On the approximate cohomology of quasi holomorphic line bundles

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On the approximate cohomology of quasi holomorphic line bundles

Jean-Pierre Demailly

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

Virtual Conference Geometry and TACoS hosted at Universit`a di Firenze

July 7 – 21, 2020

J.-P. Demailly, virtual conference Geom. & TACoS, July 7, 2020 Cohomology of quasi holomorphic line bundles 1/28

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Quasi holomorphic line bundles

Let X be a compact complex manifold, and let H_BC^p,q(X,C) = Ker∂ ∩Ker∂

Im∂∂ in bidegree (p,q) be the corresponding Bott-Chern cohomology groups.

Basic observation (cf. Laurent Laeng, PhD thesis 2002)

Given a class γ ∈H_BC^1,1(X,R)and a (1,1)-form u representing γ, there exists an infinite subset S ⊂Nand C^∞ Hermitian line bundles (L_k,h_k)k∈S equipped with Hermitian connections ∇_k, such that the curvature 2-forms θ_k = _2πⁱ ∇²_k satisfy θ_k =ku+β_k and

β_k =O(k^−1/b²), b₂ =b₂(X).

Proof. This is a consequence of Kronecker’s approximation theorem applied to the lattice H²(X,Z),→H_DR² (X,R). In fact βk can be chosen in a finite dimensional space of C^∞ closed 2-forms isomorphic to H_DR² (X,R).

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Quasi holomorphic line bundles

Given a class γ ∈H_BC^1,1(X,R)and a (1,1)-form u representing γ, there exists an infinite subset S ⊂N and C^∞ Hermitian line bundles (L_k,h_k)k∈S equipped with Hermitian connections ∇_k,

such that the curvature 2-forms θ_k = _2πⁱ ∇²_k satisfy θ_k =ku+β_k and

β_k =O(k^−1/b²), b₂ =b₂(X).

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Quasi holomorphic line bundles

Given a class γ ∈H_BC^1,1(X,R)and a (1,1)-form u representing γ, there exists an infinite subset S ⊂N and C^∞ Hermitian line bundles (L_k,h_k)k∈S equipped with Hermitian connections ∇_k, such that the curvature 2-forms θ_k = _2πⁱ ∇²_k satisfy θ_k =ku+β_k and

β_k =O(k^−1/b²), b₂ =b₂(X).

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Quasi holomorphic line bundles

β_k =O(k^−1/b²), b₂ =b₂(X).

Proof. This is a consequence of Kronecker’s approximation theorem applied to the lattice H²(X,Z),→H_DR² (X,R).

In fact βk can be chosen in a finite dimensional space of C^∞ closed 2-forms isomorphic to H_DR² (X,R).

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Quasi holomorphic line bundles

β_k =O(k^−1/b²), b₂ =b₂(X).

Proof. This is a consequence of Kronecker’s approximation theorem applied to the lattice H²(X,Z),→H_DR² (X,R).

In fact β_k can be chosen in a finite dimensional space of C^∞ closed 2-forms isomorphic to H_DR² (X,R).

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Approximate holomorphic structure

0 θ₁

θ₅

{θ_k} ∈H²(X,Z) ku, k = 1,2,3,4,5

Consequence

Let ∇_k =∇^1,0_k +∇^0,1_k . Then θ_k =ku+β_k implies (∇^0,1_k )² =θ^0,2_k =β_k^0,2 =O(k^−1/b²).

Thus the Lk are “closer and closer” to be holomorphic as k →+∞.

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Approximate holomorphic structure

0 θ₁

θ₅

{θ_k} ∈H²(X,Z) ku, k = 1,2,3,4,5

Consequence

Thus the Lk are “closer and closer” to be holomorphic as k →+∞.

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Approximate holomorphic structure

0 θ₁

θ₅

{θ_k} ∈H²(X,Z) ku, k = 1,2,3,4,5

Consequence

Thus the L_k are “closer and closer” to be holomorphic as k →+∞.

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Spectrum of the Laplace-Beltrami operator

Let k =∂_k∂^∗_k +∂^∗_k∂_k be the complex Laplace-Beltrami operator of (Lk,hk,∇k) with respect to some Hermitian metric ω on X.

Let ^p,qk,E the operator acting on C^∞(X,Λ^p,qT_X^∗ ⊗L_k ⊗E), where (E,h_E) is a holomorphic Hermitian vector bundle of rankr. We are interested in analyzing the (discrete) spectrum of the elliptic operator ^p,qk,E. Since the curvature is θ_k 'ku, it is better to renormalize and to consider instead _2πk¹ ^p,qk,E. For λ ∈R, we define

N_k^p,q(λ) = dimM

eigenspaces of 1

2πk^p,qk,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

|u₁(x)| ≥ ··· ≥ |u_s(x)|>|u_s+1(x)|=···=|u_n(x)|= 0. For a multi-index J ={j1 <j2 < . . . <jq} ⊂ {1, . . . ,n}, set

u_J(x) =X

j∈Ju_j(x), x ∈X.

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Spectrum of the Laplace-Beltrami operator

Let k =∂_k∂^∗_k +∂^∗_k∂_k be the complex Laplace-Beltrami operator of (Lk,hk,∇k) with respect to some Hermitian metric ω on X. Let ^p,qk,E the operator acting on C^∞(X,Λ^p,qT_X^∗ ⊗L_k ⊗E), where (E,h_E) is a holomorphic Hermitian vector bundle of rankr.

We are interested in analyzing the (discrete) spectrum of the elliptic operator ^p,qk,E. Since the curvature is θ_k 'ku, it is better to renormalize and to consider instead _2πk¹ ^p,qk,E. For λ ∈R, we define

N_k^p,q(λ) = dimM

eigenspaces of 1

u_J(x) =X

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Spectrum of the Laplace-Beltrami operator

Let k =∂_k∂^∗_k +∂^∗_k∂_k be the complex Laplace-Beltrami operator of (Lk,hk,∇k) with respect to some Hermitian metric ω on X. Let ^p,qk,E the operator acting on C^∞(X,Λ^p,qT_X^∗ ⊗L_k ⊗E), where (E,h_E) is a holomorphic Hermitian vector bundle of rankr. We are interested in analyzing the (discrete) spectrum of the elliptic operator ^p,qk,E.

Since the curvature is θ_k 'ku, it is better to renormalize and to consider instead _2πk¹ ^p,qk,E. For λ ∈R, we define

N_k^p,q(λ) = dimM

eigenspaces of 1

u_J(x) =X

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Spectrum of the Laplace-Beltrami operator

Let k =∂_k∂^∗_k +∂^∗_k∂_k be the complex Laplace-Beltrami operator of (Lk,hk,∇k) with respect to some Hermitian metric ω on X. Let ^p,qk,E the operator acting on C^∞(X,Λ^p,qT_X^∗ ⊗L_k ⊗E), where (E,h_E) is a holomorphic Hermitian vector bundle of rankr. We are interested in analyzing the (discrete) spectrum of the elliptic operator ^p,qk,E. Since the curvature is θ_k 'ku, it is better to renormalize and to consider instead _2πk¹ ^p,qk,E.

Forλ ∈R, we define N_k^p,q(λ) = dimM

eigenspaces of 1

u_J(x) =X

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Spectrum of the Laplace-Beltrami operator

Let k =∂_k∂^∗_k +∂^∗_k∂_k be the complex Laplace-Beltrami operator of (Lk,hk,∇k) with respect to some Hermitian metric ω on X. Let ^p,qk,E the operator acting on C^∞(X,Λ^p,qT_X^∗ ⊗L_k ⊗E), where (E,h_E) is a holomorphic Hermitian vector bundle of rankr. We are interested in analyzing the (discrete) spectrum of the elliptic operator ^p,qk,E. Since the curvature is θ_k 'ku, it is better to renormalize and to consider instead _2πk¹ ^p,qk,E. For λ ∈R, we define

N_k^p,q(λ) = dimM

eigenspaces of 1

2πk^p,qk,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_J(x) =X

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Spectrum of the Laplace-Beltrami operator

N_k^p,q(λ) = dimM

eigenspaces of 1

|u₁(x)| ≥ ··· ≥ |u_s(x)|>|u_s+1(x)|=···=|u_n(x)|= 0.

For a multi-index J ={j1 <j2 < . . . <jq} ⊂ {1, . . . ,n}, set u_J(x) =X

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Spectrum of the Laplace-Beltrami operator

N_k^p,q(λ) = dimM

eigenspaces of 1

|u₁(x)| ≥ ··· ≥ |u_s(x)|>|u_s+1(x)|=···=|u_n(x)|= 0.

For a multi-index J ={j1 <j2 < . . . <jq} ⊂ {1, . . . ,n}, set u_J(x) =X

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Fundamental spectral theory results

Consider the “spectral density functions” ν_u, ν_u defined by ν_u(λ)

ν_u(λ)

= 2^s−n|u₁| ··· |u_s| Γ(n−s + 1)

X

(p1,...,ps)∈N^s

h

λ−X

(2p_j + 1)|u_j|in−s +

. (where 0⁰ = 0 for ν_u, resp. 0⁰ = 1 for ν_u).

Theorem ([D] 1985)

The spectrum of _2πk¹ ^p,qk on C^∞(X,Λ^p,qT_X^∗ ⊗L_k ⊗E) has an asymptotic distribution of eigenvalues such that ∀λ∈R

r n

p

X

|J|=q

Z

X

ν_u(2λ+u_{J−u_J)dV_ω ≤lim inf

k→+∞k⁻ⁿN_k^p,q(λ)≤

≤lim sup

k→+∞

k⁻ⁿN_k^p,q(λ)≤r n

p

X

|J|=q

Z

X

ν_u(2λ+u_{J −u_J)dV_ω where r = rank(E). By monotonicity, as ν_u(λ) = limλ→0+ν_u(λ), all four terms are equal for λ∈R rD with D countable.

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Fundamental spectral theory results

ν_u(λ)

= 2^s−n|u₁| ··· |u_s| Γ(n−s + 1)

X

(p1,...,ps)∈N^s

h

λ−X

(2p_j + 1)|u_j|in−s +

Theorem ([D] 1985)

r n

p

X

|J|=q

Z

X

≤lim sup

k→+∞

p

X

|J|=q

Z

X

ν_u(2λ+u_{J −u_J)dV_ω where r = rank(E).

By monotonicity, as ν_u(λ) = limλ→0+ν_u(λ), all four terms are equal for λ∈R rD with D countable.

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Fundamental spectral theory results

ν_u(λ)

= 2^s−n|u₁| ··· |u_s| Γ(n−s + 1)

X

(p1,...,ps)∈N^s

h

λ−X

(2p_j + 1)|u_j|in−s +

Theorem ([D] 1985)

r n

p

X

|J|=q

Z

X

≤lim sup

k→+∞

p

X

|J|=q

Z

X

ν_u(2λ+u_{J −u_J)dV_ω where r = rank(E). By monotonicity, asν_u(λ) = limλ→0+ν_u(λ), all four terms are equal for λ∈R rD with D countable.

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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 ' ¹₂∆_k,E + curvature terms) for each (p,q). Important special case λ= 0 (harmonic forms)

X

|J|=q

ν_u(u_{J−u_J)dV_ω = (−1)^quⁿ n!.

Corollary (Laurent laeng, 2002)

For λ_k →0 slowly enough, i.e. with k^2+2/b²λ_k →+∞, one has lim inf

k→+∞k⁻ⁿN_k,E^0,0(λk)≥ r n!

Z

X(u,0)

uⁿ+ Z

X(u,1)

uⁿ

where X(u,q) =q-index set=

x ∈X/u(x) has signature (n−q,q) .

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Approximate cohomology lower bounds

∆_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 ' ¹₂∆_k,E + curvature terms) for each (p,q).

Important special case λ= 0 (harmonic forms) X

|J|=q

k→+∞k⁻ⁿN_k,E^0,0(λk)≥ r n!

Z

X(u,0)

uⁿ+ Z

X(u,1)

uⁿ

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Approximate cohomology lower bounds

|J|=q

k→+∞k⁻ⁿN_k,E^0,0(λk)≥ r n!

Z

X(u,0)

uⁿ+ Z

X(u,1)

uⁿ

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Approximate cohomology lower bounds

|J|=q

k→+∞k⁻ⁿN_k,E^0,0(λk)≥ r n!

Z

X(u,0)

uⁿ+ Z

X(u,1)

uⁿ

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Proof of the lower bound

Proof. One uses the fact that for δ⁰ > δ >0 and k 1, the composition Π◦∂_k with an eigenspace projection yields an injection

M

λ∈]λ_k,δ]

eigenspace^0,0_λ ,→ M

λ∈]0,δ⁰]

eigenspace^0,1_λ .

In fact, in the holomorphic case ∂²_k = 0 implies ∂k^0,0k =^0,1k ∂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, δ⁰ =δ. In the quasi holomorphic case ∂²_k =O(k^−1/b²), one can show that ^0,1k ∂_k−∂_k^0,0k =∂^∗_k∂²_k yields a small “deviation” of the eigenvalues to [λ_k −ε, δ+ε] with ε <min(λ_k, δ⁰−δ), whence the injectivity. This implies

N_k,E^0,1(δ⁰)≥N_k,E^0,0(δ)−N_k,E^0,0(λ_k) thus

N_k,E^0,0(λ_k)≥N_k,E^0,0(δ)−N_k,E^0,1(δ⁰), QED

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Proof of the lower bound

M

λ∈]λ_k,δ]

λ∈]0,δ⁰]

eigenspace^0,1_λ .

In fact, in the holomorphic case ∂²_k = 0 implies ∂k^0,0k =^0,1k ∂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, δ⁰ =δ.

In the quasi holomorphic case ∂²_k =O(k^−1/b²), one can show that ^0,1k ∂_k−∂_k^0,0k =∂^∗_k∂²_k yields a small “deviation” of the eigenvalues to [λ_k −ε, δ+ε] with ε <min(λ_k, δ⁰−δ), whence the injectivity. This implies

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Proof of the lower bound

M

λ∈]λ_k,δ]

λ∈]0,δ⁰]

eigenspace^0,1_λ .

In the quasi holomorphic case ∂²_k =O(k^−1/b²), one can show that ^0,1k ∂_k−∂_k^0,0k =∂^∗_k∂²_k yields a small “deviation” of the eigenvalues to [λ_k −ε, δ+ε] with ε <min(λ_k, δ⁰−δ), whence the injectivity.

This implies

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Proof of the lower bound

M

λ∈]λ_k,δ]

λ∈]0,δ⁰]

eigenspace^0,1_λ .

In the quasi holomorphic case ∂²_k =O(k^−1/b²), one can show that ^0,1k ∂_k−∂_k^0,0k =∂^∗_k∂²_k yields a small “deviation” of the eigenvalues to [λ_k −ε, δ+ε] with ε <min(λ_k, δ⁰−δ), whence the injectivity.

This implies

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Transcendental holomorphic Morse inequalities

Conjecture on Morse inequalities Let γ ∈H_BC^1,1(X,R). Then

Vol(γ)≥ sup

u∈γ,u∈C^∞

Z

X(u,≤1)

uⁿ.

(One could even suspect equality, an even stronger conjecture !).

If one sets by definition Vol(γ) = sup

u∈γ λ→0lim+

lim inf

k→+∞N_k^0,0(λ)

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_BC^1,1(X,R), set Vol(γ) = 0 if γ 63 any current T ≥0, and otherwise set Vol(γ) = sup

T∈γ,T=u0+i∂∂ϕ≥0

Z

X

T_acⁿ , u₀ ∈C^∞.

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Transcendental holomorphic Morse inequalities

Vol(γ)≥ sup

u∈γ,u∈C^∞

Z

X(u,≤1)

uⁿ.

u∈γ λ→0lim+

lim inf

k→+∞N_k^0,0(λ)

for the eigenspaces of the sequence (Lk,hk,∇k) approximating ku, then the above expected lower bound is a theorem!

There is however a stronger & more usual definition of the volume. Definition

Z

X

T_acⁿ , u₀ ∈C^∞.

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Transcendental holomorphic Morse inequalities

Vol(γ)≥ sup

u∈γ,u∈C^∞

Z

X(u,≤1)

uⁿ.

u∈γ λ→0lim+

lim inf

k→+∞N_k^0,0(λ)

for the eigenspaces of the sequence (Lk,hk,∇k) approximating ku, then the above expected lower bound is a theorem!

There is however a stronger & more usual definition of the volume.

Definition

Z

X

T_acⁿ , u₀ ∈C^∞.

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Transcendental holomorphic Morse inequalities (2)

The conjecture on Morse inequalities is known to be true when γ =c1(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,`, and one gets positive currents right away by putting

T_k = i

2kπ∂∂logX

`

|σ_k_,`|²_h+ i

2πΘ_L,h ≥0

(the volume estimate can be derived from there by Fujita).

In the “quasi-holomorphic” case, one only gets eigenfunctions σ_k,` with small eigenvalues, and the positivity of Tk is a priori lost. Conjectural corollary (fundamental volume estimate)

Let X be compact K¨ahler, dimX =n, and α, β ∈H^1,1(X,R) be nef cohomology classes. Then

Vol(α−β)≥αⁿ−nαⁿ⁻¹·β.

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Transcendental holomorphic Morse inequalities (2)

T_k = i

2kπ∂∂logX

`

|σ_k_,`|²_h+ i

2πΘ_L,h ≥0 (the volume estimate can be derived from there by Fujita).

In the “quasi-holomorphic” case, one only gets eigenfunctions σ_k,` with small eigenvalues, and the positivity of Tk is a priori lost. Conjectural corollary (fundamental volume estimate)

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Transcendental holomorphic Morse inequalities (2)

T_k = i

2kπ∂∂logX

`

|σ_k_,`|²_h+ i

In the “quasi-holomorphic” case, one only gets eigenfunctions σ_k,`

with small eigenvalues, and the positivity of Tk is a priori lost.

Conjectural corollary (fundamental volume estimate)

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Transcendental holomorphic Morse inequalities (2)

T_k = i

2kπ∂∂logX

`

|σ_k_,`|²_h+ i

In the “quasi-holomorphic” case, one only gets eigenfunctions σ_k,`

with small eigenvalues, and the positivity of Tk is a priori lost.

Conjectural corollary (fundamental volume estimate)

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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≥αⁿ−nαⁿ⁻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-Nyström when X is projective, using deep facts on Monge-Ampère operators and upper envelopes. Xiao and Popovici also proved in the Kähler case that

αⁿ−nαⁿ⁻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).

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Known results on holomorphic Morse inequalities

Recently (2016), the volume estimate for γ =α−β transcendental has been established by D. Witt-Nyström when X is projective, using deep facts on Monge-Ampère operators and upper envelopes. Xiao and Popovici also proved in the Kähler case that

αⁿ−nαⁿ⁻¹·β >0 ⇒ Vol(α−β)>0

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Known results on holomorphic Morse inequalities

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 αⁿ−nαⁿ⁻¹·β >0 ⇒ Vol(α−β)>0

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Known results on holomorphic Morse inequalities

and α−β contains a K¨ahler current.

(The proof is short, once the Calabi-Yau theorem is taken for granted).

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Known results on holomorphic Morse inequalities

and α−β contains a K¨ahler current.

(The proof is short, once the Calabi-Yau theorem is taken for granted).

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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. 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.

In the sequel of this lecture, we aim to investigate this construction and look for potential applications, especially to transcendental holomorphic Morse inequalities ...

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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.

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.

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Projective vs K¨ ahler vs non K¨ ahler varieties

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Projective vs K¨ ahler vs non K¨ ahler varieties

What for non K¨ahler compact complex manifolds?

Surprising facts (?)

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Projective vs K¨ ahler vs non K¨ ahler varieties

– Every compact complex manifold X carries a“very ample”

complex Hilbert bundle, produced by means of a natural Bergman space construction.

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Projective vs K¨ ahler vs non K¨ ahler varieties

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Projective vs K¨ ahler vs non K¨ ahler varieties

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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.

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Tubular neighborhoods (thanks to Grauert)

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.

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Tubular neighborhoods (thanks to Grauert)

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.

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Tubular neighborhoods (thanks to Grauert)

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.

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Tubular neighborhoods (thanks to Grauert)

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.