Bergman bundles and applications to the geometry of compact complex manifolds

Bergman bundles and applications to the geometry of compact complex manifolds

Jean-Pierre Demailly

Institut Fourier, Universit´e Grenoble Alpes & Acad´emie des Sciences de Paris

Virtual Conference in Complex Analysis and Geometry

hosted at Western University, London, Ontario May 4 – 24, 2020

Projective vs K¨ ahler vs non K¨ ahler varieties

Goal. Investigate positivity for general compact manifolds/C.

Obviously, non projective varieties do not carry any ample line bundle. In the K¨ahler case, a K¨ahler class {ω} ∈H^1,1(X,R),ω > 0, may 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.

The aim of this lecture is to investigate further this construction and explain potential applications to analytic geometry (invariance of plurigenera, transcendental holomorphic Morse inequalities...)

Projective vs K¨ ahler vs non K¨ ahler varieties

Goal. Investigate positivity for general compact manifolds/C.

Obviously, non projective varieties do not carry any ample line bundle.

In the K¨ahler case, a K¨ahler class {ω} ∈H^1,1(X,R),ω > 0, may 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.

The aim of this lecture is to investigate further this construction and explain potential applications to analytic geometry (invariance of plurigenera, transcendental holomorphic Morse inequalities...)

Projective vs K¨ ahler vs non K¨ ahler varieties

Goal. Investigate positivity for general compact manifolds/C.

Obviously, non projective varieties do not carry any ample line bundle.

In the K¨ahler case, a K¨ahler class {ω} ∈H^1,1(X,R),ω > 0, may 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.

The aim of this lecture is to investigate further this construction and explain potential applications to analytic geometry (invariance of plurigenera, transcendental holomorphic Morse inequalities...)

Projective vs K¨ ahler vs non K¨ ahler varieties

Goal. Investigate positivity for general compact manifolds/C.

Obviously, non projective varieties do not carry any ample line bundle.

In the K¨ahler case, a K¨ahler class {ω} ∈H^1,1(X,R),ω > 0, may 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.

The aim of this lecture is to investigate further this construction and explain potential applications to analytic geometry (invariance of plurigenera, transcendental holomorphic Morse inequalities...)

Projective vs K¨ ahler vs non K¨ ahler varieties

Goal. Investigate positivity for general compact manifolds/C.

Obviously, non projective varieties do not carry any ample line bundle.

In the K¨ahler case, a K¨ahler class {ω} ∈H^1,1(X,R),ω > 0, may 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.

The aim of this lecture is to investigate further this construction and explain potential applications to analytic geometry (invariance of plurigenera, transcendental holomorphic Morse inequalities...)

Projective vs K¨ ahler vs non K¨ ahler varieties

Goal. Investigate positivity for general compact manifolds/C.

Obviously, non projective varieties do not carry any ample line bundle.

In the K¨ahler case, a K¨ahler class {ω} ∈H^1,1(X,R),ω > 0, may 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.

The aim of this lecture is to investigate further this construction and explain potential applications to analytic geometry (invariance of plurigenera, transcendental holomorphic Morse inequalities...)

Projective vs K¨ ahler vs non K¨ ahler varieties

Goal. Investigate positivity for general compact manifolds/C.

Obviously, non projective varieties do not carry any ample line bundle.

In the K¨ahler case, a K¨ahler class {ω} ∈H^1,1(X,R),ω > 0, may 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.

The aim of this lecture is to investigate further this construction

Tubular neighborhoods (thanks to Grauert)

Let X be a compact complex manifold, dim_CX =n.

Denote by X its complex conjugate (X,−J), so that O_X =OX. 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 metricγ, and let exp :T_X →X ×X, (z, ξ)7→(z,exp_z(ξ)), z ∈X, ξ∈T_X,z be the associated geodesic exponential map.

Tubular neighborhoods (thanks to Grauert)

Let X be a compact complex manifold, dim_CX =n.

Denote by X its complex conjugate (X,−J), so that O_X =OX.

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 metricγ, and let exp :T_X →X ×X, (z, ξ)7→(z,exp_z(ξ)), z ∈X, ξ∈T_X,z be the associated geodesic exponential map.

Tubular neighborhoods (thanks to Grauert)

Let X be a compact complex manifold, dim_CX =n.

Denote by X its complex conjugate (X,−J), so that O_X =OX. The diagonal ofX×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 metricγ, and let exp :T_X →X ×X, (z, ξ)7→(z,exp_z(ξ)), z ∈X, ξ∈T_X,z be the associated geodesic exponential map.

Tubular neighborhoods (thanks to Grauert)

Let X be a compact complex manifold, dim_CX =n.

Denote by X its complex conjugate (X,−J), so that O_X =OX. The diagonal ofX×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 metricγ,

and let exp :T_X →X ×X, (z, ξ)7→(z,exp_z(ξ)), z ∈X, ξ∈T_X,z be the associated geodesic exponential map.

Tubular neighborhoods (thanks to Grauert)

Let X be a compact complex manifold, dim_CX =n.

Denote by X its complex conjugate (X,−J), so that O_X =OX. The diagonal ofX×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 metricγ, and let exp :T_X →X ×X, (z, ξ)7→(z,exp_z(ξ)), z ∈X, ξ∈T_X,z be the associated geodesic exponential map.

Exponential map diffeomorphism and its inverse

Lemma

Denote by exphthe “holomorphic” part of exp, so that for z ∈X and ξ∈T_X,z

exp_z(ξ) = X

α,β∈Nⁿ

a_{α β}(z)ξ^αξ^β, exph_z(ξ) = X

α∈Nⁿ

a_α0(z)ξ^α.

Then dξexp_z(ξ)ξ=0 =dξexph_z(ξ)ξ=0 =IdT_X, and so exphis a diffeomorphism from a neighborhood V of the 0 section of T_X to a neighborhood V⁰ of the diagonal inX ×X.

Notation

With the identification X '_diff X, let logh:X ×X ⊃V⁰ →T_X be the inverse diffeomorphism of exph and

U_ε ={(z,w)∈V⁰ ⊂X ×X; |logh_z(w)|_γ < ε}, ε >0. Then, for ε 1, U_ε is Stein and pr₁ :U_ε→X is a real analytic locally trivial bundle with fibers biholomorphic to complex balls.

Exponential map diffeomorphism and its inverse

Lemma

Denote by exphthe “holomorphic” part of exp, so that for z ∈X and ξ∈T_X,z

exp_z(ξ) = X

α,β∈Nⁿ

a_{α β}(z)ξ^αξ^β, exph_z(ξ) = X

α∈Nⁿ

a_α0(z)ξ^α. Then dξexp_z(ξ)ξ=0 =dξexph_z(ξ)ξ=0 =IdT_X, and soexph is a diffeomorphism from a neighborhood V of the 0 section of T_X to a neighborhood V⁰ of the diagonal inX ×X.

Notation

With the identification X '_diff X, let logh:X ×X ⊃V⁰ →T_X be the inverse diffeomorphism of exph and

U_ε ={(z,w)∈V⁰ ⊂X ×X; |logh_z(w)|_γ < ε}, ε >0. Then, for ε 1, U_ε is Stein and pr₁ :U_ε→X is a real analytic locally trivial bundle with fibers biholomorphic to complex balls.

Exponential map diffeomorphism and its inverse

Lemma

Denote by exphthe “holomorphic” part of exp, so that for z ∈X and ξ∈T_X,z

exp_z(ξ) = X

α,β∈Nⁿ

a_{α β}(z)ξ^αξ^β, exph_z(ξ) = X

α∈Nⁿ

a_α0(z)ξ^α. Then dξexp_z(ξ)ξ=0 =dξexph_z(ξ)ξ=0 =IdT_X, and soexph is a diffeomorphism from a neighborhood V of the 0 section of T_X to a neighborhood V⁰ of the diagonal inX ×X.

Notation

With the identification X '_diff X, let logh:X ×X ⊃V⁰ →T_X be the inverse diffeomorphism of exph and

U_ε ={(z,w)∈V⁰ ⊂X ×X; |logh_z(w)|_γ < ε}, ε >0.

Then, for ε 1, U_ε is Stein and pr₁ :U_ε→X is a real analytic locally trivial bundle with fibers biholomorphic to complex balls.

Exponential map diffeomorphism and its inverse

Lemma

Denote by exphthe “holomorphic” part of exp, so that for z ∈X and ξ∈T_X,z

exp_z(ξ) = X

α,β∈Nⁿ

a_{α β}(z)ξ^αξ^β, exph_z(ξ) = X

α∈Nⁿ

a_α0(z)ξ^α. Then dξexp_z(ξ)ξ=0 =dξexph_z(ξ)ξ=0 =IdT_X, and soexph is a diffeomorphism from a neighborhood V of the 0 section of T_X to a neighborhood V⁰ of the diagonal inX ×X.

Notation

With the identification X '_diff X, let logh:X ×X ⊃V⁰ →T_X be the inverse diffeomorphism of exph and

U_ε ={(z,w)∈V⁰ ⊂X ×X; |logh_z(w)|_γ < ε}, ε >0.

Then, for ε 1, U_ε is Stein and pr₁ :U_ε→X is a real analytic locally trivial bundle with fibers biholomorphic to complex balls.

Such tubular neighborhoods are Stein

In the special case X =Cⁿ, U_ε={(z,w)∈Cⁿ×Cⁿ; |z−w|< ε} is of course Stein since

|z−w|² =|z|²+|w|²−2ReP z_jw_j and (z,w)7→ReP

z_jw_j is pluriharmonic.

Such tubular neighborhoods are Stein

In the special case X =Cⁿ, U_ε={(z,w)∈Cⁿ×Cⁿ; |z−w|< ε}

is of course Stein since

|z−w|² =|z|²+|w|²−2ReP z_jw_j and (z,w)7→ReP

z_jw_j is pluriharmonic.

Bergman sheaves

Let U_ε=U_γ,ε⊂X ×X be the ball bundle as above, and p= (pr₁)|U_ε :U_ε →X, p= (pr₂)|U_ε :U_ε →X the natural projections.

Bergman sheaves

Let U_ε=U_γ,ε⊂X ×X be the ball bundle as above, and p= (pr₁)|U_ε :U_ε →X, p= (pr₂)|U_ε :U_ε →X the natural projections.

Bergman sheaves (continued)

Definition of the Bergman sheaf B_ε

The Bergman sheaf B_ε =B_γ,ε is by definition theL² direct image B_ε =p_∗^L2(p^∗O(K_X)),

i.e. the space of sections over an open subset V ⊂X defined by Bε(V) = holomorphic sections f of p^∗O(K_X) on p⁻¹(V),

f(z,w) =f₁(z,w)dw₁ ∧. . .∧dw_n, z ∈V, that are in L²(p⁻¹(K)) for all compact subsets K bV :

Z

p⁻¹(K)

iⁿ²f(z,w)∧f(z,w)∧γ(z)ⁿ<+∞, ∀K bV.

(This L² condition is the reason we speak of “L² 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.

Bergman sheaves (continued)

Definition of the Bergman sheaf B_ε

The Bergman sheaf B_ε =B_γ,ε is by definition theL² direct image B_ε =p_∗^L2(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) =f₁(z,w)dw₁∧. . .∧dw_n, z ∈V,

that are in L²(p⁻¹(K)) for all compact subsets K bV : Z

p⁻¹(K)

iⁿ²f(z,w)∧f(z,w)∧γ(z)ⁿ<+∞, ∀K bV.

(This L² condition is the reason we speak of “L² 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.

Bergman sheaves (continued)

Definition of the Bergman sheaf B_ε

The Bergman sheaf B_ε =B_γ,ε is by definition theL² direct image B_ε =p_∗^L2(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) =f₁(z,w)dw₁∧. . .∧dw_n, z ∈V, that are in L²(p⁻¹(K)) for all compact subsets K bV :

Z

p⁻¹(K)

iⁿ²f(z,w)∧f(z,w)∧γ(z)ⁿ <+∞, ∀K bV. (This L² condition is the reason we speak of “L² 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.

Bergman sheaves (continued)

Definition of the Bergman sheaf B_ε

The Bergman sheaf B_ε =B_γ,ε is by definition theL² direct image B_ε =p_∗^L2(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) =f₁(z,w)dw₁∧. . .∧dw_n, z ∈V, that are in L²(p⁻¹(K)) for all compact subsets K bV :

Z

p⁻¹(K)

iⁿ²f(z,w)∧f(z,w)∧γ(z)ⁿ <+∞, ∀K bV.

(This L² condition is the reason we speak of “L² 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.

Associated Bergman bundle and holom structure

Definition of the associated Bergman bundle B_ε

We consider the vector bundleB_ε →X whose fiber B_ε,z0consists of all holomorphic functions f onp⁻¹(z0)⊂Uε such that

kf(z₀)k² = Z

p⁻¹(z0)

iⁿ²f(z₀,w)∧f(z₀,w)<+∞.

Then B_ε is a real analytic locally trivial Hilbert bundle whose fiber Bε,z0 is isomorphic to the Hardy-Bergman space H²(B(0, ε)) of L² holomorphic n-forms on p⁻¹(z₀)'B(0, ε)⊂Cⁿ.

The Ohsawa-Takegoshi extension theorem implies that every f ∈Bε,z0 can be extended as a germ ˜f in the sheaf Bε,z0. Moreover, for ε⁰ > ε, there is a restriction mapB_ε⁰_,z0 →B_ε,z0 such that B_ε,z0 is the L² completion ofB_ε⁰_,z0/m_z0B_ε⁰_,z0. Question

Is there a “complex structure” on B_ε such that “B_ε =O(B_ε)” ?

Associated Bergman bundle and holom structure

Definition of the associated Bergman bundle B_ε

We consider the vector bundleB_ε →X whose fiber B_ε,z0consists of all holomorphic functions f onp⁻¹(z0)⊂Uε such that

kf(z₀)k² = Z

p⁻¹(z0)

iⁿ²f(z₀,w)∧f(z₀,w)<+∞.

Then B_ε is a real analytic locally trivial Hilbert bundle whose fiber Bε,z0 is isomorphic to the Hardy-Bergman space H²(B(0, ε)) of L² holomorphic n-forms on p⁻¹(z₀)'B(0, ε)⊂Cⁿ.

The Ohsawa-Takegoshi extension theorem implies that every f ∈Bε,z0 can be extended as a germ ˜f in the sheaf Bε,z0. Moreover, for ε⁰ > ε, there is a restriction mapB_ε⁰_,z0 →B_ε,z0 such that B_ε,z0 is the L² completion ofB_ε⁰_,z0/m_z0B_ε⁰_,z0. Question

Is there a “complex structure” on B_ε such that “B_ε =O(B_ε)” ?

Associated Bergman bundle and holom structure

Definition of the associated Bergman bundle B_ε

We consider the vector bundleB_ε →X whose fiber B_ε,z0consists of all holomorphic functions f onp⁻¹(z0)⊂Uε such that

kf(z₀)k² = Z

p⁻¹(z0)

iⁿ²f(z₀,w)∧f(z₀,w)<+∞.

Then B_ε is a real analytic locally trivial Hilbert bundle whose fiber Bε,z0 is isomorphic to the Hardy-Bergman space H²(B(0, ε)) of L² holomorphic n-forms on p⁻¹(z₀)'B(0, ε)⊂Cⁿ.

The Ohsawa-Takegoshi extension theorem implies that every f ∈Bε,z0 can be extended as a germ ˜f in the sheaf Bε,z0.

Moreover, for ε⁰ > ε, there is a restriction mapB_ε⁰_,z0 →B_ε,z0 such that B_ε,z0 is the L² completion ofB_ε⁰_,z0/m_z0B_ε⁰_,z0. Question

Is there a “complex structure” on B_ε such that “B_ε =O(B_ε)” ?

Associated Bergman bundle and holom structure

Definition of the associated Bergman bundle B_ε

We consider the vector bundleB_ε →X whose fiber B_ε,z0consists of all holomorphic functions f onp⁻¹(z0)⊂Uε such that

kf(z₀)k² = Z

p⁻¹(z0)

iⁿ²f(z₀,w)∧f(z₀,w)<+∞.

Then B_ε is a real analytic locally trivial Hilbert bundle whose fiber Bε,z0 is isomorphic to the Hardy-Bergman space H²(B(0, ε)) of L² holomorphic n-forms on p⁻¹(z₀)'B(0, ε)⊂Cⁿ.

The Ohsawa-Takegoshi extension theorem implies that every f ∈Bε,z0 can be extended as a germ ˜f in the sheaf Bε,z0. Moreover, for ε⁰ > ε, there is a restriction mapB_ε⁰_,z0 →B_ε,z0 such that B_ε,z0 is the L² completion ofB_ε⁰_,z0/m_z0B_ε⁰_,z0.

Question

Is there a “complex structure” on B_ε such that “B_ε =O(B_ε)” ?

Associated Bergman bundle and holom structure

Definition of the associated Bergman bundle B_ε

We consider the vector bundleB_ε →X whose fiber B_ε,z0consists of all holomorphic functions f onp⁻¹(z0)⊂Uε such that

kf(z₀)k² = Z

p⁻¹(z0)

iⁿ²f(z₀,w)∧f(z₀,w)<+∞.

Then B_ε is a real analytic locally trivial Hilbert bundle whose fiber Bε,z0 is isomorphic to the Hardy-Bergman space H²(B(0, ε)) of L² holomorphic n-forms on p⁻¹(z₀)'B(0, ε)⊂Cⁿ.

The Ohsawa-Takegoshi extension theorem implies that every f ∈Bε,z0 can be extended as a germ ˜f in the sheaf Bε,z0. Moreover, for ε⁰ > ε, there is a restriction mapB_ε⁰_,z0 →B_ε,z0 such that B_ε,z0 is the L² completion ofB_ε⁰_,z0/m_z0B_ε⁰_,z0.

Bergman Dolbeault complex

For this, consider the “Bergman Dolbeault” complex ∂ :F_ε^q→ F_ε^q+1 over X, with F_ε^q(V) = smooth (n,q)-forms

f(z,w) = X

|J|=q

f_J(z,w)dw₁∧...∧dw_n∧d z_J, (z,w)∈U_ε∩(V ×X),

such that f_J(z,w) is holomorphic in w, and for all K bV one has f(z,w)∈L²(p⁻¹(K)) and ∂_zf(z,w)∈L²(p⁻¹(K)).

An immediate consequence of this definition is: Proposition

∂ =∂_z yields a complex of sheaves (F_ε^•, ∂), and the kernel Ker∂ :F_ε⁰ → F_ε¹ coincides with B_ε.

If we define O_L²(B_ε) to be the sheaf of L²_loc sections f of B_ε such that ∂f = 0 in the sense of distributions, then we exactly have O_L²(B_ε) =B_ε as a sheaf.

Bergman Dolbeault complex

For this, consider the “Bergman Dolbeault” complex ∂ :F_ε^q→ F_ε^q+1 over X, with F_ε^q(V) = smooth (n,q)-forms

f(z,w) = X

|J|=q

f_J(z,w)dw₁∧...∧dw_n∧d z_J, (z,w)∈U_ε∩(V ×X),

such that f_J(z,w) is holomorphic in w, and for all K bV one has f(z,w)∈L²(p⁻¹(K)) and ∂_zf(z,w)∈L²(p⁻¹(K)).

An immediate consequence of this definition is:

Proposition

∂ =∂_z yields a complex of sheaves (F_ε^•, ∂), and the kernel Ker∂ :F_ε⁰ → F_ε¹ coincides with B_ε.

If we define O_L²(B_ε) to be the sheaf of L²_loc sections f of B_ε such that ∂f = 0 in the sense of distributions, then we exactly have O_L²(B_ε) =B_ε as a sheaf.

Bergman Dolbeault complex

For this, consider the “Bergman Dolbeault” complex ∂ :F_ε^q→ F_ε^q+1 over X, with F_ε^q(V) = smooth (n,q)-forms

f(z,w) = X

|J|=q

f_J(z,w)dw₁∧...∧dw_n∧d z_J, (z,w)∈U_ε∩(V ×X),

such that f_J(z,w) is holomorphic in w, and for all K bV one has f(z,w)∈L²(p⁻¹(K)) and ∂_zf(z,w)∈L²(p⁻¹(K)).

An immediate consequence of this definition is:

Proposition

∂ =∂_z yields a complex of sheaves (F_ε^•, ∂), and the kernel Ker∂ :F_ε⁰ → F_ε¹ coincides with B_ε.

If we define O_L²(B_ε) to be the sheaf of L²_loc sections f of B_ε such that ∂f = 0 in the sense of distributions, then we exactly have O_L²(B_ε) =B_ε as a sheaf.

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 H⁰(X,B_ε⊗ O(E)).

In that sense, B_ε is a “very ample holomorphic vector bundle” (as a Hilbert bundle of infinite dimension).

The proof is a direct consequence of H¨ormander’s L² estimates. Caution !!

B_ε is NOT a locally trivial holomorphic bundle.

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 Bε by soft sheaves over X (actually, byC_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 H⁰(X,B_ε⊗ O(E)).

In that sense, B_ε is a “very ample holomorphic vector bundle” (as a Hilbert bundle of infinite dimension).

The proof is a direct consequence of H¨ormander’s L² estimates. Caution !!

B_ε is NOT a locally trivial holomorphic bundle.

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 Bε by soft sheaves over X (actually, byC_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 H⁰(X,B_ε⊗ O(E)).

In that sense, B_ε is a “very ample holomorphic vector bundle”

(as a Hilbert bundle of infinite dimension).

The proof is a direct consequence of H¨ormander’s L² estimates. Caution !!

B_ε is NOT a locally trivial holomorphic bundle.

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 Bε by soft sheaves over X (actually, byC_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 H⁰(X,B_ε⊗ O(E)).

In that sense, B_ε is a “very ample holomorphic vector bundle”

(as a Hilbert bundle of infinite dimension).

The proof is a direct consequence of H¨ormander’s L² estimates.

Caution !!

B_ε is NOT a locally trivial holomorphic bundle.

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=H⁰(X,B_ε⊗ O(E)).

Then one gets a

“holomorphic embedding” into a Hilbert Grassmannian, Ψ :X →Gr(H), z 7→Sz,

mapping every point z ∈X to the infinite codimensional closed subspace S_z consisting of sectionsf ∈H such that f(z) = 0 in Bε,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,S⁰) = Hausdorff distance of the unit balls of S, S⁰.

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=H⁰(X,B_ε⊗ O(E)). Then one gets a

“holomorphic embedding” into a Hilbert Grassmannian, Ψ :X →Gr(H), z 7→S_z,

mapping every point z ∈X to the infinite codimensional closed subspace S_z consisting of sectionsf ∈H such that f(z) = 0 in Bε,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,S⁰) = Hausdorff distance of the unit balls of S, S⁰.

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=H⁰(X,B_ε⊗ O(E)). Then one gets a

“holomorphic embedding” into a Hilbert Grassmannian, Ψ :X →Gr(H), z 7→S_z,

mapping every point z ∈X to the infinite codimensional closed subspace S_z consisting of sectionsf ∈H such that f(z) = 0 in Bε,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

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 Bε can be equipped with a unique Chern connection.

Model case: X =Cⁿ, γ = standard hermitian metric. Then one sees that a orthonormal frame of B_ε is given by

e_α(z,w) = π^−n/2ε^−|α|−n s

(|α|+n)!

α₁!. . . α_n!(w −z)^α, α∈Nⁿ. It is non holomorphic! The (0,1)-connection ∇^0,1 =∂ is given by

∇^0,1eα =∂zeα(z,w) = ε⁻¹ X

1≤j≤n

q

αj(|α|+n)d zj ⊗eα−cj

where c_j = (0, ...,1, ...,0)∈Nⁿ.

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 Bε can be equipped with a unique Chern connection.

Model case: X =Cⁿ, γ = standard hermitian metric.

Then one sees that a orthonormal frame of B_ε is given by e_α(z,w) = π^−n/2ε^−|α|−n

s

(|α|+n)!

α₁!. . . α_n!(w −z)^α, α∈Nⁿ. It is non holomorphic! The (0,1)-connection ∇^0,1 =∂ is given by

∇^0,1eα =∂zeα(z,w) = ε⁻¹ X

1≤j≤n

q

αj(|α|+n)d zj ⊗eα−cj

where c_j = (0, ...,1, ...,0)∈Nⁿ.

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 Bε can be equipped with a unique Chern connection.

Model case: X =Cⁿ, γ = standard hermitian metric.

Then one sees that a orthonormal frame of B_ε is given by e_α(z,w) = π^−n/2ε^−|α|−n

s

(|α|+n)!

α₁!. . . α_n!(w −z)^α, α∈Nⁿ.

It is non holomorphic! The (0,1)-connection ∇^0,1 =∂ is given by

∇^0,1eα =∂zeα(z,w) = ε⁻¹ X

1≤j≤n

q

αj(|α|+n)d zj ⊗eα−cj

where c_j = (0, ...,1, ...,0)∈Nⁿ.

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 Bε can be equipped with a unique Chern connection.

Model case: X =Cⁿ, γ = standard hermitian metric.

Then one sees that a orthonormal frame of B_ε is given by e_α(z,w) = π^−n/2ε^−|α|−n

s

(|α|+n)!

α₁!. . . α_n!(w −z)^α, α∈Nⁿ. It is non holomorphic!

The (0,1)-connection ∇^0,1 =∂ is given by

∇^0,1eα =∂zeα(z,w) = ε⁻¹ X

1≤j≤n

q

αj(|α|+n)d zj ⊗eα−cj

where c_j = (0, ...,1, ...,0)∈Nⁿ.

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 Bε can be equipped with a unique Chern connection.

Model case: X =Cⁿ, γ = standard hermitian metric.

Then one sees that a orthonormal frame of B_ε is given by e_α(z,w) = π^−n/2ε^−|α|−n

s

(|α|+n)!

α₁!. . . α_n!(w −z)^α, α∈Nⁿ. It is non holomorphic! The (0,1)-connection ∇^0,1 =∂ is given by

∇^0,1eα =∂zeα(z,w) = ε⁻¹ X

1≤j≤n

q

αj(|α|+n)d zj ⊗eα−cj

where c_j = (0, ...,1, ...,0)∈Nⁿ.

Curvature of Bergman bundles

Let Θ_Bε,h =∇² be the curvature tensor of B_ε with its natural Hilbertian metric h.

Remember that

Θ_Bε,h=∇^1,0∇^0,1+∇^0,1∇^1,0 ∈C^∞(X,Λ^1,1T_X^? ⊗Hom(B_ε,B_ε)), and that one gets an associated quadratic Hermitian form on TX ⊗Bε such that

Θe_ε(v ⊗ξ) =hΘ_Bε,hσ(v,Jv)ξ, ξi_h for v ∈T_X and ξ=P

αξ_αe_α ∈B_ε. Definition

One says that the curvature tensor is Griffiths positive if Θeε(v ⊗ξ)>0, ∀06=v ∈TX, ∀06=ξ∈Bε, and Nakano positive if

Θe_ε(τ)>0, ∀06=τ ∈T_X ⊗B_ε.

Curvature of Bergman bundles

Let Θ_Bε,h =∇² be the curvature tensor of B_ε with its natural Hilbertian metric h. Remember that

Θ_Bε,h=∇^1,0∇^0,1 +∇^0,1∇^1,0 ∈C^∞(X,Λ^1,1T_X^? ⊗Hom(B_ε,B_ε)), and that one gets an associated quadratic Hermitian form on T_X ⊗B_ε such that

Θe_ε(v ⊗ξ) =hΘ_Bε,hσ(v,Jv)ξ, ξi_h for v ∈T_X and ξ=P

αξ_αe_α ∈B_ε.

Definition

One says that the curvature tensor is Griffiths positive if Θeε(v ⊗ξ)>0, ∀06=v ∈TX, ∀06=ξ∈Bε, and Nakano positive if

Θe_ε(τ)>0, ∀06=τ ∈T_X ⊗B_ε.

Curvature of Bergman bundles

Let Θ_Bε,h =∇² be the curvature tensor of B_ε with its natural Hilbertian metric h. Remember that

Θ_Bε,h=∇^1,0∇^0,1 +∇^0,1∇^1,0 ∈C^∞(X,Λ^1,1T_X^? ⊗Hom(B_ε,B_ε)), and that one gets an associated quadratic Hermitian form on T_X ⊗B_ε such that

Θe_ε(v ⊗ξ) =hΘ_Bε,hσ(v,Jv)ξ, ξi_h for v ∈T_X and ξ=P

αξ_αe_α ∈B_ε. Definition

One says that the curvature tensor is Griffiths positive if Θeε(v ⊗ξ)>0, ∀06=v ∈TX, ∀06=ξ∈Bε,

and Nakano positive if

Θe_ε(τ)>0, ∀06=τ ∈T_X ⊗B_ε.

Curvature of Bergman bundles

Let Θ_Bε,h =∇² be the curvature tensor of B_ε with its natural Hilbertian metric h. Remember that

Θ_Bε,h=∇^1,0∇^0,1 +∇^0,1∇^1,0 ∈C^∞(X,Λ^1,1T_X^? ⊗Hom(B_ε,B_ε)), and that one gets an associated quadratic Hermitian form on T_X ⊗B_ε such that

Θe_ε(v ⊗ξ) =hΘ_Bε,hσ(v,Jv)ξ, ξi_h for v ∈T_X and ξ=P

αξ_αe_α ∈B_ε. Definition

One says that the curvature tensor is Griffiths positive if Θeε(v ⊗ξ)>0, ∀06=v ∈TX, ∀06=ξ∈Bε, and Nakano positive if

Θe_ε(τ)>0, ∀06=τ ∈T_X ⊗B_ε.

Calculation of the curvature tensor for X = C

A simple calculation of ∇² in the orthonormal frame (e_α) leads to:

Formula

In the model case X =Cⁿ, the curvature tensor of the Bergman bundle (B_ε,h) is given by

Θe_ε(v ⊗ξ) = ε⁻²X

α∈Nⁿ

X

j

√α_j ξα−c_jv_j

2

+X

j

(|α|+n)|ξ_α|²|v_j|²

! .

Consequence

In Cⁿ, the curvature tensor Θ_ε(v ⊗ξ) is Nakano positive.

On should observe that Θe_ε(v ⊗ξ) is an unbounded quadratic form on B_ε with respect to the standard metric kξk² =P

α|ξ_α|². However there is convergence for all ξ=P

αξ_αe_α ∈B_ε⁰, ε⁰ > ε, since then P

α(ε⁰/ε)^2|α||ξ_α|² <+∞.

Calculation of the curvature tensor for X = C

A simple calculation of ∇² in the orthonormal frame (e_α) leads to:

Formula

In the model case X =Cⁿ, the curvature tensor of the Bergman bundle (B_ε,h) is given by

Θe_ε(v ⊗ξ) = ε⁻²X

α∈Nⁿ

X

j

√α_j ξα−c_jv_j

2

+X

j

(|α|+n)|ξ_α|²|v_j|²

! .

Consequence

In Cⁿ, the curvature tensor Θ_ε(v ⊗ξ) is Nakano positive.

On should observe that Θe_ε(v ⊗ξ) is an unbounded quadratic form on B_ε with respect to the standard metric kξk² =P

α|ξ_α|². However there is convergence for all ξ=P

αξ_αe_α ∈B_ε⁰, ε⁰ > ε, since then P

α(ε⁰/ε)^2|α||ξ_α|² <+∞.

Calculation of the curvature tensor for X = C

A simple calculation of ∇² in the orthonormal frame (e_α) leads to:

Formula

In the model case X =Cⁿ, the curvature tensor of the Bergman bundle (B_ε,h) is given by

Θe_ε(v ⊗ξ) = ε⁻²X

α∈Nⁿ

X

j

√α_j ξα−c_jv_j

2

+X

j

(|α|+n)|ξ_α|²|v_j|²

! .

Consequence

In Cⁿ, the curvature tensor Θ_ε(v ⊗ξ) is Nakano positive.

On should observe that Θe_ε(v ⊗ξ) is an unbounded quadratic form on B_ε with respect to the standard metric kξk² =P

α|ξ_α|².

However there is convergence for all ξ=P

αξ_αe_α ∈B_ε⁰, ε⁰ > ε, since then P

α(ε⁰/ε)^2|α||ξ_α|² <+∞.

Calculation of the curvature tensor for X = C

A simple calculation of ∇² in the orthonormal frame (e_α) leads to:

Formula

In the model case X =Cⁿ, the curvature tensor of the Bergman bundle (B_ε,h) is given by

Θe_ε(v ⊗ξ) = ε⁻²X

α∈Nⁿ

X

j

√α_j ξα−c_jv_j

2

+X

j

(|α|+n)|ξ_α|²|v_j|²

! .

Consequence

In Cⁿ, the curvature tensor Θ_ε(v ⊗ξ) is Nakano positive.

On should observe that Θe_ε(v ⊗ξ) is an unbounded quadratic form on B_ε with respect to the standard metric kξk² =P

α|ξ_α|². However there is convergence for all ξ=P

αξ_αe_α ∈B_ε⁰, ε⁰ > ε, since then P

α(ε⁰/ε)^2|α||ξ_α|² <+∞.

Curvature of Bergman bundles (general case)

Bergman curvature formula on a general hermitian manifold Let X be a compact complex manifold equipped with aC^ω

hermitian metric γ, and Bε =Bγ,ε the associated Bergman bundle.

Then its curvature is given by an asymptotic expansion Θeε(z,v ⊗ξ) =

+∞

X

p=0

ε^−2+pQp(z,v ⊗ξ), v ∈TX, ξ ∈Bε

where Q₀(z,v ⊗ξ) = Q₀(v ⊗ξ) is given by the model case Cⁿ: Q₀(v ⊗ξ) = ε⁻²X

α∈Nⁿ

X

j

√α_j ξα−c_jv_j

2

+X

j

(|α|+n)|ξ_α|²|v_j|²

! . The other terms Q_p(z,v ⊗ξ) are real analytic; Q₁ and Q₂ depend respectively on the torsion and curvature tensor of γ.

In particular Q₁ = 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.

Curvature of Bergman bundles (general case)

Bergman curvature formula on a general hermitian manifold Let X be a compact complex manifold equipped with aC^ω

hermitian metric γ, and Bε =Bγ,ε the associated Bergman bundle.

Then its curvature is given by an asymptotic expansion Θe_ε(z,v ⊗ξ) =

+∞

X

p=0

ε^−2+pQ_p(z,v⊗ξ), v ∈T_X, ξ ∈B_ε

where Q₀(z,v ⊗ξ) = Q₀(v ⊗ξ) is given by the model case Cⁿ: Q₀(v ⊗ξ) = ε⁻²X

α∈Nⁿ

X

j

√α_j ξα−c_jv_j

2

+X

j

(|α|+n)|ξ_α|²|v_j|²

! . The other terms Q_p(z,v ⊗ξ) are real analytic; Q₁ and Q₂ depend respectively on the torsion and curvature tensor of γ.

In particular Q₁ = 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.

Curvature of Bergman bundles (general case)

Bergman curvature formula on a general hermitian manifold Let X be a compact complex manifold equipped with aC^ω

hermitian metric γ, and Bε =Bγ,ε the associated Bergman bundle.

Then its curvature is given by an asymptotic expansion Θe_ε(z,v ⊗ξ) =

+∞

X

p=0

ε^−2+pQ_p(z,v⊗ξ), v ∈T_X, ξ ∈B_ε

where Q₀(z,v ⊗ξ) = Q₀(v ⊗ξ) is given by the model case Cⁿ: Q₀(v ⊗ξ) = ε⁻²X

α∈Nⁿ

X

j

√α_j ξα−c_jv_j

2

+X

j

(|α|+n)|ξ_α|²|v_j|²

! .

The other terms Q_p(z,v ⊗ξ) are real analytic; Q₁ and Q₂ depend respectively on the torsion and curvature tensor of γ.

In particular Q₁ = 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.