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L’INSTITUT FOURIER
Robert BERMAN
Corrigendum to: Holomorphic Morse inequalities on manifolds with
boundary
Tome 58, no 1 (2008), p. 377-381.
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Ann. Inst. Fourier, Grenoble
58, 1 (2008) 377-381
CORRIGENDUM TO:
HOLOMORPHIC MORSE INEQUALITIES ON
MANIFOLDS WITH BOUNDARY
Annales de l’institut Fourier, vol. 55 (2005), no 4,
p. 1055–1103
by Robert BERMAN
Abstract. — A statement in the paper “Holomorphic Morse inequalities on
manifolds with boundary” saying that the holomorphic Morse inequalities for an
hermitian line bundle L over X are sharp as long as L extends as semi-positive
bundle over a Stein-filling is corrected, by adding certain assumptions. A more
general situation is also treated.
Résumé. — Nous corrigeons l’énoncé qui affirme que les inégalités de Morse
holomorphes pour un fibré hermitien en droites L sur X sont optimales tant que
L s’étend comme un fibré semi-positif sur un remplissage Stein, paru dans l’article
“Inégalités de Morse holomorphes sur des variétés à bord”. Nous ajoutons certaines
conditions et considérons une situation plus générale.
Let (L, φ) be a hermitian holomorphic line bundle over a closed (i.e. com-
pact without boundary) complex hermitian manifold (Y, ω) of dimension n
and let X = {ρ 6 0} be a strongly pseudoconcave domain in Y with smooth
boundary. Let dc := i(−∂+∂)/4π and denote by ddcφ(= i∂∂φ/2π) the nor-
malized curvature form of φ. If η is a (1, 1)-form we will write ηp := ηp/p!
and we will denote by {η}y the hermitian linear operator on T 1,0(Y )y (or
on some specified subbundle) corresponding to η (using a fixed metric form
ω).
The purpose here is to correct a statement in [1] saying that the holo-
morphic Morse inequalities on X obtained in [1] are always sharp when
the curvature form ddcφ is semi-positive on all of Y and Y −X is a Stein
manifold. The correct statement, as shown below, is obtained by adding
Keywords: Line bundles, cohomology, harmonic forms, holomorphic sections, Bergman
kernel.
Math. classification: 32L10, 32A25, 32L20.
378 Robert BERMAN
the assumption (3) about the compatibility between the curvature of the
line bundle L and the curvature of the boundary ∂X. The (non-Stein) case
when Y − X contains an irreducible divisor will also be treated here. In
fact, in a large class of examples (where Y is a ruled manifold and X is
a disc bundle) the assumptions below are essentially necessary as will be
further investigated in a forthcoming paper.
Recall that the boundary contribution to the holomorphic Morse inequal-
ities in [1] (i.e. inequalities for lim supk k−n dim H0(X, Lk)) is obtained by
integrating the following form over ∂X:
(1) µ :=
∫ T
0
(ddcφ + tddcρ)n−1 ∧ dcρ)dt/(n− 1)!,
where T is the following function on ∂X, that will be referred to as the
slope function:
(2) T = sup
{
t > 0: (ddcφ + tddcρ)x > 0 along T 1,0(∂X)x
}
.
The assumptions
First assume that the defining function −ρ of the pseudoconvex manifold
Y −X may be chosen to be smooth with ddc(−ρ) > 0 in (Y −X)−Z, where
Z is either a point or an irreducible divisor in Y −X (1) . Moreover, assume
that there is a diffeomorphism between (Y −X)−Z and ∂(Y −X)×]−1, 0]
such that −ρ corresponds to projection on the second factor. Then the
slope function T extends to a function on (Y −X) − Z, by replacing ∂X
in (2) with ρ−1(c) for any c in [0, 1[. Now assume the following
(3) compatibility: T is a C1-function of ρ,
(in particular, T is constant on ∂X). Next, if Z is a point it is assumed
that −ρ extends to a smooth function on X − Y such that ddc(−ρ) > 0 on
all of Y −X. If Z is an irreducible divisor it is assumed that T is bounded
from above on Y −X, that
(4)
∫
Z
(ddcφ)n−1 = 0,
and that
(5) ddc(−ρ) = [Z] + β,
in the sense of currents on Y −X, where β is a semi-positive smooth form.
(1) i.e. a (possibly singular) connected compact closed complex submanifold of codimen-
sion one in Y −X. Then the integration current [Z] is well-defined.
ANNALES DE L’INSTITUT FOURIER
CORRIGENDUM: MORSE INEQUALITIES 379
Sharp Morse inequalities for fillings
Proposition 1. — Assume that the (normalized) curvature ddcφ of L
is a smooth semi-positive form and that the defining function −ρ of Y −X
satisfies the assumptions in the previous section. Then∫
∂X
µ =
∫
Y−X
(ddcφ)n.
Given the equality in the proposition above the holomorphic Morse in-
equalities on X are sharp as shown in [1]. To prove the proposition first note
that the form µ (formula (1)) is well-defined on any hypersurface ρ−1(c)
with c in [0, 1[ by using the extended function slope function T . The fol-
lowing lemma corrects a statement in section 7.1 (remark 7.4 and above)
in [1], where the integral Iε below was incorrectly assumed to vanish in the
general case. In its statement and proof we will use the convention that
the orientation on the level surfaces {ρ = c} is such that the restriction
of dc(−ρ)(ddc(−ρ))n−1 is a volume form, i.e. the orientation obtained by
realizing {ρ = c} as the boundary of the domain {(−ρ) 6 −c} in Y .
Lemma 2. — For each fixed positive number ε the following identity
holds:
(6) 0 =
∫
ρ−1(0)
µ−
∫
ρ−1(1−ε)
µ−
∫
ρ−1([0,1−ε]
(ddcφ)n − Iε,
where
Iε :=
∫
ρ−1([0,1−ε]
(ddcφ + Tddcρ)n−1 ∧ dc(−ρ) ∧ dT.
Proof. — Denote by Xε(0) the following manifold with boundary (and
corners), realized as a subset of the real cotangent bundle of Y :
Xε(0) :=
{
tdc(−ρ)y : y ∈ Y −X, 0 6 t 6 T, 0 6 ρ(y) 6 1− ε
}
.
Hence, Xε(0) is topologically a fiber bundle of closed intervals over a subset
of X−Y and when ε tends to zero, the base of Xε(0) tends to Y −X. Next,
observe that the following form on Xε(0) is closed:
η := (ddcφ + d(tdcρ))n.
Hence, Stokes theorem gives after expansion of η
(7)
∫
∂(Xε(0))
(ddcφ + tddcρ)n + (ddcφ + tddcρ)n−1 ∧ dcρ ∧ dt = 0.
Decomposing the boundary of Xε(0) into the following components
∂(Xε(0)) = {ρ = 0} ∪ {ρ = 1− ε} ∪ {t = 0} ∪ {t = T}
TOME 58 (2008), FASCICULE 1
380 Robert BERMAN
then proves the identity (6), by identifying the terms in (6) with the inte-
grals over the components above (in the corresponding order). To see this,
note that the first term in (7) gives no contribution on the first two compo-
nents (for degree reasons). Moreover, the first term in (7) vanishes on the
fourth component {t = T}. Indeed, at the point y the form (ddcφ+Tddcρ)n
is proportional to the determinant of the operator {ddcφ + Tddcρ} on
T 1,0(Y )y. The determinant vanishes since, by the definition of T , the opera-
tor has a zero-eigenvalue along the subspace T 1,0(ρ−1(ρ(y)) of T 1,0(Y ). 
Now if T is assumed to be a function of ρ the integrand in Iε, defined
above, may be written as a function times the form (ddcφ+Tddcρ)n−1∧∂ρ∧
∂ρ. Since the latter form at the point y is proportional to the determinant of
{ddcφ+Tddcρ} restricted to T 1,0(ρ−1(ρ(y))y, it follows from the definition
of T that
(8) Iε = 0
vanishes for each ε. Next, we will show that the following “residue” vanishes:
(9) lim
ε→0
∫
ρ−1(1−ε)
µ = 0.
The case when Z is a point (denoted by y0) : First observe that
(10)
∫
ρ−1(1−ε)
µ 6 T (1− ε)
∫
ρ−1(1−ε)
(ddcφ)n−1 ∧ dc(−ρ),
only using that ddcρ is negative along T 1,0(ρ−1(c)). Moreover, since the
form ddcρ is assumed to be strictly negative on all of Y −X, its eigenvalues
are bounded from above by a negative constant, which shows that T is
uniformly bounded from above by a positive constant C, when ε tends to
zero. A direct application of Stokes theorem now gives
I1−ε 6 C
∫
ρ−1[1−ε,1]
(ddcφ)n−1 ∧ ddc(−ρ).
Since, by assumption, ddcρ extends smoothly over the neighbourhood
ρ−1[1 − ε, 1] of the point x0 it follows that the right hand side tends to
zero when ε tends to zero (since ρ−1[1 − ε, 1] shrinks to the point y0),
proving (9).
The case when Z is a divisor: First observe that (10) still holds, by a
similar argument, using the assumption that T is bounded from above on
Y −X. Applying Stokes theorem again now gives, using the assumption (5)
on ddcρ, that∫
ρ−1(1−ε)
µ 6 C
∫
Z
(ddcφ)n−1 +
∫
ρ−1[1−ε,1]
(ddcφ)n−1 ∧ β.
ANNALES DE L’INSTITUT FOURIER
CORRIGENDUM: MORSE INEQUALITIES 381
Since β is assumed to be smooth on all of Y −X, it follows that
lim
ε→0
∫
ρ−1(1−ε)
µ 6 C
∫
Z
(ddcφ)n−1
and the right hand side vanishes by the assumption (4), proving (9).
Finally, combining the identity (6) in Lemma 2 with the vanishing in (8)
and (9) finishes the proof of Proposition 1.
Remark. — A further correction to [1]: In formula (7.1) BqX should be
replaced with limk→∞B
q,k
X /k
n. Accordingly, in formula (7.2) BqX should
be replaced with limk→∞B
q,k
X /k
n and the sum with an integral over t.
BIBLIOGRAPHY
[1] R. Berman, “Holomorphic Morse inequalities on manifolds with boundary”, Ann.
Inst. Fourier (Grenoble) 55 (2005), no. 4, p. 1055-1103.
Robert BERMAN
Chalmers University of Technology
Department of Mathematics
Eklandag 86
412 96 Göteborg (Sweden)
robertb@math.chalmers.se
TOME 58 (2008), FASCICULE 1


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