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IONUŢ CHIOSE
On the invariance of the total Monge–Ampère volume of Hermitian metrics
Tome XXXIII, no 3 (2024), p. 575–579.
https://doi.org/10.5802/afst.1781
© les auteurs, 2024.
Les articles des Annales de la Faculté des Sciences de Toulouse sont mis
à disposition sous la license Creative Commons Attribution (CC-BY) 4.0
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Annales de la faculté des sciences de Toulouse Volume XXXIII, no 3, 2024
pp. 575-579
On the invariance of the total Monge–Ampère volume
of Hermitian metrics (∗)
Ionuţ Chiose (1)
ABSTRACT. — In this note, we describe the Hermitian metrics that leave the total
Monge–Ampère volume invariant. In particular, we give several characterizations
of the Hermitian metrics which satisfy the comparison principle for the complex
Monge–Ampère operator.
RÉSUMÉ. — Dans cette note, nous décrivons les métriques hermitiennes qui laissent
le volume total de Monge–Ampère invariant, et nous donnons plusieurs caractéri-
sations des métriques hermitiennes qui satisfont le principe de comparaison pour
l’opérateur complexe de Monge–Ampère.
Let g be a Hermitian metric on a compact complex manifold X of di-
mension n. Suppose that it satisfies the equations
i∂∂̄gk = 0, ∀ k = 1, n − 1 (1)
(we denote by the same letter g the fundamental form of the Hermitian
metric g). These metrics were introduced by Guan and Li [7] because these
metrics leave the total Monge–Ampère volume unchanged, i.e., they satisfy∫
X
(g + i∂∂̄u)n =
∫
X
gn, ∀ u ∈ PSH(X, g) (2)
where PSH(X, g) denotes the space of g-plurisubharminic functions on X,
that is the C∞ real functions u on X such that
g + i∂∂̄u ⩾ 0.
Indeed, the equality (2) follows from (1) from Stokes’ theorem applied twice.
In this note, we prove the converse of this statement, namely we show
that a Hermitian metric which satisfies (2) has to satisfy (1). This answers
Question 29 in [5] (see also Problem 11.1 in [4]).
(*) Reçu le 13 septembre 2022, accepté le 16 novembre 2022.
Keywords: Hermitian metrics, Monge–Ampère operator, Comparison principle.
(1) Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest
014700, Romania — ionut.chiose@imar.ro
Article proposé par Vincent Guedj.
– 575 –
Ionuţ Chiose
Theorem 1. — Let (X, g) be a compact complex Hermitian manifold of
dimension n such that ∫
X
(g + i∂∂̄u)n =
∫
X
gn (3)
for any smooth g-plurisubharmonic function u on X. Then
i∂∂̄gk = 0, ∀ k = 1, n − 1. (4)
Proof. — First note that if u is an arbitrary C∞ real function on X,
then there exists ε0 = ε0(u) > 0 such that εu is g-plurisubharmonic for
ε ∈ (−ε0, ε0). Therefore, for any u ∈ C∞(X,R), we have∫
X
(g + εi∂∂̄u)n =
∫
X
gn, ∀ ε ∈ (−ε0, ε0) (5)
If we expand the expression on the left and write it as a polynomial in ε, we
obtain that ∫
X
gk ∧ (i∂∂̄u)n−k = 0, ∀ k = 1, n − 1. (6)
Now fix k ∈ {1, 2, . . . , n − 1} and if u1, u2, . . . , un−k are arbitrary C∞ func-
tions on X and t1, t2, . . . , tn−k are arbitrary real numbers, then, from∫
X
gk ∧
(
i∂∂̄
n−k∑
i=1
tiui
)n−k
= 0 (7)
it follows that ∫
X
gk ∧ i∂∂̄u1 ∧ i∂∂̄u2 ∧ · · · ∧ i∂∂̄un−k = 0. (8)
Now use Stokes’ theorem twice to obtain that, for arbitrary C∞ functions
u1, u2, . . . , un−k on X we have∫
X
u1i∂∂̄g
k ∧ i∂∂̄u2 ∧ · · · ∧ i∂∂̄un−k = 0. (9)
Since the function u1 is arbitrary on X, we obtain that
i∂∂̄gk ∧ i∂∂̄u2 ∧ · · · ∧ i∂∂̄un−k = 0 (10)
on X for any u2, . . . , un−k ∈ C∞(X,R). It is clear that we can let the func-
tions u2, . . . , un−k be complex valued. Fix x ∈ X and let z1, z2, . . . , zn be
local coordinates around x. If
i∂∂̄gk =
∑
i1<···<ik+1
j1<...jk+1
gi1...ik+1 j̄1...j̄k+1dzi1 ∧· · ·∧dzik+1 ∧dz̄j1 ∧· · ·∧dz̄jk+1 (11)
then let
{l1, . . . , ln−k−1} = {1, . . . , n} \ {i1, . . . , ik+1} (12)
– 576 –
On the invariance of the total Monge–Ampère volume of Hermitian metrics
and
{m1, . . . , mn−k−1} = {1, . . . , n} \ {j1, . . . , jk+1} (13)
and take u2 = zl1 z̄m1 , . . . , un−k = zln−k−1 z̄mn−k−1 near x to obtain
gi1...ik+1 j̄1...j̄k+1(x) = 0 (14)
hence i∂∂̄gk = 0. □
Combining the above result with [6], we obtain the following
Theorem 2. — Let X be a compact complex manifold of dimension n
with a Hermitian metric g. Then the following are equivalent:
(i) i∂∂̄g = i∂g ∧ ∂̄g = 0;
(ii) i∂∂̄g = i∂∂̄g2 = 0;
(iii) i∂∂̄gk = 0, ∀ k = 1, n − 1;
(iv) i∂∂̄g ⩾ 0, i∂g ∧ ∂̄g ⩾ 0;
(v)
∫
{u<v}
(g + i∂∂̄v)n ⩽
∫
{u<v}
(g + i∂∂̄u)n, ∀ u, v ∈ PSH(X, g);
(vi)
∫
X
(g + i∂∂̄u)n =
∫
X
gn, ∀ u ∈ PSH(X, g).
Proof. — It is clear that (i), (ii) and (iii) are equivalent and that they im-
ply (iv). Proposition 3.2 in [6] shows that (iv) implies (v). If u ∈ PSH(X, g),
then u − C < 0 < u + C for a constant C large enough. This shows that (v)
implies (vi) since u − C, 0, u + C ∈ PSH(X, g). Finally, (vi) implies (iii) from
the above Theorem 1. □
Remark 3. — (v) is called the comparison principle and it plays a crucial
role in the pluripotential study of the complex Monge–Ampère equation.
Remark 4. — The above result implies that a Hermitian metric g on a
compact complex manifold which satisfies i∂∂̄g ⩾ 0 and i∂g∧∂̄g ⩾ 0 actually
satisfies i∂∂̄g = i∂g ∧ ∂̄g = 0, and the proof goes through Proposition 3.2
in [6]. However, there is a more direct proof of this fact. Indeed, from
d(ign−2 ∧ ∂̄g) = (n − 2)gn−3 ∧ i∂g ∧ ∂̄g + gn−2 ∧ i∂∂̄g (15)
and Stokes’ theorem, it follows that
(n − 2)
∫
X
gn−3 ∧ i∂g ∧ ∂̄g +
∫
X
gn−2 ∧ i∂∂̄g = 0 (16)
and, since both integrals are positive, it follows that they have to be zero.
Therefore gn−3 ∧ i∂g ∧ ∂̄g = gn−2 ∧ i∂∂̄g = 0, that is the trace measures
of the positive currents i∂g ∧ ∂̄g and i∂∂̄g with respect to g are zero, hence
they have to be zero.
– 577 –
Ionuţ Chiose
Remark 5. — If a compact complex manifold admits a Hermitian metric
g as above, i.e., it satisfies i∂∂̄g = i∂∂̄g2 = 0, and if p : Y → X is a blowup
with a smooth center, then Y supports a Hermitian metric g′ with the same
property. Indeed, if E is the exceptional divisor, then the line bundle O(−[E])
has a metric with curvature β such that g′ = Np∗g + β is positive for N a
large enough constant. Then it is trivial to show that g′ satisfies the same
equations as g. However, it is not true that this property is a bimeromorphic
invariant, and this follows from the fact that a manifold bimeromorphic to a
Kähler manifold need not be Kähler [9], combined with the fact that a Fujiki
manifold which supports a SKT metric is Kähler [2].
Remark 6. — It is fairly easy to construct metrics as in Theorem 2. Take
(X, g) to be a manifold as in Theorem 2 and (Y, h) a Kähler manifold. If pX
and pY denote the two projections, then the metric p∗Xg + p∗Y h also satisfies
the above equations on X × Y . Now take g to be a Gauduchon metric on
a non-Kähler surface X to obtain examples of non-Kähler manifolds that
admit metrics as above.
Remark 7. — A similar question is the following:
Question 8. — Characterize the Hermitian metrics g on a compact
complex manifold X such that there exists a constant C such that∫
X
(g + i∂∂̄u)n ⩽ C, ∀ u ∈ PSH(X, g).
This question is related to the proof of Theorem 4.1 in [3]. For instance,
on surfaces, any Hermitian metric has the above property, while on 3-folds,
this class includes the Hermitian metrics g that satisfy i∂∂̄g ⩾ 0. Indeed, if
u ∈ PSH(X, g), then∫
X
(g + i∂∂̄u)3 =
∫
X
g3 + 3
∫
X
g2 ∧ i∂∂̄u + 3
∫
X
g ∧ (i∂∂̄u)2
and the second integral on the right is known to be bounded, while the third
integral is negative if i∂∂̄g ⩾ 0 since∫
X
g ∧ (i∂∂̄u)2 = −
∫
X
i∂∂̄g ∧ i∂u ∧ ∂̄u ⩽ 0
from Stokes’ theorem. In particular, as in [3], we obtain that if a Hermitian
3-fold (X, g) such that i∂∂̄g ⩾ 0 admits a nef class α of non-negative self-
intersection, then X is Kähler (see Theorem 2.3 in [2]).
Remark 9. — Partial results in this direction have been obtained recently
by Guedj and Lu in [8] and by Angella, Guedj and Lu in [1]. In [8], the
authors associated to a Hermitian metric g on a compact complex manifold
– 578 –
On the invariance of the total Monge–Ampère volume of Hermitian metrics
X of dimension n two numbers,
v+(g) = sup
{∫
X
(g + i∂∂̄u)n
∣∣∣∣u ∈ PSH(X, g) ∩ C∞(X,R)}
and
v−(g) = inf
{∫
X
(g + i∂∂̄u)n
∣∣∣∣u ∈ PSH(X, g) ∩ C∞(X,R)}.
Note that Question 8 is equivalent to finding the metrics g for which v+(g) <
∞. In [8], the authors showed that the conditions v+(g) < ∞ and v−(g) < ∞
are independent of the choice of g, therefore they are an intrinsic property
of the manifold X. In [1] the authors further study these conditions.
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[3] ——— , “The Kähler rank of compact complex manifolds”, J. Geom. Anal. 26 (2016),
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[4] S. Dinew, “Pluripotential theory on compact Hermitian manifolds”, Ann. Fac. Sci.
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[6] S. Dinew & S. Kołodziej, “Pluripotential estimates on compact Hermitian mani-
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On the invariance of the total Monge–Ampère volume of Hermitian metrics

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