A remark on covering of compact Kähler manifolds and applications
UDC 517.9 Recently, Kolodziej proved that, on a compact Kähler manifold $M,$ the solutions to the complex Monge – Ampére equation with the right-hand side in $L^p,$ $p>1,$ are Hölder continuous with the exponent depending on $M$ and $\|f\|_p$ (see [Math. Ann., 342, 379-386 (2...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512241956159488 |
|---|---|
| author | Hung, V. V. Quy, H. N. Hung, V. V. Quy, H. N. |
| author_facet | Hung, V. V. Quy, H. N. Hung, V. V. Quy, H. N. |
| author_sort | Hung, V. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2025-03-31T08:49:21Z |
| description | UDC 517.9
Recently, Kolodziej proved that, on a compact Kähler manifold $M,$ the solutions to the complex Monge – Ampére equation with the right-hand side in $L^p,$ $p>1,$ are Hölder continuous with the exponent depending on $M$ and $\|f\|_p$ (see [Math. Ann., 342, 379-386 (2008)]).Then, by the regularization techniques in[J. Algebraic Geom., 1, 361-409 (1992)], the authors in [J. Eur. Math. Soc., 16, 619-647 (2014)] have found the optimal exponent of the solutions.In this paper, we construct a cover of the compact Kähler manifold $M$ which only depends on curvature of $M.$ Then, as an application, base on the arguments in[Math. Ann., 342, 379-386 (2008)], we show that the solutions are Hölder continuous with the exponent just depending on the function $f$ in the right-hand side and upper bound of curvature of $M.$
  |
| doi_str_mv | 10.37863/umzh.v73i1.6038 |
| first_indexed | 2026-03-24T03:25:40Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i1.6038
UDC 517.9
V. V. Hung, H. N. Quy (Tay Bac Univ., Univ. Danang – Univ. Sci. and Education, Vietnam)
A REMARK ON COVERING OF COMPACT KÄHLER MANIFOLDS
AND APPLICATIONS*
ЗАУВАЖЕННЯ ЩОДО ПОКРИТТЯ КОМПАКТНИХ
КЕЛЕРОВИХ МНОГОВИДIВ ТА ЇХ ЗАСТОСУВАННЯ
Recently, Kolodziej proved that, on a compact Kähler manifold M, the solutions to the complex Monge – Ampère equation
with the right-hand side in Lp, p > 1, are Hölder continuous with the exponent depending on M and \| f\| p (see [Math.
Ann., 342, 379 – 386 (2008)]). Then, by the regularization techniques in [J. Algebraic Geom., 1, 361 – 409 (1992)], the
authors in [J. Eur. Math. Soc., 16, 619 – 647 (2014)] have found the optimal exponent of the solutions. In this paper, we
construct a cover of the compact Kähler manifold M which only depends on curvature of M. Then, as an application,
base on the arguments in [Math. Ann., 342, 379 – 386 (2008)], we show that the solutions are Hölder continuous with the
exponent just depending on the function f in the right-hand side and upper bound of curvature of M.
Нещодавно Колодзей довiв, що на компактному келеровому многовидi M розв’язки комплексного рiвняння Монжа –
Ампера iз правою частиною у Lp, p > 1, є неперервними за Гельдером з експонентою, що залежить вiд M та \| f\| p
(див. [Math. Ann., 342, 379 – 386 (2008)]). Пiсля цього, за допомогою методу регуляризацiї з [J. Algebraic Geom., 1,
361 – 409 (1992)], автори роботи [J. Eur. Math. Soc., 16, 619 – 647 (2014)] знайшли оптимальну експоненту розв’язкiв.
У цiй роботi ми будуємо покриття компактного келерового многовиду M, яке залежить лише вiд кривини M.
Далi, як застосування, використовуючи аргументацiю з [Math. Ann., 342, 379 – 386 (2008)], доводимо, що розв’язки
є неперервними за Гельдером з експонентою, що залежить лише вiд функцiї f у правiй частинi та верхньої межi
кривини M.
1. Introduction. Let M be a compact n-dimensional Kähler manifold with the fundamental form
\omega given in local coordinates by
\omega =
i
2
\Sigma k,jgk\=j dz
k \wedge d\=zj .
An upper semicontinuous function u on M is called \omega -plurisubharmonic if ddcu+ \omega \geq 0.
Consider the Monge – Ampère equation
(ddcu+ \omega )n = f\omega n, (1.1)
where the given function f \in L1(M), f \geq 0 and
\int
M
f\omega n =
\int
M
\omega n.
Now, we recall some results achieved on the equation (1.1) recently. In [20], by using the
continuous method, S. T. Yau has shown that the equation (1.1) has solutions belong to PSH \cap
\cap C\infty (M), when f \in C\infty (M), f > 0,
\int
M
f\omega n = 1, with a constant error. Then, in [10],
S. Kolodziej has proven that it has solutions belong to PSH \cap C(M), when f \in Lp(M), f \geq 0,\int
M
f\omega n = 1, p > 1. Recall that this result solves in particular the Calabi conjecture and allows to
construct Ricci flat metrics on X whenever c1(X) = 0. In [11], the author has proven that L\infty -norm
of a difference of solutions is controlled by L1-norm of the difference of functions on the right-hand
side (see Theorem 2.1 below). Continuing research the results in this direction, in [12], the author
* This research was supported by Funds for Science and Technology Development of the University of Danang (grant
number B2017-DN03-16).
c\bigcirc V. V. HUNG, H. N. QUY, 2021
138 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
A REMARK ON COVERING OF COMPACT KÄHLER MANIFOLDS AND APPLICATIONS 139
has shown that the solutions to the equation (1.1) are Hölder continuous with the exponent depending
on M, \| f\| p. A similar result was also proved in [8], when M is a bounded strongly pseudoconvex
domain. By demonstrating the opposite case of the main result in [7], P. H. Hiep gave a result that
is stronger than the result in [12] (see [14]). More exactly, P. H. Hiep proved, in a special case of
\mu measure, for every f \in Lp(\mu ) with p > 1 there exists a Hölder continuous \omega -plurisubharmonic
function u such that (ddcu+ \omega )n = f\mu . Then, by the regularization techniques in [5], the authors
in [6] have found the optimal exponent and other interesting results.
In this paper, we construct a cover for the compact Kähler manifold M which depends on the
curvature of M. Then, as an application, we show that the solutions are Hölder continuous with the
exponent just depending on Lp-norm of the function f in the right-hand side of (1.1) and upper
bound of curvature of M.
The paper is organized as follows. In Section 2, after two necessary lemmas (Lemmas 2.1 and
2.2), we present main result, that is Theorem 2.3).
In Section 3, we show that the solutions are Hölder continuous with the exponent depends only
on the Lp-norm of the function on the right-hand side of (1.1) and the upper bound of the curvature
of M in Theorem 3.1.
2. A covering on compact Kähler manifolds. First, recall that we use the normalization
d = \partial + \=\partial , dc = i
\bigl(
\=\partial - \partial
\bigr)
. According to [1, 2], the Monge – Ampère operator (ddc.)n is well
defined on the class of locally bounded plurisubharmonic functions (see also [3, 9]). Moreover, if
u \in PSH \cap L\infty
loc(M) then by [1] (ddcu)n is a non-negative Borel measure.
On a compact Kähler manifold M with fundamental form \omega , the Lp-norm of function f \in
\in Lp(M), p > 0 is defined by
\| f\| p =
\left( \int
M
| f | p\omega n
\right) 1/p
.
Here we cite the result about stability of solutions that is set up in [11].
Theorem 2.1. Given p > 1, \varepsilon > 0, c0 > 0 and \| f\| p < c0, \| g\| p < c0 satisfying the normali-
zing condition in (1.1) there exists c(\varepsilon , c0) such that
\| \varphi - \psi \| \infty \leq c(\varepsilon , c0)\| f - g\| 1/(n+3+\varepsilon )
1 .
Here \varphi , \psi are solutions of (1.1) corresponding to the functions f, g on the right-hand side.
Proof. See [11].
Let \Omega be a domain in \BbbC n. For fixed \delta > 0 we consider \Omega \delta = \{ z \in \Omega : \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z, \partial \Omega ) > \delta \} . With
u \in PSH(\Omega ), we define a function \widetilde u\delta on \Omega \delta as follows:
\widetilde u\delta (z) = \bigl[
\tau (n)\delta 2n
\bigr] - 1
\int
| \zeta | \leq \delta
u(z + \zeta )dV (\zeta ), \tau (n) =
\int
| \zeta | \leq 1
dV (\zeta ),
where dV denotes the Lebesgue measure. Then \widetilde u\delta is a plurisubharmonic in \Omega \delta . On the other hand,
by [8] we have the following inequality:\int
\Omega \delta
(\widetilde u\delta - u) (\zeta )dV (\zeta ) \leq c1\| \Delta u\| 1\delta 2 (2.1)
with the constant c1 depending only on the dimension.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
140 V. V. HUNG, H. N. QUY
The following main results on compact Kähler manifolds (Theorems 2.2 and 2.3) are the basis
for expanding the main results in [12]. Before presenting the theorems, we shall prove two lemmas
to prove the main theorems.
For each matrix A = (aij)i,j=1,n, aij \in \BbbC , we set A\ast is the conjugate transpose matrix of A
(i.e., A\ast = (\=aji)i,j=1,n). Set I is an unit matrix and \| A\| is a norm of matrix A.
Lemma 2.1. Let C be a matrix such that A = CBC\ast with \| A - I\| < \epsilon , \| B - I\| < \epsilon , \epsilon < 1/3.
Then \| CC\ast - I\| < 5\epsilon and \| C\ast C - I\| < 5\epsilon .
Proof. Set A = I + E and B = I + F with \| E\| < \epsilon , \| F\| < \epsilon . We have
\| CC\ast - I\| = \| CC\ast - A+ E\| = \| CC\ast - CBC\ast + E\| =
= \| E - CFC\ast \| \leq \| E\| + \| C\| \| F\| \| C\ast \| .
Hence
\| CC\ast \| \leq \| I\| + \| E\| + \| C\| \| F\| \| C\ast \| \leq 1 + \epsilon + \epsilon \| C\| \| C\ast \| .
Moreover, since \| C\| 2 = \| C\ast \| 2 = \| CC\ast \| , we obtain
\| C\| 2 \leq 1 + \epsilon + \epsilon \| C\| 2 \leftrightarrow \| C\| 2 \leq 1 + \epsilon
1 - \epsilon
< 4 \Rightarrow \| C\| < 2.
From this, we infer that
\| CC\ast - I\| = \| E - CFC\ast \| \leq \| E\| + \| C\| \| F\| \| C\ast \| < \| E\| + 4\| F\| < 5\epsilon .
On the other hand, since B = C - 1A
\bigl(
C - 1
\bigr) \ast
, applying the above result for C and A, B invert each
other we have
\bigm\| \bigm\| C - 1
\bigl(
C - 1
\bigr) \ast - I
\bigm\| \bigm\| < 5\epsilon . Now, from this we get \| C\ast C - I\| < 5\epsilon .
Remark 2.1. i) With z = [z1, z2, . . . , zn]
t and C be a matrix, we have
\| z\| 2 = z\ast z = z1\=z1 + . . .+ zn\=zn,
\| Cz\| 2 = (Cz)\ast (Cz) = z\ast C\ast Cz.
From these formulas, we obtain
\| Cz\| 2 - \| z\| 2 = z\ast C\ast Cz - z\ast z = z\ast (C\ast C - I) z.
So, if \| C\ast C - I\| < \epsilon , then
(1 - \epsilon )\| z\| 2 < \| Cz\| 2 < (1 + \epsilon )\| z\| 2.
ii) We denote by \BbbB r is the open ball of radius r > 0 and \BbbB r(z) is the open ball of radius r
centered at z in \BbbC n.
Lemma 2.2. Let U \subset \BbbC n and f : U - \rightarrow \BbbC n be a holomorphic function. Then we have the
following estimate:
\| f(\omega ) - f(z)\| \leq \| Df(z)(\omega - z)\| + \mathrm{s}\mathrm{u}\mathrm{p}
\BbbB r(z)
\bigm\| \bigm\| D2f(z)
\bigm\| \bigm\| \| \omega - z\| 2 (2.2)
for all z, \omega \in U, \| \omega - z\| < r and \BbbB r(z) \subset U.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
A REMARK ON COVERING OF COMPACT KÄHLER MANIFOLDS AND APPLICATIONS 141
Proof. This lemma as follows from the Taylor inequality for real functions.
Now we prove the main result about a covering for Kähler compact manifolds as follows.
Theorem 2.2. Let (M,\omega ) be a Kähler compact manifold. Then there exist charts \{ Uj\} j=1,m of
M and holomorphic bijective functions fj : Uj - \rightarrow \BbbB 3r such that
m\bigcup
j=1
f - 1
j (\BbbB r) =M,
i
2
n\sum
l=1
dzl \wedge dzl \leq
\bigl(
f - 1
j
\bigr) \ast
\omega \leq 2i
n\sum
l=1
dzl \wedge dzl on \BbbB 3r, (2.3)
and
\| D(fjk)D(fjk)
\ast - I\| < \epsilon r2 on fj(Uj \cap Uk), (2.4)
where fjk = fk \circ f - 1
j : fj(Uj \cap Uk) - \rightarrow fk(Uj \cap Uk) is the local translate function on Uj \cap Uk and
\epsilon > 0 depending on the curvature of M.
Proof. First, in order to prove (2.3), we use the techniques in the proof of Theorem 4.8 in [4].
Let TM and T \ast
M are the tangent and cotangent bundles of M. Then, for each a \in M, since \omega is
Kähler form, we can choose the local coordinates z\prime = (z\prime 1, . . . , z
\prime
n) such that (dz\prime 1, . . . , dz
\prime
n) is an
\omega -orthonormal basis of T \ast
aM. Hence,
\omega = i
n\sum
l=1
\omega lmdz
\prime
l \wedge dz\prime m,
where
\omega lm = \delta lm +O
\bigl(
\| z\prime \|
\bigr)
= \delta lm +
n\sum
j=1
\bigl(
ajlmz
\prime
j + a\prime jlmz
\prime
j
\bigr)
+O(\| z\prime \| 2).
By \omega is real and Kähler form, we have a\prime jlm = ajml and ajlm = aljm. Put
zm = z\prime m +
1
2
n\sum
j,l=1
ajlmz
\prime
jz
\prime
l, 1 \leq m \leq n.
Then (z1, . . . , zn) is a coordinates system at a and
dzm = dz\prime m +
n\sum
j=1
ajlmz
\prime
jdz
\prime
l.
It follows that
i
n\sum
m=1
dzm \wedge dzm = i
n\sum
m=1
dz\prime m \wedge dz\prime m + i
n\sum
j,l,m=1
ajlmz
\prime
jdz
\prime
l \wedge dz\prime m+
+i
n\sum
j,l,m=1
ajlmz
\prime
jdz
\prime
m \wedge dz\prime l +O(\| z\prime \| 2) =
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
142 V. V. HUNG, H. N. QUY
= i
n\sum
j,l,m=1
\omega lmdz
\prime
l \wedge dz\prime m +O(\| z\prime \| 2) = \omega +O(\| z\prime \| 2).
So we can conclude that, at every points a \in M, we have a holomorphic coordinate system
(z\prime n, . . . , z
\prime
n) centered at a such that
\omega = i
n\sum
l,m=1
\omega lmdz
\prime
l \wedge dz\prime m, \omega lm = \delta lm +O(\| z\prime \| 2). (2.5)
Assume that the coordinates (z\prime n, . . . , z
\prime
n) are chosen such that (2.5) is satisfied. Then by the Taylor
expansion we get
\omega lm = \delta lm +O(\| z\prime \| 2) = \delta lm +
n\sum
j,k=1
\bigl(
ajklmz
\prime
jz
\prime
k + a\prime jklmz
\prime
jz
\prime
k + a\prime \prime jklmz
\prime
jz
\prime
k
\bigr)
+O
\bigl(
\| z\prime \| 3
\bigr)
. (2.6)
However a\prime jklm = a\prime kjlm, a
\prime \prime
jklm = a\prime jkml, ajklm = akjml. Moreover, by Kähler condition \partial \omega lm/\partial z
\prime
j =
= \partial \omega lm/\partial z
\prime
j at z\prime = 0 we have a\prime jklm = a\prime lkjm, i.e., a\prime jklm is invariant under all permutations of j,
k, l. Next, if we put
zm = z\prime m +
1
3
n\sum
j,k,l=1
a\prime jklmz
\prime
jz
\prime
kz
\prime
l, 1 \leq m \leq n,
then by (2.6) we infer that
dzm = dz\prime m +
n\sum
j,k,l=1
a\prime jklmz
\prime
jz
\prime
kdz
\prime
l, 1 \leq m \leq n,
\omega = i
n\sum
m=1
dzm \wedge dzm + i
n\sum
j,k,l,m=1
ajklmz
\prime
jz
\prime
kdz
\prime
l \wedge dz\prime m +O
\bigl(
\| z\prime \| 3
\bigr)
,
\omega = i
n\sum
m=1
dzm \wedge dzm + i
n\sum
j,k,l,m=1
ajklmzjzkdzl \wedge dzm +O
\bigl(
\| z\| 3
\bigr)
. (2.7)
Now we have \biggl\langle
\partial
\partial zl
,
\partial
\partial zm
\biggr\rangle
= \delta lm +
n\sum
j,k=1
ajklmzjzk +O
\bigl(
\| z\| 3
\bigr)
,
\partial
\biggl\langle
\partial
\partial zl
,
\partial
\partial zm
\biggr\rangle
=
\biggl\{
D\prime \partial
\partial zl
,
\partial
\partial zm
\biggr\}
=
n\sum
j,k=1
ajklmzkdzj +O
\bigl(
\| z\| 2
\bigr)
.
Then the Chern curvature tensor \Theta (TM )a can find by
\Theta (TM )
\partial
\partial zl
= D\prime \prime D\prime
\biggl(
\partial
\partial zl
\biggr)
= -
n\sum
j,k,m=1
ajklmdzj \wedge dzk \otimes
\partial
\partial zm
+O(\| z\| ).
So - ajklm are the coefficients of the Chern curvature tensor \Theta (TM )a. On the other hand, from (2.7),
we have
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
A REMARK ON COVERING OF COMPACT KÄHLER MANIFOLDS AND APPLICATIONS 143
\omega lm =
\biggl\langle
\partial
\partial zl
,
\partial
\partial zm
\biggr\rangle
= \delta +
n\sum
j,k=1
ajklmzjzk +O
\bigl(
\| z\| 3
\bigr)
.
This gives (2.3).
In order to obtain (2.4), we proceed as follows. From the above, we can assume that, at each
a \in M, we can find a neighbourhood Va of a and a holomorphic bijective function fa : Va - \rightarrow \BbbB sa
such that \bigl(
f - 1
a
\bigr) \ast
\omega (z) = i
n\sum
l=1
dzl \wedge dzl +O
\bigl(
\| z\| 2
\bigr)
, z \in \BbbB sa ,
with O
\bigl(
\| z\| 2
\bigr)
depending on the curvature of M. Now, by compactness of M we can assume that
sa \geq s > 0 \forall a \in M and
\bigl(
f - 1
a
\bigr) \ast
\omega (z) = i
n\sum
l=1
dzl \wedge dzl +O
\bigl(
\| z\| 2
\bigr)
, z \in \BbbB s,
uniformly for a \in M. Hence, with \epsilon > 0 depending on the curvature of the M, we can choose
r = r(\varepsilon ) small enough such that\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigl( f - 1
a
\bigr) \ast
\omega - i
n\sum
l=1
dzl \wedge dzl
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| < \epsilon
5
r2
on \BbbB 3r. Set Ua = f - 1
a (\BbbB 3r), then by the compactness of M, there exist m = m(r) points
a1, a2, . . . , am \in M such that the family
\bigl\{
f - 1
aj (\BbbB r)
\bigr\}
j=1,m
is open cover of M. Set
Uj = f - 1
j (\BbbB 3r) and fj = faj .
Fixed 1 \leq j, k \leq m, we set that
\bigl(
f - 1
j
\bigr) \ast
\omega = i
n\sum
1\leq l,t\leq n
altdzl \wedge dzt,
\bigl(
f - 1
k
\bigr) \ast
\omega = i
n\sum
1\leq l,t\leq n
bltdzl \wedge dzt.
Since
\bigl(
f - 1
j
\bigr) \ast
\omega = (fjk)
\ast \bigl( \bigl( f - 1
k
\bigr) \ast
\omega
\bigr)
on fj(Uj \cap Uk), we get A = DfjkBDf
\ast
jk on fj(Uj \cap Uk),
where A =
\bigl(
f - 1
j
\bigr) \ast
\omega , B =
\bigl(
f - 1
k
\bigr) \ast
\omega . Now using Lemma 2.1 we obtain
\| D(fjk)D(fjk)
\ast - I\| < \epsilon r2 on fj(Uj \cap Uk).
Theorem 2.2 is proved.
From Lemma 2.2, Theorem 2.2 and Remark 2.1, we are ready to prove the following main result.
Theorem 2.3. Let (M,\omega ) be a Kähler compact manifolds. Then there exist charts \{ Uj\} j=1,m
of M and holomorphic bijective functions fj : Uj - \rightarrow \BbbB 3r such that
m\bigcup
j=1
f - 1
j (\BbbB r) =M,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
144 V. V. HUNG, H. N. QUY
i
2
n\sum
l=1
dzl \wedge dzl \leq
\bigl(
f - 1
j
\bigr) \ast
\omega \leq 2i
n\sum
l=1
dzl \wedge dzl on \BbbB 3r,
and
fjk (\BbbB \delta (z)) \subset \BbbB C0\delta (fjk(z)) \forall z \in fj
\bigl(
f - 1
j (\BbbB 2r) \cap f - 1
k (\BbbB 2r)
\bigr)
\forall \delta \in (0, \delta \epsilon ),
where C0 = 1 + \epsilon r2 and \epsilon > 0 depending on the curvature of M.
Proof. First of all, we use the same notation as in Theorem 2.2. Now we wish to apply the
Lemma 2.2 for fjk, with w \in \BbbB \delta (z). Indeed, by (2.2) with f replaced by fjk, we have
\| fjk(w) - fjk(z)\| \leq \| Dfjk(w - z)\| + \mathrm{s}\mathrm{u}\mathrm{p}
\BbbB \delta (z)
\bigm\| \bigm\| D2(fjk)
\bigm\| \bigm\| \| w - z\| 2.
Therefore, in view of Theorem 2.2 and Remark 2.1, we get
\| Dfjk(w - z)\| <
\sqrt{}
1 + \epsilon r2\delta <
\biggl(
1 +
\epsilon r2
2
\biggr)
\delta .
Set d = \mathrm{s}\mathrm{u}\mathrm{p}\BbbB \delta (z)
\bigm\| \bigm\| D2(fjk)
\bigm\| \bigm\| <\infty and choose \delta \epsilon = \delta (\varepsilon ) <
\epsilon r2
2d
, we conclude that
\| fjk(w) - fjk(z)\| <
\bigl(
1 + \epsilon r2
\bigr)
\delta .
Therefore,
fjk(w) \in \BbbB C0\delta (fjk(z)) with C0 = 1 + \epsilon r2 and for all \delta \in (0, \delta \epsilon ).
Theorem 2.3 is proved.
3. Applications to the complex Monge – Ampère equation. In this section, we apply the main
result to show that the solutions to the equation (1.1) are Hölder continuous with the exponent just
depending on the upper bound of the curvature of M.
Theorem 3.1. Assume that p > 1 and f \in Lp(M) satisfying the normalizing condition in (1.1).
Then the solutions to the equation (1.1) are Hölder continuous with the Hölder exponent which
depends on \| f\| p and upper bound of curvature of M.
Proof. Take \epsilon > 0 which only depends on curvature of M. By Theorem 2.3, there exist charts
\{ Uj\} j=1,m of M and holomorphic bijective functions fj : Uj - \rightarrow \BbbB 3r such that
m\bigcup
j=1
f - 1
j (\BbbB r) =M,
i
2
n\sum
l=1
dzl \wedge dzl \leq
\bigl(
f - 1
j
\bigr) \ast
\omega \leq 2i
n\sum
l=1
dzl \wedge dzl on \BbbB 3r,
and
fjk(\BbbB \delta )(z) \subset \BbbB C0\delta (fjk(z)) \forall z \in fj
\Bigl(
f - 1
j (\BbbB 2r) \cap f - 1
k (\BbbB 2r)
\Bigr)
\forall \delta \in (0, \delta \epsilon ), (3.1)
where C0 = 1 + \epsilon r2. For each j = 1, 2, . . . ,m, we set B\prime \prime
j = f - 1
j (\BbbB 3r), B
\prime
j = f - 1
j (\BbbB r), Bj =
= f - 1
j (\BbbB 2r). Choose h \in C\infty (\BbbC n) such that - 1 \leq h \leq 0 on \BbbC n, h = 0 on \BbbB 1 and h = - 1 on
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
A REMARK ON COVERING OF COMPACT KÄHLER MANIFOLDS AND APPLICATIONS 145
\BbbC n \setminus \BbbB 3
2
. Set \widehat \rho (z) = h
\Bigl( z
r
\Bigr)
. We have \widehat \rho \in C\infty (\BbbC n) such that - 1 \leq \widehat \rho \leq 0 on \BbbC n, \widehat \rho = 0 on \BbbB r,\widehat \rho = - 1 on \BbbC n \setminus \BbbB 3r
2
and
ddc\widehat \rho \geq - c(n)
2r2
i
n\sum
l=1
dzl \wedge d\=zl,
where c(n) is a constant depending on n. Set \rho j = \widehat \rho \circ fj , then we obtain \rho j \in C\infty (B\prime \prime
j ), - 1 \leq
\leq \rho j \leq 0 on B\prime \prime
j , \rho j = 0 on B\prime
j and \rho j = - 1 on the neighbourhood of \partial Bj . Since ddc\widehat \rho \geq
\geq - c(n)
r2
\bigl(
f - 1
j
\bigr) \ast
\omega on \BbbB 3r, we get ddc\rho j \geq - c(n)
r2
\omega on B\prime \prime
j . Set C =
c(n)
r2
and fixed N > 4 big
enough. Then, by \mathrm{l}\mathrm{o}\mathrm{g}C0 = \mathrm{l}\mathrm{o}\mathrm{g}
\bigl(
1 + \epsilon r2
\bigr)
\leq \epsilon r2 and C \mathrm{l}\mathrm{o}\mathrm{g}C0 \leq \epsilon c(n), we can choose r = r(\epsilon )
small enough such that 2C0 < N, \alpha <
1
q(n+ 3 + \epsilon ) + 1
(where p, q conjugate) and
2(2C\| u\| \infty + 1) \mathrm{l}\mathrm{o}\mathrm{g}C0 < N - \alpha \mathrm{l}\mathrm{o}\mathrm{g}N.
From the above it follows that
2(2C\| u\| \infty + 1) < N - \alpha \mathrm{l}\mathrm{o}\mathrm{g}N
\mathrm{l}\mathrm{o}\mathrm{g}C0
. (3.2)
On the local chart B\prime \prime
j , we define regularizations
\widehat uj,\delta (z) = \mathrm{m}\mathrm{a}\mathrm{x}
| t| <\delta
u(z + t), z \in Bj .
Set uj,\delta = \widehat uj,\delta \circ f - 1
j (function uj,\delta defined locally on the neighbourhood of 0 in \BbbC n). We also define
two auxiliary functions
\chi (\delta ) = \delta - \alpha \mathrm{m}\mathrm{a}\mathrm{x}
j
\mathrm{m}\mathrm{a}\mathrm{x}
z\in \BbbB 2r
\bigl(
uj,\delta - u \circ f - 1
j
\bigr)
(z),
\eta (\delta ) = \mathrm{m}\mathrm{a}\mathrm{x}
j
\mathrm{m}\mathrm{a}\mathrm{x}
z\in \BbbB 2r
(uj,C0\delta - uj,\delta )(z).
According to (3.1), we have
\mathrm{m}\mathrm{a}\mathrm{x}
z\in \BbbB 2r
| (uj,\delta - uk,\delta )(z)| \leq \eta (\delta ). (3.3)
We will approximate the function u by \omega -plurisubharmonic functions u\delta which are created by gluing
together the local regularization uj,\delta (see [5]). Then by (3.3) the function \eta (\delta ) plays adjustment
functions uj,\delta in the intersection of the charts when moving from local definition to global definition.
Note that, due to the continuity of the function u (see [10]) should have \mathrm{l}\mathrm{i}\mathrm{m}\delta \rightarrow 0 \eta (\delta ) = 0. Set
u\delta (z) = (1 + C1\eta (\delta ))
- 1\mathrm{m}\mathrm{a}\mathrm{x}
j
(uj,\delta (z) + \eta (\delta )\rho j(z)), C1 = 2C.
By (3.3) and the property of \rho j the maximum in the above definition should always be achieved on
\BbbB 2r, so the function u\delta is continuous on
\bigcup m
j=1 fj(B
\prime \prime
j ). Moreover, by C\eta (\delta ) < 1, we get
ddc (uj,\delta (z) + \eta (\delta )\rho j(z)) \geq - (1 + C\eta (\delta ))
\bigl(
f - 1
j
\bigr) \ast
\omega .
From the above results and the inequality approximately ddc\mathrm{m}\mathrm{a}\mathrm{x}(u, v) (see [1]) we derive
ddcu\delta + \omega > 0 (with \delta is sufficiently small). (3.4)
To finish the proof, we need to verify the following proposition.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
146 V. V. HUNG, H. N. QUY
Proposition 3.1. The function \chi is bounded on some nonempty interval (0, \widetilde \delta ).
Proof. Suppose that \chi (\delta ) > \mathrm{m}\mathrm{a}\mathrm{x}(9, \chi (N\delta )) and N\delta < r/2. Then we have
E =
\bigcup
j
\biggl\{
z \in \BbbB 2r :
\bigl(
uj,\delta - u \circ f - 1
j
\bigr)
(z) >
\biggl(
\chi (\delta )
3
- 2
\biggr)
\delta \alpha
\biggr\}
\not = \varnothing .
Choose the function g such that g \circ f - 1
j = 0 on E and g = C2f on M\setminus
\bigcup
j f
- 1
j (E) with C2 as a
constant satisfying condition
\int
M
f\omega n =
\int
M
\omega n. Set
\widetilde uj,\delta (z) = \bigl[
\tau (n)\delta 2n
\bigr] - 1
\int
| \zeta | \leq \delta
u \circ f - 1
j (z + \zeta )dV (\zeta ), \tau (n) :=
\int
| \zeta | \leq 1
dV (\zeta ).
We will compare uj,\delta and \widetilde uj,\delta as follows. Given z \in \BbbB 2r we find tz with | tz| = \delta such that
uj,\delta (z) = u \circ f - 1
j (z + tz) \leq \widetilde uj,\surd \delta (z + tz) \leq \widetilde uj,\surd \delta (z) + 2\| u\| \infty
\surd
\delta .
Since \alpha <
1
2
, we conclude from above estimate for \delta < \delta 0 and \delta 0 small enough that
E \cap \BbbB 2r \subset
\bigl\{
uj,\delta - u \circ f - 1
j > \delta \alpha
\bigr\}
\subset
\bigl\{ \widetilde uj,\surd \delta - u \circ f - 1
j > \delta \alpha /2
\bigr\}
.
As \| \Delta u\| 1 is bounded on every B\prime \prime
j , thus, applying formula (2.1) we have\int
E\cap \BbbB 2r
\omega n < C3\delta
1 - \alpha for all j.
Therefore, \int
E
\omega n < C4\delta
1 - \alpha .
So, by Hölder inequality, we get
\int
E
f\omega n \leq \| f\| p
\left( \int
E
\omega n
\right) 1/q
\leq C5\delta
(1 - \alpha )/q,
where C5 depends only on \| f\| p. So, if v is a solution of (\omega + ddcv)n = g\omega n, then by Theorem 2.1
with \delta < \delta 1 (\delta 1 small enough) we conclude that
\| u - v\| \infty \leq \| f - g\|
1
(n+3+\epsilon ) \leq C6\delta
1 - \alpha
q(n+3+\epsilon ) \leq \delta \alpha , (3.5)
where \alpha is choosen such that \alpha <
1 - \alpha
q(n+ 3 + \epsilon )
.
To end the proof of Proposition 3.1, we will prove the following lemma.
Lemma 3.1. If z0 \in \BbbB 2r such that
\bigl(
uj0,\delta - u \circ f - 1
j0
\bigr)
(z0) = \chi (\delta )\delta \alpha , then we have
\mathrm{s}\mathrm{u}\mathrm{p}
\cup jfj(Bj)\setminus E
\bigl(
u\delta - v \circ f - 1
j
\bigr)
< (u\delta - v)(z0).
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A REMARK ON COVERING OF COMPACT KÄHLER MANIFOLDS AND APPLICATIONS 147
Proof. We can assume that u > 1. Take z \in
\Bigl( \bigcup
j fj(Bj) \setminus E
\Bigr)
\cap \BbbB 2r, then
\bigl(
uj,\delta - u \circ f - 1
j
\bigr)
(z) \leq
\biggl(
\chi (\delta )
3
- 2
\biggr)
\delta \alpha .
So, by (3.5), we get \bigl(
uj,\delta - v \circ f - 1
j
\bigr)
(z) \leq
\biggl(
\chi (\delta )
3
- 1
\biggr)
\delta \alpha .
Since u > 1, we infer that
\bigl(
u\delta - v \circ f - 1
j
\bigr)
(z) \leq \mathrm{m}\mathrm{a}\mathrm{x}
j : z\in Bj
\bigl(
uj,\delta - v \circ f - 1
j
\bigr)
(z) \leq
\biggl(
\chi (\delta )
3
- 1
\biggr)
\delta \alpha . (3.6)
Again, by (3.5) we conclude similarly that\bigl(
uj0,\delta - v \circ f - 1
j0
\bigr)
(z0) \geq (\chi (\delta ) - 1) \delta \alpha .
So, by definition of the functions, we have\bigl(
u\delta - v \circ f - 1
j0
\bigr)
(z0) \geq (\chi (\delta ) - 1) \delta \alpha - \eta (\delta )(2C\| u\| \infty + 1). (3.7)
With \delta small enough and by the three circles theorem we get
\bigl(
uj,N\delta - uj,\delta
\bigr)
\geq \mathrm{l}\mathrm{o}\mathrm{g}N
\mathrm{l}\mathrm{o}\mathrm{g}C0
(uj,C0\delta - uj,\delta ) .
Choose j such that
\eta (\delta ) = \mathrm{m}\mathrm{a}\mathrm{x}
z\in \BbbB 2r
(uj,C0\delta - uj,\delta ) (z).
From this we obtain
(N\delta )\alpha \chi (N\delta ) \geq \mathrm{l}\mathrm{o}\mathrm{g}N
\mathrm{l}\mathrm{o}\mathrm{g}C0
\eta (\delta ).
On the other hand, since \chi (\delta ) \geq \chi (N\delta ) then, from (3.2), we get the following result:
\delta \alpha \chi (\delta ) \geq \mathrm{l}\mathrm{o}\mathrm{g}N
\mathrm{l}\mathrm{o}\mathrm{g}C0
\eta (\delta )N - \alpha > 2\eta (\delta )(2C\| u\| \infty + 1).
The above result combined with (3.7), we obtain
\bigl(
u\delta - v \circ f - 1
j0
\bigr)
(z0) \geq
\biggl(
\chi (\delta )
2
- 1
\biggr)
\delta \alpha .
From this and (3.6), we complete the proof of Lemma 3.1.
We proceed to finish the proof of the theorem.
Applying the Lemma 3.1, we can find C7 such that
z0 \in U =
\bigl\{
v \circ f - 1
j < u\delta - C7
\bigr\}
\subset E.
By the comparison principle (see [10]) and (3.4), we lead to a contradiction because
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
148 V. V. HUNG, H. N. QUY
0 <
\int
U
\Bigl(
ddcu\delta +
\bigl(
f - 1
j
\bigr) \ast
\omega
\Bigr) n
\leq
\int
U
(ddcv + \omega ) \leq
\int
E
(ddcv + \omega )n =
\int
E
g\omega n = 0.
This contradiction shows that the choice of small enough \delta such that
\chi (\delta ) > \mathrm{m}\mathrm{a}\mathrm{x}(9, \chi (N\delta ))
is impossible. Thus, the proof of Proposition 3.1 and, so, of Theorem 3.1 is completed.
Finally, we get the following corollary as a special case of Theorem 3.1 when M be a compact
Kähler manifold of zero curvature (such as in [17 – 19]).
Corollary 3.1. Let M be a compact Kähler manifold of zero curvature. Then the solutions of
(1.1) in Theorem 3.1 are Hölder continuous with the Hölder exponent which depends only on \| f\| p.
References
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(1976).
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3. U. Cegrell, The general definition of the complex Monge – Ampère operator, Ann. Inst. Fourier, 54, 159 – 179 (2004).
4. J.-P. Demailly, Complex analytic and differential geometry, http://www-fourier.ujf-grenoble.fr/demailly/books.html.
5. J. -P. Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom., 1, 361 – 409
(1992).
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Ampère equation, J. Eur. Math. Soc., 16, 619 – 647 (2014).
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Received 05.03.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
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| id | umjimathkievua-article-6038 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:40Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/82/45982a66a1c8c09db54e8f433223e182.pdf |
| spelling | umjimathkievua-article-60382025-03-31T08:49:21Z A remark on covering of compact Kähler manifolds and applications A remark on covering of compact Kähler manifolds and applications A remark on covering of compact Kähler manifolds and applications Hung, V. V. Quy, H. N. Hung, V. V. Quy, H. N. Complex Monge-Amp`ere operator ω-plurisubharmonic functions compact K¨ahler manifolds Complex Monge-Amp`ere operator ω-plurisubharmonic functions compact K¨ahler manifolds UDC 517.9 Recently, Kolodziej proved that, on a compact Kähler manifold $M,$ the solutions to the complex Monge – Ampére equation with the right-hand side in $L^p,$ $p&gt;1,$ are Hölder continuous with the exponent depending on $M$ and $\|f\|_p$ (see [Math. Ann.,&nbsp;342, 379-386 (2008)]).Then, by the regularization techniques in[J. Algebraic Geom., 1, 361-409 (1992)], the authors in [J. Eur. Math. Soc., 16, 619-647 (2014)] have found the optimal exponent of the solutions.In this paper, we construct a cover of the compact Kähler manifold $M$ which only depends on curvature of $M.$ Then, as an application, base on the arguments in[Math. Ann., 342, 379-386 (2008)], we show that the solutions are Hölder continuous with the exponent just depending on the function $f$ in the right-hand side and upper bound of curvature of $M.$ &nbsp; УДК 517.9 Зауваження щодо покриття компактних келерових многовидiв та їх застосування Нещодавно Колодзей довів, що на компактному келеровому многовиді $M$ розв'язки комплексного рівняння Монжа – Ампера із правою частиною у $L^p,$ $p&gt;1,$ є неперервними за Гельдером з експонентою, що залежить від $M$ та $\|f\|_p$ (див. [Math. Ann., 342, 379-386 (2008)]). Після цього, за допомогою методу регуляризації з [J. Algebraic Geom., 1, 361-409 (1992)], автори роботи [J. Eur. Math. Soc., 16, 619-647 (2014)] знайшли оптимальну експоненту розв'язків.У цій роботі ми будуємо покриття компактного келерового многовиду $M,$ яке залежить лише від кривини $M.$ Далі, як застосування, використовуючи аргументацію з [Math. Ann., 342, 379-386 (2008)], доводимо, що розв'язки є неперервними за Гельдером з експонентою, що залежить лише від функції $f$ у правій частині та верхньої межі кривини $M.$ Institute of Mathematics, NAS of Ukraine 2021-01-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6038 10.37863/umzh.v73i1.6038 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 1 (2021); 138 - 148 Український математичний журнал; Том 73 № 1 (2021); 138 - 148 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6038/8910 |
| spellingShingle | Hung, V. V. Quy, H. N. Hung, V. V. Quy, H. N. A remark on covering of compact Kähler manifolds and applications |
| title | A remark on covering of compact Kähler manifolds and applications |
| title_alt | A remark on covering of compact Kähler manifolds and applications A remark on covering of compact Kähler manifolds and applications |
| title_full | A remark on covering of compact Kähler manifolds and applications |
| title_fullStr | A remark on covering of compact Kähler manifolds and applications |
| title_full_unstemmed | A remark on covering of compact Kähler manifolds and applications |
| title_short | A remark on covering of compact Kähler manifolds and applications |
| title_sort | remark on covering of compact kähler manifolds and applications |
| topic_facet | Complex Monge-Amp`ere operator ω-plurisubharmonic functions compact K¨ahler manifolds Complex Monge-Amp`ere operator ω-plurisubharmonic functions compact K¨ahler manifolds |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6038 |
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