Some non-pseudoconvex domains with explicitly computable non-Hausdorff Dolbeault cohomology

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by  Debraj Chakrabarti

SOME NON-PSEUDOCONVEX DOMAINS WITH EXPLICITLY
COMPUTABLE NON-HAUSDORFF DOLBEAULT COHOMOLOGY
5
1
DEBRAJCHAKRABARTI
0
2

n
a
J Abstract. We explicitly compute the Dolbeault cohomologies of certain domains in complex

6 space generalizing the classical Hartogs figure. The cohomology groups are non-Hausdorff topo-
1 logicalvectorspaces,anditispossibletoidentifythereduced(Hausdorff)andtheindiscretepart
of the cohomology.
]
V
C 1. Introduction
.
h 1.1. Topology of Dolbeault groups. TheDolbeaultcohomology groupsofacomplex manifold
t
a areamongitsfundamentalinvariants. Usually, incomplexanalysis,motivated bythetwoclassical
m nineteenth century examples of planar domains and compact Riemann surfaces, we consider man-
[ ifolds which have “nice” cohomology. Importantresults in this direction (under extra “positivity”

or “convexity” conditions such as pseudoconvexity, Stein-ness, q-convexity, Ka¨hlerness, existence
1
v of a positive line bundle, etc.) assert that the cohomology in a certain degree vanishes or is of
0 finite dimension.
3
For a general complex manifold, there is very little that one can say about the cohomology.
1
4 Typically,thenaturallineartopologyoftheDolbeaultgroup(arisingasthequotientoftheFr´echet
0 topology on the space of smooth ∂-closed forms of a particular degree by the subspace of ∂-exact
.
1 forms) is not Hausdorff, since there is no reason in general why the ∂-operator should have closed
0 range. Manifolds with non-Hausdorff Dolbeault cohomology, of which many examples are known,
5
arise primarily as counter-examples (e.g. [Ser55, Cas71, Mal75, Dem78].) Among the simplest
1
: examples of such domains is the classical Hartogs figure, the venerable non-pseudoconvex domain
v
contained in the unit bidisc ∆2 ⊂ C2 represented as
i
X
1 1
r H = (z ,z )∈ ∆2: |z | > or |z |< . (1)
a 1 1 2 1 2 2 2
(cid:26) (cid:27)
There are partial results, especially for domains in C2 or in Stein surfaces, which give sufficient
condition for the Dolbeault cohomology in certain degrees to be Hausdorff (e.g. [Tra86, LTS13].)
Many aspects of the structure of the cohomology groups are not well-understood. For example,
recall that a (possibly non-Hausdorff) topological vector space E can be written as a topological
direct sum of subspaces as E = E ⊕E , where, in the subspace topology, the linear subspace
ind red
E is indiscrete (the only non-empty open set is the whole space), and the linear subspace E
ind red
is Hausdorff (see, e.g. [KN76, Theorem 5.11]). Therefore, the cohomology group has an indiscrete
and an Hausdorff direct summand, the latter being referred to as the “reduced” cohomology (see
[Cas71].) It is natural to ask what features of the geometry of the domain are reflected in this
decomposition of the cohomology into the “reduced” and “indiscrete” parts.
This work was partially supported by a grant from the Simons Foundation (#316632), and also by an Early
Career internal grant from Central Michigan University.
1
2 DEBRAJCHAKRABARTI
As a step towards understanding such phenomena, we compute explicitly the cohomology of
a class of domains in Stein manifolds generalizing the Hartogs figure (1). Let Z be a connected
Stein manifold, and let Z be a open Stein domain in Z with Z 6= Z. We call (Z ,Z) a Stein
0 0 0
pair. For Stein pairs (X ,X) and (Y ,Y) we consider the domain H( X ×Y, defined as
0 0
H = (X ×Y )∪(Y ×X ), (2)
0 0
which we may refer to as a generalized Hartogs figure. Denoting the unit disc in the plane by
∆, the domain H in (1) corresponds to X = Y = ∆, X = z ∈ X: |z|> 1 , and Y =
1 0 2 0
z ∈Y : |z| < 1 .
2 (cid:8) (cid:9)
(cid:8)1.2. Results. W(cid:9)e adopt the following notation: if Z is a domain in a manifold Z, and Q is a
0
subspace of O(Z), then Q| denotes the space of restrictions to Z of functions in Q. We think
Z0 0
of Q| as a subspace of O(Z ) with the subspace topology. Recall that the Stein pair (Z ,Z)
Z0 0 0
is called Runge, if O(Z)| is dense in O(Z ). Also recall that a topological space is said to be
Z0 0
indiscrete if the only nonempty open set in it is the whole space. A simple class of domains with
non-Hausdorff cohomology is given by the following result:
Proposition 1.1. If at least one of the pairs (X ,X) and (Y ,Y) is Runge, then for each p with
0 0
0 ≤ p ≤ dimH, the space Hp,1(H) is either the zero space, or an indiscrete topological vector
space of uncountably infinite dimension. Further, there is at least one p in this range for which
the latter option holds.
For q > 1, we have Hp,q(H) = 0, and this last fact holds whenever (X ,X) and (Y ,Y) are
0 0
Stein pairs, without the hypothesis that one of them is Runge.
In the important special case when X,Y are domains in complex vector spaces we can say
more:
Corollary 1.2. Suppose that the Stein manifolds X,Y are parallelizable. Then for each p with
0 ≤ p ≤ dimH the cohomology Hp,1(H) is an indiscrete topological vector space of uncountable
dimension.
We now introduce further hypotheses on the pairs (X ,X) and (Y ,Y) in (2) so that we can
0 0
explicitly compute the cohomology in certain degrees. We say that the Stein pair (Z ,Z) is
0
(1) Split if O(Z)| is a closed subspace of O(Z ), and there is a closed subspace Q(Z ,Z) of
Z0 0 0
O(Z ) such that we have a direct sum representation
0
O(Z ) =O(Z)| ⊕Q(Z ,Z). (3)
0 Z0 0
(2) Quasi-split if O(Z)| is neither dense nor closed in O(Z ), and there is a closed subspace
Z0 0
Q (Z ,Z) of O(Z ) such that
r 0 0
O(Z ) =O(Z)| ⊕Q (Z ,Z). (4)
0 Z0 r 0
To illustrate this somewhat uncommon situation, let Z be the unit disc {|z| < 1} ⊂ C, and
consider the annuli Z = 1 < |z|< 1 and Z = 1 < |z| < 3 . Then it is not difficult to see
1 2 2 2 4
that the pair (Z ,Z) is split and the the pair (Z ,Z) is quasi-split. In Propositions 4.1 and 4.2
1 2
(cid:8) (cid:9) (cid:8) (cid:9)
below, a wider class of examples generalizing these are given.
Note also given a split pair (Z ,Z) (resp. a quasi-split pair (W ,W)) the space Q(Z ,Z) (resp.
0 0 0
Q (W ,W))isnotunique,butallsuchspacesareisomorphictothequotientTVSO(Z )/O(Z)|
r 0 0 Z0
(resp. to O(W )/O(W)| .)
0 W0
The main result of this paper is the following:
EXPLICITLY COMPUTABLE COHOMOLOGY 3
Theorem 1.3. Assume that the pair (X ,X) is split. Then we have an isomorphism of TVS:
0
Q(X ,X)⊗O(Y )
H0,1(H)∼= 0 0 . (5)
Q(X ,X)⊗O(Y) |
0 X0×Y0
b
Consequently:
(cid:0) (cid:1)
(1) If (Y ,Y) is Runge, H0,1(H) is an indiscretbe topological vector space of uncountable di-
0
mension. (cf. Proposition 1.1 above.)
(2) If (Y ,Y) is also split, then H0,1(H) is Hausdorff and infinite-dimensional, and is isomor-
0
phic to Q(X ,X)⊗Q(Y ,Y).
0 0
(3) If (Y ,Y) is quasi-split, then H0,1(H) is non-Hausdorff, but not indiscrete. There is a
0
topological direct sbum decomposition into subspaces
H0,1(H)= H0,1(H)⊕H0,1(H) (6)
ind red
where H0,1(H) is an indiscrete topological vector space of uncountable dimension, and
ind
0,1
H (H) is a non-zero Hausdorff topological vector space, and are given explicitly as:
red
H0,1(H) ∼= Q(X ,X)⊗Q (Y ,Y) (7)
red 0 r 0
and
Q(X ,X)b⊗O(Y)|
H0,1(H)∼= 0 Y0 . (8)
ind Q(X ,X)⊗O(Y) |
0 X0×Y0
b
In Section 5 below, we use these resu(cid:0)lts to compute th(cid:1)e cohomologies of several elementary
Reinhardt domains in C2. b
1.3. Acknowledgments. The author gratefully acknowledges the inspiration and encourage-
ment of Christine Laurent-Thi´ebaut and Mei-Chi Shaw. Also, many thanks to Franc Forstneric
and Bo Berndtsson for interesting conversations and comments.
2. Preliminaries
2.1. Natural topology on the cohomology. We collect here a few facts which we will use
(see [GR09]) . Let Ωp denote the sheaf of germs of holomorphic p-forms on a complex manifold
M. By Dolbeault’s theorem we can naturally identify the Sheaf cohomology group Hq(M,Ωp)
with the Dolbeault group Hp,q(M). Further, if U is an open cover of M by countably many Stein
open subsets, by Leray’s theorem, there is a natural isomorphism of the Cˇech cohomology group
Hq(U,Ωp) with the cohomology group Hq(M,Ωp). Each of the three isomorphic cohomology
groups Hp,q(M),Hq(M,Ωp) and Hq(U,Ωp) is a topological vector space in a natural way. Recall
that for the Dolbeault group Hp,q(M) this topology arises as the quotient topology of the Fr´echet
topology of the space Zp,q(M) of ∂-closed (p,q)-forms by the subspace ∂(Ap,q−1(M)) of exact
∂
forms. Note also that the space of Cˇech cochains Cq(U,Ωp) with respect to the cover U is the
direct sum of Fr´echet spaces, and therefore carries a natural topology. Denoting by Zq(U,Ωp)
the subspace of cocycles, the Cˇech group Hq(U,Ωp) is the quotient Zq(U,Ωp)/δ(Cq−1(U,Ωp),
and therefore carries a natural quotient topology. Finally, the group Hq(M,Ωp) is the direct
limit (with respect to refinement of covers) of the Cˇech groups Hq(U,Ωp), and therefore can be
endowed with the strong topology, i.e. the finest topology with respect to which the natural map
Hq(U,Ωp)֒→ Hq(M,Ωp) is continuous for each open cover U of M.
It is a fundamental fact that these natural topologies are essentially the same. More precisely
we have the following:
4 DEBRAJCHAKRABARTI
Result 2.1 ([Lau67, Section 2]). With respect to the natural topologies, described above, the
Dolbeault and Leray isomorphisms are linear homeomorphisms of topological vector spaces.
Thismeans, in particular, thatthedecomposition into indiscreteandreducedpartshas analogs
for the Cˇech cohomology, and further the Dolbeault and Leray isomorphisms respect this decom-
position. For details, see [Cas71].
2.2. Laufer and Siu Theorems. We now recall an important general fact regarding Dolbeault
groups of domains in Stein manifolds, which will allow us to conclude that certain topological
vector spaces are of uncountable dimension by knowing that these spaces are non-zero.
Result 2.2 ([Lau75, Siu67]). Let Ω be a non-empty domain in a Stein manifold of positive
dimension. Then the cohomology group Hp,q(Ω) is either zero or uncountably infinite dimension.
IfHp,q(Ω)= Hp,q(Ω)⊕Hp,q(Ω)isthedecomposition ofthecohomology intoitsreduced(Hausdorff)
red ind
and indiscrete part, then each of these parts is either zero or of uncountably infinite dimension.
2.3. Tensor products. For the general theory of nuclear spaces and topological tensor products
we refer to [Tr`e67]. However, in our application, the tensor products used will be of a very
simple type and can be easily described in elementary terms. Let X,Y be complex manifolds,
and let Q ⊂ O(X) and R ⊂ O(Y) be linear subspaces. The algebraic tensor product Q⊗R is
the linear span of the functions f ⊗g on X ×Y, where by definition (f ⊗g)(z,w) = f(z)g(w)
for z ∈ X,w ∈ Y. The topological tensor product Q⊗R, is the closure of the algebraic tensor
product Q⊗R in the space O(X ×Y) (this is equivalent to the abstract definition in [Tr`e67] in
this particular case.) Therefore, whatever Q,R might bbe, Q⊗R is a Fr´echet space.
3. Proofs
b
3.1. Proof of Proposition 1.1 and its Corollary 1.2.
Lemma 3.1. The manifold H in (2) is not Stein. This result holds even if the complex manifolds
X,X ,Y,Y are not assumed to be Stein, all we need is that X ( X and Y ( Y are proper
0 0 0 0
domains.
This is a consequence of the fact that if we think of H as a domain in X ×Y, there is a point
on the boundary of H to which each holomorphic function on H extends holomorphically. Indeed
with more work, we can show that each holomorphic function on H extends to a neighborhood of
∂X ×∂Y , but this will not be needed for our application.
0 0
Proof of Lemma 3.1. Let m = dimX,n = dimY. We first consider the special case in which
X = ∆m,Y = ∆n, and for some 0< ρ,r < 1, we have X = ∆m,Y = ∆n, where ∆m denotes the
0 ρ 0 r ρ
polydisc {|z |< ρ for j = 1,...,m} and ∆n is similarly defined. Then the logarithmic image
j r
L ={(log|z |,...,log|z |) ∈Rm+n: (z ,...,z )∈ H}
H 1 m+n 1 m+n
is notconvex, andconsequently byclassical results offunction theory, theenvelope of holomorphy
of H is the smallest domain in ∆m+n containing H whose logarithmic image is convex (see e.g.
[FG02, page 83 ff.].) In particular, it follows that each holomorphic function on H extends to a
neighborhood of ∂X ×∂Y .
0 0
In the general situation, let P ∈ X be a point near the boundary ∂X in the sense that there
0 0
is a neighborhood V of P in X which is biholomorphic to the polydisc ∆m by a biholomorphic
map which carries P to 0, and V has nonempty intersection with ∂X . Identifying V with ∆m,
0
let 0 < ρ < 1 be the smallest radius such that the intersection ∂∆m∩∂X is nonempty. Similarly,
ρ 0
EXPLICITLY COMPUTABLE COHOMOLOGY 5
let W be a coordinate neighborhood on Y biholomorphic to ∆n centered at a point of Y , and let
0
r be the smallest radius for which the intersection ∂∆n∩∂Y is nonempty. Then define
r 0
H = ∆m×∆n∪∆m×∆n ⊂ H.
r ρ
By the previous paragraph, each holomorphic function on h extends to a neighborhood of ∂∆m×
ρ
∂∆n. Choosing P′ ∈ ∂∆m∩∂X and Q′ ∈ ∂∆n∩∂Y , we see that each holomorphic function on
r ρ 0 r 0
H extends to a neighborhood of (P′,Q′)∈ ∂H, so that H is not Stein. (cid:3)
Proof of Proposition 1.1. Let
U = X ×Y, and U = X ×Y (9)
1 0 2 0
ThenU = {U ,U }isanopencoverofofHbySteinopensets,andisconsequentlybyLeray’stheo-
1 2
rem, the Cˇech group Hq(U,Ωp)is naturally isomorphicto the sheaf cohomology group Hq(H,Ωp),
where Ωp denotes the sheaf of germs of holomorphic p-forms on H. Since the manifold H admits
an open cover by two Stein domains, it follows that if q > 1, then the the space of Cˇech cochains
Cq(U,Ωp) vanishes, and consequently, we have Hq(U,Ωp) = 0. By the Dolbeault isomorphism
theorem, we have that Hp,q(H)∼= Hq(H,Ωp) ∼= Hq(U,Ωp) = 0.
If it was true that Hp,1(H)= 0 for each p, then we will have Hp,q(H)= 0 for q > 0. Therefore
we would have that H is Stein, which contradicts Lemma 3.1. Therefore, there is at least one p,
for which Hp,1(H)6= 0.
Denote by U the intersection U ∩U , and recall that Ωp is the sheaf of germs of holomorphic
12 1 2
p-forms. By Result 2.1, we have a linear homeomorphism:
Z1(U,Ωp)
Hp,1(H)∼=
δ(C0(U,Ωp))
Ωp(U )
12
= , (10)
Ωp(U )| +Ωp(U )|
1 U12 2 U12
where Ωp(U )| means the space of restrictions of holomorphic p-forms on U to the subset U
1 U12 1 12
(and similarly for the other term in the denominator), and the sum of two subspaces of a vector
space is as usual the linear span of their union. Note that till this point we have not made any
use of the hypothesis that one of the Stein pairs (X ,X) and (Y ,Y) is Runge.
0 0
By hypothesis atleast one of thepairs (X ,X) and(Y ,Y) is Runge. Let ussay fordefiniteness
0 0
that (Y ,Y) is Runge, i.e., Y is a properStein domain in the Stein manifold Y such that O(Y)|
0 0 Y0
is dense in O(Y ). It follows that the pair of Stein manifolds (X ×Y ,X ×Y)= (U ,U ) is also
0 0 0 0 12 1
Runge. ItnowfollowsfromaclassicalresultofSteintheory(see[GR04,page170])thatΩp(U )|
1 U12
is dense in Ωp(U ) for each p. Consequently, the space of exact Cˇech cochains δ(C0(U,Ωp)) =
12
Ωp(U )| +Ωp(U )| is dense in the space of closed cochains Z1(U,Ωp) = Ωp(U ). For a
1 U12 2 U12 12
particular p, we therefore have two possibilities:
(1) Ωp(U )| +Ωp(U )| = Ωp(U ) so that Hp,1(H)= 0.
1 U12 2 U12 12
(2) Ωp(U )| +Ωp(U )| 6= Ωp(U ) so that Hp,1(H) is a nonzero indiscrete topological
1 U12 2 U12 12
vector space, and we have already shown that there is at least one p for which this option
must hold. An appeal to Result 2.2 completes the proof.
(cid:3)
6 DEBRAJCHAKRABARTI
Proof of Corollary 1.2. Let N = dimH. From the triviality of the tangent bundle of H, we see
(N)
p
that on H, we have an isomorphism of sheaves Ωp ∼= Ω0, which gives rise to an isomorphism
j=1
M
at the cohomology level
(N)
p
Hp,1(H) ∼= H0,1(H). (11)
j=1
M
Thereisap = p forwhichthelefthandsideisanindiscreteTVSofuncountabledimension. Since
0
H0,1(H) is linearly homeomorphic to a subspace of Hp0,1(H), it follows from (11) that H0,1(H)
is also an indiscrete TVS of uncountable dimension. For for any p with 1 ≤ p ≤ N, the result
follows from (11), since the direct sum of indiscrete TVS is clearly indiscrete. (cid:3)
3.2. Proof of Theorem 1.3. Recall from (9) that the two-element Leray cover U that we are
using to compute cohomologies consists of U = X × Y, and U = X × Y . We also have
1 0 2 0
U = X ×Y .
12 0 0
By hypothesis, the pair (X ,X) is split. It is easy to see that this implies that the pairs
0
(U ,U ) = (X ×Y ,X ×Y ) and (U ,X ×Y) = (X ×Y,X ×Y) are also split, are if Q(X ,X)
12 2 0 0 0 1 0 0
is a complement of O(X)| in O(X ), then we have:
X0 0
O(U ) = O(U )| ⊕Q (X ,X)⊗O(Y ). (12)
12 2 U12 0 0 0
Similarly, we have
b
O(U ) = O(X ×Y)| ⊕Q (X ,X)⊗O(Y). (13)
1 U1 0 0
Restricting each side of (13) to the subset U ⊂ U , we obtain
12 1
b
O(U )| = O(X ×Y)| ⊕ Q (X ,X)⊗O(Y) | (14)
1 U12 U12 0 0 U12
We analyze the relation of the terms of (14) with O(cid:0)(U )| . Observe(cid:1)that we have
2 U12 b
Q(X ,X)⊗O(Y) | ∩O(U )| = Q(X ,X)⊗O(Y) | ∩O(X ×Y )|
0 U12 2 U12 0 U12 0 X0×Y0
(cid:0) (cid:1) ⊆ Q(cid:0) (X0,X)⊗O(Y0)∩(cid:1) O(X)|X0⊗O(Y0)
b b
⊆ (Q(X ,X)∩O(X)| )⊗O(Y )
0 X0 0
b b
= {0}⊗O(Y )
0
b
= {0}. (15)
b
Also, we have
O(X ×Y)| ⊆O(X)| ⊗O(Y )
U12 X0 0
=O(U )| . (16)
2 U12
b
Using (14), (15) and (16), we can compute
O(U )| +O(U )| = O(X ×Y)| ⊕ Q(X ,X)⊗O(Y) | +O(U )|
1 U12 2 U12 U12 0 U12 2 U12
= (cid:0)Q(X0,X)⊗O(Y)(cid:0)|U12 +O(U2)|U12(cid:1) (cid:1) (17)
b
= (cid:0)Q(X0,X)⊗O(Y)(cid:1)|U12 ⊕O(U2)|U12, (18)
b
(cid:0) (cid:1)
b
EXPLICITLY COMPUTABLE COHOMOLOGY 7
where in (17) we use (16) to absorb the term O(X×Y)| in O(U )| , and in (18), we use (15)
U12 2 U12
to conclude that the sum is actually a direct sum of closed subspaces. Therefore we have
O(U)|
H0,1(H) ∼= U12 (cf. (10))
O(U )| +O(U )|
1 U12 2 U12
Q(X ,X)⊗O(Y )⊕O(U )|
= 0 0 2 U12 using (12) and (18)
Q(X ,X)⊗O(Y) | ⊕O(U )|
0 U12 2 U12
b
Q(X ,X)⊗O(Y )
∼ (cid:0) 0 (cid:1)0
= b ,
Q(X ,X)⊗O(Y) |
0 U12
b
(cid:0) (cid:1)
Which establishes (5), since U b= X ×Y . We now consider what happens undervarious further
12 0 0
hypotheses on the pair (Y ,Y):
0
(1) First assume that (Y ,Y) is Runge. We claim that in this case Q(X ,X)⊗O(Y) |
0 0 X0×Y0
is dense in Q(X ,X)⊗O(Y ) so that H0,1(H) is an indiscrete space of infinite dimension. Since
0 0 (cid:0) (cid:1)
the algebraic tensor product Q(X ,X) ⊗ O(Y ) is dense in Q(X ,X)⊗O(Y ), itbsuffices to see
0 0 0 0
that the algebraic tenbsor product Q(X0,X)⊗O(Y)|Y0 is dense in Q(X0,X)⊗O(Y0). But this is
obvious since O(Y)|Y0 is dense by hypothesis in O(Y0). b
(2) Now assume that (Y ,Y) is split, and we have a direct sum decomposition into closed
0
subspaces
O(Y ) = O(Y)| ⊕Q(Y ,Y).
0 Y0 0
Also, we have
Q(X ,X)⊗O(Y) | = Q(X ,X)⊗O(Y)| .
0 X0×Y0 0 Y0
(cid:0) (cid:1)
Therefore b b
Q(X ,X)⊗O(Y )
H0,1(H) ∼= 0 0
Q(X ,X)⊗O(Y) |
0 U12
b
Q(X ,X)⊗(O(Y)| ⊕Q(Y ,Y))
= (cid:0) 0 b (cid:1)Y0 0
Q(X ,X)⊗O(Y) |
0 X0×Y0
b
Q(X ,X)⊗O(Y)| ⊕Q(X ,X)⊗Q(Y ,Y)
= (cid:0)0 b Y0 (cid:1) 0 0
Q(X ,X)⊗O(Y)|
0 Y0
b b
= Q(X ,X)⊗Q(Y ,Y)
0 0
b
which completes the proof in this case. Nbote that since Q(X ,X) ∼= Q(X ,X) and Q(Y ,Y) ∼=
0 0 0
Q(Y ,Y), it follows we have the invariant representation
0
H0,1(H)∼= Q(X ,X)⊗Q(Y ,Y). (19)
0 0
(3) Now suppose that the pair (Y ,Y) is quasi-split, abnd we have a direct sum decomposition
0
O(Y ) = O(Y)| ⊕Q (Y ,Y).
0 Y0 r 0
8 DEBRAJCHAKRABARTI
Therefore, using (5), we obtain
Q(X ,X)⊗O(Y )
H0,1(H) ∼= 0 0
Q(X ,X)⊗O(Y) |
0 X0×Y0
b
(cid:0)Q(X ,X)⊗ O(Y)(cid:1)| ⊕Q (Y ,Y)
0 b Y0 r 0
=
Q(X ,(cid:16)X)⊗O(Y) | (cid:17)
0 X0×Y0
b
= Q(X0(cid:0),X)⊗O(Yb)|Y0 ⊕Q(cid:1)(X0,X)⊗Qr(Y0,Y)
Q(X ,X)⊗O(Y) |
0 X0×Y0
b b
= Q(X0,(cid:0)X)⊗O(Y)b|Y0 ⊕(cid:1) Q(X0,X)⊗Qr(Y0,Y).
Q(X ,X)⊗O(Y) |
0 X0×Y0
b
Note that the first term of t(cid:0)his direct sum de(cid:1)composition is the qbuotient of the Fr´echet space
b
Q(X ,X)⊗O(Y)| by a dense subspace, and hence is indiscrete, and the second term is the
0 Y0
tensor product of two subspaces of Fr´echet spaces, and hence a Hausdorff (indeed a Fr´echet)
topologicabl vector space. The expressions (6), (7) and (8) now follow.
4. Split and Quasi-split pairs
In this section we give some simple sufficient conditions guaranteeing that a pair is split or
quasi-split.
Proposition 4.1. (a) Let X ⊂ Cn be a bounded star-shaped domain, and X = U ∩X, where
0
U is a neighborhood in Cn of the Sˇilov boundary of X. Then the image of the restriction map
O(X) → O(X ) is closed.
0
(b) Assume further in (a) that X is complete Reinhardt and X is Reinhardt. Then the pair
0
(X ,X) is split.
0
Proof. It is easy to see that the conclusion of part (a) of the proposition is equivalent to the
following: for each compact K ⊂ X, there is a C > 0 and a compact K ⊂ X such that for
K 0 0
each f ∈ O(X), we have
kfk ≤ C kfk , (20)
K K K0
wherek·k ,k·k denotethe supnorm on K,K . Let ΣbetheSˇilov boundaryof X, andwithout
K K0 0
loss of generality assume that X is star-shaped with respect to the origin. Let K be a compact
subset of X. Then there is a λ with 0 < λ < 1 such that K is still compactly contained in λX.
Note that we have λΣ∩K = ∅, and setting K = λΣ, we see that K is the Sˇilov boundary of
0 0
λX. Therefore, we have for each f ∈ O(X) that kfk ≤ kfk , which proves part (a).
K K0
For part (b), let A = {α ∈ Zn|zα ∈ O(X )}. Then any f ∈ O(X ) has a Laurent expansion
0 0
f(z) = c (f)zα, where c (f)∈ C. We can write
α∈A α α
P f(z)= c (f)zα
α
α∈A
X
= c (f)zα+ c (f)zα
α α
α∈Nn∩A α∈A\Nn
X X
= f (z)+f (z).
1 2
EXPLICITLY COMPUTABLE COHOMOLOGY 9
From estimate (20) it follows that f extends to a holomorphic function in O(X). If we consider
1
the closed subspace of O(X ) given by
0
Q(X ,X) = {f ∈ O(X )|c (f)= 0 if α∈ Nn},
0 0 α
then clearly f ∈ Q(X ,X), so that we obtain a direct sum decomposition into closed subspaces
2 0
O(X ) =O(X)| ⊕Q(X ,X),
0 X0 0
which shows that (X ,X) is split. (cid:3)
0
Proposition 4.2. Suppose that X ,X are Stein domains in a Stein manifold X such that X ⊂
0 1 0
X , the pair (X ,X ) is split and the pair (X ,X) is Runge. Then the pair (X ,X) is quasi-split.
1 0 1 1 0
Proof. Since the pair (X ,X ) is split, there is a closed subspace Q(X ,X ) of O(X ) such that
0 1 0 1 0
O(X )= O(X )| ⊕Q(X ,X ). (21)
0 1 X0 0 1
DefineQ (X ,X) = Q(X ,X ),andnotethatsince(X ,X)isRunge,wehaveO(X ) = O(X)| .
r 0 0 1 1 1 X1
Therefore:
O(X )| = O(X)| |
1 X0 X1 X0
(cid:16) (cid:17)
= O(X)| .
X0
Therefore, (21) takes the form:
O(X ) =O(X)| ⊕Q (X ,X),
0 X0 r 0
which shows that (X ,X) is quasi-split.
0
(cid:3)
5. Examples
We now apply Proposition 1.1 and Theorem 1.3 to study the cohomology of some well-known
domainsofelementarymulti-variablecomplexanalysis. Eachoftheseisadomainintheunitbidisc
∆2 ⊂ C2. It suffices to find H0,1(H), as H1,1(H) ∼= H0,1(H)⊕H0,1(H), and H2,1(H) ∼= H0,1(H).
We already know that Hp,2(H) = 0 for each p.
Let ∆ be the unit disc in the plane, and for r > 0, let ∆ = {z ∈ C: |z| < r} be the disc of
r
radius r. Also, for 0 ≤ r < R ≤ ∞, let A(r,R) = {z ∈ C: r < |z| < R} be an annulus of inner
radius r and outer radius R. We begin by observing the following:
(1) if 0 < r < 1, the pair (∆ ,∆) is Runge. Indeed, the holomorphic polynomials are dense
r
in both the domains.
(2) if 0 < r < 1, then the pair (A(r,1),∆) is split, as one can see from Proposition 4.1. We
can take
−1
Q(A(r,1),∆) = f ∈ O(A(r,1)) f(z) = a zν . (22)
ν
( (cid:12) )
(cid:12) ν=X−∞
(cid:12)
(3) If 0 < r < R < 1, then the pair (A(r,R),∆(cid:12)) is quasi-split. This follows from Propo-
(cid:12)
sition 4.2, since (X ,X ) = (A(r,R),∆ ) is split and (X ,X) = (∆ ,∆) is Runge. In
0 1 R 1 R
particular, we can take
Q (A(r,R),∆) = Q(A(r,R),∆ ). (23)
r R
10 DEBRAJCHAKRABARTI
We now consider several special cases of the domain H as defined in (2). Let 0 < r ,r < 1 be
1 2
fixed in what follows.
(1)LetH = z ∈ ∆2||z | < r , or |z | < r .H correspondstothechoice(X ,X) = (∆ ,∆)
0 1 1 2 2 0 0 r1
and (Y ,Y) = (∆ ,∆) in (2). By Corollary 1.2 to Proposition 1.1, it follows that H0,1(H ) and
0 (cid:8)r2 (cid:9) 0
H1,1(H )∼= H0,1(H )⊕H0,1(H ) are both indiscrete.
0 0 0
(2) Let H = z ∈ ∆2||z | > r , or |z |< r , generalizing slightly the domain in (1). This
1 1 1 2 2
corresponds to taking (X ,X) = (A(r,1),∆) and (Y ,Y) = (∆ ,∆) in (2). Since (Y ,Y) is
(cid:8) 0 (cid:9) 0 r2 0
Runge, by Corollary 1.2 we know that H0,1(H ) and H1,1(H ) are indiscrete. Since (X ,X) is
1 1 0
split, we can obtain more information using Theorem 1.3. We have
Q(A(r ,1),∆)⊗O(∆ )
H0,1(H ) ∼= 1 r2 .
1
(Q(A(r ,1),∆)⊗O(∆))|
1 A(r1,1)×∆r2
b
From the explicit description of Q(A(r1,1),∆) in (22b) it follows that an element of the numerator
may be represented by a Laurent series
−1 ∞
a zµzν,
µ,ν 1 2
µ=−∞ν=0
X X
convergentinthedomainA(r ,∞)×∆ ,andthedenominatoristhesubspaceofseriesconverging
1 r2
in the larger domain A(r ,∞)×∆. This allows us finally to parametrize the cohomology classes
1
in H0,1(H ) by the double sequence of coefficients {a }.
1 µ,ν
(3) Let H = z ∈ ∆2||z |> r , or |z |> r , which corresponds to (X ,X) = (A(r ,1),∆)
2 1 1 2 2 0 1
and (Y ,Y) = (A(r ,1),∆) in (2). Note that H is the difference of two polydiscs H = ∆2 \
0 (cid:8) 2 (cid:9)2 2
(∆ ×∆ ). Since the compact set ∆ ×∆ has a basis of Stein neighborhoods, it is known
r1 r2 r1 r2
that ∂ has closed range in A(0,1)(H ), and H0,1(H ) is Hausdorff (see [Tra86, Teorema 3].) This
2 2
also follows from our computations: thanks to Theorem 1.3, part (2), we know that H0,1(H ) is
2
Hausdorff, and using (19):
H0,1(H ) ∼= Q(X ,X)⊗Q(Y ,Y)
2 0 0
= Q(A(r ,1),∆)⊗Q(A(r ,1),∆).
1 2
b
This can be identified with the collection of Laurent series
b
−1 −1
a zµzν, (24)
µ,ν 1 2
µ=−∞ν=−∞
X X
convergent in the domain A(r ,∞)×A(r ,∞), i.e., the cohomology can beidentified with a space
1 2
of holomorphic functions. This may be compared with [GH78, page 49, example 4].
(4) Let R be such that r < R < 1, and set H = {z ∈ ∆2||z | > r , or r < |z | < R}. This
2 3 1 1 2 2
corresponds to the choice (X ,X) = (A(r ,1),∆) (which is split) and (Y ,Y) = (A(r ,R),∆)
0 1 0 2
(which is quasi-split.) We may think of this domain as a Hartogs figure from which a closed
polydisc has been excised. Using [Tra86, Teorema 3], we see that H0,1(H ) is non-Hausdorff, and
3
we will show that this is a domain for which there is a non-zero reduced cohomology, as well as a
non-zero indiscrete part of the cohomology, and this is in some sense the combination of the two
previous examples.