$L^2$-Serre duality on singular complex spaces and rational singularities

0.38 MB·

by  Jean Ruppenthal

L2-SERRE DUALITY ON SINGULAR COMPLEX SPACES AND
RATIONAL SINGULARITIES
J. RUPPENTHAL
Abstract. In the present paper, we devise a version of topological L2-Serre
duality for singular complex spaces with arbitrary singularities. This duality is
2
used to deduce various new L -vanishing theorems for the ∂-equation on singular
spaces. It is shown that complex spaces with rational singularities behave quite
5 tame with respect to the ∂-equation in the L2-sense. More precisely: a singular
1 2
point is rational if and only if the L -∂-complex is exact in this point. So, we
0
2
obtain an L -∂-resolutionof the structure sheaf in rational singular points.
2

r
p
A
1. Introduction

5
1 The Cauchy-Riemann operator ∂ plays a fundamental role in Complex Analysis
and Complex Geometry. On complex manifolds, functions – or more generally distri-

]
V butions – are holomorphic if and only if they are in the kernel of the ∂-operator, and
C the same holds in a certain sense on normal complex spaces. For forms of arbitrary
. degree, the importance of the ∂-operator appearsstrikingly for example in the notion
h
t of ∂-cohomology which can be used to represent the cohomology of complex mani-
a
m folds by the Dolbeault isomorphism. Solving ∂-equations, i.e., ∂-vanishing theorems,
play a central role in a vast number of problems in Complex Analysis and Geometry,
[

let us just mention the Cousin problems.
4
v However, a huge part of the ∂-theory is still restricted to the smooth setting.
3 In the present paper, we introduce a version of topological L2-Serre duality for ∂-
6
cohomology classes on singular complex spaces with arbitrary singularities, which
5
4 allows to deduce some new L2-vanishing theorems for the ∂-operator on singular
1. spaces. We obtain an explicit L2-∂-resolution of the structure sheaf of spaces with
0 rational singularities (and an L2-∂-characterization of rationality).
4
The L2-theory for the ∂-operator is of particular importance in Complex Analysis
1
: and Geometry and has become indispensable for the subject after the fundamental
v
i work of H¨ormander on L2-estimates and existence theorems for the ∂-operator [H1]
X
and the related work of Andreotti and Vesentini [AV]. Important applications of the
r
a L2-theory are e.g. the Ohsawa-Takegoshi extension theorem [OT], Siu’s analyticity
of the level sets of Lelong numbers [S2] or the invariance of plurigenera [S3] – just to
name some.
On the other hand, it is almost dispensable to mention the importance of Serre
duality at all. It is one of the most important tools in Complex Algebraic Geometry
and Complex Analysis. On singular spaces, the classical Serre duality has to be
replaced by the more involved Grothendieck duality (developed by Ramis and Ruget
intheanalyticsetting[RR]). Inthepresent paper,weproposeaversionoftopological
L2-Serre duality which is well-adapted to singular spaces.
Date: April 16, 2015.
2000 Mathematics Subject Classification. 32J25, 14F10, 14E15,32C35,32C37, 32W05.
Key words and phrases. Cauchy-Riemann equations, L2-theory, Serre duality, vanishing theo-
rems, singular complex spaces, canonical singularities,rational singularities.
1
2 J. RUPPENTHAL
The first problem one has to face when studying the ∂-equation on singular spaces
isthatitisnotclearwhatkindofdifferentialformsandoperatorsoneshouldconsider.
Recently, there has been some progress by different approaches.
Andersson and Samuelsson developed in [AS] Koppelman integral formulas for the
∂-equation on arbitrary singular complex spaces which allow for a ∂-resolution of the
structure sheaf in terms of certain fine sheaves of currents, called A-sheaves. These
A-sheaves are defined by an iterative procedure of repeated application of singular
integral operators.
A second, somewhat more explicit approach is as follows: consider differential
forms which are defined on the regular part of a singular variety and which are
square-integrable up to the singular set. This setting seems to be quite fruitful and
has some history by now (see [PS]). Also in this direction, some progress has been
made recently. Øvrelid–Vassiliadou and the author obtained in [OV2] and [R4] a
pretty complete description of the L2-cohomology of the ∂-operator (in the sense of
distributions) at isolated singularities.
Inthissetting, weunderstandtheclassofobjectswithwhichwedealverywell(just
L2-forms), but the disadvantage is a different one. Whereas the ∂-equation is locally
solvable for ∂-closed (0,q)-forms in the category of A-sheaves by the Koppelman
formulasin[AS], therearelocalobstructionstosolvingthe∂-equationintheL2-sense
at singular points (see e.g. [FOV], [OV2], [R4]). So, there can be no L2-∂-resolution
for the structure sheaf in general.
The starting point of the present paper was the idea that the ∂-operator in the
L2-sense may behave very well on spaces with canonical singularities. There are
several results which substantiate this idea. First, holomorphic differential forms are
reflexive on such spaces (see [GKKP]). It follows that holomorphic differential forms
on the regular part of the variety are in L2. Second, it is known that the ∂-equation
is solvable in the L2-sense for (0,q)-forms on affine cones over projective varieties of
low degree1 (see [R3] and [OV2], Section 7, Example 3). Third, there is a long list
of related positive results on the ∂-equation at rational double points2 (see [AZ1],
[AZ2], [AS], and also [R3], [LR] for the A -singularity).
1
The underlying idea is that canonical singularities are rational (see e.g. [K], The-
orem 11.1), i.e., we expect that the singularities do not contribute to the local coho-
mology in a certain sense. Pursuing this idea, it turned out that there is a notion
of L2-∂-cohomology for (0,q)-forms which can be described completely in terms of
a resolution of singularities (see Theorem 1.1 below). A singular point is rational
if and only if the L2-∂-complex is exact in this point. If the underlying space has
rational singularities, then we obtain an L2-∂-resolution of the structure sheaf, i.e., a
resolution of the structure sheaf in terms of a well-known class of differential forms.
This gives even more hope for the idea that spaces with canonical singularities may
allow for a comprehensive L2-theory for the ∂-operator.
One of our main tools is a version of topological L2-Serre duality for singular
complex spaces with arbitrary singularities. In order to establish and apply this kind
of Serre duality, we have to overcome several difficulties which do not appear in the
smooth setting. We have to establish some new duality relations between different
closed L2-extensions of the ∂-operator at arbitrary singularities, and to provide new
Hausdorffness results for some cohomology groups on singular spaces.
1Such varieties are canonical.
2 Rational double points are precisely the possible canonical singularities in dimension two.
L2-SERRE DUALITY AND RATIONAL SINGULARITIES 3
To explain our results more precisely, let us introduce some notation (see Sec-
tion 2.1 for the details). Let X be a Hermitian complex space3 of pure dimension n
and F → X a Hermitian holomorphic line bundle. We denote by Lp,q(F) the sheaf of
germs of F-valued (p,q)-forms on the regular part of X which are square-integrable
on K \SingX for compact sets K in their domain of definition.4
Due to the incompleteness of the metric on X \ SingX, there are different rea-
sonable definitions of the ∂-operator on Lp,q(F)-forms. To be more precise, let ∂
cpt
be the ∂-operator on smooth forms with support away from the singular set SingX.
Then ∂ can be considered as a densely defined operator Lp,q(F) → Lp,q+1(F). One
cpt
can now consider various closed extensions of this operator. The two most important
are the maximal closed extension, i.e., the ∂-operator in the sense of distributions
which we denote by ∂ , and the minimal closed extension, i.e., the closure of the
w
graph of ∂ which we denote by ∂ . Let Cp,q(F) be the domain of definition of ∂
cpt s w
which is a subsheaf of Lp,q(F), and Fp,q(F) the domain of definition of ∂ which in
s
turn is a subsheaf of Cp,q(F). We obtain complexes of fine sheaves
Cp,0(F) −∂→w Cp,1(F) −∂→w Cp,2(F) −∂→w ... (1)
and
Fp,0(F) −∂→s Fp,1(F) −∂→s Fp,2(F) −∂→s ... (2)
If F is just the trivial line bundle, then K := ker∂ ⊂ Cn,0 is the canonical sheaf
X w
of Grauert–Riemenschneider and Ks := ker∂ ⊂ Fn,0 is the sheaf of holomorphic
X s
n-forms with a certain boundary condition that was introduced in [R4]. We will see
below that O = ker∂ ⊂ F0,0 for the sheaf of weakly holomorphic functions O .
X s X
It is clear that (1) and (2) are exact in regular points of X. Exactness in singular
points is equbivalent to the difficult problem of solving ∂-equations locally in thbe L2-
sense at singularities, which is not possible in general (see e.g. [FOV], [OV1], [OV2],
[R2], [R3], [R4]). However, it is known that (1) is exact for p = n, and that (2) is
exact for p = n if X has only isolated singularities ([PS] and [R4]; see Section 2.2).
In these cases, the complexes (1) and (2) are fine resolutions of the canonical sheaves
K and Ks , respectively.
X X
For an open set Ω ⊂ X, we denote by Hp,q (Ω,F) the cohomology of the com-
w,loc
plex (1), and by Hp,q (Ω,F) the cohomology of (1) with compact support. Anal-
w,cpt
ogously, let Hp,q (Ω,F) and Hp,q (Ω,F) be the cohomology groups of (2). These
s,loc s,cpt
L2-cohomology groups inherit the structure of topological vector spaces, which are
locally convex Hausdorff spaces if the corresponding ∂-operators have closed range.5
We can now formulate the main results of the present paper.
Theorem 1.1. Let X be a Hermitian complex space, π : M → X a resolution of
singularities and Ω ⊂ X a holomorphically convex domain. Then push-forward of
forms induces for all q ≥ 0 a natural topological isomorphism
Hq π−1(Ω),O −∼=→ H0,q (Ω). (3)
M s,loc
3AHermitiancomplexspace((cid:0)X,g)isareduc(cid:1)edcomplexspaceX withametricg onthe regular
part such that the following holds: If x ∈ X is an arbitrary point there exists a neighborhood
U = U(x) and a biholomorphic embedding of U into a domain G in CN and an ordinary smooth
Hermitian metric in G whose restriction to U is g|U.
4This is what we mean by square-integrableup to the singular set.
5 Note that different Hermitian metrics lead to ∂-complexes which are equivalent on relatively
compact subsets. So, one can put any Hermitian metric on X in many of the results below.
4 J. RUPPENTHAL
From that we obtain for the local L2-∂ -cohomology immediately:
s
Theorem 1.2. Let X be a Hermitian complex space, π : M → X a resolution of
singularities and q ≥ 0. Then:
Hq(F0,∗) ∼= Rqπ O ∀x ∈ X. (4)
x ∗ M x
Particularly, for q =(cid:0)0, (cid:1) (cid:0) (cid:1)
ker∂0,0 = H0(F0,∗) = O , (5)
s X
where O denotes the sheaf of germs of weakly holomorphic functions.
X
b
0,0
It follows that x ∈ X is a normal point exactly if ker∂ = O . As x ∈ X is
b s x X,x
a rational point if it is normal and (Rqπ O ) = 0 for q > 0, we get also:
∗ M x
(cid:0) (cid:1)
Theorem 1.3. Let X be a Hermitian complex space. Then the L2-∂-complex
0 → O −→ F0,0 −∂→s F0,1 −∂→s F0,2 −∂→s F0,3 −∂→s ... (6)
X
is exact in a point x ∈ X if and only if x is a rational point.
Hence, if X has only rational singularities, then (6) is a fine resolution of the
structure sheaf O .
X
If X has only rational singularities, then Theorem 1.3 yields directly further finite-
ness and vanishing results, e.g. if X is q-convex or q-complete. An essential tool in
the proof of Theorem 1.1 is the following new version of topological L2-Serre duality
on singular complex spaces with arbitrary singularities:
Theorem 1.4 (Serre duality). Let X be a Hermitian complex space of pure di-
mension n, Ω ⊂ X an open set, F → X a Hermitian holomorphic line bundle, and
let 0 ≤ p,q ≤ n. If Hp,q (Ω,F) and Hp,q+1(Ω,F) are Hausdorff, then the mapping
w,loc w,loc
Lp,q(Ω,F)×Ln−p,n−q(Ω,F∗) → C , (η,ω) 7→ η ∧ω,
cpt
ZΩ∗
induces a non-degenerate pairing of topological vector spaces
Hp,q (Ω,F)×Hn−p,n−q(Ω,F∗) → C
w,loc s,cpt
such that Hn−p,n−q(Ω,F∗) is the topological dual of Hp,q (Ω,F) and vice versa.
s,cpt w,loc
The same statement holds with the indices {s,w} in place of {w,s}. Then there
is a non-degenrate pairing
Hp,q (Ω,F)×Hn−p,n−q(Ω,F∗) → C.
s,loc w,cpt
If the topological vector spaces Hp,q (Ω,F), Hp,q+1 (Ω,F) are non-Hausdorff,
w/s,loc w/s,loc
then the statement of Theorem 1.4 holds at least for the separated cohomology
groups H = ker∂ /Im ∂ (Theorem 3.5).6 Though Theorem 1.4 looks at first
w/s w/s w/s
glance like a standard version of Serre duality, there are two essential difficulties in
its proof which do usually not appear. First, the ∂-operators under consideration
are just closed densely defined operators in the Fr´echet spaces Lp,q(Ω,F) and the
(LF)-spaces Ln−p,n−q(Ω,F∗). Second, we have to show that the operators ∂ and
cpt w
∂ are topologically dual at arbitrary singularities. This point is the main technical
s
difficulty in this paper.
6The notation w/s refers either to the index w or the index s in the whole statement.
L2-SERRE DUALITY AND RATIONAL SINGULARITIES 5
Note that Hp,q (Ω,F) is Hausdorff if and only if ∂ has closed range in
w/s,loc w/s
Lp,q(Ω,F), and to decide whether this is the case is usually as difficult as solving the
corresponding ∂-equation. Using local L2-∂-solution results for singular spaces and
the theory of Fr´echet sheaves, we will show at least:
Theorem 1.5. Let X be a Hermitian complex space of pure dimension n, F → X
a Hermitian holomorphic line bundle, and let 0 ≤ p,q ≤ n. Let Ω ⊂ X be a
holomorphically convex open subset. Then the topological vector spaces
Hn,q (Ω,F) , Hn,q (Ω,F) , H0,n−q(Ω,F∗) , H0,n−q(Ω,F∗)
w,loc w,cpt s,cpt s,loc
are Hausdorff for all 0 ≤ q ≤ n.
If X has only isolated singularities, then the topological vector spaces
Hn,q (Ω,F) , H0,n−q(Ω,F∗)
s,loc w,cpt
are Hausdorff for all 0 ≤ q ≤ n, too.
If X has only homogeneous (conical) isolated singularities, then the topological
vector spaces
Hn,q (Ω,F) , H0,n−q(Ω,F∗)
s,cpt w,loc
are Hausdorff for all 0 ≤ q ≤ n, too.
A main point in the proof of Theorem 1.5 is to show that the canonical Fr´echet
sheafstructureofcompactconvergenceonthecoherentanalyticcanonicalsheavesK
X
and Ks , respectively, coincides with the Fr´echet sheaf structure of L2-convergence
X
on compact subsets (Theorem 3.6). This allows then to show also the topological
equivalence of Cˇech cohomology and L2-cohomology (Theorem 3.8).
As a direct application of our Serre duality, Theorem 1.4, we deduce again by use
of local ∂-solution results for (n,q)-forms and the equivalence of Fr´echet structures,
Theorem 3.8, another main result:
Theorem 1.6. Let X be a Hermitian complex space of pure dimension n, F → X
a Hermitian holomorphic line bundle and Ω ⊂ X a cohomologically q-complete open
subset, q ≥ 1. Then
Hn,r (Ω,F) = H0,n−r(Ω,F∗) = 0 for all r ≥ q.
w,loc s,cpt
If X has only isolated singularities, then also
Hn,r (Ω,F) = H0,n−r(Ω,F∗) = 0 for all r ≥ q.
s,loc w,cpt
Note that Ω is cohomologically q-complete if it is q-complete by the Andreotti-
Grauert vanishing theorem [AG]. So, Theorem 1.6 allows to solve the ∂ -equation
s
withcompact support for (0,n−q)-formsonq-completespaces, which is of particular
interest for 1-complete spaces, i.e., Stein spaces.
It is thus interesting to understand the ∂ -operator better. In contrast to the
s
∂-operator in the sense of distributions (the ∂ -operator), ∂ comes with a certain
w s
kind of boundary (respectively growth) condition at the singular set. It follows by
an argument of N. Sibony from [S1] that locally bounded forms in the domain of the
∂ -operator are also in the domain of the ∂ -operator (see Theorem 4.1). This is a
w s
direct way to see that weakly holomorphic functions are ∂ -closed (cf. Theorem 1.2).
s
To exemplify the use of ∂-equations on singular spaces, we give as an application
a short proof of the Hartogs’ extension theorem in its most general form:
6 J. RUPPENTHAL
Theorem 1.7. Let X be a connected normal complex space of dimension n ≥ 2
which is cohomologically (n − 1)-complete. Furthermore, let D be a domain in X
and K ⊂ D a compact subset such that D\K is connected. Then each holomorphic
function f ∈ O(D \K) has a unique holomorphic extension to the whole set D.
In this generality, Hartogs’ extension theorem is due to Merker and Porten [MP],
who proved also extension of meromorphic functions. Merker and Porten gave an
involved geometrical proof by using a finite number of parameterized families of
holomorphic discs and Morse-theoretical tools for the global topological control of
monodromy, but no ∂-theory. Shortly after that, Coltoiu and Ruppenthal obtained a
∂-theoretical proof by the Ehrenpreis-∂-technique (see [CR]). This approach involves
Hironaka’s deep results on resolution of singularities. It is thus a natural question
to ask whether the extension can be achieved by working on the original singular
space only. In the present paper, we obtain a very short proof of Theorem 1.7 by
the Ehrenpreis-∂-technique without needing a resolution of singularities. We just
use the ∂-vanishing H0,1 (X) = 0 for an (n−1)-complete space X and the fact that
s,cpt
bounded ∂-closed forms are in the kernel of ∂ (see Section 4.3).
s
Let us also mention that one can show H0,1 (X) = 0 for an (n − 1)-complete
L∞,cpt
space X by using a resolution of singularities and Takegoshi’s vanishing theorem (see
Theorem 4.3).
Let us point out also the following interesting fact. Let X be a Gorenstein space
with canonical singularities. By exactness of (6) and exactness of (1) for p = n, the
non-degenerate L2-Serre duality pairing
H0,q (Ω)×Hn,n−q(Ω) → C , ([η],[ω]) 7→ η ∧ω,
s,loc w,cpt
ZΩ∗
is for 0 ≤ q ≤ n then an explicit realization of Grothendieck duality after Ramis-
Ruget [RR],
Hq(Ω,O ) ∼= Hn−q(Ω,ω ),
X cpt X
given the cohomology groups under consideration are Hausdorff. Here, ω denotes
X
the Grothendieck dualizing sheaf which coincides with the Grauert-Riemenschneider
canonical sheaf K as X has canonical Gorenstein singularities.
X
The present paper is organized as follows. In Section 2 we provide the necessary
preliminaries: the∂ -andthe∂ -complex, someL2-∂-solutionresultsandtheirdirect
w s
consequences, L2-∂-Hilbert space theory, topological preliminaries, Fr´echet sheaves.
In Section 3, we prove Serre duality, Theorem 1.4, and study the equivalence of the
topology of compact convergence and L2-topology which leads to Theorem 1.5 and
Theorem 1.6. Section 4 is then devoted to the study of the ∂ -operator and Hartogs’
s
extension theorem. Finally, we prove Theorem 1.1 and its consequences (Theorem
1.2 and Theorem 1.3) in the last section.
2. Preliminaries
2.1. Two ∂-complexes on singular spaces. Let us recall some of the essential
constructions from [R4].
Let X be a (singular) Hermitian complex space of pure dimension n. For any
subset S ⊂ X, we use the notation S∗ for S \SingX. Let F → X∗ be a Hermitian
holomorphic line bundle and U ⊂ X an open subset. On a singular space, it is
L2-SERRE DUALITY AND RATIONAL SINGULARITIES 7
fruitful to consider forms that are square-integrable up to the singular set. Hence,
we use the following concept of locally square-integrable forms:
Lp,q(U,F) := {f ∈ Lp,q(U∗,F) : f| ∈ Lp,q(K∗,F) ∀K ⊂⊂ U}.
loc loc K
It is easy to check that the presheaves given as
Lp,q(U,F) := Lp,q(U,F)
loc
are already sheaves Lp,q(F) → X. On Lp,q(U,F), we denote by
loc
∂ (U) : Lp,q(U,F) → Lp,q+1(U,F)
w loc loc
the ∂-operator in the sense of distributions on U∗ = U \SingX which is closed and
densely defined. When there is no danger of confusion, we will simply write ∂ for
w
∂ (U). The subscript refers to ∂ as an operator in a weak sense. Since ∂ is a local
w w w
operator, i.e. ∂ (U)| = ∂ (V) for open sets V ⊂ U, we can define the presheaves
w V w
of germs of forms in the domain of ∂ ,
w
Cp,q(F) := Lp,q(F)∩∂−1Lp,q+1(F),
w
given by
Cp,q(U,F) = Lp,q(U,F)∩Dom∂ (U).
w
These are actually already sheaves because the following is also clear: If U = U
µ
is a union of open sets, f = f| and
µ Uµ S
f ∈ Dom∂ (U ),
µ w µ
then
f ∈ Dom∂ (U) and ∂ (U)f | = ∂ (U )f .
w w Uµ w µ µ
Moreover, it is easy to see that the sheaves Cp,q(F) admit partitions of unity, and so
(cid:0) (cid:1)
we obtain a complex of fine sheaves
Cp,0(F) −∂→w Cp,1(F) −∂→w Cp,2(F) −∂→w ... (7)
We use simply Cp,q to denote the sheaves of forms with values in the trivial line
bundle. We define
K (F) := ker∂ ⊂ Cn,0(F). (8)
X w
Using a resolution of singularities, one sees that K := ker∂ ⊂ Cn,0 is just the
X w
canonical sheaf of Grauert and Riemenschneider because the L2-property of (n,0)-
forms remains invariant under modifications (see [R4], Section 2.2).
The L2,loc-Dolbeault cohomology for forms with values in F with respect to the
∂ -operator on an open set U ⊂ X is by definition the cohomology of the complex
w
(7) which is denoted by Hq(Γ(U,Cp,∗(F))). The cohomology with compact support
is Hq(Γ (U,Cp,∗(F))). Note that this is the cohomology of forms with compact
cpt
support in U, not with compact support in U∗ = U \SingX.
We use also the following notation for the ∂ -cohomology:
w
Definition 2.1. For an open set Ω ⊂ X and a Hermitian holomorphic line bundle
F → X, let
Hp,q (Ω,F) := Hq Γ(Ω,Cp,∗(F)) ,
w,loc
Hp,q (Ω,F) := Hq Γ (Ω,Cp,∗(F)) .
w,cpt (cid:0) cpt (cid:1)
(cid:0) (cid:1)
8 J. RUPPENTHAL
Secondly, we introduce a suitable local realization of a minimal version of the ∂-
operator. This is the ∂-operator with a certain boundary condition at the singular
set SingX of X. Let
∂ (U) : Lp,q(U,F) → Lp,q+1(U,F)
s loc loc
be defined as follows.7 We say that f ∈ Dom∂ is in the domain of ∂ if there exists
w s
a sequence of forms {f } ⊂ Dom∂ ⊂ Lp,q(U,F) with essential support away from
j j w loc
the singular set,
suppf ∩SingX = ∅,
j
such that
f → f in Lp,q(K∗,F), (9)
j
∂ f → ∂ f in Lp,q+1(K∗,F) (10)
w j w
for each compact subset K ⊂⊂ U. The subscript refers to ∂ as an extension in a
s
strong sense. Note that we can assume without loss of generality (by use of cut-off
functions and smoothing with Dirac sequences) that the forms f are smooth with
j
compact support in U∗ = U −SingX. This is the equivalent definition that we used
in [R4] where we denoted the operator by ∂ .
s,loc
It is now clear that ∂ (U)| = ∂ (V) for open sets V ⊂ U, and we can define the
s V s
presheaves of germs of forms in the domain of ∂ ,
s
Fp,q(F) := Lp,q(F)∩∂−1Lp,q+1(F),
s
given by
Fp,q(U,F) = Lp,q(U,F)∩Dom∂ (U).
s
Here, weshallcheckabitmorecarefullythatthesearealreadysheaves: LetU = U
µ
be a union of open sets, f ∈ Lp,q(U,F) and f = f| ∈ Dom∂ (U ) for all µ. We
loc µ Uµ s µ S
claim that f ∈ Dom∂ (U). To see this, we can assume (by taking a refinement if
s
necessary) that the open cover U := {U } is locally finite, and choose a partition of
µ µ
unity {ϕ } for U. On U choose a sequence {fµ} ⊂ Lp,q(U ,F) as in (9), (10), and
µ µ µ j j loc µ
consider f := ϕ fµ. It is clear that {f } ⊂ Lp,q(U,F). If K ⊂⊂ U is compact,
j µ µ j j j loc
then K ∩ suppϕ is a compact subset of U for each µ, so that {fµ} and {∂fµ}
Pµ µ j j j j
converge in the L2-sense to f resp. ∂ f on K∩suppϕ . But then {f } and {∂f }
µ w µ µ j j j j
converge in the L2-sense to f resp. ∂ f on K (recall that the cover is locally finite)
w
and that is what we had to show.
AsforCp,q(F), itisclearthatthesheaves Fp,q(F)arefine, andweobtainacomplex
of fine sheaves
Fp,0(F) −∂→s Fp,1(F) −∂→s Fp,2(F) −∂→s ... (11)
Again, we use simply Fp,q to denote the sheaves of forms with values in the trivial
line bundle. We define the canonical sheaf of holomorphic n-forms with a Dirichlet
boundary condition:
Ks (F) := ker∂ ⊂ Fn,0(F).
X s
Definition 2.2. For Ω ⊂ X open, we use the notation:
Hp,q (Ω,F) := Hq Γ(Ω,Fp,∗(F)) ,
s,loc
Hp,q (Ω,F) := Hq Γ (Ω,Fp,∗(F)) .
s,cpt (cid:0) cpt (cid:1)
(cid:0) (cid:1)
7Again, we write simply ∂s for ∂s(U) if there is no danger of confusion.
L2-SERRE DUALITY AND RATIONAL SINGULARITIES 9
2.2. LocalL2-solvability for (n,q)-forms. Itisclearlyinterestingtostudywhether
the sequences (7) and (11) are exact, which is well-known to be the case in regular
points of X where the ∂ - and the ∂ -operator coincide. In singular points, the
w s
situation is quite complicated for forms of arbitrary degree and not completely un-
derstood. However, the ∂ -equation is locally solvable in the L2-sense at arbitrary
w
singularities for forms of degree (n,q), q > 0 ([PS], Proposition 2.1), and for forms
of degree (p,q), p+q > n, at isolated singularities ([FOV], Theorem 1.2). We may
restrict ourselves here to the case of (n,q)-forms and have (see [R4], Theorem 3.1):
Theorem 2.3. Let X be a Hermitian complex space of pure dimension n, and F →
X∗ = X \SingX a Hermitian holomorphic line bundle which is locally semi-positive
with respect to X, i.e. for each point x ∈ X there is a neighborhood U ⊂ X such
x
that F is semi-positive on U∗ = U \SingX. Then the complex
x x
0 → K (F) −→ Cn,0(F) −∂→w Cn,1(F) −∂→w Cn,2(F) −∂→w ... (12)
X
is exact, i.e. it is a fine resolution of K (F). For an open set U ⊂ X, it follows that
X
Hq(U,K (F)) ∼= Hn,q (U,F) , Hq (U,K (F)) ∼= Hn,q (U,F).
X w,loc cpt X w,cpt
Note that the assumption on F is trivially fulfilled if F extends to a holomorphic
line bundle over X. For the case of the trivial line bundle, F = X ×C, Theorem 2.3
is due to Pardon-Stern ([PS], Proposition 2.1).
Concerning the ∂ -equation, local L2-solvability for forms of degree (n,q) is known
s
to hold on spaces with isolated singularities (see [R4], Lemma 5.4 and Lemma 6.3),
but the problem is open at arbitrary singularities.
So, let X have only isolated singularities. Then the ∂ -equation is locally exact on
s
(n,q)-formsfor 1 ≤ q ≤ n−1 by [R4], Lemma 5.4, and for q ≥ 2 by [R4], Lemma 6.3.
Both statements were deduced from the results of Fornæss, Øvrelid and Vassiliadou
[FOV]. The case of dimX = n = 1 is treated in [RS]. Hence:
Theorem 2.4. Let X be a Hermitian complex space of pure dimension n with only
isolated singularities. Then
0 → Ks ֒→ Fn,0 −∂→s Fn,1 −∂→s Fn,2 −∂→s ... −→ Fn,n → 0 (13)
X
is a fine resolution. For an open set U ⊂ X, it follows that
Hq(U,Ks ) ∼= Hn,q (U) , Hq (U,Ks ) ∼= Hn,q (U).
X s,loc cpt X s,cpt
2.3. Resolution of singularities and Takegoshi’s vanishing theorem. We
need to recall some more material from [R4], Section 2.2. For X as above and
F → X a Hermitian holomorphic line bundle, let π : M → X be a resolution of sin-
gularities (which exists due to Hironaka), and give M an arbitrary positive definite
Hermitian metric σ. Then we denote analogously to (7) by
0 → K (π∗F) −→ Cn,0(π∗F) −∂→w Cn,1(π∗F) −∂→w Cn,2(π∗F) −∂→w ... (14)
M σ σ σ
the fine L2,loc-resolution of the canonical sheaf K (π∗F) on M with values in π∗F (it
σ M
is well known that (14) is exact). As K is the Grauert-Riemenschneider canonical
X
sheaf on X, we have (see [R4], Theorem 2.1):
K (F) = π K (π∗F). (15)
X ∗ M
10 J. RUPPENTHAL
Takegoshi’s vanishing theorem [T] yields the vanishing of the higher direct image
sheaves:
Rqπ K (π∗F) = 0 , q > 0. (16)
∗ M
Moreover, square-integrable (n,q)-forms remain square-integrable under pull-back
by π and this pull-back commutes with the ∂ -operator. The exceptional set of the
w
resolution π : M → X does no harm as the ∂-equation in the L2-sense extends over
hypersurfaces. So, π induces a natural mapping of complexes
π∗ : Cn,∗(F),∂ −→ π Cn,∗(π∗F),π ∂ , (17)
w ∗ σ ∗ w
andby Theorem2.3 an(cid:0)d(16), both(cid:1)compl(cid:0)exes in(17) arefine(cid:1)resolutionsof KX(F) =
π K (π∗F). As π∗ commutes with the ∂ -operator, it induces isomorphisms
∗ M w
Hn,q Ω,F −∼=→ Hn,q π−1(Ω),π∗F ∼= Hq π−1(Ω),K (π∗F) , (18)
w,loc w,loc M
Hn,q (cid:0)Ω,F(cid:1) −∼=→ Hn,q (cid:0)π−1(Ω),π∗F(cid:1) ∼= Hq(cid:0) π−1(Ω),K (π∗F(cid:1)) (19)
w,cpt w,cpt cpt M
for any open (cid:0)set Ω(cid:1)⊂ X and 0 ≤(cid:0) q ≤ n (se(cid:1)e [R4], T(cid:0)heorem 2.1). We(cid:1) will see
later (Section 3.7) that the algebraic isomorphisms in (18), (19) are also topological
isomorphisms.
For the rest of this section, assume that X has only homogeneous isolated singu-
larities. In this situation, we have Ks ∼= K by [R4], Theorem 1.10 (with D = ∅),
X X
because homogeneous isolated singularities can be resolved by a single blow-up.
As the ∂ -operator is stronger than the ∂ -operator, π induces also a natural
s w
mapping of complexes
π∗ : Fn,∗(F),∂ −→ π Cn,∗(π∗F),π ∂ . (20)
s ∗ σ ∗ w
By Theorem 2.4 and ((cid:0)16), both co(cid:1)mplex(cid:0)es in (20) are fine (cid:1)resolutions of Ks (F) =
X
π K (π∗F) and so π∗ induces also isomorphisms
∗ M
Hn,q Ω,F −∼=→ Hn,q π−1(Ω),π∗F ∼= Hq π−1(Ω),K (π∗F) , (21)
s,loc w,loc M
Hn,q (cid:0)Ω,F(cid:1) −∼=→ Hn,q (cid:0)π−1(Ω),π∗F(cid:1) ∼= Hq(cid:0) π−1(Ω),K (π∗F(cid:1)) (22)
s,cpt w,cpt cpt M
for any open s(cid:0)et Ω ⊂(cid:1) X and 0 ≤ q(cid:0)≤ n. Again,(cid:1)we will(cid:0)see in Section 3.7 t(cid:1)hat (21),
(22) are also topological isomorphisms.
2.4. Metrizable topology of Lp,q(Ω,F). Let Ω ⊂ X be an open subset. We give
loc
Lp,q(Ω,F) the structure of a Fr´echet space with the topology of L2-convergence on
loc
compact subsets. This topology is obtained as follows. Let K ⊂ K ⊂ K ⊂ ... ⊂ Ω
1 2 3
be a compact exhaustion of Ω, and define the separating family of seminorms
1/2
p (η) := |η|2dV (23)
j F X
ZKj∗ !
for η ∈ Lp,q(Ω,F) and j = 1,2,..., where dV is the volume form on X∗ induced by
loc X
the Hermitian metric of the Hermitian space X. Lp,q(Ω,F) is then a Fr´echet space
loc
with the metric
∞
p (η−ω)
d(η,ω) := 2−j j , η,ω ∈ Lp,q(Ω,F)
1+p (η −ω) loc
j
j=1
X