• date post

10-Jul-2022
• Category

## Documents

• view

0

0

Embed Size (px)

### Transcript of The Bergman Kernel on Riemann Surfaces

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