Bergman kernels on punctured Riemann surfaces Hugues Auvray joint - PowerPoint PPT Presentation

Bergman kernels on punctured Riemann surfaces Hugues Auvray — joint work with X. Ma and G. Marinescu — December 16, 2019 2019 Taipei Conference on Complex Geometry Hugues Auvray Bergman kernels on punctured Riemann surfaces 1 / 17

Content Bergman kernels on complete manifolds 1 Landscape General results Punctured Riemann surfaces 2 Setting Application of Theorem 0 Results Proofs 3 Corollary 4 Theorem 1 Hugues Auvray Bergman kernels on punctured Riemann surfaces 2 / 17

I-Bergman kernels on complete manifolds a) Landscape ◮ Start with an hermitian holomorphic line bundle ( L , h ) over a complete ahler manifold ( X n , ω X ) K¨ ( h might not polarize ω X ). ◮ Consider, for p ≥ 1 , the Hilbert space L p σ ∈ L 2 ( X , L p ) � ∂ H 0 (2) ( X , L p ) = � � � σ = 0 (here and below, L p is a shortcut for ( L ⊗ p , h p ) ). It might be of infinite dimension when X is non-compact. ◮ To these data, associate the Bergman kernels ( y ) ∗ ∈ L p s ( p ) ( x ) ⊗ s ( p ) � x ⊗ ( L p y ) ∗ B p : ( x , y ) �− → ℓ ℓ ℓ ≥ 0 for some (any) orthonormal basis ( s ( p ) ) ℓ ≥ 0 of H 0 (2) ( X , L p ) . More particularly, ℓ ℓ ≥ 0 | s ( p ) ( x ) | 2 look at the density functions B p ( x ) = B p ( x , x ) = � h p ≥ 0 . ℓ | σ ( x ) | 2 h p ◮ Alternatively: B p ( x ) = sup . � σ � 2 σ ∈ H 0 (2) , p ,σ � =0 L 2 Hugues Auvray Bergman kernels on punctured Riemann surfaces 3 / 17

I-Bergman kernels on complete manifolds b) B p asymptotics: general results Theorem 0 (Ma-Marinescu, 2007) With previous notations, assume that: i) (” uniform ampleness” ) there exists ε > 0 such that: iR h = loc − i ∂∂ log( | σ | 2 h ) ≥ εω X on X ; ii) (” bounded geometry” ) Ric( ω X ) ≥ − C ω X on X , for some C ≥ 0 . Then: for all j ≥ 0 , there exists b j ∈ C ∞ ( X ) such that: ∀ K ⋐ X , ∀ k , m ≥ 0 , ∃ Q = Q ( K , k , m , ε, C , n ) , ∀ p ≥ 1 , k � b j p − j � � p − n B p ( x ) − � C m ( K ) ≤ Qp − k − 1 . � � � j =0 More precisely, b 0 = ω n 2 π R h ) and i X (with ω h = h ω n b 1 = b 0 � � scal( ω h ) − 2∆ ω h log b 0 . 8 π Hugues Auvray Bergman kernels on punctured Riemann surfaces 4 / 17

I-b) B p asymptotics: general results A few remarks: ⊲ Long history; many names associated to this result: Tian (1990, k = 0 , m = 2 ), Bouche (1990), Catlin-Zelditch (1999-98, compact X ), ... ⊲ Quantization of Kodaira embedding theorem / scalar curvature in K¨ ahler geometry. ⊲ The proof requires two steps: 1- localization on B p ; 2- computations of the asymptotics with geometric data brought to C n (scaling techniques). ⊲ This statement does not say what happens to the Bergman density functions on neighbourhoods of infinity... Hugues Auvray Bergman kernels on punctured Riemann surfaces 5 / 17

II-Punctured Riemann surfaces a) Setting ” The most elementary class of complete non-compact K¨ ahler manifolds.” ◮ Take: • Σ = ¯ Σ � D , where D = { a 1 , . . . , a N } is the puncture divisor inside a compact Riemann surface ¯ Σ , and ω Σ a smooth K¨ ahler form on Σ ; • an hermitian line bundle ( L | Σ , h ) , with L holomorphic on ¯ Σ . ◮ Suppose moreover that there are trivializations ∼ − − → C z j × D r L | V j ( 0 < r < 1 ) around the a j ’s, such that: ( α ) | 1 | 2 � � log( | z j | 2 ) � h ( z j ) = � ; ( β ) i ( R h ) | j = ω Σ | j . V ∗ V ∗ In particular, ω Σ = ω D ∗ ( z j ) on V ∗ j , idz ∧ d ¯ z e metric on D ∗ ). where ω D ∗ = | z | 2 log 2 ( | z | 2 ) (Poincar´ Hugues Auvray Bergman kernels on punctured Riemann surfaces 6 / 17

II-a) Setting An arithmetic class of examples. — These (notably, properties ( α ) and ( β ) ) are natural hypotheses, as revealed by the following class of examples. If Γ ⊂ Psl(2 , R ) is a Fuchsian group of the first kind, which is geometrically finite and contains no elliptic element, then Σ = Γ \ H can be compactified by adjunction of finitely many points. Conversely, if Σ = ¯ Σ � { a 1 , . . . , a N } is such that (equivalently): • ˜ Σ = H , • 2 g ¯ Σ − 2 + N > 0 , • Σ admits a K¨ ahler-Einstein metric with negative scalar curvature, or • K ¯ Σ [ D ] ( D = { a 1 , . . . , a N } ) is ample, then: Γ = π 1 (Σ) is Fuchsian, first kind, geometrically finite, with no elliptic element. Hugues Auvray Bergman kernels on punctured Riemann surfaces 7 / 17

II-a) Setting An arithmetic class of examples. — Easy case: the principal congruence subgroup of level 2 Γ = ¯ Γ(2) = ker { Psl(2 , Z ) → Sl(2 , Z / 2 Z ) } ; Γ(2) \ H = P 1 � { 0 , 1 , ∞} . then as Riemann surfaces, ¯ In this context, K ¯ Σ [ D ] is ample, and (the formal square root) of ( K ¯ Σ [ D ] | Σ , π ∗ ω H ⊗ h D ) verifies ( α ) and ( β ) — here, ω H descends to Σ , and h D is defined on Σ by: | σ D | 2 h D ≡ 1 for some σ D ∈ O ([ D ]) such that D = { σ D = 0 } . Hugues Auvray Bergman kernels on punctured Riemann surfaces 8 / 17

II-Punctured Riemann surfaces b) Application of Theorem 0 Assume (Σ , ω Σ , L , h ) verify ( α ) and ( β ) ; then, for p ≥ 2 , H 0 Σ , L p → H 0 � ¯ Σ , L p � � � ֒ , (2) | Σ and more precisely, by Skoda’s theorem: H 0 Σ , L p σ ∈ H 0 � ¯ � σ ( a j ) = 0 , j = 0 , . . . , N � � � Σ , L p � � � ≃ ; (2) | Σ (2) (Σ , L p in particular, H 0 | Σ ) is of finite dimension, denoted by d p . Thus: p ( x ) = � d p j =1 | σ ( p ) 1- as B Σ ( x ) | 2 h p , for any fixed p , j B Σ p ( x ) → 0 as x → D ; 2- whereas for all m ≥ 1 and all compact subsets K of Σ , � 2 π � � p B Σ p ( x ) − 1 C m ( K ) → 0 as p → ∞ � � � by Theorem 0. What happens in the transition region? How to describe it? Hugues Auvray Bergman kernels on punctured Riemann surfaces 9 / 17

II-Punctured Riemann surfaces c) Results First, two localization results (comparison with the model D ∗ ): Theorem 1 For any m ≥ 0 , ℓ ≥ 0 and δ > 0 , there exists Q = Q ( m , ℓ, δ ) such that for all p ≫ 1 , � δ � p ( z ) − B D ∗ ∀ z ∈ V ∗ 1 ∪ . . . ∪ V ∗ � � log( | z | 2 ) � � B Σ � C m ( ω Σ ) ≤ Qp − ℓ , N , p ( z ) � where B D ∗ � D ∗ , ω D ∗ , C , � log( | z | 2 ) � � � is computed from the data � | · | . p Hugues Auvray Bergman kernels on punctured Riemann surfaces 10 / 17

And more recently: Theorem 2 The quotient B Σ p / B D ∗ can be extended smoothly through the origin, and, for any p m ≥ 0 and ℓ ≥ 0 , there exists Q = Q ( m ) such that for all p ≫ 1 , � B Σ � �� p � ≤ Qp − ℓ � � � D 1 · · · D m − 1 � � B D ∗ p ∂ ∂ where each D j represents ∂ z or z . ∂ ¯ Which tranlstes geometrically as: Theorem 3 Fix a neighbourhood of coordinate z around a puncture of Σ such that conditions ( α ) and ( β ) are verified. Then the difference of the pull-backs of the Fubini-Study metrics by the embeddings respectively induced by orthonormal bases of (2) (Σ , L p | Σ ) and H 0 , p H 0 (2) ( D ∗ ) can be written as η p idz ∧ dz , with: D 1 · · · D m η p = O ( p −∞ ) for all m ≥ 0 . Hugues Auvray Bergman kernels on punctured Riemann surfaces 11 / 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 4 For p ≫ 1 , � p | σ ( x ) | 2 � 3 / 2 h p sup = sup B p ( x ) = + O ( p ) . � σ � 2 2 π x ∈ Σ , σ ∈ H 0 x ∈ Σ (2) , p � { 0 } L 2 In the arithmetic situation evoked above, for non-cocompact Γ , this translates as: (2Im z ) 2 p | f ( z ) | 2 � p � 3 / 2 sup = + O ( p ) , � f � 2 π z ∈ H , f ∈S Γ 2 p � { 0 } Pet where S Γ 2 p is the space of cusp modular forms (Spitzenformen) of weight 2 p . Remarks: ⊲ If Γ were cocompact, the sup above would be p π + O (1) . ⊲ In the line of results by Abbes-Ullmo, Michel-Ullmo, Friedman-Jorgenson-Kramer. ⊲ Version with Γ admitting elliptic elements. Hugues Auvray Bergman kernels on punctured Riemann surfaces 12 / 17

III-Proofs a) Corollary 4 By Theorems 0 and 1, enough to establish the same result for B D ∗ (close to p 0 ∈ D ). Observe that { z ℓ } ℓ ≥ 1 is a complete orthogonal family of � p | · | H 0 D ∗ , ω D ∗ , C , � � log( | z | 2 ) � � � ; direct computations then lead to: (2) � p ∞ � � log( | z | 2 ) � B D ∗ � ℓ p − 1 | z | 2 ℓ . p ( z ) = 2 π ( p − 1)! ℓ =1 This is explicit enough to: i) confirm the convergence given by Theorem 0, even near ∂ D , and with exponential rate; e.g. on annuli { a ≤ | z | < 1 } ( a ∈ (0 , 1) ), p ( x ) − p − 1 � � � B Σ C m ( { a ≤| z | < 1 } ) = O ( e − cp ) for some c = c ( a ) > 0; � � 2 π � up to 0 : setting x = | z | 2 / p and f p ( x ) = B D ∗ ii) analyze B D ∗ p +1 ( z ) , one gets: p ∞ � 2 π � 3 / 2 [ Gaussian functions centered at e − 1 /ℓ , of height 1 � f p = ℓ ] . p ℓ =1 Hugues Auvray Bergman kernels on punctured Riemann surfaces 13 / 17

III-a) Corollary 4 ii) � 2 π � 3 / 2 f p on (0 , 1) The scaled functions p � p � 3 / 2 + O ( p ) , and this sup is reached near From this, we infer sup [0 , 1] f p = 2 π x = e − 1 (which corresponds to | z | = e − p / 2 ). Hugues Auvray Bergman kernels on punctured Riemann surfaces 14 / 17