Optimal extension of sections from subvarieties in weakly pseudoconvex manifolds
Abstract.
In this paper, we obtain optimal extension of holomorphic sections of a holomorphic vector bundle from subvarieties in weakly pseudoconvex Kähler manifolds. Moreover, in the case of line bundle the Hermitian metric is allowed to be singular .
Key words and phrases:
optimal extension, plurisubharmonic function, multiplier ideal sheaf, strong openness, weakly pseudoconvex manifold, Kähler manifold1. Introduction and main results
The extension problem is an important topic in several complex variables and complex geometry. Many generalizations and applications have been obtained since the original work of Ohsawa and Takegoshi ([25]). A recent progress is about the optimal extension and its applications.
Most recently, several general extension theorems with optimal estimates were proved in [14] for holomorphic sections defined on subvarieties in Stein or projective manifolds. In [11], several extension theorems were obtained for holomorphic sections defined on subvarieties in weakly pseudoconvex Kähler manifolds.
In this paper, we prove an optimal extension theorem, which generalizes the main theorems in [14] to weakly pseudoconvex Kähler manifolds and optimizes a main theorem in [11] (cf. Theorem 2.8 and Remark 2.9 in [11]).
Let us recall some definitions in [11].
Definition 1.1.
A function on a complex manifold is said to be quasiplurisubharmonic if is locally the sum of a plurisubharmonic function and a smooth function. In addition, we say that has neat analytic singularities if every point possesses an open neighborhood on which can be written as
where is a nonnegative number, and .
Definition 1.2.
If is a quasiplurisubharmonic function on a complex manifold , the multiplier ideal sheaf is the coherent analytic subsheaf of defined by
where is an open coordinate neighborhood of , and is the Lebesgue measure in the corresponding open chart of . We say that the singularities of are log canonical along the zero variety if for every .
If is a Kähler metric on , we let be the corresponding Kähler volume element, where . In case has log canonical singularities along , one can associate in a natural way a measure on the set of regular points of as follows.
Definition 1.3.
If is a compactly supported nonnegative continuous function on and is a compactly supported nonnegative continuous extension of to such that , then we set
Remark 1.1.
Remark 1.2.
We will define a class of functions before the statement of our main theorem.
Definition 1.4.
Let and . When , let be the class of functions defined by
When , we replace with and in the above definition of .
Remark 1.3.
The number , and the function are equal to the number , and the function defined just before the main theorems in [14]. If and is decreasing on , the longest inequality in the definition of holds for all . If , the longest inequality in the definition of implies that for all . Therefore, , i.e., or .
Theorem 1.1 (The main theorem).
Let . Let be a weakly pseudoconvex complex dimensional manifold possessing a Kähler metric , and be a quasiplurisubharmonic function on with neat analytic singularities. Let be the analytic subvariety of defined by and assume that has log canonical singularities along . Let (resp. ) be a holomorphic line bundle (resp. a holomorphic vector bundle) over equipped with a singular Hermitian metric (resp. a smooth Hermitian metric ), which is written locally as for some quasiplurisubharmonic function with respect to a local holomorphic frame of . Assume that

is semipositive on in the sense of currents (resp. in the sense of Nakano),
and that there is a continuous function on such that the following two assumptions hold:

is semipositive on in the sense of currents (resp. in the sense of Nakano),

,
where is the function
(1.1) 
Then for every section (resp. ) on such that
(1.2) 
there exists a section (resp. ) such that on and
(1.3) 
Remark 1.4.
The case of Theorem 1.1 when is Stein or projective was proved in [14] (see also Proposition 4.1 in [31] for a simplified version). Hence Theorem 1.1 can be regarded as a generalization of the main theorems in [14] to weakly pseudoconvex Kähler manifolds. Then it is easy to see from Remark 1.2 and the main theorems in [14] that the constant in is optimal. Hence Theorem 1.1 gives an optimal version of a main theorem in [11] (cf. Theorem 2.8 and Remark 2.9 in [11]).
Remark 1.5.
In [31], Theorem 1.1 was proved for in the special case when , and is decreasing on , where is a global holomorphic section of some holomorphic vector bundle of rank over equipped with a smooth Hermitian metric, and is transverse to the zero section. Similarly as in [31], a global plurisubharmonic negligible weight can be added to Theorem 1.1 by adding another regularization process to Step 2 in Section 4.
Remark 1.6.
In order to deal with the singular metric on the weakly pseudoconvex Kähler manifold , not only the regularization theorem 2.2 and the error term method of solving equations (Lemma 2.1) are needed, but also a limit problem about integrals with singular weights needs to be solved. We solve the limit problem in Proposition 3.2. Then by using Proposition 3.1, Proposition 3.2 and the strong openness property of multiplier ideal sheaves (Theorem 2.6) as the key tools, we construct a family of smooth extensions of satisfying some uniform estimates, and overcome the difficulty in dealing with the singular metric (see also [31] for the special case).
The rest sections of this paper are organized as follows. First, we give some results used in the proof of Theorem 1.1 in Section 2. Then, we prove two key propositions in Section 3 which will be used to deal with the singular metric . Finally, we prove Theorem 1.1 in Section 4 by using the results in Section 2 and Section 3.
2. Some results used in the proof of Theorem 1.1
In this section, we give some results which will be used in the proof of Theorem 1.1.
Lemma 2.1 ([9], [11]).
Let be a complete Kähler manifold equipped with a (non necessarily complete) Kähler metric , and let be a holomorphic vector bundle over equipped with a smooth Hermitian metric . Assume that and are smooth and bounded positive functions on and let
Assume that is a nonnegative number such that is semipositive definite everywhere on for some . Then given a form such that and
there exists an approximate solution and a correcting term such that and
Theorem 2.2 (Theorem 6.1 in [8]).
Let be a complex manifold equipped with a Hermitian metric , and be an open subset. Assume that is a closed current on , where is a smooth real form and is a quasiplurisubharmonic function. Let be a continuous real form such that . Suppose that the Chern curvature tensor of satisfies
for some continuous nonnegative form on . Then there is a family of closed currents defined on a neighborhood of ( and for some positive number ) independent of , such that

is quasiplurisubharmonic on a neighborhood of , smooth on , increasing with respect to and on , and converges to on as ,

on ,
where is the upperlevel set of Lelong numbers, and is an increasing family of positive numbers such that .
Remark 2.1.
Lemma 2.3 (Theorem 1.5 in [7]).
Let be a Kähler manifold, and be an analytic subset of . Assume that is a relatively compact open subset of possessing a complete Kähler metric. Then carries a complete Kähler metric.
Lemma 2.4 (Theorem 4.4.2 in [19]).
Let be a pseudoconvex open set in , and be a plurisubharmonic function on . For every with there is a solution of the equation such that
where is the dimensional Lebesgue measure on .
Lemma 2.5 (Lemma 6.9 in [7]).
Let be an open subset of and be a complex analytic subset of . Assume that is a form with coefficients and is a form with coefficients such that on (in the sense of currents). Then on .
Theorem 2.6 (Strong openness property of multiplier ideal sheaves, [15]).
Let be a negative plurisubharmonic function on the unit polydisk . Assume that is a holomorphic function on satisfying
Then there exists and such that
where .
Theorem 2.7 (Hironaka’s desingularization theorem, [18], [4]).
Let be a complex manifold, and be an analytic subvariety in . Then there is a local finite sequence of blowups with smooth centers such that:

Each component of lies either in or in , where , denotes the strict transform of by , denotes the singular set of , and denotes the exceptional divisor .

Let and denote the final strict transform of and the exceptional divisor respectively. Then:

The underlying pointset is smooth.

and simultaneously have only normal crossings.

Remark 2.2.
We say that and simultaneously have only normal crossings if, locally, there is a coordinate system in which is a union of coordinate hyperplanes, and is a coordinate subspace.
3. Key propositions used to deal with the singular metric
In order to deal with the singular metric , we will prove two key propositions in this section, which are generalizations of the key propositions in [31].
Proposition 3.1.
Let be a positive continuous function defined on such that and . Let be a bounded pseudoconvex domain, be a plurisubharmonic function on , and be a quasiplurisubharmonic function defined on a neighborhood on . Assume that has neat analytic singularities and the singularities of are log canonical along the zero variety . Set
Furthermore, assume that
on for some nonnegative number , where is the coordinate vector in . Then for every and every holomorphic form on satisfying
there exists a holomorphic form on satisfying on ,
(3.1) 
and
(3.2) 
Proof.
This proposition is a modification of a theorem in [12].
Since is a pseudoconvex domain, there is a sequence of pseudoconvex subdomains such that . Then for fixed , by convolution we can get a decreasing family of smooth plurisubharmonic functions defined on a neighborhood of such that .
Let be a smooth function such that on , on and on .
Fix and . Set . Then the construction of implies that is smooth on and on .
Set . Then on .
Let . Lemma 2.3 implies that is a complete Kähler manifold. Let be endowed with the Euclidean metric and let be the trivial line bundle on equipped with the metric
Then we want to solve a equation on by applying Lemma 2.1 to the case , and (in fact, the case and is the non twisted version of Lemma 2.1). The key step in applying Lemma 2.1 is to estimate the term
where .
Set . Then on .
Since
we get
on , where denotes the operator and is its Hilbert adjoint operator. Then we get and
Hence it follows from Lemma 2.1 that there exists such that on and
Thus
Hence we have . Since , Lemma 2.5 implies that holds on .
Let . Then on . Thus is holomorphic on . Hence is smooth on . Then the nonintegrability of along implies that on . Therefore, on .
It follows from that
Since
we get
Since
(3.5) 
for any inner product space , where , we get
Then
Since , we get
Hence it follows from the two inequalities above and that
Since on , the desired holomorphic form on and the estimates and can be obtained from and by applying Montel’s theorem and extracting weak limits of , first as and then as .
∎
Proposition 3.2.
Let , , and be as in Theorem 1.1. Let be three local coordinate balls in , be a plurisubharmonic function on such that , and be a nonnegative continuous function on with . Let , , and be positive numbers, and let be a small enough positive number. Assume that is a holomorphic function on satisfying
(3.7) 
and that are a family of holomorphic functions such that for all , on ,
(3.8) 
and
(3.9) 
Then
(3.10) 
Remark 3.1.
One of the key points in the proof of Proposition 3.2 is to verify that the upper limit in produces the zero measure on the singular set of , i.e., we have . Then the key uniform estimates in Step 2 of the proof are obtained.
In order to prove Proposition 3.2, we prove the following lemma at first.
Lemma 3.3.
Let , and be positive numbers such that . Let be a bounded negative subharmonic function on , where . Assume that are nonnegative continuous functions defined on such that
(3.11) 
where and . Let
Then
(3.12) 
Proof.
Put