The Bergman Kernel on Riemann Surfaces
Transcript of The Bergman Kernel on Riemann Surfaces
The Bergman Kernel on Riemann Surfaces
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
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 =
1
n!
∫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 =
1
3∆ρ+
1
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
(1
m
).
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
γ
Figure : a C (γ) on a compact Riemann surface with two genus
The Bergman Kernel on Riemann Surfaces
Introduction
On Riemann Surfaces
γ
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
8
)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
D
(1 + 1√
mδ2x0e
πδx0
) ,where δx0 is the injective radius of x0.
The Bergman Kernel on Riemann Surfaces
Introduction
On Riemann Surfaces
x0
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
D
(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.
w1
w2
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.
w1
w2
w1w2 = 0
The Bergman Kernel on Riemann Surfaces
Singular Riemann Surfaces
Deformation on the Collars
Figure : Twist to be a double point
x0
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.
The Bergman Kernel on Riemann Surfaces
Uniform estimate of the Bergman kernel on the collars
Thank you for your attention!