Bergman kernels on punctured Riemann surfaces
Hugues Auvray— joint work with X. Ma and G. Marinescu —October 9, 2018
International Workshop on Geometric Quantization and Applications

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Content1

Bergman kernels on complete manifolds
LandscapeGeneral results2

Punctured Riemann surfaces
Setting
Application of Theorem 0Results

3

ProofsCorollary 2
Theorem 1
Corollary 3

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I-Bergman kernels on complete manifoldsa) Landscape
I Start with an hermitian holomorphic line bundle (L, h) over a completeKähler manifold (X n , ωX )(h does not necessarily polarize ωX ).
I Consider, for p ≥ 1, the Hilbert space

Lp0
H(2)
(X , Lp ) = σ ∈ L2 (X , Lp ) ∂ σ = 0(here and below, Lp is a shortcut for (L⊗p , h p )).
It might be of infinite dimension when X is non-compact.
I To these data, associate the Bergman kernels
X (p)
(p)
Bp : (x , y) 7−→s` (x ) ⊗ s` (y)∗ ∈ Lpx ⊗ (Lpy )∗`≥0(p)0for some (any) orthonormal basis (s` )`≥0 of H(2)(X , Lp ). More particularly,P(p)look at the density functions Bp (x ) = Bp (x , x ) = `≥0 |s` (x )|2h p ≥ 0.
|σ(x )|2h p.
I Alternatively: Bp (x ) =sup2σ∈H 0 ,σ6=0 kσkL2(2),pHugues AuvrayBergman kernels on punctured Riemann surfaces

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I-Bergman kernels on complete manifoldsb) Bp asymptotics: general results
Theorem 0 (Ma-Marinescu, 2007)
With previous notations, assume that:i) (”uniform ampleness”) there exists ε > 0 such that:iR h = −i ∂∂ log(|σ|2h ) ≥ εωX on X ;locii) (”bounded geometry”) Ric(ωX ) ≥ −C ωX on X , for some C ≥ 0.
Then: for all j ≥ 0, there exists b j ∈ C ∞ (X ) such that:∀K b X , ∀k , m ≥ 0, ∃Q = Q(K , k , m, ε, C , n), ∀p ≥ 1,p −n Bp (x ) −kXb j p −jj =0

C m (K )≤ Qp −k −1 .

ωniMore precisely, b 0 = ωnh (with ωh = 2πR h ) andXb0b 1 = 8πscal(ωh ) − 2∆ωh log b 0 .

Hugues AuvrayBergman kernels on punctured Riemann surfaces

4 / 17I-b) Bp asymptotics: general results

A few remarks:
B Long history; many names associated to this result: Tian (1990,k = 0, m = 2), Bouche (1990), Catlin-Zelditch (1999-98, compact X ), ...
B Quantization of Kodaira embedding theorem / scalar curvature in Kähler
geometry.
B The proof requires two steps:
1- localization on Bp ;2- computations of the asymptotics with geometric data brought to Cn (scalingtechniques).

B This statement does not say what happens to the Bergman density functions
on neighbourhoods of infinity...

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II-Punctured Riemann surfacesa) Setting”The most elementary class of complete non-compact Kähler manifolds.”
I Take:• Σ = Σ̄ \ D, where D = {a1 , . . . , aN } is the puncture divisor inside a compact
Riemann surface Σ̄, and ωΣ a smooth Kähler form on Σ;• an hermitian line bundle (L|Σ , h), with L holomorphic on Σ̄.

I Suppose moreover that there are trivializations∼L|Vj −−→ Czj × Dr
(0 < r < 1) around the aj ’s, such that:(α) |1|2h (zj ) = log(|zj |2 ) ;(β) i(R h )|Vj∗ = ωΣ|Vj∗ .

In particular,ωΣ = ωD∗ (zj ) on Vj∗ ,∧d z̄
where ωD∗ = |z |2idz(Poincaré metric on D∗ ).
log2 (|z |2 )Hugues AuvrayBergman kernels on punctured Riemann surfaces

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II-a) Setting
An arithmetic class of examples. —These (notably, properties (α) and (β)) are natural hypotheses, as revealed by thefollowing class of examples.
If Γ ⊂ Psl(2, R) is a Fuchsian group of the first kind, which is geometrically finiteand contains no elliptic element, thenΣ = Γ\Hcan be compactified by adjunction of finitely many points.
Conversely, if Σ = Σ̄ \ {a1 , . . . , aN } is such that (equivalently):• Σ̃ = H,• 2gΣ̄ − 2 + N > 0,
• Σ admits a Kähler-Einstein metric with negative scalar curvature, or• KΣ̄ [D] (D = {a1 , . . . , aN }) is ample,then: Γ = π1 (Σ) is Fuchsian, first kind, geometrically finite, with no elliptic
element.
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7 / 17II-a) Setting
An arithmetic class of examples. —Easy case: the principal congruence subgroup of level 2
Γ = Γ̄(2) = ker{Psl(2, Z) → Sl(2, Z/2Z)};
then as Riemann surfaces, Γ̄(2)\H = P1 \ {0, 1, ∞}.

In this context, KΣ̄ [D] is ample, and (the formal square root) of(KΣ̄ [D]|Σ , π∗ ωH ⊗ hD ) verifies (α) and (β) —here, ωH descends to Σ, and hD is defined on Σ by: |σD |2hD ≡ 1 for someσD ∈ O([D]) such that D = {σD = 0}.

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II-Punctured Riemann surfacesb) Application of Theorem 0
Assume (Σ, ωΣ , L, h) verify (α) and (β); then, for p ≥ 2,0Σ, Lp|Σ ,→ H 0 Σ̄, Lp ,H(2)and more precisely, by Skoda’s theorem: 0Σ, Lp|Σ ' σ ∈ H 0 Σ̄, Lp σ(aj ) = 0, j = 0, . . . , N ;H(2)0in particular, H(2)(Σ, Lp|Σ ) is of finite dimension, denoted by dp .
Thus:Pdp(p)1- as BpΣ (x ) = j =1|σj (x )|2h p , for any fixed p,BpΣ (x ) → 0

asx → D;

2- whereas for all m ≥ 1 and all compact subsets K of Σ,2π Σ
B (x ) − 1 m→0p p
C (K )

asp→∞by Theorem 0.
What happens in the transition region? How to describe it?
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II-Punctured Riemann surfacesc) Results

First, a localization result (comparison with the model D∗ ):Theorem 1For any m ≥ 0, ` ≥ 0 and δ > 0, there exists Q = Q(m, δ) such that for allp  1,∀z ∈ V1∗ ∪ . . . ∪ VN∗ ,log(|z |2 )δ∗BpΣ (z ) − BpD (z ) C m (ωΣ ) ≤ Qp −` ,

∗where BpD is computed from the data D∗ , ωD∗ , C, log(|z |2 ) | · | .

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10 / 17II-c) Results

Then, from Theorems 0, 1, and an explicit computation on the model D∗ , one
can, among others, estimate precisely the distorsion factor :Corollary 2For p  1,sup0x ∈Σ, σ∈H(2),p p 3/2|σ(x )|2h p+ O(p).
=supB(x)=p22πx ∈Σ\{0} kσkL2

In the arithmetic situation evoked above, for non-cocompact Γ, this translates as: p 3/2(2Imz )2p |f (z )|2sup=+ O(p),2kf kPetπ
Γ \{0}z ∈H, f ∈S2p
Γ
where S2p
is the space of cusp modular forms (Spitzenformen) of weight 2p.

Remarks: B If Γ were cocompact, the sup above would be πp + O(1).
B In the line of results by Abbes-Ullmo, Michel-Ullmo,
Friedman-Jorgenson-Kramer.
B Version with Γ admitting elliptic elements.
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II-c) Results

With a 2-variable version of Theorem 1, one can moreover get the following sharp
asymptotics for BpΣ :Corollary 3
With the same notations as above, for and ` ≥ 0, there exists C = C (`) such thatfor all p  1,BpΣ
sup− 1 ≤ Cp −` .
D∗z ∈V1∗ ∪...∪VN∗ Bp∗(sharp indeed: we’ll soon see that BpD can take extremely small values).

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III-Proofsa) Corollary 2∗

By Theorems 0 and 1, enough to establish the same result for BpD (close to0 ∈ D). Observe that {z ` }`≥1 is a complete orthogonal family ofp0H(2)D∗ , ωD∗ , C, log(|z |2 ) | · | ; direct computations then lead to:p∗

BpD (z ) =∞log(|z |2 ) X p−1 2``|z | .
2π(p − 1)!
`=1This is explicit enough to:i) confirm the convergence given by Theorem 0, even near ∂D, and withexponential rate; e.g. on annuli {a ≤ |z | < 1} (a ∈ (0, 1)),
BpΣ (x ) −p−1= O(e −cp )2π C m ({a≤|z |<1})

for some c = c(a) > 0;∗∗Dii) analyze BpD up to 0: setting x = |z |2/p and fp (x ) = Bp+1(z ), one gets: 2π 3/2
pHugues Auvray

fp =∞X1[Gaussian functions centered at e −1/` , of height ].
``=1Bergman kernels on punctured Riemann surfaces

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III-a) Corollary 2ii)

The scaled functions

2π 3/2
fp on (0, 1)
pp 3/2
From this, we infer sup[0,1] fp = 2π+ O(p), and this sup is reached nearx = e −1 (which corresponds to |z | = e −p/2 ).
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III-a) Corollary 2
For the translation to modular forms, recall:
I the definition of the space of modular forms of weight 2p:
MΓ2p = f ∈ O(H) ∀γ = ac db , f (γ · z ) = (cz + d )2p f (z ) ;I Mumford’s isomorphism:∼Φ : MΓ2p −→ H 0 Σ̄, L2p7−→ f (dz )⊗pf

,which restricts to an isometry∼Γ
0−→ H(2)Σ, L2pS2pΓ
where S2p
= {f ∈ MΓ2p | (Φf )(aj ) = 0, j = 1, . . . , N } is endowed withPetersson’s inner product:Z
hf , giPet =f (z )g(z )(2y)2p d volH (z ).
fdmtl dmn


Hugues AuvrayBergman kernels on punctured Riemann surfaces

15 / 17III-Proofsb) Theorem 1I Relies on Ma-Marinescu’s technology, inspired by Bismut-Lebeau, andcentered at the singularity!
I Based on:i) finite propagation speed for the wave equations for Kodaira Laplacians;ii) spectral gap for the Kodaira Laplacians.

I First get the estimate∀z ∈ V1∗ ∪ . . . ∪ VN∗ ,log(|z |2 )δ∗BpΣ (z ) − BpD (z ) C m (ωΣ ) ≤ Qp −` ,but with δ < − 21 !
I Then improve to δ > 0 with help of the holomorphicity of the sections.

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III-Proofsc) Corollary 3
Assume N = 1 for simplicity, and then:(p)

I truncate the orthonormal family {c` z ` }1≤`≤δp (δp ≤ dp ) far from 0, anduse the trivialization near a1 to see it as an orthogonal family on Σ;
I correct it into an orthonormal family, and complete it into an orthonormal(p)0(Σ, Lp );basis (σ` )1≤`≤dp of H(2)I carefully compare
BpΣ (z ) =2log(|z |)

dppX

(p)

|σ` |2h p0∗

and BpD (z ) = log(|z 2 |)∞pX(p)(c` )2 |z |2``=1`=1on a punctured disc of shape {0 < |z | ≤ cp −A } (the estimate on the annulus{cp −A < |z | < r } follows at once from Theorem 1 and refined analysis on∗BpD );I for the comparison, adjust δp (linear in p) and A, and use a localization
result on BpΣ (analogous to Theorem 1 for BpΣ ) to estimate the error terms
(p)

σ`(p)

− c` z ` , ` ∈ {1, . . . , δp }.

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