On the approximate cohomology of quasi holomorphic line bundles - PowerPoint PPT Presentation

Spectrum of the Laplace-Beltrami operator ∗ ∗ Let � k = ∂ k ∂ k + ∂ k ∂ k be the complex Laplace-Beltrami operator of ( L k , h k , ∇ k ) with respect to some Hermitian metric ω on X . p , q Let � k , E the operator acting on C ∞ ( X , Λ p , q T ∗ X ⊗ L k ⊗ E ), where ( E , h E ) is a holomorphic Hermitian vector bundle of rank r . We are interested in analyzing the (discrete) spectrum of the p , q elliptic operator � k , E . Since the curvature is θ k ≃ ku , it is better to p , q 1 2 π k � renormalize and to consider instead k , E . For λ ∈ R , we define � 1 p , q N p , q 2 π k � k ( λ ) = dim eigenspaces of k , E of eigenvalues ≤ λ. Let u j ( x ), 1 ≤ j ≤ n , be the eigenvalues of u ( x ) with respect to ω ( x ) at any point x ∈ X , ordered so that if s = rank( u ( x )), then | u 1 ( x ) | ≥ ··· ≥ | u s ( x ) | > | u s +1 ( x ) | = ··· = | u n ( x ) | = 0. For a multi-index J = { j 1 < j 2 < . . . < j q } ⊂ { 1 , . . . , n } , set � u J ( x ) = j ∈ J u j ( x ), x ∈ X . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 4/28

Fundamental spectral theory results Consider the “spectral density functions” ν u , ν u defined by � � � n − s = 2 s − n | u 1 | ··· | u s | � � ν u ( λ ) λ − (2 p j + 1) | u j | + . ν u ( λ ) Γ( n − s + 1) ( p 1 ,..., p s ) ∈ N s (where 0 0 = 0 for ν u , resp. 0 0 = 1 for ν u ). J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 5/28

Fundamental spectral theory results Consider the “spectral density functions” ν u , ν u defined by � � � n − s = 2 s − n | u 1 | ··· | u s | � � ν u ( λ ) λ − (2 p j + 1) | u j | + . ν u ( λ ) Γ( n − s + 1) ( p 1 ,..., p s ) ∈ N s (where 0 0 = 0 for ν u , resp. 0 0 = 1 for ν u ). Theorem ([D] 1985) p , q 2 π k � 1 on C ∞ ( X , Λ p , q T ∗ The spectrum of X ⊗ L k ⊗ E ) has an k asymptotic distribution of eigenvalues such that ∀ λ ∈ R � n � � � k → + ∞ k − n N p , q r ν u (2 λ + u ∁ J − u J ) dV ω ≤ lim inf k ( λ ) ≤ p X | J | = q � n � � � k − n N p , q ≤ lim sup k ( λ ) ≤ r ν u (2 λ + u ∁ J − u J ) dV ω p k → + ∞ X | J | = q where r = rank( E ). J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 5/28

Fundamental spectral theory results Consider the “spectral density functions” ν u , ν u defined by � � � n − s = 2 s − n | u 1 | ··· | u s | � � ν u ( λ ) λ − (2 p j + 1) | u j | + . ν u ( λ ) Γ( n − s + 1) ( p 1 ,..., p s ) ∈ N s (where 0 0 = 0 for ν u , resp. 0 0 = 1 for ν u ). Theorem ([D] 1985) p , q 2 π k � 1 on C ∞ ( X , Λ p , q T ∗ The spectrum of X ⊗ L k ⊗ E ) has an k asymptotic distribution of eigenvalues such that ∀ λ ∈ R � n � � � k → + ∞ k − n N p , q r ν u (2 λ + u ∁ J − u J ) dV ω ≤ lim inf k ( λ ) ≤ p X | J | = q � n � � � k − n N p , q ≤ lim sup k ( λ ) ≤ r ν u (2 λ + u ∁ J − u J ) dV ω p k → + ∞ X | J | = q where r = rank( E ). By monotonicity, as ν u ( λ ) = lim λ → 0 + ν u ( λ ), all four terms are equal for λ ∈ R � D with D countable. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 5/28

Approximate cohomology lower bounds Proof. One first estimates the spectrum of the total Laplacian ∆ k , E = ∇ k , E ∇ ∗ k , E + ∇ ∗ k , E ∇ k , E (harmonic oscillator with magnetic and electric fields), J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 6/28

Approximate cohomology lower bounds Proof. One first estimates the spectrum of the total Laplacian ∆ k , E = ∇ k , E ∇ ∗ k , E + ∇ ∗ k , E ∇ k , E (harmonic oscillator with magnetic and electric fields), and then one uses a Bochner formula to relate � k , E and ∆ k , E ( � k , E ≃ 1 2 ∆ k , E + curvature terms) for each ( p , q ). J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 6/28

Approximate cohomology lower bounds Proof. One first estimates the spectrum of the total Laplacian ∆ k , E = ∇ k , E ∇ ∗ k , E + ∇ ∗ k , E ∇ k , E (harmonic oscillator with magnetic and electric fields), and then one uses a Bochner formula to relate � k , E and ∆ k , E ( � k , E ≃ 1 2 ∆ k , E + curvature terms) for each ( p , q ). Important special case λ = 0 (harmonic forms) � ν u ( u ∁ J − u J ) dV ω = ( − 1) q u n n ! . | J | = q J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 6/28

Approximate cohomology lower bounds Proof. One first estimates the spectrum of the total Laplacian ∆ k , E = ∇ k , E ∇ ∗ k , E + ∇ ∗ k , E ∇ k , E (harmonic oscillator with magnetic and electric fields), and then one uses a Bochner formula to relate � k , E and ∆ k , E ( � k , E ≃ 1 2 ∆ k , E + curvature terms) for each ( p , q ). Important special case λ = 0 (harmonic forms) � ν u ( u ∁ J − u J ) dV ω = ( − 1) q u n n ! . | J | = q Corollary (Laurent laeng, 2002) For λ k → 0 slowly enough, i.e. with k 2+2 / b 2 λ k → + ∞ , one has � � � � k , E ( λ k ) ≥ r u n + k → + ∞ k − n N 0 , 0 u n lim inf where n ! X ( u , 0) X ( u , 1) � � X ( u , q ) = q -index set = x ∈ X / u ( x ) has signature ( n − q , q ) . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 6/28

Proof of the lower bound Proof. One uses the fact that for δ ′ > δ > 0 and k ≫ 1, the composition Π ◦ ∂ k with an eigenspace projection yields an injection � � eigenspace 0 , 0 eigenspace 0 , 1 ֒ → λ . λ λ ∈ ] λ k ,δ ] λ ∈ ]0 ,δ ′ ] J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 7/28

Proof of the lower bound Proof. One uses the fact that for δ ′ > δ > 0 and k ≫ 1, the composition Π ◦ ∂ k with an eigenspace projection yields an injection � � eigenspace 0 , 0 eigenspace 0 , 1 ֒ → λ . λ λ ∈ ] λ k ,δ ] λ ∈ ]0 ,δ ′ ] 2 0 , 0 0 , 1 k = 0 implies ∂ k � = � In fact, in the holomorphic case ∂ k ∂ k , k hence ∂ k maps the (0 , 0)-eigenspaces to the (0 , 1)-eigenspaces for the same eigenvalues, and one can even take λ k = 0, δ ′ = δ . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 7/28

Proof of the lower bound Proof. One uses the fact that for δ ′ > δ > 0 and k ≫ 1, the composition Π ◦ ∂ k with an eigenspace projection yields an injection � � eigenspace 0 , 0 eigenspace 0 , 1 ֒ → λ . λ λ ∈ ] λ k ,δ ] λ ∈ ]0 ,δ ′ ] 2 0 , 0 0 , 1 k = 0 implies ∂ k � = � In fact, in the holomorphic case ∂ k ∂ k , k hence ∂ k maps the (0 , 0)-eigenspaces to the (0 , 1)-eigenspaces for the same eigenvalues, and one can even take λ k = 0, δ ′ = δ . 2 k = O ( k − 1 / b 2 ), one can show that In the quasi holomorphic case ∂ 0 , 1 0 , 0 ∗ 2 � k ∂ k − ∂ k � = ∂ k ∂ k yields a small “deviation” of the eigenvalues k to [ λ k − ε, δ + ε ] with ε < min( λ k , δ ′ − δ ), whence the injectivity. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 7/28

Proof of the lower bound Proof. One uses the fact that for δ ′ > δ > 0 and k ≫ 1, the composition Π ◦ ∂ k with an eigenspace projection yields an injection � � eigenspace 0 , 0 eigenspace 0 , 1 ֒ → λ . λ λ ∈ ] λ k ,δ ] λ ∈ ]0 ,δ ′ ] 2 0 , 0 0 , 1 k = 0 implies ∂ k � = � In fact, in the holomorphic case ∂ k ∂ k , k hence ∂ k maps the (0 , 0)-eigenspaces to the (0 , 1)-eigenspaces for the same eigenvalues, and one can even take λ k = 0, δ ′ = δ . 2 k = O ( k − 1 / b 2 ), one can show that In the quasi holomorphic case ∂ 0 , 1 0 , 0 ∗ 2 � k ∂ k − ∂ k � = ∂ k ∂ k yields a small “deviation” of the eigenvalues k to [ λ k − ε, δ + ε ] with ε < min( λ k , δ ′ − δ ), whence the injectivity. This implies N 0 , 1 k , E ( δ ′ ) ≥ N 0 , 0 k , E ( δ ) − N 0 , 0 k , E ( λ k ) thus N 0 , 0 k , E ( λ k ) ≥ N 0 , 0 k , E ( δ ) − N 0 , 1 k , E ( δ ′ ), QED J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 7/28

Transcendental holomorphic Morse inequalities Conjecture on Morse inequalities Let γ ∈ H 1 , 1 BC ( X , R ). Then � u n . Vol ( γ ) ≥ sup u ∈ γ, u ∈ C ∞ X ( u , ≤ 1) (One could even suspect equality, an even stronger conjecture !). J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 8/28

Transcendental holomorphic Morse inequalities Conjecture on Morse inequalities Let γ ∈ H 1 , 1 BC ( X , R ). Then � u n . Vol ( γ ) ≥ sup u ∈ γ, u ∈ C ∞ X ( u , ≤ 1) (One could even suspect equality, an even stronger conjecture !). If one sets by definition k → + ∞ N 0 , 0 Vol ( γ ) = sup lim lim inf k ( λ ) λ → 0 + u ∈ γ for the eigenspaces of the sequence ( L k , h k , ∇ k ) approximating ku , then the above expected lower bound is a theorem! J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 8/28

Transcendental holomorphic Morse inequalities Conjecture on Morse inequalities Let γ ∈ H 1 , 1 BC ( X , R ). Then � u n . Vol ( γ ) ≥ sup u ∈ γ, u ∈ C ∞ X ( u , ≤ 1) (One could even suspect equality, an even stronger conjecture !). If one sets by definition k → + ∞ N 0 , 0 Vol ( γ ) = sup lim lim inf k ( λ ) λ → 0 + u ∈ γ for the eigenspaces of the sequence ( L k , h k , ∇ k ) approximating ku , then the above expected lower bound is a theorem! There is however a stronger & more usual definition of the volume. Definition For γ ∈ H 1 , 1 BC ( X , R ), set Vol ( γ ) = 0 if γ �∋ any current T ≥ 0, � T n ac , u 0 ∈ C ∞ . and otherwise set Vol ( γ ) = sup T ∈ γ, T = u 0 + i ∂∂ϕ ≥ 0 X J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 8/28

Transcendental holomorphic Morse inequalities (2) The conjecture on Morse inequalities is known to be true when γ = c 1 ( L ) is an integral class ([D-1985]). In fact, one then gets a Hermitian holomorphic line bundle ( L , h ) and its multiples L ⊗ k . The spectral estimates provide many holomorphic sections σ k ,ℓ , and one gets positive currents right away by putting � i h + i | σ k ,ℓ | 2 T k = 2 k π∂∂ log 2 π Θ L , h ≥ 0 ℓ J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 9/28

Transcendental holomorphic Morse inequalities (2) The conjecture on Morse inequalities is known to be true when γ = c 1 ( L ) is an integral class ([D-1985]). In fact, one then gets a Hermitian holomorphic line bundle ( L , h ) and its multiples L ⊗ k . The spectral estimates provide many holomorphic sections σ k ,ℓ , and one gets positive currents right away by putting � i h + i | σ k ,ℓ | 2 T k = 2 k π∂∂ log 2 π Θ L , h ≥ 0 ℓ (the volume estimate can be derived from there by Fujita). J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 9/28

Transcendental holomorphic Morse inequalities (2) The conjecture on Morse inequalities is known to be true when γ = c 1 ( L ) is an integral class ([D-1985]). In fact, one then gets a Hermitian holomorphic line bundle ( L , h ) and its multiples L ⊗ k . The spectral estimates provide many holomorphic sections σ k ,ℓ , and one gets positive currents right away by putting � i h + i | σ k ,ℓ | 2 T k = 2 k π∂∂ log 2 π Θ L , h ≥ 0 ℓ (the volume estimate can be derived from there by Fujita). In the “quasi-holomorphic” case, one only gets eigenfunctions σ k ,ℓ with small eigenvalues, and the positivity of T k is a priori lost. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 9/28

Transcendental holomorphic Morse inequalities (2) The conjecture on Morse inequalities is known to be true when γ = c 1 ( L ) is an integral class ([D-1985]). In fact, one then gets a Hermitian holomorphic line bundle ( L , h ) and its multiples L ⊗ k . The spectral estimates provide many holomorphic sections σ k ,ℓ , and one gets positive currents right away by putting � i h + i | σ k ,ℓ | 2 T k = 2 k π∂∂ log 2 π Θ L , h ≥ 0 ℓ (the volume estimate can be derived from there by Fujita). In the “quasi-holomorphic” case, one only gets eigenfunctions σ k ,ℓ with small eigenvalues, and the positivity of T k is a priori lost. Conjectural corollary (fundamental volume estimate) ahler, dim X = n , and α, β ∈ H 1 , 1 ( X , R ) be Let X be compact K¨ nef cohomology classes. Then Vol ( α − β ) ≥ α n − n α n − 1 · β . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 9/28

Known results on holomorphic Morse inequalities The conjectural corollary is derived from the main conjecture by an elementary symmetric function argument. In fact, one has a pointwise inequality of forms 1 X ( α − β, ≤ 1) ( α − β ) n ≥ α n − n α n − 1 · β . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 10/28

Known results on holomorphic Morse inequalities The conjectural corollary is derived from the main conjecture by an elementary symmetric function argument. In fact, one has a pointwise inequality of forms 1 X ( α − β, ≤ 1) ( α − β ) n ≥ α n − n α n − 1 · β . Again, the corollary is known for γ = α − β when α, β are integral classes (by [D-1993] and independently [Trapani, 1993]). J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 10/28

Known results on holomorphic Morse inequalities The conjectural corollary is derived from the main conjecture by an elementary symmetric function argument. In fact, one has a pointwise inequality of forms 1 X ( α − β, ≤ 1) ( α − β ) n ≥ α n − n α n − 1 · β . Again, the corollary is known for γ = α − β when α, β are integral classes (by [D-1993] and independently [Trapani, 1993]). Recently (2016), the volume estimate for γ = α − β transcendental has been established by D. Witt-Nystr¨ om when X is projective, using deep facts on Monge-Amp` ere operators and upper envelopes. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 10/28

Known results on holomorphic Morse inequalities The conjectural corollary is derived from the main conjecture by an elementary symmetric function argument. In fact, one has a pointwise inequality of forms 1 X ( α − β, ≤ 1) ( α − β ) n ≥ α n − n α n − 1 · β . Again, the corollary is known for γ = α − β when α, β are integral classes (by [D-1993] and independently [Trapani, 1993]). Recently (2016), the volume estimate for γ = α − β transcendental has been established by D. Witt-Nystr¨ om when X is projective, using deep facts on Monge-Amp` ere operators and upper envelopes. Xiao and Popovici also proved in the K¨ ahler case that α n − n α n − 1 · β > 0 ⇒ Vol ( α − β ) > 0 and α − β contains a K¨ ahler current. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 10/28

Known results on holomorphic Morse inequalities The conjectural corollary is derived from the main conjecture by an elementary symmetric function argument. In fact, one has a pointwise inequality of forms 1 X ( α − β, ≤ 1) ( α − β ) n ≥ α n − n α n − 1 · β . Again, the corollary is known for γ = α − β when α, β are integral classes (by [D-1993] and independently [Trapani, 1993]). Recently (2016), the volume estimate for γ = α − β transcendental has been established by D. Witt-Nystr¨ om when X is projective, using deep facts on Monge-Amp` ere operators and upper envelopes. Xiao and Popovici also proved in the K¨ ahler case that α n − n α n − 1 · β > 0 ⇒ Vol ( α − β ) > 0 and α − β contains a K¨ ahler current. (The proof is short, once the Calabi-Yau theorem is taken for granted). J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 10/28

Projective vs K¨ ahler vs non K¨ ahler varieties Problem. Investigate positivity for general compact manifolds/ C . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 11/28

Projective vs K¨ ahler vs non K¨ ahler varieties Problem. Investigate positivity for general compact manifolds/ C . Obviously, non projective varieties do not carry any ample line bundle. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 11/28

Projective vs K¨ ahler vs non K¨ ahler varieties Problem. Investigate positivity for general compact manifolds/ C . Obviously, non projective varieties do not carry any ample line bundle. ahler class { ω } ∈ H 1 , 1 ( X , R ), ω > 0, may In the K¨ ahler case, a K¨ sometimes be used as a substitute for a polarization. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 11/28

Projective vs K¨ ahler vs non K¨ ahler varieties Problem. Investigate positivity for general compact manifolds/ C . Obviously, non projective varieties do not carry any ample line bundle. ahler class { ω } ∈ H 1 , 1 ( X , R ), ω > 0, may In the K¨ ahler case, a K¨ sometimes be used as a substitute for a polarization. What for non K¨ ahler compact complex manifolds? J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 11/28

Projective vs K¨ ahler vs non K¨ ahler varieties Problem. Investigate positivity for general compact manifolds/ C . Obviously, non projective varieties do not carry any ample line bundle. ahler class { ω } ∈ H 1 , 1 ( X , R ), ω > 0, may In the K¨ ahler case, a K¨ sometimes be used as a substitute for a polarization. What for non K¨ ahler compact complex manifolds? Surprising facts (?) – Every compact complex manifold X carries a “very ample” complex Hilbert bundle, produced by means of a natural Bergman space construction. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 11/28

Projective vs K¨ ahler vs non K¨ ahler varieties Problem. Investigate positivity for general compact manifolds/ C . Obviously, non projective varieties do not carry any ample line bundle. ahler class { ω } ∈ H 1 , 1 ( X , R ), ω > 0, may In the K¨ ahler case, a K¨ sometimes be used as a substitute for a polarization. What for non K¨ ahler compact complex manifolds? Surprising facts (?) – Every compact complex manifold X carries a “very ample” complex Hilbert bundle, produced by means of a natural Bergman space construction. – The curvature of this bundle is strongly positive in the sense of Nakano, and is given by a universal formula. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 11/28

Projective vs K¨ ahler vs non K¨ ahler varieties Problem. Investigate positivity for general compact manifolds/ C . Obviously, non projective varieties do not carry any ample line bundle. ahler class { ω } ∈ H 1 , 1 ( X , R ), ω > 0, may In the K¨ ahler case, a K¨ sometimes be used as a substitute for a polarization. What for non K¨ ahler compact complex manifolds? Surprising facts (?) – Every compact complex manifold X carries a “very ample” complex Hilbert bundle, produced by means of a natural Bergman space construction. – The curvature of this bundle is strongly positive in the sense of Nakano, and is given by a universal formula. In the sequel of this lecture, we aim to investigate this construction and look for potential applications, especially to transcendental holomorphic Morse inequalities ... J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 11/28

Tubular neighborhoods (thanks to Grauert) Let X be a compact complex manifold, dim C X = n . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 12/28

Tubular neighborhoods (thanks to Grauert) Let X be a compact complex manifold, dim C X = n . Denote by X its complex conjugate ( X , − J ), so that O X = O X . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 12/28

Tubular neighborhoods (thanks to Grauert) Let X be a compact complex manifold, dim C X = n . Denote by X its complex conjugate ( X , − J ), so that O X = O X . The diagonal of X × X is totally real, and by Grauert, we know that it possesses a fundamental system of Stein tubular neighborhoods. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 12/28

Tubular neighborhoods (thanks to Grauert) Let X be a compact complex manifold, dim C X = n . Denote by X its complex conjugate ( X , − J ), so that O X = O X . The diagonal of X × X is totally real, and by Grauert, we know that it possesses a fundamental system of Stein tubular neighborhoods. Assume that X is equipped with a real analytic hermitian metric γ , J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 12/28

Tubular neighborhoods (thanks to Grauert) Let X be a compact complex manifold, dim C X = n . Denote by X its complex conjugate ( X , − J ), so that O X = O X . The diagonal of X × X is totally real, and by Grauert, we know that it possesses a fundamental system of Stein tubular neighborhoods. Assume that X is equipped with a real analytic hermitian metric γ , and let exp : T X → X × X , ( z , ξ ) �→ ( z , exp z ( ξ )), z ∈ X , ξ ∈ T X , z be the associated geodesic exponential map. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 12/28

Exponential map diffeomorphism and its inverse Lemma Denote by exph the “holomorphic” part of exp, so that for z ∈ X and ξ ∈ T X , z � � β , a α β ( z ) ξ α ξ a α 0 ( z ) ξ α . exp z ( ξ ) = exph z ( ξ ) = α,β ∈ N n α ∈ N n J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 13/28

Exponential map diffeomorphism and its inverse Lemma Denote by exph the “holomorphic” part of exp, so that for z ∈ X and ξ ∈ T X , z � � β , a α β ( z ) ξ α ξ a α 0 ( z ) ξ α . exp z ( ξ ) = exph z ( ξ ) = α,β ∈ N n α ∈ N n Then d ξ exp z ( ξ ) ξ =0 = d ξ exph z ( ξ ) ξ =0 = Id T X , and so exph is a diffeomorphism from a neighborhood V of the 0 section of T X to a neighborhood V ′ of the diagonal in X × X . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 13/28

Exponential map diffeomorphism and its inverse Lemma Denote by exph the “holomorphic” part of exp, so that for z ∈ X and ξ ∈ T X , z � � β , a α β ( z ) ξ α ξ a α 0 ( z ) ξ α . exp z ( ξ ) = exph z ( ξ ) = α,β ∈ N n α ∈ N n Then d ξ exp z ( ξ ) ξ =0 = d ξ exph z ( ξ ) ξ =0 = Id T X , and so exph is a diffeomorphism from a neighborhood V of the 0 section of T X to a neighborhood V ′ of the diagonal in X × X . Notation With the identification X ≃ diff X , let logh : X × X ⊃ V ′ → T X be the inverse diffeomorphism of exph and U ε = { ( z , w ) ∈ V ′ ⊂ X × X ; | logh z ( w ) | γ < ε } , ε > 0 . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 13/28

Exponential map diffeomorphism and its inverse Lemma Denote by exph the “holomorphic” part of exp, so that for z ∈ X and ξ ∈ T X , z � � β , a α β ( z ) ξ α ξ a α 0 ( z ) ξ α . exp z ( ξ ) = exph z ( ξ ) = α,β ∈ N n α ∈ N n Then d ξ exp z ( ξ ) ξ =0 = d ξ exph z ( ξ ) ξ =0 = Id T X , and so exph is a diffeomorphism from a neighborhood V of the 0 section of T X to a neighborhood V ′ of the diagonal in X × X . Notation With the identification X ≃ diff X , let logh : X × X ⊃ V ′ → T X be the inverse diffeomorphism of exph and U ε = { ( z , w ) ∈ V ′ ⊂ X × X ; | logh z ( w ) | γ < ε } , ε > 0 . Then, for ε ≪ 1, U ε is Stein and pr 1 : U ε → X is a real analytic locally trivial bundle with fibers biholomorphic to complex balls. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 13/28

Such tubular neighborhoods are Stein J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 14/28

Such tubular neighborhoods are Stein In the special case X = C n , U ε = { ( z , w ) ∈ C n × C n ; | z − w | < ε } . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 14/28

Such tubular neighborhoods are Stein In the special case X = C n , U ε = { ( z , w ) ∈ C n × C n ; | z − w | < ε } . It is of course Stein since | z − w | 2 = | z | 2 + | w | 2 − 2 Re � z j w j and ( z , w ) �→ Re � z j w j is pluriharmonic. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 14/28

Bergman sheaves Let U ε = U γ,ε ⊂ X × X be the ball bundle as above, and p = ( pr 1 ) | U ε : U ε → X , p = ( pr 2 ) | U ε : U ε → X the natural projections. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 15/28

Bergman sheaves Let U ε = U γ,ε ⊂ X × X be the ball bundle as above, and p = ( pr 1 ) | U ε : U ε → X , p = ( pr 2 ) | U ε : U ε → X the natural projections. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 15/28

Bergman sheaves (continued) Definition of the Bergman sheaf B ε The Bergman sheaf B ε = B γ,ε is by definition the L 2 direct image B ε = p L 2 ∗ ( p ∗ O ( K X )) , J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 16/28

Bergman sheaves (continued) Definition of the Bergman sheaf B ε The Bergman sheaf B ε = B γ,ε is by definition the L 2 direct image B ε = p L 2 ∗ ( p ∗ O ( K X )) , i.e. the space of sections over an open subset V ⊂ X defined by B ε ( V ) = holomorphic sections f of p ∗ O ( K X ) on p − 1 ( V ), f ( z , w ) = f 1 ( z , w ) dw 1 ∧ . . . ∧ dw n , z ∈ V , J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 16/28

Bergman sheaves (continued) Definition of the Bergman sheaf B ε The Bergman sheaf B ε = B γ,ε is by definition the L 2 direct image B ε = p L 2 ∗ ( p ∗ O ( K X )) , i.e. the space of sections over an open subset V ⊂ X defined by B ε ( V ) = holomorphic sections f of p ∗ O ( K X ) on p − 1 ( V ), f ( z , w ) = f 1 ( z , w ) dw 1 ∧ . . . ∧ dw n , z ∈ V , that are in L 2 ( p − 1 ( K )) for all compact subsets K ⋐ V : � i n 2 f ( z , w ) ∧ f ( z , w ) ∧ γ ( z ) n < + ∞ , ∀ K ⋐ V . p − 1 ( K ) (This L 2 condition is the reason we speak of “ L 2 direct image”). J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 16/28

Bergman sheaves (continued) Definition of the Bergman sheaf B ε The Bergman sheaf B ε = B γ,ε is by definition the L 2 direct image B ε = p L 2 ∗ ( p ∗ O ( K X )) , i.e. the space of sections over an open subset V ⊂ X defined by B ε ( V ) = holomorphic sections f of p ∗ O ( K X ) on p − 1 ( V ), f ( z , w ) = f 1 ( z , w ) dw 1 ∧ . . . ∧ dw n , z ∈ V , that are in L 2 ( p − 1 ( K )) for all compact subsets K ⋐ V : � i n 2 f ( z , w ) ∧ f ( z , w ) ∧ γ ( z ) n < + ∞ , ∀ K ⋐ V . p − 1 ( K ) (This L 2 condition is the reason we speak of “ L 2 direct image”). Clearly, B ε is an O X -module over X , but since it is a space of functions in w , it is of infinite rank. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 16/28

Associated Bergman bundle and holom structure Definition of the associated Bergman bundle B ε We consider the vector bundle B ε → X whose fiber B ε, z 0 consists of all holomorphic functions f on p − 1 ( z 0 ) ⊂ U ε such that � � f ( z 0 ) � 2 = i n 2 f ( z 0 , w ) ∧ f ( z 0 , w ) < + ∞ . p − 1 ( z 0 ) J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 17/28

Associated Bergman bundle and holom structure Definition of the associated Bergman bundle B ε We consider the vector bundle B ε → X whose fiber B ε, z 0 consists of all holomorphic functions f on p − 1 ( z 0 ) ⊂ U ε such that � � f ( z 0 ) � 2 = i n 2 f ( z 0 , w ) ∧ f ( z 0 , w ) < + ∞ . p − 1 ( z 0 ) Then B ε is a real analytic locally trivial Hilbert bundle whose fiber B ε, z 0 is isomorphic to the Hardy-Bergman space H 2 ( B (0 , ε )) of L 2 holomorphic n -forms on p − 1 ( z 0 ) ≃ B (0 , ε ) ⊂ C n . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 17/28

Associated Bergman bundle and holom structure Definition of the associated Bergman bundle B ε We consider the vector bundle B ε → X whose fiber B ε, z 0 consists of all holomorphic functions f on p − 1 ( z 0 ) ⊂ U ε such that � � f ( z 0 ) � 2 = i n 2 f ( z 0 , w ) ∧ f ( z 0 , w ) < + ∞ . p − 1 ( z 0 ) Then B ε is a real analytic locally trivial Hilbert bundle whose fiber B ε, z 0 is isomorphic to the Hardy-Bergman space H 2 ( B (0 , ε )) of L 2 holomorphic n -forms on p − 1 ( z 0 ) ≃ B (0 , ε ) ⊂ C n . The Ohsawa-Takegoshi extension theorem implies that every f ∈ B ε, z 0 can be extended as a germ ˜ f in the sheaf B ε, z 0 . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 17/28

Associated Bergman bundle and holom structure Definition of the associated Bergman bundle B ε We consider the vector bundle B ε → X whose fiber B ε, z 0 consists of all holomorphic functions f on p − 1 ( z 0 ) ⊂ U ε such that � � f ( z 0 ) � 2 = i n 2 f ( z 0 , w ) ∧ f ( z 0 , w ) < + ∞ . p − 1 ( z 0 ) Then B ε is a real analytic locally trivial Hilbert bundle whose fiber B ε, z 0 is isomorphic to the Hardy-Bergman space H 2 ( B (0 , ε )) of L 2 holomorphic n -forms on p − 1 ( z 0 ) ≃ B (0 , ε ) ⊂ C n . The Ohsawa-Takegoshi extension theorem implies that every f ∈ B ε, z 0 can be extended as a germ ˜ f in the sheaf B ε, z 0 . Moreover, for ε ′ > ε , there is a restriction map B ε ′ , z 0 → B ε, z 0 such that B ε, z 0 is the L 2 completion of B ε ′ , z 0 / m z 0 B ε ′ , z 0 . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 17/28

Associated Bergman bundle and holom structure Definition of the associated Bergman bundle B ε We consider the vector bundle B ε → X whose fiber B ε, z 0 consists of all holomorphic functions f on p − 1 ( z 0 ) ⊂ U ε such that � � f ( z 0 ) � 2 = i n 2 f ( z 0 , w ) ∧ f ( z 0 , w ) < + ∞ . p − 1 ( z 0 ) Then B ε is a real analytic locally trivial Hilbert bundle whose fiber B ε, z 0 is isomorphic to the Hardy-Bergman space H 2 ( B (0 , ε )) of L 2 holomorphic n -forms on p − 1 ( z 0 ) ≃ B (0 , ε ) ⊂ C n . The Ohsawa-Takegoshi extension theorem implies that every f ∈ B ε, z 0 can be extended as a germ ˜ f in the sheaf B ε, z 0 . Moreover, for ε ′ > ε , there is a restriction map B ε ′ , z 0 → B ε, z 0 such that B ε, z 0 is the L 2 completion of B ε ′ , z 0 / m z 0 B ε ′ , z 0 . Question Is there a “complex structure” on B ε such that “ B ε = O ( B ε )” ? J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 17/28

Bergman Dolbeault complex For this, consider the “Bergman Dolbeault” complex ∂ : F q ε → F q +1 ε over X , with F q ε ( V ) = smooth ( n , q )-forms � f ( z , w ) = f J ( z , w ) dw 1 ∧ ... ∧ dw n ∧ dz J , ( z , w ) ∈ U ε ∩ ( V × X ) , | J | = q such that f J ( z , w ) is holomorphic in w , and for all K ⋐ V one has f ( z , w ) ∈ L 2 ( p − 1 ( K )) and ∂ z f ( z , w ) ∈ L 2 ( p − 1 ( K )) . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 18/28

Bergman Dolbeault complex For this, consider the “Bergman Dolbeault” complex ∂ : F q ε → F q +1 ε over X , with F q ε ( V ) = smooth ( n , q )-forms � f ( z , w ) = f J ( z , w ) dw 1 ∧ ... ∧ dw n ∧ dz J , ( z , w ) ∈ U ε ∩ ( V × X ) , | J | = q such that f J ( z , w ) is holomorphic in w , and for all K ⋐ V one has f ( z , w ) ∈ L 2 ( p − 1 ( K )) and ∂ z f ( z , w ) ∈ L 2 ( p − 1 ( K )) . An immediate consequence of this definition is: Proposition ∂ = ∂ z yields a complex of sheaves ( F • ε , ∂ ), and the kernel Ker ∂ : F 0 ε → F 1 ε coincides with B ε . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 18/28

Bergman Dolbeault complex For this, consider the “Bergman Dolbeault” complex ∂ : F q ε → F q +1 ε over X , with F q ε ( V ) = smooth ( n , q )-forms � f ( z , w ) = f J ( z , w ) dw 1 ∧ ... ∧ dw n ∧ dz J , ( z , w ) ∈ U ε ∩ ( V × X ) , | J | = q such that f J ( z , w ) is holomorphic in w , and for all K ⋐ V one has f ( z , w ) ∈ L 2 ( p − 1 ( K )) and ∂ z f ( z , w ) ∈ L 2 ( p − 1 ( K )) . An immediate consequence of this definition is: Proposition ∂ = ∂ z yields a complex of sheaves ( F • ε , ∂ ), and the kernel Ker ∂ : F 0 ε → F 1 ε coincides with B ε . If we define O L 2 ( B ε ) to be the sheaf of L 2 loc sections f of B ε such that ∂ f = 0 in the sense of distributions, then we exactly have O L 2 ( B ε ) = B ε as a sheaf. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 18/28

Bergman sheaves are “very ample” Theorem Assume that ε > 0 is taken so small that ψ ( z , w ) := | logh z ( w ) | 2 is strictly plurisubharmonic up to the boundary on the compact set U ε ⊂ X × X . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 19/28

Bergman sheaves are “very ample” Theorem Assume that ε > 0 is taken so small that ψ ( z , w ) := | logh z ( w ) | 2 is strictly plurisubharmonic up to the boundary on the compact set U ε ⊂ X × X . Then the complex of sheaves ( F • ε , ∂ ) is a resolution of B ε by soft sheaves over X (actually, by C ∞ X -modules ), and for every holomorphic vector bundle E → X we have H q ( X , B ε ⊗ O ( E )) = H q � � Γ( X , F • ε ⊗ O ( E )) , ∂ = 0 , ∀ q ≥ 1 . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 19/28

Bergman sheaves are “very ample” Theorem Assume that ε > 0 is taken so small that ψ ( z , w ) := | logh z ( w ) | 2 is strictly plurisubharmonic up to the boundary on the compact set U ε ⊂ X × X . Then the complex of sheaves ( F • ε , ∂ ) is a resolution of B ε by soft sheaves over X (actually, by C ∞ X -modules ), and for every holomorphic vector bundle E → X we have H q ( X , B ε ⊗ O ( E )) = H q � � Γ( X , F • ε ⊗ O ( E )) , ∂ = 0 , ∀ q ≥ 1 . Moreover the fibers B ε, z ⊗ E z are always generated by global sections of H 0 ( X , B ε ⊗ O ( E )). In that sense, B ε is a “very ample holomorphic vector bundle” (as a Hilbert bundle of infinite dimension). J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 19/28

Bergman sheaves are “very ample” Theorem Assume that ε > 0 is taken so small that ψ ( z , w ) := | logh z ( w ) | 2 is strictly plurisubharmonic up to the boundary on the compact set U ε ⊂ X × X . Then the complex of sheaves ( F • ε , ∂ ) is a resolution of B ε by soft sheaves over X (actually, by C ∞ X -modules ), and for every holomorphic vector bundle E → X we have H q ( X , B ε ⊗ O ( E )) = H q � � Γ( X , F • ε ⊗ O ( E )) , ∂ = 0 , ∀ q ≥ 1 . Moreover the fibers B ε, z ⊗ E z are always generated by global sections of H 0 ( X , B ε ⊗ O ( E )). In that sense, B ε is a “very ample holomorphic vector bundle” (as a Hilbert bundle of infinite dimension). ormander’s L 2 estimates. The proof is a direct consequence of H¨ Caution !! B ε is NOT a locally trivial holomorphic bundle. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 19/28

Embedding into a Hilbert Grassmannian Corollary of the very ampleness of Bergman sheaves Let X be an arbitrary compact complex manifold, E → X a holomorphic vector bundle (e.g. the trivial bundle). Consider the Hilbert space H = H 0 ( X , B ε ⊗ O ( E )). J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 20/28

Embedding into a Hilbert Grassmannian Corollary of the very ampleness of Bergman sheaves Let X be an arbitrary compact complex manifold, E → X a holomorphic vector bundle (e.g. the trivial bundle). Consider the Hilbert space H = H 0 ( X , B ε ⊗ O ( E )). Then one gets a “holomorphic embedding” into a Hilbert Grassmannian, Ψ : X → Gr ( H ) , z �→ S z , mapping every point z ∈ X to the infinite codimensional closed subspace S z consisting of sections f ∈ H such that f ( z ) = 0 in B ε, z , i.e. f | p − 1 ( z ) = 0. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 20/28

Embedding into a Hilbert Grassmannian Corollary of the very ampleness of Bergman sheaves Let X be an arbitrary compact complex manifold, E → X a holomorphic vector bundle (e.g. the trivial bundle). Consider the Hilbert space H = H 0 ( X , B ε ⊗ O ( E )). Then one gets a “holomorphic embedding” into a Hilbert Grassmannian, Ψ : X → Gr ( H ) , z �→ S z , mapping every point z ∈ X to the infinite codimensional closed subspace S z consisting of sections f ∈ H such that f ( z ) = 0 in B ε, z , i.e. f | p − 1 ( z ) = 0. The main problem with this “holomorphic embedding” is that the holomorphicity is to be understood in a weak sense, for instance the map Ψ is not even continuous with respect to the strong metric topology of Gr ( H ), given by d ( S , S ′ ) = Hausdorff distance of the unit balls of S , S ′ . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 20/28

Chern connection of Bergman bundles Since we have a natural ∇ 0 , 1 = ∂ connection on B ε , and a natural hermitian metric as well, it follows from the usual formalism that B ε can be equipped with a unique Chern connection. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 21/28

Chern connection of Bergman bundles Since we have a natural ∇ 0 , 1 = ∂ connection on B ε , and a natural hermitian metric as well, it follows from the usual formalism that B ε can be equipped with a unique Chern connection. Model case: X = C n , γ = standard hermitian metric. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 21/28

Chern connection of Bergman bundles Since we have a natural ∇ 0 , 1 = ∂ connection on B ε , and a natural hermitian metric as well, it follows from the usual formalism that B ε can be equipped with a unique Chern connection. Model case: X = C n , γ = standard hermitian metric. Then one sees that a orthonormal frame of B ε is given by � ( | α | + n )! e α ( z , w ) = π − n / 2 ε −| α |− n α 1 ! . . . α n ! ( w − z ) α , α ∈ N n . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 21/28

Chern connection of Bergman bundles Since we have a natural ∇ 0 , 1 = ∂ connection on B ε , and a natural hermitian metric as well, it follows from the usual formalism that B ε can be equipped with a unique Chern connection. Model case: X = C n , γ = standard hermitian metric. Then one sees that a orthonormal frame of B ε is given by � ( | α | + n )! e α ( z , w ) = π − n / 2 ε −| α |− n α 1 ! . . . α n ! ( w − z ) α , α ∈ N n . This frame is non holomorphic! J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 21/28

Chern connection of Bergman bundles Since we have a natural ∇ 0 , 1 = ∂ connection on B ε , and a natural hermitian metric as well, it follows from the usual formalism that B ε can be equipped with a unique Chern connection. Model case: X = C n , γ = standard hermitian metric. Then one sees that a orthonormal frame of B ε is given by � ( | α | + n )! e α ( z , w ) = π − n / 2 ε −| α |− n α 1 ! . . . α n ! ( w − z ) α , α ∈ N n . This frame is non holomorphic! The (0 , 1)-connection ∇ 0 , 1 = ∂ is given by � ∇ 0 , 1 e α = ∂ z e α ( z , w ) = ε − 1 � α j ( | α | + n ) dz j ⊗ e α − c j 1 ≤ j ≤ n where c j = (0 , ..., 1 , ..., 0) ∈ N n . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 21/28

Curvature of Bergman bundles Let Θ B ε , h = ∇ 2 be the curvature tensor of B ε with its natural Hilbertian metric h . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 22/28

Curvature of Bergman bundles Let Θ B ε , h = ∇ 2 be the curvature tensor of B ε with its natural Hilbertian metric h . Remember that Θ B ε , h = ∇ 1 , 0 ∇ 0 , 1 + ∇ 0 , 1 ∇ 1 , 0 ∈ C ∞ ( X , Λ 1 , 1 T ⋆ X ⊗ Hom ( B ε , B ε )) , J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 22/28

Curvature of Bergman bundles Let Θ B ε , h = ∇ 2 be the curvature tensor of B ε with its natural Hilbertian metric h . Remember that Θ B ε , h = ∇ 1 , 0 ∇ 0 , 1 + ∇ 0 , 1 ∇ 1 , 0 ∈ C ∞ ( X , Λ 1 , 1 T ⋆ X ⊗ Hom ( B ε , B ε )) , and that one gets an associated quadratic Hermitian form on T X ⊗ B ε such that � Θ ε ( v ⊗ ξ ) = � Θ B ε , h σ ( v , Jv ) ξ, ξ � h for v ∈ T X and ξ = � α ξ α e α ∈ B ε . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 22/28

Curvature of Bergman bundles Let Θ B ε , h = ∇ 2 be the curvature tensor of B ε with its natural Hilbertian metric h . Remember that Θ B ε , h = ∇ 1 , 0 ∇ 0 , 1 + ∇ 0 , 1 ∇ 1 , 0 ∈ C ∞ ( X , Λ 1 , 1 T ⋆ X ⊗ Hom ( B ε , B ε )) , and that one gets an associated quadratic Hermitian form on T X ⊗ B ε such that � Θ ε ( v ⊗ ξ ) = � Θ B ε , h σ ( v , Jv ) ξ, ξ � h for v ∈ T X and ξ = � α ξ α e α ∈ B ε . Definition One says that the curvature tensor is Griffiths positive if � Θ ε ( v ⊗ ξ ) > 0 , ∀ 0 � = v ∈ T X , ∀ 0 � = ξ ∈ B ε , J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 22/28

Curvature of Bergman bundles Let Θ B ε , h = ∇ 2 be the curvature tensor of B ε with its natural Hilbertian metric h . Remember that Θ B ε , h = ∇ 1 , 0 ∇ 0 , 1 + ∇ 0 , 1 ∇ 1 , 0 ∈ C ∞ ( X , Λ 1 , 1 T ⋆ X ⊗ Hom ( B ε , B ε )) , and that one gets an associated quadratic Hermitian form on T X ⊗ B ε such that � Θ ε ( v ⊗ ξ ) = � Θ B ε , h σ ( v , Jv ) ξ, ξ � h for v ∈ T X and ξ = � α ξ α e α ∈ B ε . Definition One says that the curvature tensor is Griffiths positive if � Θ ε ( v ⊗ ξ ) > 0 , ∀ 0 � = v ∈ T X , ∀ 0 � = ξ ∈ B ε , and Nakano positive if � Θ ε ( τ ) > 0 , ∀ 0 � = τ ∈ T X ⊗ B ε . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 22/28

Calculation of the curvature tensor for X = C n A simple calculation of ∇ 2 in the orthonormal frame ( e α ) leads to: Formula In the model case X = C n , the curvature tensor of the Bergman bundle ( B ε , h ) is given by �� � � Θ ε ( v ⊗ ξ ) = ε − 2 � � 2 � � � √ α j ξ α − c j v j � � � ( | α | + n ) | ξ α | 2 | v j | 2 + . � � α ∈ N n j j J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 23/28

Calculation of the curvature tensor for X = C n A simple calculation of ∇ 2 in the orthonormal frame ( e α ) leads to: Formula In the model case X = C n , the curvature tensor of the Bergman bundle ( B ε , h ) is given by �� � � Θ ε ( v ⊗ ξ ) = ε − 2 � � 2 � � � √ α j ξ α − c j v j � � � ( | α | + n ) | ξ α | 2 | v j | 2 + . � � α ∈ N n j j Consequence In C n , the curvature tensor Θ ε ( v ⊗ ξ ) is Nakano positive. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 23/28

Calculation of the curvature tensor for X = C n A simple calculation of ∇ 2 in the orthonormal frame ( e α ) leads to: Formula In the model case X = C n , the curvature tensor of the Bergman bundle ( B ε , h ) is given by �� � � Θ ε ( v ⊗ ξ ) = ε − 2 � � 2 � � � √ α j ξ α − c j v j � � � ( | α | + n ) | ξ α | 2 | v j | 2 + . � � α ∈ N n j j Consequence In C n , the curvature tensor Θ ε ( v ⊗ ξ ) is Nakano positive. On should observe that � Θ ε ( v ⊗ ξ ) is an unbounded quadratic form on B ε with respect to the standard metric � ξ � 2 = � α | ξ α | 2 . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 23/28

Calculation of the curvature tensor for X = C n A simple calculation of ∇ 2 in the orthonormal frame ( e α ) leads to: Formula In the model case X = C n , the curvature tensor of the Bergman bundle ( B ε , h ) is given by �� � � Θ ε ( v ⊗ ξ ) = ε − 2 � � 2 � � � √ α j ξ α − c j v j � � � ( | α | + n ) | ξ α | 2 | v j | 2 + . � � α ∈ N n j j Consequence In C n , the curvature tensor Θ ε ( v ⊗ ξ ) is Nakano positive. On should observe that � Θ ε ( v ⊗ ξ ) is an unbounded quadratic form on B ε with respect to the standard metric � ξ � 2 = � α | ξ α | 2 . However there is convergence for all ξ = � α ξ α e α ∈ B ε ′ , ε ′ > ε , since then � α ( ε ′ /ε ) 2 | α | | ξ α | 2 < + ∞ . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 23/28

Curvature of Bergman bundles (general case) Bergman curvature formula on a general hermitian manifold Let X be a compact complex manifold equipped with a C ω hermitian metric γ , and B ε = B γ,ε the associated Bergman bundle. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 24/28

Curvature of Bergman bundles (general case) Bergman curvature formula on a general hermitian manifold Let X be a compact complex manifold equipped with a C ω hermitian metric γ , and B ε = B γ,ε the associated Bergman bundle. Then its curvature is given by an asymptotic expansion + ∞ � � ε − 2+ p Q p ( z , v ⊗ ξ ) , Θ ε ( z , v ⊗ ξ ) = v ∈ T X , ξ ∈ B ε p =0 J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 24/28

Curvature of Bergman bundles (general case) Bergman curvature formula on a general hermitian manifold Let X be a compact complex manifold equipped with a C ω hermitian metric γ , and B ε = B γ,ε the associated Bergman bundle. Then its curvature is given by an asymptotic expansion + ∞ � � ε − 2+ p Q p ( z , v ⊗ ξ ) , Θ ε ( z , v ⊗ ξ ) = v ∈ T X , ξ ∈ B ε p =0 where Q 0 ( z , v ⊗ ξ ) = Q 0 ( v ⊗ ξ ) is given by the model case C n : �� � � Q 0 ( v ⊗ ξ ) = ε − 2 � � 2 � � � √ α j ξ α − c j v j � � ( | α | + n ) | ξ α | 2 | v j | 2 + . � � α ∈ N n j j J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 24/28

Curvature of Bergman bundles (general case) Bergman curvature formula on a general hermitian manifold Let X be a compact complex manifold equipped with a C ω hermitian metric γ , and B ε = B γ,ε the associated Bergman bundle. Then its curvature is given by an asymptotic expansion + ∞ � � ε − 2+ p Q p ( z , v ⊗ ξ ) , Θ ε ( z , v ⊗ ξ ) = v ∈ T X , ξ ∈ B ε p =0 where Q 0 ( z , v ⊗ ξ ) = Q 0 ( v ⊗ ξ ) is given by the model case C n : �� � � Q 0 ( v ⊗ ξ ) = ε − 2 � � 2 � � � √ α j ξ α − c j v j � � ( | α | + n ) | ξ α | 2 | v j | 2 + . � � α ∈ N n j j The other terms Q p ( z , v ⊗ ξ ) are real analytic; Q 1 and Q 2 depend respectively on the torsion and curvature tensor of γ . J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 24/28

Curvature of Bergman bundles (general case) Bergman curvature formula on a general hermitian manifold Let X be a compact complex manifold equipped with a C ω hermitian metric γ , and B ε = B γ,ε the associated Bergman bundle. Then its curvature is given by an asymptotic expansion + ∞ � � ε − 2+ p Q p ( z , v ⊗ ξ ) , Θ ε ( z , v ⊗ ξ ) = v ∈ T X , ξ ∈ B ε p =0 where Q 0 ( z , v ⊗ ξ ) = Q 0 ( v ⊗ ξ ) is given by the model case C n : �� � � Q 0 ( v ⊗ ξ ) = ε − 2 � � 2 � � � √ α j ξ α − c j v j � � ( | α | + n ) | ξ α | 2 | v j | 2 + . � � α ∈ N n j j The other terms Q p ( z , v ⊗ ξ ) are real analytic; Q 1 and Q 2 depend respectively on the torsion and curvature tensor of γ . In particular Q 1 = 0 is γ is K¨ ahler. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 24/28

Curvature of Bergman bundles (general case) Bergman curvature formula on a general hermitian manifold Let X be a compact complex manifold equipped with a C ω hermitian metric γ , and B ε = B γ,ε the associated Bergman bundle. Then its curvature is given by an asymptotic expansion + ∞ � � ε − 2+ p Q p ( z , v ⊗ ξ ) , Θ ε ( z , v ⊗ ξ ) = v ∈ T X , ξ ∈ B ε p =0 where Q 0 ( z , v ⊗ ξ ) = Q 0 ( v ⊗ ξ ) is given by the model case C n : �� � � Q 0 ( v ⊗ ξ ) = ε − 2 � � 2 � � � √ α j ξ α − c j v j � � ( | α | + n ) | ξ α | 2 | v j | 2 + . � � α ∈ N n j j The other terms Q p ( z , v ⊗ ξ ) are real analytic; Q 1 and Q 2 depend respectively on the torsion and curvature tensor of γ . In particular Q 1 = 0 is γ is K¨ ahler. A consequence of the above formula is that B ε is strongly Nakano positive for ε > 0 small enough. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 24/28

Idea of proof of the asymptotic expansion The formula is in principle a special case of a more general result proved by Wang Xu, expressing the curvature of weighted Bergman bundles H t attached to a smooth family { D t } of strongly pseudoconvex domains. J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 25/28