Asymptotic Tian-Yau-Zelditch Expansions on Singular Riemann Surfaces

PREPRINT

國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University

www.math.ntu.edu.tw/ ~ mathlib/preprint/2011- 13.pdf

Asymptotic Tian-Yau-Zelditch Expansions on Singular Riemann Surfaces

Chiung-Ju Liu and Zhiqin Lu

April, 2011

ASYMPTOTIC TIAN-YAU-ZELDITCH EXPANSIONS ON SINGULAR RIEMANN SURFACES

CHIUNG-JU LIU AND ZHIQIN LU

Abstract. We give both lower and upper bound estimates of the Bergman kernel for a degeneration of Riemann surfaces with constant curvature −1. As a result, we gave a geometric proof of the Riemann-Roch Theorem for singular Riemann surface.

Contents

1. Introduction 1

2. The holomorphic plumbing fixture 2

3. Subharmonic functions 4

4. Main results 6

4.1. Riemann-Roch Theorem on singular surface 8

4.2. The upper bound of the Bergman Kernel 9

References 10

1. Introduction

Let M be a compact complex manifold of dimension n and let (L, h) → M be a positive Hermitian line bundle. Let g be the K¨ahler metric on M corresponding to the K¨ahler form ωg = Ric(h). For each m ∈ N, h induces a Hermitian metric h^m on L^m. Let {S₀, · · · , S_dm−1} be any orthonormal basis of H⁰(M, L^m) with respect to the inner product

(S_i, S_j) = Z

M

hS_i(x), S_j(x)i_hmdV_g,

where d_m = dimH⁰(M, L^m) and dV_g = _n!¹ω_gⁿ is the volume form of g. The sequence of functions,

B_m(x) =

dm−1

X

i=0

kS_i(x)k²_hm, is called the Bergman kernel of the manifold.

Catlin [2] and Zelditch [12] proved that for any given point x ∈ M there is a complete asymptotic expansion for the Bergman kernel:

B_m(x) ∼ a₀(x)mⁿ+ a₁(x)mⁿ⁻¹+ a₂(x)mⁿ⁻²+ · · · (1.1)

1Subject Classification. Primary 53C55; Secondary 32S30 Date: March 31, 2011.

Key words and phrases. Asymptotic Expansion, Singular Riemann Surface.

1

for certain smooth coefficients a_j(x) with a₀ = 1. The asymptotic expansion is conver- gent in the following sense: for any positive integers k and l, there exists a constant C, depending on k, l and the manifold M , such that

B_m(x) − mⁿ+ a₁(x)mⁿ⁻¹+ · · · + a_k(x)m^n−k
  

_{Cl(M )} ≤ Cm^n−k−1. (1.2) In [8], the first several coefficients of the above expansion were computed. Moreover, aj(x), for any j, is a polynomial of the curvature and their covariant derivatives at x. Thus, the expansion of Catlin and Zelditch is local. On the other hand, the Bergman kernel itself contains global information of the manifold M . Thus the asymptotic expansion, in general, is not uniform. The constant in (1.2) may go to infinity when the metric varies.

One of the main applications of the above expansion is the theorem of Tian [9], which is equivalent to

∂ ¯∂ log B_m(x) = O 1 m

 .

A necessary condition for the above equality to be true uniformly is the uniform lower bound of the Bergman kernel. For general line bundle L, if for some positive integer m, B_m(x) > 0, then L^m is called base point free. By the work of Demailly and Siu, for an ample line bundle L there is a constant c(n), depending only on the dimension n of M , such that for any m > c(n), L^m is base point free. For a family of manifolds, it is interesting to see whether there is a constant ε > 0, depending only on the geometry of the family, such that

dm−1

X

i=0

kS_i(x)k²_hm > ε.

In K¨ahler-Einstein geometry, the above estimate is very important. Similar estimate is called strong partial C⁰ estimate in [10] and is called tamed in the paper of Chen-Wang [1].

In this paper, we study the behavior of the Bergman kernels on a degeneration family of Riemann surfaces, polarized by their canonical bundles. This is the prototype case for the volume collapsing family of K¨ahler manifolds. We proved that there is no strong partial C⁰ estimate in this case and we believe that the same statement is true for the Bergman kernels in all volume collapsing cases.

We give both lower and upper bounds estimates of the Bergman kernel for a family of Riemann surfaces in this paper (Theorem 4.1 and Theorem 4.3). Using the techniques developed, we are able to study the limiting behaviors of the holomorphic sections of the pluri-canonical bundles. Let X_t be a degeneration of smooth Riemann surfaces such that X₀ is a single Riemann surface with one ordinary double point. We obtain

dim H_L⁰2(X0, K_X^m₀) = dim H⁰(Xt, K_X^m_t) − 1.

We are able to single out one of the holomorphic section whose L² normal blows up at t = 0.

Throughout this paper, a constant C is a positive real number which is independent to t but may differ line by line.

2. The holomorphic plumbing fixture

Consider a holomorphic degeneration family π : X → ∆ of Riemann surfaces, where

∆ = {t ∈ C : |t| < 1} and for any 0 6= t ∈ ∆, Xt= π⁻¹(t) is a smooth Riemann surface of

w1

w2

w1w2= t

γ

w1

w2

w1w2= 0

x0

Figure 1. Algebra pictures vs. geometric pictures of Xt near the singu- larity point x₀

genus greater than 1. We assume that X is smooth and the central fiber X₀ is a singular Riemann surface which has only one ordinary double point x₀.

Near x0, Xt can be represented by the curve {w1w2 = t} within the unit ball of C². In particular, X0 is the zero locus of {w1w2 = 0}.

On the other hand, from the differential geometric point of view, near x₀, X₀ is the union of two cusps, each of which is biholomorphic to the punctured unit disk with the Poincar´e metric

4dz ⊗ d¯z

|z|²(log |z|⁻²)². (2.1)

Please see Figure 1. To unify the above two view points, we start with the following Collar Theorem of Keen [5, page 264]:

Theorem 2.1 (Keen). Consider the region T of U , the upper half plane, bounded by the curve r = 1, r = e^l, θ = θ₀ and θ = π − θ₀. Let γ be a closed geodesic on M with length l. Then there is a conformal isometric mapping ϕ : T → M such that ϕ(iy) = γ. The image ϕ(T ) of T is called a collar. Then we can choose θ₀ small enough such that the area of the Collar is at least ^√8

5.

Topologically, a collar C_R is the cylinder (−R, R) × S¹, where 2R is the length of the collar. For the rest of the paper, we write CR = (−R, R) × [0, 2π) for the sake of convenience. There is a holomorphic embedding

p_t: C_R→ X_t for each t. We identify C_R with its image p_t(C_R).

The Gaussian curvature of C_R is −1 with respect to the above Poincar´e metric. Let γ be the shortest closed geodesic line of C_R in X_t and let 4πσ₀ be its length. Then the

Riemannian metric on C_R, with the natural coordinates (ρ, θ), can be written as ds² = dρ²+ (2σ0cosh ρ)²(dθ)².

By Wolpert [11, page 13], we have (up to a constant) σ₀ = π

log |t|⁻². Let

x = θ + 1

2θ₀, y = 1 σ0



arctan e^ρ− π 4

 ,

where t = |t|e^iθ0 and let z = x + iy. Then under coordinate z, the metric ds² can be written as

ds² = (2σ₀cosh ρ)²(dz ⊗ d¯z). (2.2) By the conformal structure of Riemann surface, we conclude that z = x + iy is either holomorphic or anti-holomorphic. With the suitable orientation, we assume that z is holomorphic.

Let

p(ρ, θ) = (w1, w2) be the embedding. Then we have

w1 = e^{−σ01 (π2−arctan eρ)−iθ}; w₂ = t

w₁, (2.3)

The area of the collar C_R is

A = 8πσ₀sinh R = ε₁ = 8

√5.

As t goes to zero, the sequences of collars C_R preserves the areas while the length 2R goes to infinity and the injective radius 2πσ₀ goes to zero. For the rest of the paper, the real number R is related to σ₀ (also in terms of t) by the above equation, and ε₁ is the absolute constant 8/√

5.

3. Subharmonic functions

The injectivity radius of the singular Riemann surface X₀ is 0. In order to make use of the ¯∂-estimate, we need to construct suitable subharmonic functions on X₀ (Note that on Riemann surfaces, subharmonic functions and plurisubharmonic functions are the same thing). The main result of this section is Proposition 3.2, which is a generalization of [7, Proposition 4.2] in the singular case.

For t 6= 0, let x₁ = (R − ρ₁, θ) for some fixed 4 ≤ ρ₁ < R and p₀ = (R − 1, 0) in C_R. As t approaches zero, let w_x1, w_p0 be the corresponding limit points in

D1\ {0} = {0 < |w1| < e^−4πε1} ⊂ X0, respectively. By taking the limit in (2.3), We have

w_x1 = e^{−4πε1eρ1−iθ} w_p0 = e^−4eπε1

In [7], the following result was proved (with slightly different notations)

Proposition 3.1. Let X be a compact Riemann surface of genus g ≥ 2 and constant curvature −1. Then for any x₁ ∈ M , there is a function ϕ_t = ϕ_t,x1

ϕ_t(w₁) = log    

wx1

w₁ − 1    

−2 arctan e^R−ρ1

π log

wp0

w₁ − 1     , such that

(1) In a neighborhood Ux1 of x1, ϕt can be written as ϕ_t= 2 log d(x) + ψ,

where ψ is a smooth function on U_x1 and d(x) is the distance function to x₁. Consequently

Z

U_x1

e^−ϕt = +∞;

(2) There is a constant C such that

√−1

2π ∂ ¯∂ϕ_t≥ −Cω_g on M \{x₁}, where ω_g is the K¨ahler form of M ; (3) ϕ_t satisfies

ϕ_t ≤ C on X and

ϕt ≥ 2 log d(x) − 2π δ_x1 − C for d(x) ≤ δ_x1 and the injectivity radius δ_x1 at x₁.

 Since ρ₁ is fixed, we have

2 arctan e^R−ρ1

π → 1

as t → 0. Thus point-wisely, the functions ϕ_t have the following limit

ϕ = log |w_x1 − w₁| − log |w_p0 − w₁| (3.1) on X₀.

Since the Poincar´e metric

ds²₀ = 4

|w1|²(log |w1|⁻²)²dw₁⊗ dw₁, is the limit of the metric in (2.2), we concluded the following result.

Proposition 3.2. Suppose X₀ is a singular Riemann surface with singular point x₀ of genus g ≥ 2 and constant curvature −1 except at the singular point. Then for any x₁ ∈ X₀, there is a function ϕ on X₀\ {x₀, x₁} such that

(1) In a neighborhood U_x1 of x₁, ϕ can be written as ϕ = 2 log d + ψ, where ψ is a smooth function on U_x1 and

Z

U_x1

e^−ϕ = +∞; (3.2)

(2) There is a constant C such that

√−1

2π ∂ ¯∂ϕ ≥ −Cω_g0 (3.3)

on X₀\ {x₀, x₁}, where g₀ is the complete K¨ahler metric of X₀; (3) ϕ satisfies

ϕ ≤ C (3.4)

on X₀ and

ϕ ≥ 2 log d − 2π δ_x1 − C for d ≤ δ_x1.

Remark 3.1. Since the (global) injectivity radius of X₀ is zero, the existence of the sub- harmonic function is not trivial. We don’t know how to use the method in high dimensions.

On the other hand, by our construction, we obtained a smooth family of subharmonic func- tions over X . The functions ϕ_t,x1 naturally extends to the punctured unit ball of C² such

that √

−1∂ ¯∂ϕ_t,x1 ≥ −Cω_t, for sufficiently large constant C.

 4. Main results

We need the following H¨ormander’s L₂ estimate.

Proposition 4.1. Suppose X₀ is a singular Riemann surface of genus g ≥ 2 with ordinary double point x₀ . Let L be a line bundle over X₀ with the Hermitian metric h such that Ric(h) is a complete metric. Let ψ be plurisubharmonic function on X₀, which can be approximated by a decreasing sequence of smooth functions {ψ`}1≤`≤∞. If

∂ ¯∂ψ_`+ 2

√−1(Ric(h) + Ric(ω_g0)), v ∧ ¯v

g0

≥ Ckvk².

for any tangent vector v of type (1, 0) at X0 \ {x0} for each `, where C is a constant independent of `, and h·, ·i_g0 is the inner product induced by g₀, then for any C^∞ L-valued (0, 1)-form τ on X₀ with ¯∂τ = 0 and R

X0kτ k²e^−ψdV_g0 < +∞, there exists a C^∞ L-valued u on X₀ such that ¯∂u = τ and

Z

X0

kuk²e^−ψdV_g0 ≤ 1 C

Z

X0

kτ k²e^−ψdV_g0, where dV_g0 is the volume form of g₀.

 With the above estimate and the existence of the subharmonic function in Proposi- tion 3.2, we are able to generalize [7, Theorem 1.3] into the singular case in Theorem 4.1 below.

We apply Proposition 4.1 to the case that L = K_X^m₀, h = g₀^−m for m ≥ 2, where g₀ is the complete K¨ahler metric on X₀ with scalar curvature −1. By Proposition 3.2, we have

∂ ¯∂ϕ + 2

√−1(Ric(h) + Ric(ω_g0)), v ∧ ¯v

≥ (m − C − 1)kvk².

We choose m ≥ 2C.

Theorem 4.1. Suppose X₀ is a singular Riemann surface of genus g ≥ 2 with ordinary double point x₀ and constant scalar curvature −1. Then there exists constant D such that for any x₁ ∈ X₀\ x₀, the Bergman kernel B_m(x) satisfies

B_m(x₁) ≥ m D

 1 + ^e

2π δx1

mδ⁴_x1

 , (4.1)

where δx1 is the injectivity radius of x1.

Proof. The result is obviously true if x₁ is away from the singular point. Thus we only need to prove the theorem under the assumption that |w_x1| < e^−4πε1.

Let V_x1 = {x | dist(x, x₁) < δ_x1}. Let z₁ be the holomorphic function on V_x1 under which the K¨ahler metric can be represented as

ds² = 2

(1 − |z₁|²)²dz₁⊗ d¯z₁. (4.2) For sufficiently large m, let

τ = ¯∂η 2|z₁| δ_x1



(dz₁)^m, where the smooth cut-off function

η(t) =

( 1 for t ≤ ¹₂ 0 for t ≥ 1 such that |η⁰| < 4 and |η⁰⁰| < 4. Then ¯∂τ = 0 and

kτ k² ≤ 16 δ²_x

1

1 − |z₁|²
2m+2

for ¹₄δ_x1 ≤ |z₁| ≤ ¹₂δ_x1. The L²-norm of τ with respect to the weight ϕ given in Proposition 3.2 is

Z

X0

kτ k²e^−ϕdV_g0 ≤ Ce

2π δx1

mδ_x⁴

1

. By Proposition 4.1, for m large, there exists u ∈ C^∞(X₀, K_X^m

0) such that ¯∂u = τ and Z

X0

kuk²e^−ϕdV_g0 ≤ Ce

2π δx1

m²δ_x⁴

1

. (4.3)

Let S = η

2|z|

δ_x1



(dz₁)^m− u. Thus S ∈ H⁰(X₀, K_X^m₀). SinceR

X0e^−ϕdV_g0 = +∞, u(x₁) = 0.

Therefore

kSk²(x₁) = 1. (4.4)

By parallelogram law, we have kSk²_L

2 ≤ 2

Z

X0

kuk²dV_g0 + Z

X0

kη · (dz₁)^mk²dV_g0

 .

By (4.3) and Proposition 3.2, we have Z

X0

kuk²dV_g0 ≤ Ce

2π δx1

m²δ_x⁴₁. (4.5)

Z

X0

kη · (dz₁)^mk²dV_g0 ≤ Z

V_x1

k(dz₁)^mkdV_g0 ≤ π

m − ¹₂. (4.6)

By (4.4)-(4.6), there exists a constant D such that kSk²(x₁)

kSk²_L2

≥ m

D

 1 + ^e

2π δx1

mδ_x1⁴

 . The theorem is proved.

 4.1. Riemann-Roch Theorem on singular surface. The canonical line bundle K_X0 of X₀ is defined as follows. Let U_j, j = 1, · · · , N be an open cover of X₀. We assume that U₀ is a neighborhood of x₀ , and

U₁ = {(w₁, w₂) | w₁ 6= 0; w₂ = 0}; U₂ = {(w₁, w₂) | w₁ = 0; w₂ 6= 0}

Moreover, we assume that x₁ ∈ U/ _j for j > 2. Let V be an open set of X₀. A section f of K_X0 over V is define as

(1) On Uj∩ V 6= ∅, f is a holomorphic 1 form;

(2) On U₁, f is represented by the meromorphic function f₁ of the form f₁(w₁) = a

w₁ + ˜f₁(w₁),

where ˜f1(w1) is a holomorphic function of w1 in the unit disk;

(3) Similarly, on U2 f is represented by the meromorphic function f2 of the form f₂(w₂) = − a

w₂ + ˜f₂(w₂),

where ˜f₂(w₂) is a holomorphic function of w₂ in the unit disk.

By the flatness of the family X , we concluded that

dim H⁰(X, K_X^m₀) = dim H⁰(X_t, K_X^m_t) = (2m − 1)(g − 1).

We prove the following result.

Theorem 4.2. Let H_L⁰2(X₀, K_X^m

0) be the space of L² holomorphic sections of K_X^m

0. Then dim H_L⁰2(X₀, K_X^m₀) = dim H⁰(X₀, K_X^m₀) − 1 = (2m − 1)(g − 1) − 1.

Proof. Define a local section u of K_X^m

0 on U₁∪ U₂ by f₁(w₁) = 1

w₁^m on U₁; f2(w2) = − 1

w₂^m on U2.

Let ρ be a cut-off function such that in a neighborhood of x₀, ρ is identically equal to 1.

Let

τ = ¯∂ρ · u.

By Proposition 4.1, there exists a u₁ ∈ C^∞(X₀, K_X^m₀) such that

∂u¯ ₁ = ¯∂ρ · u.

We claim that

S = ρu − u₁ is not in L². To see this, we first observe that

Z

X0

||u₁||²dV_g0 ≤ C m

Z

X0

||τ ||²dV_g0 < +∞.

On the other hand, we have Z

X0

||ρu||²dV_g0 = Z

|w1|<δ

|w₁|^2m(log |w₁|⁻²)^2m· 1

|w₁|^2m · 2√

−1dw₁∧ d ¯w₁

|w₁|²(log |w₁|⁻²)² = +∞.

Thus we have

dim H_L⁰2(X₀, K_X^m

0) ≤ dim H⁰(X₀, K_X^m

0) − 1.

To prove the other direction of the inequality, let T be an arbitrary section of K_X^m₀. Then for a suitable choice of complex number α, T − αS has the pole of order at most m − 1 near w₁ = 0. Therefore we have

Z

X0

||T − αS||²dV_g0 ≤ C₁

+ C₂ Z

|w1|<δ

|w₁|^2m(log |w₁|⁻²)^2m· 1

|w₁|^2m−2 · 2√

−1dw₁∧ d ¯w₁

|w₁|²(log |w₁|⁻²)² < +∞

for some positive constants C₁ and C₂. The theorem is proved.

 4.2. The upper bound of the Bergman Kernel. We end this paper by proving the family version of upper bounds of the Bergman kernel.

Theorem 4.3. Let X be a Riemann surface with genus g ≥ 2 and constant scalar curva- ture −1. Let K_X be the canonical line bundle of X and h be a positive Hermitian metric on K_X. Then we have

Bm(x) ≤ 2m − 1 π (1 − e^{−(2m−1)δ2x}). Proof. Given any x ∈ X, we have

B_m(x) ≤ sup

S∈H⁰(X,K_X^m)

kSk²(x) kSk²_L2

.

Let z₁ be the holomorphic coordinate on V_x = {y ∈ X | dist(x, y) < δ_x} centered at x and let the metric under z₁ be defined as in (4.2). Any section S ∈ H⁰(X, K_X^m) restricted to V_x can be written as

S|_Vx = f (z₁)(dz₁)^m

for some holomorphic function f (z₁) defined on U_x. Thus we have kSk²(x)

kSk²_L2

≤ kSk²(x) kSk²_L2(Bx(δx))

= |f (0)|²

R

Bx(δx)|f (z)|²h(z)^m−1

√−1

2 dz ∧ d¯z, where

h(z) = (1 − |z|²)². In polar coordinates, we have

Z

Bx(δx)

|f |²h^m−1

√−1

2 dz ∧ d¯z = Z δx

0

Z

|z|=r

|f |²h^m−1rdθdr.

By the mean value inequality, we have Z

Bx(δx)

|f |²h^m−1

√−1

2 dz ∧ d¯z ≥ 2π|f (0)|² Z δx

0

(1 − r²)^2m−2rdr

≥ π|f (0)|² 2m − 1



1 − e^(2m−1)δx2 , which proves the theorem.

 Acknowledgement. The first author is partially supported by the NSC grant 100- 2115-M-002-010- in Taiwan. The second author is partially supported by the NSF grant DMS-09-04653. The authors thank Zheng Huang for his help for explaining the concept holomorphic plumbing structure in the book of Wolpert [11].

References

[1] X.X. Chen and B. Wang, K¨ahler-Ricci flow on Fano Manifolds (I). arXiv:0909.2391v2, to appear in Journal of European Mathematical Society.

[2] David Catlin, The Bergman kernel and a theorem of Tian, Analysis and geometry in several complex variables (Katata, 1997), Trends Math., Birkh¨auser Boston, Boston, MA, 1999, pp. 1–23. MR1699887 (2000e:32001)

[3] Jean-Pierre Demailly, Holomorphic Morse inequalities, Several complex variables and complex ge- ometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 93–114. MR1128538 (93b:32048)

[4] S. K. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), no. 3, 479–522. MR1916953 (2003j:32030)

[5] Linda Keen, Collars on Riemann surfaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Princeton Univ. Press, Princeton, N.J., 1974, pp. 263–268.

Ann. of Math. Studies, No. 79. MR0379833 (52 #738)

[6] Chiung-ju Liu, The asymptotic Tian-Yau-Zelditch expansion on Riemann surfaces with constant curvature, Taiwanese J. Math. 14 (2010), no. 4, 1665–1675. MR2663940

[7] Zhiqin Lu, On the lower bound estimates of sections of the canonical bundles over a Riemann sur- face, Internat. J. Math. 12 (2001), no. 8, 891–926, DOI 10.1142/S0129167X01001064. MR1863285 (2002j:32018)

[8] , On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math.

122(2000), no. 2, 235–273. MR1749048 (2002d:32034)

[9] Gang Tian, On a set of polarized K¨ahler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99–130. MR1064867 (91j:32031)

[10] G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math.

101(1990), no. 1, 101–172, DOI 10.1007/BF01231499. MR1055713 (91d:32042)

[11] Scott A. Wolpert, Families of Riemann surfaces and Weil-Petersson geometry, CBMS Regional Conference Series in Mathematics, vol. 113, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2010. MR2641916 (2011c:32020)

[12] Steve Zelditch, Szeg˝o kernels and a theorem of Tian, Internat. Math. Res. Notices 6 (1998), 317–331, DOI 10.1155/S107379289800021X. MR1616718 (99g:32055)

(Chiung-ju Liu) Department of Mathematics, National Taiwan University, Taipei, Taiwan 106

E-mail address: cjliu4@ntu.edu.tw

(Zhiqin Lu) Department of Mathematics, University of California, Irvine, CA 92697-3875 E-mail address: zlu@uci.edu