.

In fact, let dv

_Tx0X

be the Riemannian volume form on (T

_x0

X, g

^Tx0X

), and κ be the function defined by

(20) dv

X

(Z ) = κ(x

0

, Z)dv

Tx0X

(Z).

Then (also cf. [4, (1.31)])

(21) κ(x

₀

, Z) = 1 + 1 6

R

^{T X}_x0

(Z, e

_i

)Z, e

_i

x⁰

+ O ( | Z |

³

).

As we only work on C

∞

(X, L

^p

⊗ E), by [2, (4.115)], we get the first equation in (19).

Recall that in the normal coordinate, after the rescaling Z → Z/t with t =

√¹p

, we get an operator L

_t

from the restriction of D

_p²

on C

∞

(X, L

^p

⊗ E) which has the following formal expansion (cf. [2, (1.104)], [4, Theorem 1.4]),

(22) L

_t

= L +

∞

X

r=1

Q

^r

t

^r

.

Now, from [2, Theorem 5.1] (or [4, (1.87), (1.97)]), L =

n

X

j=1

( − 2

_∂z^∂

i

+ πz

i

)(2

_∂z^∂

i

+ πz

i

), Q

¹

= 0.

(23)

(In fact, P

^N

(Z, Z

^′

) is the smooth kernel of the orthogonal projection from L

²

( R ) onto Ker( L )). Thus from [2, (4.107)] (cf. [4, (1.111)]), (21) and (23) we get the second equation of (19).

Note that | P

ω,p,x0

(Z, Z

^′

) |

²

= P

ω,p,x0

(Z, Z

^′

)P

ω,p,x0

(Z, Z

^′

), thus from (9), (18) and (19), there exist J

_r^′

(Z, Z

^′

) polynomials in Z, Z

^′

such that

(24)

1 p

²ⁿ⁺¹

∆

Z

K

ω,p,x0

(Z, Z

^′

) − 1 +

k

X

r=2

p

^−r/2

J

_r^′

( √ pZ, √ pZ

^′

)

e

^{−πp|Z−Z′|2}

≤ Cp

^−(k+1)/2

(1 + | √ pZ | + | √ pZ

^′

| )

^N

exp( − C

₀

√ p | Z − Z

^′

| ) + O (p

^−∞

).

For a function f ∈ C

∞

(X), we denote it as f (x

0

, Z) a family (with parameter x

0

) of

function on Z in the normal coordinate near x

0

. Now, for any polynomial Q

x⁰

(Z

^′

), we

define the operator

( Q

^p

f)(x

0

) = p

ⁿ

Z

|Z^′|≤ε

Q

x0

( √ pZ

^′

)e

^{−πp|Z′|2}

f (x

0

, Z

^′

)dv

X

(x

0

, Z

^′

).

(25)

Then we observe that there exists C

₁

> 0 such that for any p ∈ N , f ∈ C

^∞

(X ), we have

|Q

^p

f |

^L2

≤ C

1

| f |

^L2

. (26)

In fact,

(27) |Q

^p

f |

²L²

≤ Z

X

dv

X

(x

0

) n p

ⁿ

Z

|Z^′|≤ε

| Q

x⁰

( √ pZ

^′

) | e

^{−πp|Z′|2}

dv

X

(x

0

, Z

^′

)

× p

ⁿ

Z

|Z^′|≤ε

| Q

x0

( √ pZ

^′

) | e

^{−πp|Z′|2}

| f(x

0

, Z

^′

) |

²

dv

X

(x

0

, Z

^′

) o

≤ C

^′

Z

X

dv

_X

(x

₀

)p

ⁿ

Z

|Z^′|≤ε

| Q

_x0

( √ pZ

^′

) | e

^{−πp|Z′|2}

| f (x

₀

, Z

^′

) |

²

dv

_X

(x

₀

, Z

^′

)

≤ C

1

| f |

²L²

. Observe that in the normal coordinate, at Z = 0, ∆

Z

= − P

2n

j=1 ∂²

∂Z_j²

. Thus (28) (∆

_Z

e

^{−πp|Z−Z′|2}

) |

Z=0

= 4πp(n − πp | Z

^′

|

²

)e

^{−πp|Z′|2}

.

Thus from (16), (18), (19), (24) and (26), we get

p

⁻ⁿ

K

ω,p

f − p

ⁿ

Z

|Z^′|≤ε

e

^{−πp|Z′|2}

f (x

0

, Z

^′

)dv

X

(x

0

, Z

^′

)

L²

≤ C p | f |

L²

,

p

⁻ⁿ⁻¹

∆K

_ω,p

f − 4πp

ⁿ

Z

|Z^′|≤ε

(n − πp | Z

^′

|

²

)e

^{−πp|Z′|2}

f(x

₀

, Z

^′

)dv

_X

(x

₀

, Z

^′

)

L²

≤ C p | f |

L²

. (29)

Set

K

η,ω,p

(x, y) = h dη(x), d

x

K

ω,p

(x, y) i

g^T∗X

, (K

η,ω,p

f )(x) =

Z

X

K

η,ω,p

(x, y)f(y)dv

X

(y).

(30)

Then from (18), (19) and (26), we get

p

⁻ⁿ⁻¹

K

η,ω,p

f − 2πp

ⁿ

Z

|Z^′|≤ε 2n

X

i=1

( ∂

∂Z

_i

η)(x

0

, 0)Z

_i^′

e

^{−πp|Z′|2}

f (x

0

, Z

^′

)dv

X

(x

0

, Z

^′

)

L²

≤ C p | f |

L²

. (31)

Let e

^−u∆

(x, x

^′

) be the smooth kernel of the heat operator e

^−u∆

with respect to dv

X

(x

^′

).

Let d(x, y) be the Riemannian distance from x to y on (X, g

^{T X}

). By the heat kernel expansion in [1, Theorems 2.23, 2.26], there exist Φ

i

(x, y) smooth functions on X × X such that when u → 0, we have the following asymptotic expansion

∂

^l

∂u

^l

e

^−u∆

(x, y) − (4πu)

⁻ⁿ

k

X

i=0

u

ⁱ

Φ

i

(x, y )e

^{−41ud(x,y)2}

_Cm

(X×X)

= O (u

^{k−n−l−m2+1}

),

(32)

6 KEFENG LIU AND XIAONAN MA

and

Φ

0

(x, y) = 1.

(33)

If we still use the normal coordinate, then by (32), there exist φ

i,x⁰

(Z

^′

) := Φ

i

(0, Z

^′

) such that uniformly for x

0

∈ X, Z

^′

∈ T

x⁰

X, | Z

^′

| ≤ ε, we have the following asymptotic expansion when u → 0,

∂

^l

∂u

^l

e

^−u∆

(0, Z

^′

) − (4πu)

⁻ⁿ

1 +

k

X

i=1

u

ⁱ

φ

i,x0

(Z

^′

)

e

^{−41u|Z′|2}

= O (u

^k−n−l+1

), (34)

and (35)

h dη(x

0

), d

x0

e

^−u∆

i

g^T∗X

(0, Z

^′

)

− (4πu)

⁻ⁿ

2n

X

i=1

( ∂

∂Z

i

η)(x

0

, 0) Z

_i^′

2u

1 +

k

X

i=1

u

ⁱ

φ

i,x0

(Z

^′

)

e

^{−41u|Z′|2}

− (4πu)

⁻ⁿ

k

X

i=1

u

ⁱ

h dη(x

0

), (d

x⁰

Φ

i

)(0, Z

^′

) i e

^{−41u|Z′|2}

= O (u

^k−n+21

).

Observe that

1

p ∆ exp

− ∆ 4πp

= − 1

p (

_∂u^∂

e

^−u∆

) |

u=₄¹_πp

. (36)

Now from (26), (29)–(36), we get

p

⁻ⁿ

K

ω,p

− exp

− ∆ 4πp

f

L²

≤ C p | f |

L²

,

1 p

p

⁻ⁿ

∆K

ω,p

− ∆ exp

− ∆ 4πp

f

L²

≤ C p | f |

L²

. (37)

and

1 p

p

⁻ⁿ

K

η,ω,p

− h dη, d exp( − ∆ 4πp ) i

f

L²

≤ C p | f |

L²

. (38)

Note that

(39) (∆ηK

ω,p

)(x, y) = (∆η)(x)K

ω,p

(x, y ) + η(x)∆

x

K

ω,p

(x, y)

− 2 h dη(x), d

x

K

ω,p

(x, x

^′

) i

g^T∗X

, and R

_p

=

^Vol(X,dv_Vol(X,ν)^X)

p

ⁿ

+ O (p

ⁿ⁻¹

). From (12), (37)-(39), we get (6).

To get the last part of Theorem 1, as we noticed in [2, § 4.5], the constants in (18) will be uniformly bounded under our condition, thus we can take C in (6), (37)and (38)

independent of h

^L

, dν.

Acknowledgments. We thank Professor Simon Donaldson for useful communications.

References

[1] N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Grundl. Math. Wiss.

Band 298, Springer-Verlag, Berlin, 1992.

[2] X. Dai, K. Liu, and X. Ma, On the asymptotic expansion of Bergman kernel, C. R. Math.

Acad. Sci. Paris 339 (2004), no. 3, 193–198. The full version: J. Differential Geom. to appear, math.DG/0404494.

[3] S. K. Donaldson, Some numerical results in complex differential geometry, math.DG/0512625.

[4] X. Ma and G. Marinescu, Generalized Bergman kernels on symplectic manifolds, C. R. Math. Acad.

Sci. Paris 339 (2004), no. 7, 493–498. The full version: math.DG/0411559.

[5] , Holomorphic Morse Inequalities and Bergman Kernels, book in preparation, (2006).

Center of Mathematical Science, Zhejiang University and Department of Mathemat- ics, UCLA, CA 90095-1555, USA (liu@math.ucla.edu)

Centre de Math´ ematiques Laurent Schwartz, UMR 7640 du CNRS, Ecole Polytech-

nique, 91128 Palaiseau Cedex, France (ma@math.polytechnique.fr)