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The Bergman Kernel on Riemann Surfaces

Chiung-ju Liu (Joint work with Zhiqin Lu)

National Taiwan University

2013 AMMSDept. of Applied Math., NSYU

December 6, 2013

The Bergman Kernel on Riemann Surfaces

Outline

Introduction

On smooth and singular Riemann surfaces

Partial Results

The Bergman Kernel on Riemann Surfaces

Introduction

The pair (M, L) is a polarized Kahler manifold if there is an ampleHermitian line bundle (L, h) over M such that it defines a Kahlermetric

Ric(h) = ωg

on M. That is, [ωg ] ∈ c1(L).

The Bergman Kernel on Riemann Surfaces

Introduction

Inner Product

For each positive integer m, h induces a Hermitian metric hm onLm. Consider the space H0(M, Lm) of all global holomorphicsections for large m.

For U,V ∈ H0(M, Lm), the L2 inner product

(U,V ) =

∫M〈U(x),V (x)〉hmdVg ,

where dVg =ωng

n! is the volume form of g .

The Bergman Kernel on Riemann Surfaces

Introduction

The Bergman Kernel

DefinitionLet {S1, · · · , Sd} be an orthonormal basis of H0(M, Lm). For anypoint x ∈ M, define the Bergman kernel

Bm(x) =d∑

i=1

‖Si‖2hm(x).

The Bergman Kernel on Riemann Surfaces

Introduction

Let φm be the Kodaira embedding and ωFS be the Fubini-Studymetric on CPd−1. Define the Bergman metric

ωm =1

mφ∗m(ωFS)

√−1

2πm∂∂ log

( d∑i=1

|Si |2)

= ωg +

√−1

2πm∂∂ log

d∑i=1

‖Si‖2hm ,

The Bergman Kernel on Riemann Surfaces

Introduction

History

Theorem (Tian ’90)

Let M be an algebraic manifold with a polarization L and let g bethe corresponding polarized Kahler metric on M. Then∥∥∥gm − g

∥∥∥C2

= O

(1√m

The Bergman Kernel on Riemann Surfaces

Introduction

History

Past Results

Theorem (Zelditch, Catlin)

For any x ∈ M, there is an asymptotic expansion:

Bm(x) ∼ a0(x)mn + a1(x)mn−1 + a2(x)mn−2 + · · ·

for certain smooth coefficients aj(x) with a0 = 1. More precisely,for any s

||Bm(x)−s∑

k=0

aj(x)mn−k ||Cµ ≤ Cs,µmn−s−1,

where Cs,µ depends on s, µ and the manifold M.

The Bergman Kernel on Riemann Surfaces

Introduction

History

Riemann-Rock Formula

d =

∫MBmdVg ∼ A0m

n + A1mn−1 + · · · ,

where

A0 =

∫MdVg =

∫M

(c1(L))n, A1 =

∫Ma1dVg .

The Bergman Kernel on Riemann Surfaces

Introduction

History

Application

Theorem (Donaldson(2001))

Suppose that Aut(M, L) is discrete and (M, Lm) is balanced forsufficiently large m. Suppose that Bergman metrics ωm converge toω∞ in C∞ as m→∞. Then ω∞ has a constant scalar curvature.

The Bergman Kernel on Riemann Surfaces

Introduction

History

Coefficients

Theorem (Lu)

With the same notation as above, each coefficient aj(x) is apolynomial of the curvature and its covariant derivatives at x .Such polynomial can be found by finitely many steps of algebraicoperations. We have

a0 = 1, a1 =1

2ρ, a2 =

3∆ρ+

24(|R|2 − 4|Ric |2 + 3ρ2), · · · ,

where ρ, R, and Ric represent the scalar curvature, the curvaturetensor, and the Ricci curvature.

The Bergman Kernel on Riemann Surfaces

Introduction

History

Higher Order Coefficients

In particular,

Theorem (Lu-Tian)

With the same notation as above, for any k ≥ 1, there exists aconstant C = C (k , n) 6= 0 such that

ak = C∆k−1ρ+ · · · ,

where ρ is the scalar curvature and ∆ is the Laplace operator of M.

The Bergman Kernel on Riemann Surfaces

Introduction

History

Recent Results

Shiffman and Zelditch (2002) generalized the result tosymplectic manifold for ”almost” holomorphic sections

Dai, Liu and Ma (2006) gave the asymptotic expansion of theBergman kernel of the spinc Dirac operator

Berman, Berndtsson and Sjostrand (2008) gave a directapproach, avoiding using the paramatrix of Bergman kernel.

Ma and Marinescu (2008) gave a full off-diagonal asymptoticof the generalized Bergman kernels of renomalizedBouchner-Laplacian on high powers of a positive line bundleover a compact symplectic manifolds.

Ross and Thomas (2009) gave an asymptotic expansion of theweighted Bergman kernel on orbifolds and showed that if apolarized orbifold is balanced, then it has a metric withconstant scalar curvature.

The Bergman Kernel on Riemann Surfaces

Introduction

History

Xu (2011) gave a graph-theoretic interpretation of the coefficientsof the expansion.

The Bergman Kernel on Riemann Surfaces

Introduction

History

Theorem (L-Lu (2012))

Suppose that the Hermitian metrics hL is real analytic at a fixedpoint x . Then for m big enough, the expansion of the Bergmankernel

∞∑j=0

aj(x)mn−j

is convergent in Cµ. Moreover, we have

‖Bm(x)−∞∑j=0

ajmn−j‖Cµ ≤ mne−ε(logm)3

for some absolute constant ε > 0. There is a C > 1 such that

‖aj(x)‖Cµ < C j

for all j ≥ 0.

The Bergman Kernel on Riemann Surfaces

Introduction

History

In general, the asymptotic expansion is local not uniform. Theequivalent statement of Tian’s theorem is

∂∂ logBm(x) = 0

A necessary condition for the above is the uniformly lower boundof the Bergman kernel.

The Bergman Kernel on Riemann Surfaces

Introduction

History

For a family of Kahler manifolds, the property that there exists aconstant ε > 0 such that

d∑i=1

‖Si (x)‖2hm > ε

is called strong partial C 0 estimate.

The Bergman Kernel on Riemann Surfaces

Introduction

On Riemann Surfaces

Theorem (Lu 2000)

Let M be a Riemann surface of genus g ≥ 2 and KM be thecanonical line bundle endowed with a Hermitian metric h such thatthe curvature Ric(h) of h defines a Kahler metric g on M. Let theGauss curvature G of g satisfy

−C1 ≤ G ≤ C2

for some nonnegative constants C1 and C2 and let δ′ be theinjective radius of M. Let

δ = min{δ′, 1√C1 + C2

Then there is an absolute constant C > 0 such that for m ≥ 2,

Bm ≥ e−Cg3

δ6 .

The Bergman Kernel on Riemann Surfaces

Introduction

On Riemann Surfaces

Collar Theorem

TheoremLet M be a compact Riemann surface of genus g ≥ 2. Then(i)There exists simple closed geodesics γ1, · · · , γ3g−3 whichdecompose M into pairs of pants.(ii)The collars

C (γi ) = {p ∈ M|dist(p, γi ) ≤ w(γi )}

with widths w(γi ) = arcsinh{ 1sinh( 1

2`(γi ))} are pairwise disjoint for

i = 1, · · · , 3g − 3.(iii) Each C (γi ) is isometric to the cylinder [−w(γi ),w(γi )]× S1

with the Riemannian metric ds2 = dρ2 + `2(γi ) cosh2(ρ)dt2.

The Bergman Kernel on Riemann Surfaces

Introduction

On Riemann Surfaces

Figure : a C (γ) on a compact Riemann surface with two genus

The Bergman Kernel on Riemann Surfaces

Introduction

On Riemann Surfaces

Figure : Collar C (γ)

The Bergman Kernel on Riemann Surfaces

Introduction

On Riemann Surfaces

Theorem (Liu)

Let M be a regular compact Riemann surface and KM be thecanonical line bundle endowed with a Hermitian metric h such thatthe curvature Ric(h) of h defines a Kahler metric g on M.Suppose that this metric g has constant scalar curvature ρ. Thenthere is a complete asymptotic expansion:

Bm(x) ∼ m(1 +ρ

2m) + O

(e−

(logm)2

)for some m large enough.

The Bergman Kernel on Riemann Surfaces

Introduction

On Riemann Surfaces

Theorem (Lu)

For any ε > 0 and m ≥ 2, there exists a Riemann surface M ofgenus g ≥ 2 with the constants Gauss curvature −1 such that

infx∈M

Bm(x) ≤ ε.

The Bergman Kernel on Riemann Surfaces

Introduction

On Riemann Surfaces

Partial Uniform Estimate

Theorem (Lu)

Let M be a Riemann surface of genus g ≥ 2 and constantcurvature −1. Then there are absolute constants m0 and D > 0such that for any m > m0 and any x0 ∈ M, there exists a sectionS ∈ H0(M,Km

M ) with ‖S‖L2 = 1 such that

‖S‖(x0) ≥√m

(1 + 1√

mδ2x0e

πδx0

) ,where δx0 is the injective radius of x0.

The Bergman Kernel on Riemann Surfaces

Introduction

On Riemann Surfaces

Figure : Collar

The Bergman Kernel on Riemann Surfaces

Singular Riemann Surfaces

Semi-Partial Lower Estimate

Theorem (Liu-Lu(2011))

Suppose X0 is a singular Riemann surface of genus with ordinarydouble point x0 and scalar constant curvature −1. Then thereexists constant D such that for any x1 ∈ X0 \ x0, the Bergmankernel

Bm(x1) ≥ m

(1 + e

2πδx1

mδ4x1

) . (1)

The Bergman Kernel on Riemann Surfaces

Singular Riemann Surfaces

Deformation

From the algebraic geometry point of view, consider a holomorphicdegeneration family π : χ→ ∆ for ∆ = {t ∈ C : |t| < 1} suchthat Xt = π−1(t) are smooth Riemann surfaces except for t = 0.By Mumford’s semi-stable reduction theorem, one can assume thatχ is smooth and the central fiber X0 = D1 ∪ D2 with normalcrossing divisors.

The Bergman Kernel on Riemann Surfaces

Singular Riemann Surfaces

From algebraic geometry point of view, consider Xt as a sequenceof algebraic curves w1w2 = t that degenerate to the central fiberX0.

w1w2 = t

The Bergman Kernel on Riemann Surfaces

Singular Riemann Surfaces

Let D1 = {w1 = 0}, D2 = {w2 = 0} and D1 ∩ D2 = {x0}. Let X0

be a Riemann surface with a singular point x0.

w1w2 = 0

The Bergman Kernel on Riemann Surfaces

Singular Riemann Surfaces

Deformation on the Collars

Figure : Twist to be a double point

Figure : Singular Collar

The Bergman Kernel on Riemann Surfaces

Singular Riemann Surfaces

Proposition

Suppose X0 is a singular Riemann surface with singular point x0 ofgenus g ≥ 2 and constant curvature −1 accept at the singularpoint. Then for any x1 ∈ X0, there is a function ψ such that ψ issmooth on X0 \ {x0, x1} and

√−1∂∂ψ ≥ −Cωg .

x1x0

Figure : Singular Collar

The Bergman Kernel on Riemann Surfaces

Singular Riemann Surfaces

The Bergman kernel does not have strong partial C 0 estimate inall volume-collapsing cases.

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

Bergman kernel on the Collars

TheoremLet X be a compact Riemann surface of genus g ≥ 2 withconstant scalar curvature −1. Suppose that x1 is a point in onecollar with closed geodesic of arc length 4πσ0 and the distance ofx1 to the geodesic is ρ1. Then the Bergman kernel has a uniformlyestimate on that collar∣∣∣Bm(x1)− Bm(x1)

∣∣∣ < e−εm

for some ε > 0.

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

On the collar CR , we define the following holomorphic sections ofKmX for any m > 0:

Tj = w j1(dz)m, (2)

where w1 =√te−√−1z for 0 ≤ j ≤ d − 1 on X = w1w2 = t.

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

DefinitionThe pseudo-Bergman kernel on CR is defined as

Bm(x1) =d−1∑j=0

||Tj ||2(x1)/‖Tj‖2L2(CR). (3)

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

LemmaThere exist holomorphic global sections {Sj}Nj=0 of H0(X ,Km

X )such that Sj

Sj = ηTj − uj

for 0 ≤ j ≤ N, uj ∈ ΓC∞(X ,KmX ). Here η is a smooth cut-off

function defined by

η =

{1 ρ ≤ R − 1

0 ρ ≥ R,

|η′| ≤ 4 and |η′′| ≤ 4. Moreover, we have the following estimate

‖uj‖2L2 ≤ ||Tj ||2L2e− 1

4m (4)

for 0 ≤ j ≤ N = [(2m − 1)σ0 sinh(R − 2)].

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

We define

V = {S ∈ H0(X ,KmX ) | (S ,Tj)

∣∣CR

= 0 for j = 0, · · · ,N}.

Let SN+1, · · · ,Sd−1 be an orthonormal basis of V such that

Sj(x1) = 0

for j > N + 1.

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

In this setting, the Bergman kernel can be written as

Bm(x1) =N+1∑i ,j=0

(F−1)ijSi (x1)Sj(x1),

where

Fij =

(Si , Sj) 0 ≤ i , j ≤ N

(ui , Sj) 0 ≤ i ≤ N; j > N

(Si , uj) i > N; 0 ≤ j ≤ N

δij i , j > N

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

We have ∣∣∣Bm(x1)− Bm(x1)∣∣∣ ≤ 3e−εm

for some 0 < ε < 18 .

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

An explicit formula for the pseudo-Bergman kernel

Theorem (Liu-Lu)

We have

|Bm(x1)− 1

cosh2m ρ1

×d−1∑j=0

j( j2

σ20

+ (2m − 2)2) · · · ( j2

σ20

+ 22) · e2jσ0

(arctan eρ1−π2)

4πσ20(2m − 2)!

(1− e

− 8jπε1

)−1|

≤ e−cm.

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

Thus {S0, ·,Sd−1} forms a regular basis and almost orthonormalH0(X ,Km

X ) and the expansion Bm(x1) is in C 0. Then we can getthe expansion ∣∣∣∣∣∣Bm(x1)− Bm(x1)

∣∣∣∣∣∣Cµ≤ e−εm

for small enough ε > 0.

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

A recent result by Lu and Zelditch provided a formula to computeBm(x1) from the Szego kernel of the universal covering X of X .Together with the Agmon estimate, we have∣∣∣∣Bm(x1, x1)−

∑γ∈π1(X )

d(x1,γ·x1)≤1

˜Πm(x1, γ · x1)]

∣∣∣∣ ≤ e−β√m, (5)

where ˜Πm be the Szego kernel on the unit circle of K ∗X

β = β(X ,KX ), and d(x , y) is the distance function of M.