Increasing unions of Stein spaces with singularities

Let X be a Stein space and D ⊂ X an open subset which is the union of an increasing sequence of Stein open subsets of X .

Does it follow that D is necessarily Stein?

It is known from a classical theorem due to Behnke and Stein [1] that if D 1 ⊂ D 2 ⊂ · · · ⊂ D n C ... is an increasing sequence of Stein open sets in Cⁿ , then their union Uj>i D j is Stein.

In 1977, Markoe [2] proved the following:

Let X be a reduced complex space which the union of an increasing sequence X 1 ⊂ X 2 ⊂ · · · ⊂ X _n ⊂ . . . of Stein domains.

Then X is Stein if and only if the 1 ^th cohomology group of X with values in the structure sheaf O x vanishes ( H 1 (X, O x ) = 0 ).

Similarly, it is known (see [3]) that in an arbitrary complex space X an increasing union of Stein spaces (X n ) _n >0 is itself Stein if H 1 (X, O x ) is separated.

It has been proved earlier by Fornaess in [4], [5] and [6] that, if an additional condition is not imposed on H 1 (X, O x ) , the space X is not necessarily holomorphically-convex or holomorphically-separate.

( 2021 Alaoui, Y.

It was shown in [7] that if (D j ) j >i is an increasing sequence of Stein domains in a normal Stein space X , then D = Uj >i D j is a domain of holomorphy (i. e. for each x G dD there is f G O(D) which is not holomorphically extendable through x ).

It was proved in [8] that if X is a complex space and (D j ) j >i is an increasing sequence of Stein open subsets of X , then D = U D j is 2 -complete. We recall that a complex space X is said to be q -complete if there exists an exhaustion function ф G C ^ro (X, R ) which is q -convex on the whole space X , that is every point x ∈ X has an open neighborhood U isomorphic to a closed analytic set in a domain D ⊂ C ⁿ such that the restriction φ | U has an extension ф G C ^ (D) whose Levi form L(ф, z) has at most q — 1 negative or zero eingenvalues at any point z of D .

Here we solve affirmatively the above problem in the general case. We show that if X is a Stein space and, if Q is an increasing sequence of Stein open subsets of X , then there exists an increasing sequence (Q V ) _V>i of open subsets of Q such that Q = U_v>i Q V and there are continuous strictly psh functions ф ’’ ^: Q V ^ ]0, + ^ [ with the following properties

• (a)    ф " > 2 ^V+2 on Q V+2 \ Q V+i for every j >  v + 1 .

• (b)    (Ф V ‘ ) _V>1 is stationary on every compact subset of Q .

• 2.    The Union Problem

This implies that the function ф : Q ^ R defined by ф = lim ф ’’ is a continuous strictly psh exhaustion function on Q .

In order to solve the problem in dimension 2 , it is sufficient to show

Theorem 1. Every domain of holomorphy D which is relatively compact in a 2-dimensional normal Stein space X is Stein.

• <1 By the theorem of Andreotti—Narasimhan [9] we have only to prove that D is locally Stein and, we may of course assume that X is connected.

Let p G dD П Sing(X) , and choose a connected Stein open neighborhood U of p with U П Sing(X) = { p } and such that U is biholomorphic to a closed analytic set in a domain M in some C ^N . Let E be a complex affine subspace of C ^N of maximal dimension such that p is an isolated point of E ∩ U .

By a coordinate transformation, one can obtain that z (p) = 0 for all i G { 1, 2, • • • , N } and we may assume that there is a connected Stein open neighborhood V of p in M such that U П V П { z _i (x) = z ₂ (x) = 0 } = { p } .

We may suppose that N >  4 and, let E i = V П { z 2 (x) = • • • = z n - i (x) = 0 } , E 2 = { x G E i : z i (x) = 0 } . Then A = (U П V) U E i is a Stein closed analytic set in V as the union of two closed analytic subsets of V .

Let £ : A ^ A be a normalization of A . Then £ : A \ £ ^-i (p) ^ A \{ p } is biholomorphic and, clearly £ ^-i (A П E 2 ) = { x G A : z _i (£(x)) = ••• = z n _-i (£(x)) = 0 } is everywhere 1 dimensional. It follows from a theorem of Simha [10] that A \ £ ^-i (A П E ₂ ) is Stein. Hence A \ E 2 = ^(^4 \ ^ ^-i (A П E 2 )) itself is Stein.

Since p ∈ E 2 is the unique singular point of A , then U ∩ V ∩ D is Stein, being a domain of holomorphy in the Stein manifold A \ E ₂ . >

Let now X be a Stein space of dimension n >  2 and Q С X an open subset which is the union of an increasing sequence Q i С Q 2 С • • • С Q _n С ... of Stein open sets in X . Let ф _V : Q _V ^ ]0, +to[ be a smooth strictly psh exhaustion function on Q _V , and let (d _V ) _V>i be a sequence with d _V <  d _V+i , and Sup d _V = +to . One may assume that if Q V = { x G Q _V : ф _V (x) <  d _V } , then Q V СС Q V_+i .

Lemma 1. There exist for each v ^ 1 an exhaustion function ^ _v E C ” (fi _v ) which is strictly psh in a neighborhood of fi ‘ _v \ fi ^ - 1 , a locally finite covering (U _v ) _v^1 of fi by open sets U _v C fi V+1 , and constants c _v E R, v ^ф 1 , with the following properties:

• (a)    For each v ^ 1 there exists a function ^ _v : fi ^+1 ^ ]0, + ^ [ such that ^ _v | u _v is strictly psh and ^ _v = ^ _v-1 on { x E U _v : ^ _v+1 (x) <  c _v } П U _{v - 1} .

• (b)    For every index v ^ 1 , there exists E _v > 0 such that

fiv-1 \ fi‘v—2 C {x E Uv : ^v+1(x) < cv Ev } and

{ x ^E U v : ^ v+1 ^(x)< c v ^{+ E} v } ^C U v - 1 .

• <1 There exists a C ” exhaustion function ^ _v +1 on fi _v+1 which is strictly plurisubharmonic in a neighborhood of fi ‘ _v+1 \ fi ‘ _v such that, if m _v+1 = min ^ y ₊₁ \ q / W+1 and M _v+1 = max ^ ‘ _{v - i} ^ v+1 , then m v+1 > M v+1 .

In fact, we choose 9 _v E C Q ” (fi _v +1 ) with compact support in fi _v+1 \ Q ' _v-1 so that 0 C 9 _v C 1 and 9 _v (x) = 1 when x E fi ‘ _v+1 \ fi ‘ _v . Let £ be a point of dQ ‘ _v-1 such that ф _v+1 (£) = max Q 1 ф _v+1 . Then it is clear that

Vv+1 = фv + 1 + Фv+1(^)9v satisfies the requirements.

We now assume that fi o = 0 and put

U 1 = Q ‘ 2 , and U _v = (Q ‘ _v+1 \ fi ⁷ _v-2 ) for v >  2.

Then (U _v ) _v^1 is a locally finite covering of fi . Moreover, if we set

Cv = mv+1 = Inf |^v+1(x), x E (fi‘v+1 \ fi’v)}, then

( 0 ⁷v - 1 \ Q ⁷v - 2 ) C { x E U v : ^ v+1 (x) C . } C ( fi t \ ^- 2 ) C U v - 1 .

Furthermore, there exist c _v > 0 and E _v > 0 such that c _v + E _v = C _v and (fi ‘ _v-1 \ fi ‘ _v- 2) C {x E U v : ^ v+1 (x) < C v - E v }.

Moreover, if the function 9 _v E C Q ” ( fi _v+1 \ fi ^ - 1 ) is chosen so that 9 _v = 1 on

(fi L+1 \ fi t ) U {x E fi ’ v \ fi ‘ v - 1 : Inf _Q ‘v+1 \ QL Ф v+1 - E ^

C фv+1(x) C InfQ'v+1\QL фv+1 + Mv+11, then clearly we obtain (x E Uv : cv + ^^ C ^v+1(x) C cv + Ev} C {9v = 1}. Therefore with such a choice of 9v there exists for each v a function ^v : fi^+1 ^]0, +to[ such that ^v|uv is strictly plurisubharmonic and, ^v = ^v-1 on (x E Uv : ^v+1(x) < cv + ^^}.

In fact, if v = 1 , then it is obvious that ^ф 1 = Ф 2 has the required properties for fi 1 = 0, since U 1 = fi 2 and (x E U 1 : ^ 2 (x) < C 1 + "2 ¹} is contained in fi ( .

We now assume that v ^ 2 and, that ^ 1 ,..., ^ _v-1 have been constructed. Let x _v (t) = a _v (t - c _v - ^ r) where a _v is a positive constant, and consider the function ^ _v : Q ^+1 ^ ]0, + ^ [ defined by

^ v - 1 on { V v+1 <  C v - E v } ,

^ v = max(^ v - 1 ,X v (V v+1 ))

on { C v - E v ^ V v+1 <  C v + E v } ,

₄X v (Ф v+1 + Ф v+1O on { V v+1 >  C v + E v } .

Since on U v = {x E U _v : ^ _v+1 (x) < c _v + ^} C U _v-1 we have ^ _v-1 > 0 > X _v (v _v+1 ) and ^ _v-1 is strictly psh on U _v-1 , then ^ _v l u ^ = ^ _v-1 | u v is strictly psh on U ‘ . On the other hand, the subset {c _v + ^- ^ у _v+1 ^ c _v + e _v} C U _v - 1 is contained in { 9 _v = 1 } , which implies that ^ v = max(^ v - 1 ,X v (Ф v +1 + Ф v +1 (€))) on {c v + ^ f ^ V v +1 <  C v + E_v }. Then clearly the function ψ is well-defined and satisfies the required conditions, if a is taken so that a v — > ^max {^ +i =c v ■ v }n Qt ^ v - 1 . ▻

Theorem 2. If X is a Stein space and Q an open subset of X which is an increasing union of Stein open sets in X, then Q is Stein.

• <1 We shall prove that there exists for each v ^ 1 a continuous strictly psh function ^ '7 in a neighborhood of Q^ such that ^ j⁷ > 2 ^v+1 on Q ’ +2 \ Q^ +1 for every j ^ v + 2 and (- 0 7’ ) _v^1 is stationary on every compact set in Q .

In fact, let ϕ ^′ be the function defined by

ϕ ^′

(^ v on Q '‘, +1 \ Q ' v - 1 ,

[^_M on { x E U ^+1 : VM 2 (x) <  c ^ +1 - E _y+1 } for д ^ v.

Then, by Lemma 1 , V v is a continuous strictly plurisubharmonic function on Q ^ +1 .

Moreover, we have V _v = vL — 1 on { x E U ^+1 : V _y+2 (x) < C _y+1 - E _y+1 } for all ^ <  v - 1 .

Let now K be a compact set in Q and v ^ 2 such that K C Q^ — 1 . Since ^_v = Vv- 1 on K П (Q ⁷^ \ Q ‘ ^ — 1 ) C { x E U ^+1 : VM 2 (x) <  c ^ +1 - E _y+1 } for all д <  v - 1 , then ^v = V — 1 on K. This implies that the sequence (^_v ) _v>1 is stationary on every compact subset of Q .

Let now v ^ 1 be an arbitrary natural number. Then there exists a smooth function ^ '_v E C ^( X ) which is strictly plurisubharmonic in a neighborhood of (X \ Q ^+1 ) U Q ⁷ _v such that ^> 2 ^v+2 in Q ⁷_v+2 \ Q^ +1 but ^ 7 < 0 in Q ⁷ _v .

In fact, let h E Cro(X) be a strictly plurisubharmonic exhaustion function such that h < 0 in Q7v, and let xv E C^(X) be a smooth function with compact support in Q^+1 such that Xv = 1 in Q7v. Then it is clear that hv = h + bvXv, where bv = minxeQ7 +2\q/ h(x), is a smooth exhaustion function on X which is strictly plurisubharmonic in a neighborhood of (X\Q^ +1) U Q‘v such that if m'v = minyGQ‘v+2\Q‘?+1 hv(У) and M‘ = miixy -, hv(У), then mL > M‘.

Let E’, > 0 be such that m^ >  M ‘ + E ’,. Then we can choose a sufficiently big constant C_v >  1 so that

^ ‘ (x) = C v ( h v (x) - M ‘ - E_v^

is > 2 ^v+2 in (Q ‘ _v+2 \ Q^ +1 ) , ^v <  0 in Q ‘ _v , and strictly plurisubharmonic in a neighborhood of (X \ Qt +1 ) U Q ⁷v .

If now we consider the following function defined in Lemma 1

^ф v - 1

^ V = max (^ v - 1 ,X v C^ v+1 ))

₄ X v (^ф V + 1 ^{+ ф} V + 1 (€⁾)

on {^v+1 < Cv — Ev}, on {Cv — Ev ^ ^v+1 < Cv + Ev }, on {^v+1 > Cv + Ev}

and the fact that c _V + e _v = Inf {^ _V+1 (x)

x € (Q ‘ _v+1 \ Q V)}, we find that

(QV+1\Q‘v) C {x E Uv : ^v+1(x) > Cv + Ev} and, on the set (QV+1\^‘V) we have

^ v = ^ V = X v ^{( ф} V +1 ^{+ ф} v+1 ⁽^⁾⁾ >  a V

( ^v +1

- c _ν

-

ε ν

2/

>

εν aV "2" ■

We can therefore choose a _V again big enough so that a _V^ > ^ V on (Q' _v+1 \ QV) ■ Moreover, by suitable choice of the constants a . we can also achieve that ^ _V > ф . on (Q .₊₁ \ Q.) for all ц < v ■ In fact, since (Q . \ Q ' ._{- 1}) C }x € U .+1 : y .+₂ (x) < c .+1 — Е .+1} , then, for every 2 <  ц ^ v , ^ v = Ф . on ( Q . \ Q ' . - 1 ) ■ If we set A . = ( Q . \ Q ' . - 1 ) П { x € U . : ^_M1 (x) <  c . — e^} , then ф . = ф . - 1 on A . Since in addition (Q . \ Q ' . - 1) C {x € U . - 1 : ^ . (x) >  c . - 1 + e ^ - 1} , then on the set A . we have y V = ф . = ф . - 1 >  X . - 1 (^ . ) >  a . - 1 ^- 1 ■ Let now x € (Q . \ Q ._{- 1})■ If x € A . , since x € U . , then y .+1 (x) >  c . — e . Because (Q . \ Q ' ._{- 1}) C {x € U . - 1 : ^ . (x) >  c . - 1 + Е д - x}, we obtain, if ^ .₊1 (x) <  c . + e . ,

^ v ^(x) = ФД^ =^max ( Ф д — 1 ^(x),X . ⁽^ д+1 ^(x)) ) >  Ф д - 1 ^(x) = X . - 1 ⁽^ . ^(x)) > a . - 1 —J— .

Or ^ V ^(x) = ФДД >  x4^. + 1 ^)(x) > a . ^, ^if ^+1 ^(x) Ф ^c . ^{+ Е} д .

So we may of course take the constants a . sufficiently large so that a . — 1 ^ т ^{- 1} > ^ф . - 1 and a.^ 2 > ^ф . - 1 on (Q ^,. \ Q ._-1 ) for all ц ^ v ■ Since only finitely many conditions are required to get ^ V > ^ф . on (Q .+1 \ Q . ) for ц ^ v , it follows that the function ^ V ‘ ^: ^ L+1 ^ R given by ^ V ‘ = max(^ V ,^ф V ,^ V - 1 ,---,^ 1 ) is obviously continuous and strictly plurisubharmonic in ^ V+1 - Also it is clear that for every j ^ v + 1 , ^ j ‘ ^ ^ V >  2 ^{v +2} on (^ V+2 \ ^ ‘ _V+1 ) ^

Let now K C Q be a compact subset and v ^ 2 such that K C ^ V - 1 - Since ^ _V > 0 > ^ V on Q ‘ v - 1 ^and ^ V = ^ v - 1 ^on K , ^{then max(}У ^,v -1/ ^ф V -1/ ^ф V - 2 ^, ••• ^'l ) =^max(^ ‘ v ^,ф V ^,C - 1 , ••• ^,ф 1 ) on K , which implies that the sequence (^ф V ' ) _V>1 is stationary on every compact subset of Q

This proves that the limit ф" of (^ V ' ) is a continuous strictly plurisubharmonic exhaustion function on Q , which shows that Q is Stein, >