# The Bergman Kernel on Riemann Surfaces

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The Bergman Kernel on Riemann SurfacesChiung-ju Liu (Joint work
with Zhiqin Lu)

National Taiwan University

December 6, 2013

Outline

Introduction

Partial Results

Introduction

Introduction

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

Ric(h) = ωg

Introduction

Inner Product

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

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

(U,V ) =

The Bergman Kernel on Riemann Surfaces

Introduction

The Bergman Kernel

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

Bm(x) = d∑

i=1

Si2hm(x).

Introduction

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

ωm = 1

Introduction

History

Theorem (Tian ’90)

Let M be an algebraic manifold with a polarization L and let g be the corresponding polarized Kahler metric on M. Thengm − g

C2

= O

Introduction

History

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

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

The Bergman Kernel on Riemann Surfaces

Introduction

History

Introduction

History

Application

Theorem (Donaldson(2001))

Suppose that Aut(M, L) is discrete and (M, Lm) is balanced for sufficiently large m. Suppose that Bergman metrics ωm converge to ω∞ 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 a polynomial of the curvature and its covariant derivatives at x . Such polynomial can be found by finitely many steps of algebraic operations. We have

a0 = 1, a1 = 1

24 (|R|2 − 4|Ric |2 + 3ρ2), · · · ,

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

The Bergman Kernel on Riemann Surfaces

Introduction

History

In particular,

Theorem (Lu-Tian)

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

ak = Ck−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 to symplectic manifold for ”almost” holomorphic sections

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

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

Ma and Marinescu (2008) gave a full off-diagonal asymptotic of the generalized Bergman kernels of renomalized Bouchner-Laplacian on high powers of a positive line bundle over a compact symplectic manifolds.

Ross and Thomas (2009) gave an asymptotic expansion of the weighted Bergman kernel on orbifolds and showed that if a polarized orbifold is balanced, then it has a metric with constant scalar curvature.

The Bergman Kernel on Riemann Surfaces

Introduction

History

Xu (2011) gave a graph-theoretic interpretation of the coefficients of 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 fixed point x . Then for m big enough, the expansion of the Bergman kernel

∞∑ j=0

aj(x)mn−j

Bm(x)− ∞∑ j=0

ajm n−jCµ ≤ 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.

Introduction

History

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

∂∂ logBm(x) = 0

) .

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

The Bergman Kernel on Riemann Surfaces

Introduction

History

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

d∑ i=1

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 the canonical line bundle endowed with a Hermitian metric h such that the curvature Ric(h) of h defines a Kahler metric g on M. Let the Gauss curvature G of g satisfy

−C1 ≤ G ≤ C2

for some nonnegative constants C1 and C2 and let δ′ be the injective 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

Introduction

Collar Theorem

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

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

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

γ

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

The Bergman Kernel on Riemann Surfaces

Introduction

Introduction

Theorem (Liu)

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

Bm(x) ∼ m(1 + ρ

The Bergman Kernel on Riemann Surfaces

Introduction

Theorem (Lu)

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

inf x∈M

Introduction

Theorem (Lu)

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

M ) with SL2 = 1 such that

S(x0) ≥ √ m

The Bergman Kernel on Riemann Surfaces

Introduction

Singular Riemann Surfaces

Semi-Partial Lower Estimate

Theorem (Liu-Lu(2011))

Suppose X0 is a singular Riemann surface of genus with ordinary double point x0 and scalar constant curvature −1. Then there exists constant D such that for any x1 ∈ X0 \ x0, the Bergman kernel

Bm(x1) ≥ m

Singular Riemann Surfaces

Deformation

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

The Bergman Kernel on Riemann Surfaces

Singular Riemann Surfaces

From algebraic geometry point of view, consider Xt as a sequence of algebraic curves w1w2 = t that degenerate to the central fiber X0.

w1

w2

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.

w1

w2

Singular Riemann Surfaces

x0

Singular Riemann Surfaces

Proposition

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

√ −1∂∂ψ ≥ −Cωg .

Singular Riemann Surfaces

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

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

Bergman kernel on the Collars

Theorem Let X be a compact Riemann surface of genus g ≥ 2 with constant scalar curvature −1. Suppose that x1 is a point in one collar with closed geodesic of arc length 4πσ0 and the distance of x1 to the geodesic is ρ1. Then the Bergman kernel has a uniformly estimate on that collarBm(x1)− Bm(x1)

< e−εm

Uniform estimate of the Bergman kernel on the collars

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

Tj = w j 1(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

Definition The pseudo-Bergman kernel on CR is defined as

Bm(x1) = d−1∑ j=0

||Tj ||2(x1)/Tj2L2(CR) . (3)

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

Lemma There 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 ,Km X ). Here η is a smooth cut-off

function defined by

0 ρ ≥ R ,

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

uj2L2 ≤ ||Tj ||2L2e − 1

4 m (4)

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

We define

CR

= 0 for j = 0, · · · ,N}.

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

Sj(x1) = 0

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),

(ui , Sj) 0 ≤ i ≤ N; j > N

(Si , uj) i > N; 0 ≤ j ≤ N

δij i , j > N

Uniform estimate of the Bergman kernel on the collars

We have Bm(x1)− Bm(x1) ≤ 3e−εm

for some 0 < ε < 1 8 .

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

Uniform estimate of the Bergman kernel on the collars

Thus {S0, ·,Sd−1} forms a regular basis and almost orthonormal H0(X ,Km

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

Cµ ≤ e−εm

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 compute Bm(x1) from the Szego kernel of the universal covering X of X . Together with the Agmon estimate, we haveBm(x1, x1)−

∑ γ∈π1(X )

≤ e−β √ m, (5)

,

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

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

Thank you for your attention!

Introduction

History

National Taiwan University

December 6, 2013

Outline

Introduction

Partial Results

Introduction

Introduction

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

Ric(h) = ωg

Introduction

Inner Product

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

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

(U,V ) =

The Bergman Kernel on Riemann Surfaces

Introduction

The Bergman Kernel

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

Bm(x) = d∑

i=1

Si2hm(x).

Introduction

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

ωm = 1

Introduction

History

Theorem (Tian ’90)

Let M be an algebraic manifold with a polarization L and let g be the corresponding polarized Kahler metric on M. Thengm − g

C2

= O

Introduction

History

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

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

The Bergman Kernel on Riemann Surfaces

Introduction

History

Introduction

History

Application

Theorem (Donaldson(2001))

Suppose that Aut(M, L) is discrete and (M, Lm) is balanced for sufficiently large m. Suppose that Bergman metrics ωm converge to ω∞ 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 a polynomial of the curvature and its covariant derivatives at x . Such polynomial can be found by finitely many steps of algebraic operations. We have

a0 = 1, a1 = 1

24 (|R|2 − 4|Ric |2 + 3ρ2), · · · ,

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

The Bergman Kernel on Riemann Surfaces

Introduction

History

In particular,

Theorem (Lu-Tian)

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

ak = Ck−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 to symplectic manifold for ”almost” holomorphic sections

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

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

Ma and Marinescu (2008) gave a full off-diagonal asymptotic of the generalized Bergman kernels of renomalized Bouchner-Laplacian on high powers of a positive line bundle over a compact symplectic manifolds.

Ross and Thomas (2009) gave an asymptotic expansion of the weighted Bergman kernel on orbifolds and showed that if a polarized orbifold is balanced, then it has a metric with constant scalar curvature.

The Bergman Kernel on Riemann Surfaces

Introduction

History

Xu (2011) gave a graph-theoretic interpretation of the coefficients of 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 fixed point x . Then for m big enough, the expansion of the Bergman kernel

∞∑ j=0

aj(x)mn−j

Bm(x)− ∞∑ j=0

ajm n−jCµ ≤ 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.

Introduction

History

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

∂∂ logBm(x) = 0

) .

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

The Bergman Kernel on Riemann Surfaces

Introduction

History

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

d∑ i=1

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 the canonical line bundle endowed with a Hermitian metric h such that the curvature Ric(h) of h defines a Kahler metric g on M. Let the Gauss curvature G of g satisfy

−C1 ≤ G ≤ C2

for some nonnegative constants C1 and C2 and let δ′ be the injective 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

Introduction

Collar Theorem

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

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

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

γ

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

The Bergman Kernel on Riemann Surfaces

Introduction

Introduction

Theorem (Liu)

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

Bm(x) ∼ m(1 + ρ

The Bergman Kernel on Riemann Surfaces

Introduction

Theorem (Lu)

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

inf x∈M

Introduction

Theorem (Lu)

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

M ) with SL2 = 1 such that

S(x0) ≥ √ m

The Bergman Kernel on Riemann Surfaces

Introduction

Singular Riemann Surfaces

Semi-Partial Lower Estimate

Theorem (Liu-Lu(2011))

Suppose X0 is a singular Riemann surface of genus with ordinary double point x0 and scalar constant curvature −1. Then there exists constant D such that for any x1 ∈ X0 \ x0, the Bergman kernel

Bm(x1) ≥ m

Singular Riemann Surfaces

Deformation

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

The Bergman Kernel on Riemann Surfaces

Singular Riemann Surfaces

From algebraic geometry point of view, consider Xt as a sequence of algebraic curves w1w2 = t that degenerate to the central fiber X0.

w1

w2

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.

w1

w2

Singular Riemann Surfaces

x0

Singular Riemann Surfaces

Proposition

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

√ −1∂∂ψ ≥ −Cωg .

Singular Riemann Surfaces

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

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

Bergman kernel on the Collars

Theorem Let X be a compact Riemann surface of genus g ≥ 2 with constant scalar curvature −1. Suppose that x1 is a point in one collar with closed geodesic of arc length 4πσ0 and the distance of x1 to the geodesic is ρ1. Then the Bergman kernel has a uniformly estimate on that collarBm(x1)− Bm(x1)

< e−εm

Uniform estimate of the Bergman kernel on the collars

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

Tj = w j 1(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

Definition The pseudo-Bergman kernel on CR is defined as

Bm(x1) = d−1∑ j=0

||Tj ||2(x1)/Tj2L2(CR) . (3)

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

Lemma There 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 ,Km X ). Here η is a smooth cut-off

function defined by

0 ρ ≥ R ,

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

uj2L2 ≤ ||Tj ||2L2e − 1

4 m (4)

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

We define

CR

= 0 for j = 0, · · · ,N}.

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

Sj(x1) = 0

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),

(ui , Sj) 0 ≤ i ≤ N; j > N

(Si , uj) i > N; 0 ≤ j ≤ N

δij i , j > N

Uniform estimate of the Bergman kernel on the collars

We have Bm(x1)− Bm(x1) ≤ 3e−εm

for some 0 < ε < 1 8 .

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

Uniform estimate of the Bergman kernel on the collars

Thus {S0, ·,Sd−1} forms a regular basis and almost orthonormal H0(X ,Km

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

Cµ ≤ e−εm

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 compute Bm(x1) from the Szego kernel of the universal covering X of X . Together with the Agmon estimate, we haveBm(x1, x1)−

∑ γ∈π1(X )

≤ e−β √ m, (5)

,

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

The Bergman Kernel on Riemann Surfaces

Uniform estimate of the Bergman kernel on the collars

Thank you for your attention!

Introduction

History