### 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

J.-P. Demailly, Virtual Conference CAG at UWO, May 7, 2020 Bergman bundles and applications to geometry 1/22

### 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...)

### 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) 7→ (z,exp_{z}(ξ)), z ∈ X, ξ ∈ T_{X}_{,z} be the
associated geodesic exponential map.

J.-P. Demailly, Virtual Conference CAG at UWO, May 7, 2020 Bergman bundles and applications to geometry 3/22

### 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}

exp_{z}(ξ) = X

α,β∈N^{n}

a_{α β}(z)ξ^{α}ξ^{β}, exph_{z}(ξ) = X

α∈N^{n}

a_{α}_{0}(z)ξ^{α}.
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^{0} of the diagonal in X × X.

Notation

With the identification X '_{diff} X, let logh : X × X ⊃ V^{0} → T_{X} be the
inverse diffeomorphism of exph and

U_{ε} = {(z,w) ∈ V^{0} ⊂ 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.

### Such tubular neighborhoods are Stein

In the special case X = C^{n}, U_{ε} = {(z,w) ∈ C^{n} × C^{n} ; |z − w| < ε}

is of course Stein since

|z − w|^{2} = |z|^{2} + |w|^{2} − 2ReP
z_{j}w_{j}
and (z,w) 7→ ReP

z_{j}w_{j} is pluriharmonic.

J.-P. Demailly, Virtual Conference CAG at UWO, May 7, 2020 Bergman bundles and applications to geometry 5/22

### 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.

### 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 b V :

Z

p^{−1}(K)

i^{n}^{2}f(z,w)∧f (z,w) ∧γ(z)^{n} < +∞, ∀K b V.

(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 CAG at UWO, May 7, 2020 Bergman bundles and applications to geometry 7/22

### 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

kf(z_{0})k^{2} =
Z

p^{−1}(z0)

i^{n}^{2}f(z_{0},w) ∧f(z_{0},w) < +∞.

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^{0} > ε, there is a restriction map B_{ε}^{0}_{,z}_{0} → B_{ε,z}_{0} such that
B_{ε,z}_{0} is the L^{2} completion of B_{ε}^{0}_{,z}_{0}/m_{z}_{0}B_{ε}^{0}_{,z}_{0}.

Question

Is there a “complex structure” on B_{ε} such that “B_{ε} = O(B_{ε})” ?

### 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)dw_{1} ∧...∧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 b 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_{ε}^{0} → F_{ε}^{1} 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 CAG at UWO, May 7, 2020 Bergman bundles and applications to geometry 9/22

### 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).

The proof is a direct consequence of H¨ormander’s L^{2} 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 = H^{0}(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 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^{0}) = Hausdorff distance of the unit balls of S, S^{0}.

J.-P. Demailly, Virtual Conference CAG at UWO, May 7, 2020 Bergman bundles and applications to geometry 11/22

### 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
e_{α}(z,w) = π^{−n/2}ε^{−|α|−n}

s

(|α|+n)!

α_{1}!. . . α_{n}! (w − z)^{α}, α ∈ N^{n}.
It is non holomorphic! The (0,1)-connection ∇^{0,1} = ∂ is given by

∇^{0,1}e_{α} = ∂_{z}e_{α}(z,w) = ε^{−1} X

1≤j≤n

q

α_{j}(|α|+ n) d z_{j} ⊗e_{α−c}_{j}
where c_{j} = (0, ...,1, ...,0) ∈ N^{n}.

### 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

Θe_{ε}(v ⊗ ξ) = hΘ_{B}_{ε}_{,h}σ(v,Jv)ξ, ξi_{h}
for v ∈ T_{X} and ξ = P

αξ_{α}e_{α} ∈ B_{ε}.
Definition

One says that the curvature tensor is Griffiths positive if
Θe_{ε}(v ⊗ξ) > 0, ∀0 6= v ∈ T_{X}, ∀0 6= ξ ∈ B_{ε},
and Nakano positive if

Θeε(τ) > 0, ∀0 6= τ ∈ T_{X} ⊗Bε.

J.-P. Demailly, Virtual Conference CAG at UWO, May 7, 2020 Bergman bundles and applications to geometry 13/22

### Calculation of the curvature tensor for X = C

^{n}

A simple calculation of ∇^{2} in the orthonormal frame (eα) leads to:

Formula

In the model case X = C^{n}, the curvature tensor of the Bergman bundle
(B_{ε},h) is given by

Θe_{ε}(v ⊗ξ) = ε^{−2}X

α∈N^{n}

X

j

√α_{j} ξ_{α−c}_{j}v_{j}

2

+ X

j

(|α|+n) |ξ_{α}|^{2}|v_{j}|^{2}

! .

Consequence

In C^{n}, 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 kξk^{2} = P

α|ξ_{α}|^{2}.
However there is convergence for all ξ = P

αξ_{α}e_{α} ∈ B_{ε}^{0}, ε^{0} > ε, since
then P

α(ε^{0}/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 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
Θe_{ε}(z,v ⊗ ξ) =

+∞

X

p=0

ε^{−2+p}Q_{p}(z,v ⊗ ξ), v ∈ T_{X}, ξ ∈ B_{ε}

where Q_{0}(z,v ⊗ξ) = Q_{0}(v ⊗ ξ) is given by the model case C^{n} :
Q_{0}(v ⊗ξ) = ε^{−2}X

α∈N^{n}

X

j

√α_{j} ξ_{α−c}_{j}v_{j}

2

+ X

j

(|α|+n) |ξ_{α}|^{2}|v_{j}|^{2}

!
.
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 CAG at UWO, May 7, 2020 Bergman bundles and applications to geometry 15/22

### 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.

Wang’s formula is however in integral form and not completely explicit.

Here, one simply uses the real analytic Taylor expansion of
logh : X × X → T_{X} (inverse diffeomorphism of exph)

logh_{z}(w) = w −z + X

z_{j}a_{j}(w − z) +X

z_{j}a^{0}_{j}(w − z)

+X

z_{j}z_{k}b_{jk}(w −z) +X

z_{j}z_{k}b^{0}_{jk}(w − z)

+X

z_{j}z_{k}c_{jk}(w − z) +O(|z|^{3}),

which is used to compute the difference with the model case C^{n}, for
which logh_{z}(w) = w − z.

### Potential application: invariance of plurigenera for polarized families of compact K¨ ahler manifolds ?

Conjecture

Let π : X → S be a proper holomorphic map defining a family of
smooth compact K¨ahler manifolds over an irreducible base S. Assume
that the family admits a polarization, i.e. a closed smooth (1,1)-form ω
such that ω_{|X}_{t} is positive definite on each fiber X_{t} := π^{−1}(t). Then the
plurigenera

p_{m}(X_{t}) = h^{0}(X_{t},mK_{X}_{t}) are independent of t for all m ≥ 0.

The conjecture is known to be true for a projective family X → S:

• Siu and Kawamata (1998) in the case of varieties of general type

• Siu (2000) and P˘aun (2004) in the arbitrary projective case

No algebraic proof is known in the latter case; one deeply uses the L^{2}
estimates of the Ohsawa-Takegoshi extension theorem.

J.-P. Demailly, Virtual Conference CAG at UWO, May 7, 2020 Bergman bundles and applications to geometry 17/22

### Invariance of plurigenera: strategy of proof (1)

It is enough to consider the case of a family X → ∆ over the disc, such that there exists a relatively ample line bundle A over X.

Given s ∈ H^{0}(X_{0},mK_{X}_{0}), the point is to show that it extends into
es ∈ H^{0}(X,mK_{X}), and for this, one only needs to produce a hermitian
metric h = e^{−ϕ} on K_{X} such that:

• Θ_{h} = i∂∂ϕ ≥ 0 in the sense of currents

• |s|^{2}_{h} = |s|^{2}e^{−ϕ} ≤ 1, i.e. ϕ ≥ log|s| on X0.

The Ohsawa-Takegoshi theorem then implies the existence of se.

To produce h = e^{−ϕ}, one produces inductively (also by O-T !) sections
of σ_{p}_{,j} of L_{p} := A+pK_{X} such that:

• (σ_{p,j}) generates L_{p} for 0 ≤ p < m

• σ_{p,j} extends (σ_{p−m,j}s)_{|X}_{0} to X for p ≥ m

•

Z

X

P

j |σ_{p,j}|^{2}
P

j |σ_{p−1,j}|^{2} ≤ C for p ≥ 1.

### Invariance of plurigenera: strategy of proof (2)

By H¨older, the L^{2} estimates imply R

X

P

j |σ_{p,j}|^{2}1/p

≤ C for all p, and
using the fact that lim ^{1}_{p}Θ_{A} = 0, one can take

ϕ = lim sup_{p→+∞}ϕ_{p}, ϕ_{p} := _{p}^{1} logP

j |σ_{p,j}|^{2}.

Idea. In the polarized K¨ahler case, use the Bergman bundle B_{ε} → X
instead of an ample line bundle A → X. This amounts to applying the
Ohsawa-Takegoshi L^{2} extension on Stein tubular neighborhoods

U_{ε} ⊂ X × X, with projections pr_{1} : U_{ε} → X and π : X → ∆.

Proposition

In the polarized K¨ahler case (X, ω), shrinking from U_{ε} to U_{ρε} with
ρ < 1, the B_{ε} curvature estimate gives

ϕ_{p} := 1

p logX

j

kσ_{p,j}k^{2}_{U}_{ρε} ⇒ i∂∂ϕ_{p} ≥ − C

ε^{2}ρ^{2}(C^{0} − ϕ_{p})ω.

This implies that ϕ = lim supϕ_{p} satisfies ψ := −log(C^{00} − ϕ)
quasi-psh, but yields invariance of plurigenera only for ε → +∞.

J.-P. Demailly, Virtual Conference CAG at UWO, May 7, 2020 Bergman bundles and applications to geometry 19/22

### Transcendental holomorphic Morse inequalities

Conjecture

Let X be a compact n-dimensional complex manifold and
α ∈ H_{BC}^{1,1}(X,R) a Bott-Chern class, represented by closed real
(1,1)-forms modulo ∂∂ exact forms. Set

Vol(α) = sup

T=α+i∂∂ϕ≥0

Z

X

T_{ac}^{n} , T ≥ 0 current.

Then

Vol(α) ≥ sup

u∈{α}, u∈C^{∞}

Z

X(u,0)

u^{n}
where

X(u,0) = 0-index set of u =

x ∈ X ; u(x) positive definite .

Conjectural corollary (fundamental volume estimate)

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

Vol(α− β) ≥ α^{n} − nα^{n−1} ·β.

### Transcendental Morse: known facts & beyond

The conjecture on Morse inequalities is known to be true when

α = c_{1}(L) is the class of a line bundle ([D-1985]), and the corollary can
be derived from this when α, β are integral classes (by [D-1993] and
independently by [Trapani, 1993]).

Recently, the volume estimate for α, β transcendental has been

established by D. Witt-Nystr¨om when X is projective, and Xiao-Popovici
even proved in general that Vol(α −β) > 0 if α^{n} − nα^{n−1} ·β > 0.

Idea. In the general case, one can find a sequence of non holomorphic
hermitian line bundles (L_{m},h_{m}) such that

mα = Θ_{L}_{m}_{,h}_{m} + γ_{m}^{2,0} + γ^{0,2}_{m} , γ_{m} → 0.

As U_{ε} is Stein, γ^{0,2}_{m} = ∂v_{m}, v_{m} → 0, and pr^{∗}_{1}L_{m} becomes a holomorphic
line bundle with curvature form Θ_{pr}^{∗}

1 Lm ' mpr^{∗}_{1}α.

Then apply L^{2} direct image (pr_{1})^{L}_{∗}^{2} and use Bergman estimates instead
of dimension counts in Morse inequalities.

J.-P. Demailly, Virtual Conference CAG at UWO, May 7, 2020 Bergman bundles and applications to geometry 21/22