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q-convexity properties of locally semi-proper morphisms of complex spaces.

1 introduction

According to Grauert [11] and Narasimhan [14], [15] and their solution to the Levi problem, a complex space is Stein if and only if it admits a continuous strongly plurisubharmonic exhaustion function (see Definitions 3 and 4).

In [18], Stein showed that if X and Z are two complex spaces and if [pi] : Z [right arrow] X is an unramified covering such that X is Stein, then Z is Stein. This result was generalized to ramified coverings by Le Barz in [13].

The notion of a Stein space was generalized by Andreotti and Grauert in [1], where they defined q-convex and q-complete complex spaces. They extended Cartan's Theorem B and proved finiteness and vanishing theorems for the cohomology of a q-convex and of a q-complete space with values in a coherent analytic sheaf.

Also, in [3], Ballico generalized Stein's result to arbitrary q-complete spaces instead of Stein spaces and in [2], he proved the same type of result for finite morphisms of complex spaces.

In [5], Coltoiu and Vajaitu considered locally trivial analytic fibrations [pi] : E [right arrow] B such that the fiber is a Stein curve and B is q-complete. In this way they improved the result of [3].

Further generalizations of the results of Ballico in [2] were obtained by Vajaitu in [19].

The purpose of this paper is to generalize the results in [2], [3], [13], [18] and [19]. This is contained in Theorem 7.

Acknowledgments: I am very grateful to Professor Mihnea Coltoiu for suggesting me this problem and for his helpful advice.

2 Preliminaries

All complex spaces are assumed to be reduced, finite dimensional and with countable topology.

2.1

Definition 1. A complex space X is said to be a Stein space if the following hold:

(a) X is holomorphically convex, i.e.,for every compact set K [subset] X the holomorphically convex hull

[[??].sub.X] = {x [member of] X : [absolute value of (f (x))] [less than or equal to] [[parallel]f[parallel].sub.k], [for all]f [member of] O(X)}

is also compact;

(b) For every x [member of] X there are global functions [f.sub.1], ..., [f.sub.N] [member of] O (X) which give a local holomorphic embedding of a neighbourhood of x into [C.sup.N];

(c) For every pair of distinct points x [not equal to] y in X there is a holomorphic function f [member of] O (X) such that f (x) [not equal to] f (y).

Definition 2. Let D be an open neighbourhood of a point [z.sub.0] [member of] [C.sup.n] and f [member of] [C.sup.[infinity]] (D, R). We define the Levi form L(f, [z.sub.0]) off at [z.sub.0] as follows: for arbitrary [xi], [eta] [member of] [C.sup.n] set

L(f, [z.sub.0])([xi], [eta]) := [n.summation over (i,j=1)] [[partial derivative].sup.2]/[partial derivative][z.sub.i][partial derivative][[bar.z].sub.j] ([z.sub.0]) [[xi].sub.i] [[bar.[eta]].sub.j].

Also we set L(f, [z.sub.0])[xi] = L(f, [z.sub.0])([xi], [xi]), [xi] [member of] [C.sup.n].

Definition 3. 1) A real valued [C.sup.2]-function [phi] : D [right arrow] R, where D is an open set in [C.sup.n], is said to be plurisubharmonic (respectively strongly plurisubharmonic) if and only if its Levi form is positive-semidefinite (respectively positive definite), that is for each [z.sub.0] [member of] D and for every [xi] [member of] [C.sup.n] the inequality L(f, [z.sub.0])[xi] [greater than or equal to] 0 (respectively > 0 on [C.sup.n]\{0}) holds.

2) Let X be a complex space. A function y : X [right arrow] R is said to be (strongly) plurisubharmonic at a point x [member of] X if there is a local chart i : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of X, U [member of] x and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that:

1. [??] [omicron] i = [phi][|.sub.U];

2. The function [??] is (strongly) plurisubharmonic on [??].

The function [phi] is said to be (strongly) plurisubharmonic on a subset W [subset] X if it is (strongly) plurisubharmonic at every point of W.

Definition 4. Let X be a complex space. An upper semi-continuous function y : X [right arrow] R is said to be an exhaustion function on X if the sublevel sets {x [member of] X : [phi](x) < c} are relatively compact in X for any c [member of] R.

We have the following result (see [11] and [14], [15]):

Theorem 1. A complex space X is Stein if and only if there exists [phi] : X [right arrow] R a continuous strongly plurisubharmonic exhaustion function on X.

Definition 5. Let X and Z be two complex spaces. A morphism [pi] : Z [right arrow] X is said to be proper if for every compact set K in X the preimage [[pi].sup.-1] (K) is compact. A morphism [pi] : Z [right arrow] X is said to be finite if it is proper and it has finite fibers.

Remark 1. Let [pi] : Z [right arrow] X be a finite morphism of complex spaces. Then Z is Stein if and only if X is Stein.

We recall the following theorem of Stein [18]:

Theorem 2. Let [pi] : Z [right arrow] X be an unramified covering of complex spaces. If X is Stein, then Z is Stein

Le Barz [13] extended Stein's result to locally semi-finite morphisms of complex spaces (see Definition 6 and Theorem 3).

Definition 6. Let X and Z be two complex spaces. We say that a morphism [pi] : Z [right arrow] X is

(a) semi-finite if Z is the disjoint union of some open spaces [([W.sup.m]).sub.m[member of]N] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a finite morphism;

(b) locally semi-finite if for all x [member of] X, there exists a neighbourhood U [member of] x such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a semi-finite morphism.

Theorem 3. Let [pi] : Z [right arrow] X a locally semi-finite morphism of complex spaces. If X is Stein, then Z is Stein.

2.2 As we mentioned in the introduction, the notions of a q-convex and of a q-complete complex space were introduced in [1].

Definition 7. 1) A function [phi] [member of] [C.sup.[infinity]] (D, R), where D is an open subset of [C.sup.n] is said to be q-convex (q [member of] N, 1 [less than or equal to] q [less than or equal to] n) if its Levi form has at least n - q + 1 positive (> 0) eigenvalues at every point of U.

2) Let Xbea complex space. A function [phi] : X [right arrow] R is said to be q-convex at a point x [member of] X if there is a local chart i : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of X, U [member of] x and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that:

1. [??] [omicron] i = [phi][|.sub.U];

2. The function [??] is q-convex on [??].

The second condition can be replaced by the following:

2'. There exists a complex linear space E [subset] [C.sup.n], dim E [greater than or equal to] n - q + 1 such that the Levi form L([??], i (x)) is positive definite when restricted to E.

The function [phi] is said to be q-convex on a subset W [subset] X if it is q-convex at every point of W.

Definition 8. A complex space X is said to be q-convex, if there exists a compact subset K of X and a smooth exhaustion function [phi] : X [right arrow] R, which is q-convex on X\K. If we can choose K = [empty set], then X is said to be q-complete.

Remark 2. From [14] and [15] we have that a complex space X is Stein if and only if is 1-complete.

Ballico [3] improved Theorem 2 in another direction.

Theorem 4. Let [pi] : Z [right arrow] X be an unramified covering. If X is q-complete, then Z is q-complete.

Also, in [2], Ballico showed that if [pi] : Z [right arrow] X is a finite morphism between complex spaces and X is q-complete or q-convex, then Z is q-complete or q-convex, respectively.

Coltoiu and Vajaitu [5] proved that if [pi] : E [right arrow] B is a locally analytic fibration of complex spaces such that the fiber is a Stein curve and B is q-complete, then E is q-complete. The case when E is a topological covering of B was already done in [3].

Vajaitu [19] generalized Ballico's results from [2] and showed the following:

Theorem 5. Let n : Z [right arrow] X be a proper holomorphic map between finite dimensional complex spaces. If X is q-complete, then Z is (q + r)-complete, where r is the dimension of the fiber.

Let X be a complex space of complex dimension n and q an integer with 1 [less than or equal to] q [less than or equal to] n. For q > 1 the sum and the maximum of two q-convex functions on X is not q-convex as they might have different directions of positivity. It was proved in [7] and [8] that every q-convex function with corners (i.e., a function which locally is equal to the maximum of a finite family of q-convex functions) can be approximated by a q-convex function, where [??] = n - [n/q] + 1 (here [n/q] denotes as usual the largest integer [less than or equal to] n/q). Diederich and Fornaess also showed that this [??] is optimal. As a consequence, a finite intersection of q-convex open sets is [??]-convex. The optimality of this [??] is proved by Chiriacescu, Coltoiu and Joita in [4] in the case of quasi-projective varieties in a cohomological context.

To overcome this type of problem, M. Peternell [16] introduced the notion of convexity with respect to a linear set M.

As before X is a reduced, finite dimensional and with countable topology complex space. For any x [member of] X we denote by [T.sub.x]X the Zariski tangent space of X to x. Set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Consider an arbitrary subset M [subset] TX and for any point x [member of] X put [M.sub.x] = M [intersection] [T.sub.x]X. Then M is said to be a linear set over X if [M.sub.x] is a complex vector subspace of [T.sub.x]X for any x [member of] X.

Let now [OMEGA] [subset] X be an open subset. We define:

(i) [codim.sub.[OMEGA]] M = [sup.sub.x[member of][OMEGA]] codim [M.sub.x];

(ii) M[|.sub.[OMEGA]] as [(M [|.sub.[OMEGA]]).sub.x] = [M.sub.x] for every x [member of] [OMEGA].

Let [pi] : Y [right arrow] X be an analytic morphism of complex spaces and M a linear set over X. For every y [member of] Y we have the tangent map which is a C-linear map of complex vector spaces [[pi].sub.*,y] : [T.sub.y]Y [right arrow] [T.sub.x]X, where x = [pi](y). We set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We have that [[pi].sup.*]M is a linear set over Y. Moreover, if codim M [less than or equal to] q - 1, then codim [[pi].sup.*] M [less than or equal to] q - 1.

The following are due to M. Peternell (see [16]).

Definition 9. Let X be a complex space, W [subset] X an open subset, M a linear set over W and [phi] : W [right arrow] R a smooth function.

(a) Let x [member of] W. Then f is said to be weakly 1-convex with respect to [M.sub.x] if there are a local chart i: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of X with x [member of] U [subset] W and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

[??] [omicron] i = [phi][|.sub.U] and L([??], i(x))[i.sub.*][xi] [greater than or equal to] 0 for any [xi] [member of] [M.sub.x].

Furthermore, f is said to be weakly 1-convex with respect to M if f is weakly 1-convex with respect to [M.sub.x] for every x [member of] W.

(b) We say that f is 1-convex with respect to M, if for any x [member of] W there exist an open neighbourhood U [subset] W of x and a 1-convex function [psi] [member of] [C.sup.[infinity]] (U, R) such that [phi] [|.sub.U] - [psi] is weakly 1-convex with respect to M [|.sub.U].

Definition 10. Let X be a complex space and M a linear set over X. We denote by B(X, M) the set of all continuous functions [phi] : X [right arrow] R such that every point of X admits an open neighbourhood D on which there are functions [f.sub.1], ..., [f.sub.k] [member of] [C.sup.[infinity]] (D, R) which are 1-convex with respect to M[|.sub.D] and

[phi][|.sub.D] = max([f.sub.1], ..., [f.sub.k]).

We need also the following results of M. Peternell (see [16]):

Lemma 1. Suppose that [phi] is a q-convex function on a complex space X. Then there exists a linear set M over X of codimension [less than or equal to] q - 1 such that f is 1-convex with respect to M.

Lemma 2. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a local chart of the complex space X and p : U [right arrow] R a smooth function. Then [phi] is 1-convex with respect to some linear set M if and only if for every compact subset K [subset] U there exists [delta] > 0 and for each x [member of] K there exist [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [??] [omicron] i = [phi] and

L([??], i(x))[i.sub.*] ([xi]) [greater than or equal to] [delta] [[parallel][i.sub.*]([xi])[parallel].sup.2]

for all [xi] [member of] [M.sub.x].

In general, an increasing union of Stein open subsets [{[X.sub.i]}.sub.i[member of]N] of a complex space X is not Stein, even if X is smooth (see [9] and [10]). However, if ([X.sub.i+1], [X.sub.i]) is Runge, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is Stein. We recall that if [gamma] is a Stein open subset of a Stein space X, then (X, [gamma]) is said to be a Runge pair if for any compact subset K of [gamma], the set [[??].sub.X] [intersection] [gamma] is compact. Using the approximation theorem of Oka-Weil, (X, [gamma]) is a Runge pair if and only if [gamma] is a Stein space and every holomorphic function on [gamma] can be approximated uniformly on compact subset of [gamma] by holomorphic functions on X.

The following result follows from Theorem 3 proved by Coltoiu and Vaja-itu in [6]; it gives us a criterion for testing the q-completeness of a complex space. The same kind of result as Theorem 3 in [6] was obtained in the q-concave case in [12].

Theorem 6. Let X be a complex space and M a linear set over X. Let [{[X.sub.i]}.sub.i[member of]N] be an increasing sequence of open subsets of X such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and there are functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and constants [C.sub.i], [D.sub.i] [member of] R, [C.sub.i] < [D.sub.i], i [member of] N with the following properties:

(a) {x [member of] [X.sub.i] : [u.sub.i] (x) < [D.sub.i]} [subset][subset] [X.sub.i] for every i [member of] N

(b) {x [member of] [X.sub.i+1] : [u.sub.i+1](x) < [C.sub.i]} [subset] {x [member of] [X.sub.i] : [u.sub.i] (x) < [D.sub.i]} for every i [member of] N;

(c) for every compact set K [subset] X there is j = j(K) [member of] N such that

K [subset] {x [member of] [X.sub.i+1] : [u.sub.i+1] (x) < [C.sub.i]} for every i [greater than or equal to] j.

Then there exists an exhaustion function v [member of] B(X, M). In particular, if codim M [less than or equal to] q - 1, then X is q-complete.

2.3 Let X be a complex space and A an analytic subset of X. The Andreotti function will help us to get some positive eigenvalues in the "normal direction" at the regular points of A. Denote by [J.sub.A] the coherent ideal sheaf of germs of holomorphic functions vanishing along A.

Choose a locally finite covering [{[U.sub.j]}.sub.j] of X by relatively compact open subsets of X such that on each [U.sub.j] there are functions [h.sup.(j).sub.1], ..., [h.sup.(j).sub.q(j)] [member of] O([U.sub.j]) with [J.sub.A] | [U.sub.j] = ([h.sup.(j).sub.1], ..., [h.sup.(j).sub.q(j)]).

Let [{[[rho].sub.j]}.sub.j] be a partition of unity subordinated to the covering [{[U.sub.j]}.sub.j] of X. We define the Andreotti function [f.sub.A] : X [right arrow] R by setting:

[f.sub.A](x) = [summation over (j)] [[rho].sub.j] (x) [[parallel][h.sup.(j)](x)[parallel].sup.2], x [member of] X,

where

[[parallel][h.sup.(j)][parallel].sup.2] = [q(j).summation over (i=1)] [[absolute value of ([h.sup.(j).sub.i])].sup.1].

We remark that [f.sub.A] [greater than or equal to] 0, [f.sub.A] [member of] [C.sup.[infinity]] (X) and A = {[f.sub.A] = 0}.

Suppose that [x.sub.0] [member of] A and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a local chart around [x.sub.0]. Extend [[rho].sub.j] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Locally, [f.sub.A] has an extension [[??].sub.A] defined on [??] such that

1. the Levi form L([[??].sub.A], i(x)) is positive semidefinite for all x [member of] U [intersection] A;

2. for x [member of] U [intersection] Reg(A) we have that the Levi form L([[??].sub.A], i(x))(v) = 0 iff v [member of] [i.sub.*,x] ([T.sub.x]A).

The Andreotti function is used in the next result which follows from Lemma 3 and Lemma 4 in [19]:

Lemma 3. Let [pi] : Z [right arrow] X be a holomorphic map between finite dimensional reduced complex spaces and A an analytic subset of Z. Put r = max{dim [[pi].sup.-1] (x) : x [member of] X} and let B [subset] A be an analytic subset such that Sing (A) [subset or equal to] B and suppose that the restriction map [pi][|.sub.A\B] : A\B [right arrow] X has locally constant rank.

Assume also that there exists a locally finite covering [{[V'.sub.l]}.sub.l] of X by relatively compact open subsets and 1-convex functions [[phi].sub.l] : [V'.sub.l] [right arrow] [R.sub.+]. Let [V.sub.l] [subset] [V'.sub.l] be open subsets, [[bar.V].sub.l] [subset] [V'.sub.l] and [union] [V.sub.l] = X. Denote [U.sub.l] := [[pi].sup.-1]([V.sub.l]), [U'.sub.l] := [[pi].sup.-1] and put

[[psi].sub.l] = [f.sub.A] + [[phi].sub.l] [omicron] [pi] : [U'.sub.l] [right arrow] [R.sub.+],

where [f.sub.A] is the Andreotti function of the analytic subset A of Z.

Then there is an open neighbourhood [OMEGA] of A\B in Z and a linear set M over [OMEGA], codim M [less than or equal to] r such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1-convex with respect to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any l.

3 The main result

Following the ideas of Le Barz [13] we give the next definition.

Definition 11. Let X and Z be two complex spaces. We say that a morphism n : Z [right arrow] X is

(a) semi-proper if Z is the disjoint union of some open spaces [([W.sup.m]).sub.m[member of]N] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is proper;

(b) locally semi-proper if for all x [member of] X, there exists a neighbourhood U [there exists] x such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a semi-proper morphism.

Now we are ready to state the main result.

Theorem 7. Let X and Z be two complex spaces and [pi] : Z [right arrow] X a locally semiproper morphism and r = max{dim [[pi].sup.-1] (x) : x [member of] X}. If X is q-complete, then Z is (q + r)-complete.

Proof. Since X is q-complete there exists a smooth q-convex exhaustion function [phi] : X [right arrow] R on X. Due to Lemma 1 there exists a linear set M over X of codimension [less than or equal to] q - 1 such that [phi] is 1-convex with respect to M. The idea is to use the q-completeness criterion provided by Theorem 6.

Now we need the following result from [19]:

Proposition 1. Let [pi] : Z [right arrow] X be a holomorphic map. Then there exists a decreasing chain of p + 1 analytic subsets [A.sub.k] of Z, where p [less than or equal to] dim Z, Z = [A.sub.p] [contains] [A.sub.p-1] [contains] ... [contains] [A.sub.1] [contains] [A.sub.0] = [empty set] such that for every k [member of] {1, 2, ..., p} we have dim [A.sub.k-1] < dim [A.sub.k], Sing ([A.sub.k]) [subset] [A.sub.k-1] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

has locally constant rank.

The above decomposition of Z with respect to n is called the singular filtration of [pi] (see also [17]).

So, consider [A.sub.1] [contains] [A.sub.2] [contains] ... [contains] [A.sub.p] the analytic subsets of Z given by Proposition 1 and the corresponding Andreotti functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The next ingredient that we need is a lemma. This lemma was proved by Le Barz [13] in the case of 0-dimensional fibers, but the proof in the general case (the dimension of the fiber is > 0) goes exactly the same way.

Lemma 4. Let X and Z be two complex spaces and [pi] : Z [right arrow] X a locally semi-proper morphism. Then there exists a locally finite covering [{[U.sub.j]}.sub.j] of Z and a locally finite covering [{[V.sub.l]}.sub.l] of X such that the following conditions hold:

1. for all j, there exists a positive integer [m.sub.j] and a local chart [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[??].sub.j] is an open subset of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

2. for all l, there exists a positive integer n and a local chart [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[??].sub.l] is an open subset of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

3. for all j, there exists l(j) such that we have [pi]([U.sub.j]) [subset] [V.sub.l(j)] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] extends to a holomorphic map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

Also there exists a [C.sup.[infinity]] function f : Z [right arrow] R such that:

* {z [member of] Z : f (z) < [c.sub.1]} [intersection] {z [member of] Z : ([phi] [omicron] [pi])(z) < [c.sub.2]} [subset][subset] Z, [for all][c.sub.1], [c.sub.2] [member of] R;

* for all j, there exists a map [g.sub.j]: [V.sub.l(j)] [right arrow] R such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

* [g.sub.j] has a [C.sup.[infinity]] extension, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

* for all compact sets K [subset] X,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We choose [{[W.sup.1.sub.j]}.sub.j] a locally finite covering of Z and [{[W.sup.2.sub.k]}.sub.k] a locally finite covering of X such that the conditions in Lemma 4 hold. We denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a q-convex extension of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also consider the function f : Z [right arrow] R, the function [g.sub.j] : [W.sup.2k.sub.(j)] [right arrow] R and its extension [[??].sub.j] given by Lemma 4.

Using the boundedness condition for the second derivatives of the function f, on every compact set, there exists a convex and strictly increasing function [chi] so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is q-convex for all j.

Because [z [member of] Z : f (z) < [c.sub.1]} [intersection] [z [member of] Z : ([phi] [omicron] [pi])(z) < [c.sub.2]} [subset][subset] Z, [for all][c.sub.1], [c.sub.2] [member of] R we get that [chi] [omicron] [phi] [omicron] [pi] + f is an exhaustion function. We denote by [Z.sub.i] the sublevel sets {[chi] [omicron] [phi] [omicron] [pi] + f < i} which are relatively compact in Z. This increasing sequence of open sets that cover Z will be the one that is needed in Theorem 6.

Now we have to build the functions [u.sub.i] in Theorem 6. For this we will make use of a lemma which is based upon Lemma 3. For details one should consult the Main Lemma of [19] and the Remark that follows.

Lemma 5. Let [pi] : Z [right arrow] X be a holomorphic map between reduced complex spaces with r = max[dim [[pi].sup.-1] (x) : x [member of] X}. Then there exists N a linear set of codimension [less than or equal to] r over Z such that for any relatively compact open subset U of Z, there exists a finite covering [{[V.sub.l]}.sub.l] of [bar.[pi](U)] by relatively compact open subsets and smooth functions [[phi].sub.l] : [U.sub.l] [right arrow] [R.sub.+] such that [[psi].sub.l] is 1-convex with respect to N over [U.sub.l] [intersection] U, where [U.sub.l] = [[pi].sup.-1] ([V.sup.l]).

Now we go back to the proof. Since [Z.sub.i] [subset][subset] Z, there exists a linear set N of codimension [less than or equal to] r over Z, a finite covering [{[V.sup.i.sub.l]}.sub.l] of [bar.[pi]([Z.sub.i])] by relatively compact open subsets and smooth functions [[phi].sup.i.sub.l] : [U.sup.i.sub.l] [right arrow] [R.sub.+] such that [[psi].sup.i.sub.l] is 1-convex with respect to N over [U.sup.i.sub.l] [intersection] [Z.sub.i], where [U.sup.i.sub.l] = [[pi].sup.-1] ([V.sup.i.sub.l]). The functions [[psi].sup.i.sub.l] may be taken > 0.

Let [{[[rho].sup.i.sub.l]}.sub.l] be a partition of unity subordinated to the covering [{[V.sup.i.sub.l]}.sub.l] and we define a smooth function [u.sub.i] on [Z.sub.i] as follows:

[u.sub.i] = [chi] [omicron] [phi] [omicron] [pi] + f + [summation over (l)] [[epsilon].sup.i.sub.l] * [([[rho].sup.il] [omicron] [pi]).sup.2] * [[psi].sup.i.sub.l],

where [[epsilon].sup.i.sub.l] > 0 are sufficiently small constants to be chosen later in the proof. Since the above sum is > 0, there exists [[delta].sub.i] > 0 such that for all z [member of] [Z.sub.i] we have [summation] [[epsilon].sup.i.sub.l] * [([[rho].sup.i.sub.l] [omicron] [pi]).sup.2] * [[psi].sup.i.sub.l] [greater than or equal to] [[delta].sub.i]. By choosing the constants [[epsilon].sup.i.sub.l] to be sufficiently small we may assume that [summation] [[epsilon].sup.i.sub.l] * [([[rho].sup.i.sub.l] [omicron] [pi]).sup.2] * [[psi].sup.i.sub.l] < 1.

First we show that the functions [u.sub.i] satisfy the conditions (a), (b) and (c) from Theorem 6. We define [C.sub.i] := i - 1 and [D.sub.i] := i. For simplicity we denote [summation] [[epsilon].sup.i.sub.l] * [([[rho].sup.i.sub.l] [omicron] [pi]).sup.2] * [[psi].sup.i.sub.l] by [[summation].sub.i]. Since [[summation].sub.i] [greater than or equal to] [[delta].sub.i], we have that {[u.sub.i] < i} [subset][subset] {x [omicron] [phi] [omicron] [pi] + f < i}, so this proves (a). For the second condition, let z [member of] [Z.sub.i+1] such that [u.sub.i+1] (z) < i - 1. We get that [chi] [omicron] [phi] [omicron] [pi] + f + [[summation].sup.i+1] < i - 1, thus x [chi] [omicron] [phi] [omicron] [pi] + f < i - 1 and z [member of] [Z.sub.i]. Adding [[summation].sup.i] to the last inequality, we obtain [chi] [omicron] [phi] [omicron] [pi] + f + [[summation].sup.i] < i - 1 + [[summation].sup.i] < i, since [[summation].sup.i] < 1. Now for condition (c), since [union]{[chi] [omicron] [phi] [omicron] [pi] + f < i - 2} = Z, it is enough to prove that {z [member of] [Z.sub.i+1] : [chi] [omicron] [phi] [omicron] [pi] + f < i - 2} [subset] {z [member of] [Z.sub.i+1] : [u.sub.i+1] < i - 1}. Adding [[summation].sup.i] to [chi] [omicron] [phi] [omicron] [pi] + f and using the fact that [[summation].sup.i] < 1, we easily get the claim.

Now we prove that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where P := [[pi].sup.*] M [intersection] N. We have that P is a linear set over Z and codim P [less than or equal to] q + r - 1. It is enough to show that every point z [member of] [Z.sub.i] admits an open neighbourhood D such that [u.sub.i] is 1-convex with respect to P[|.sub.D]. Using Lemma 2, this is equivalent to proving that for every compact K [subset] [Z.sub.i] there exists [delta] > 0 and for all z [member of] K there exists an extension [[??].sub.i] of [u.sub.i] such that

L([[??].sub.i], i(z))[i.sub.*] ([xi]) [greater than or equal to] [delta] [[parallel][i.sub.*]([xi])[parallel].sup.2]

for all [xi] [member of] [P.sub.z].

This is a local statement. So, without any loss of generality, we may suppose that there are local charts [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that:

(i) [pi](U) [subset] V and there exists an extension [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(ii) there exists A > 0 and smooth extensions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all [zeta] [member of] [M.sub.x] and x [member of] [pi](K) (this is true due to Lemma 4);

(iii) there are constants [a.sub.l] > 0 and smooth extensions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all [xi] [member of] [N.sub.z] and z [member of] K (this is true due to Lemma 5).

Let [[??].sub.l] be smooth extensions of [[??].sub.l] to [??].

So we get an extension [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of [u.sub.i][|.sub.U] given as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, using the same computations as in [19] (see Theorem A, pages 231-232), we get, for a sufficiently small positive [epsilon], that for any choice of the constants [[epsilon].sub.l], with 0 < [[epsilon].sub.l] [less than or equal to] [epsilon], the Levi form of [[??].sub.i] at i(z) in direction [i.sub.*] ([xi]) is strictly positive for [xi] [member of] [P.sub.z]. This means that there exists [delta] > 0 such that L([[??].sub.i], i(z))[i.sub.*]([xi]) [greater than or equal to] [delta] [[parallel][i.sub.*]([xi])[parallel].sup.2] for all [xi] [member of] [P.sub.z].

References

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[2] Ballico, E.: Morfismifiniti tra spazi complessi e q-convessita, Ann. Univ. Ferrara, sez. VII--Sc. Mat. 26 (1980), 29-31.

[3] Ballico, E.: Rivestimenti di spazi complessi e q-completezza, Riv. Mat. Univ. Parma 7 (1981), 443-452.

[4] Chiriacescu, G., Coltoiu, M., Joita, C.: Analytic cohomology groups in top degrees of Zariski open sets in Pn, Math. Z. 264 (2010), 671-677.

[5] Coltoiu, M., Vajaitu, V.: Locally trivial fibrations with singular 1-dimensional Stein fiber over q-complete spaces, Nagoya Math. J. 157 (2000), 1-13.

[6] Coltoiu, M., Vajaitu, V.: On the n-completeness of covering spaces with parameters, Math. Z. 237 (2001), 815-831.

[7] Diederich, K., Fornaess, J. E.: Smoothing q-convex functions and vanishing theorems, Invent. Math. 82 (1985), 291-305.

[8] Diederich, K., Fornaess, J. E.: Smoothing q-Convex Functions in the Singular Case, Math. Ann. 273 (1986), 665-671.

[9] Fornaess, J. E.: An Increasing Sequence of Stein Manifolds whose Limit is not Stein, Math. Ann. 223 (1976), 275-277.

[10] Fornaess, J. E.: 2 Dimensional Counterexamples to Generalizations of the Levi Problem, Math. Ann. 230 (1977), 169-173.

[11] Grauert, H.: On Levi's Problem and the Imbedding of Real Analytic Manifolds, Ann. Math. 68 (1958), 460-472.

[12] Joita, C.: On the n-concavity of covering spaces with parameters, Math. Z. 245 (2003), 221-231.

[13] Le Barz, P.: A propos des revetements ramifies d'espaces de Stein, Math. Ann. 222 (1976), 63-69.

[14] Narasimhan, R.: The Levi Problem for Complex Spaces, Math. Ann. 129 (1961), 355-365.

[15] Narasimhan, R.: The Levi Problem for Complex Spaces II, Math. Ann. 146 (1962), 195-216.

[16] Peternell, M.: Algebraische Varietaten und q-vollstandige komplexe Raume, Math. Z. 200 (1989), 547-581.

[17] Sommese, A. J.: A convexity theorem, Proc. Symp. Pure Math. 40 Part 2 (1983), 497-505.

[18] Stein, K.: Uberlagerung holomorph vollstandiger komplexer Raume, Arch. Mat. 7 (1956), 354-361.

[19] Vajaitu, V.: Some convexity properties of morphisms of complex spaces, Math. Z. 217 (1994), 215-245.

Simion Stoilow Institute of Mathematics of the Romanian Academy Research group of the project ID-3-0269 P.O. Box 1-764, Bucharest 014700, Romania and Department of Mathematics and Computer Science, "Politehnica" University of Bucharest, 313 Splaiul Independentei, Bucharest 060042, Romania. E-mail address: georgeionutionita@yahoo.com

George-Ionut Ioniza *

* The author was supported by CNCS grant PN-II-ID-PCE-2011-3-0269.

Received by the editors in October 2013--In revised form in September 2014.

Communicated by H. De Schepper.

2010 Mathematics Subject Classification : 32F10, 32C15.
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