The Canonical 2-Gerbe of a Complex Manifold

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by  Markus Upmeier

THE CANONICAL 2-GERBE OF A COMPLEX MANIFOLD
6 MARKUSUPMEIER
1
0
2 Abstract. Wepresenttheconstructionofaholomorphicbundle2-gerbefor
eachcomplexmanifoldX,ahigheranalogofthecanonicallinebundle. Itisa
n
geometricrepresentative ofthesecondBeilinson-Chernclass. Also,anAtiyah
a
classforgerbesisintroducedandaKoszul-Malgrangetypetheoremisproven.
J

9
1
A fundamental object associated to complex manifolds X is its canonical line

] bundle Λtop(T∗X). The present paper deals with an extension of this concept. To
G
explaintheanalogy,recallthatthecanonicalbundlemaybeviewedasarepresenta-
D
tive of the first Chernclass. In Theorem3 we construct a geometric representative
. of the second Be˘ılinson-Chern class, introduced in [Be˘ı84]. This is a 2-gerbe and
h
t maybecomputedfromtheeigenvaluesandeigenspacesofcharttransitionfunctions,
a
just like the canonical bundle depends only on the product of these eigenvalues.
m
We are motivated by the relationship between 2-gerbes and line bundles on the
[ space of curves, via transgression[CJM+05, Bry99b, Bry08, Wal12]. According to

1 the paradigm of string topology, structure on the infinite-dimensional loop space
v maybestudiedthroughhigher-categorical,finite-dimensionalstructureontheorig-
9
inalspace. Thisis theviewpointofthe presentpaper. Havingapplicationstocom-
1
plex geometry in mind, we shall be particularly concerned with the holomorphic
8
4 structure on our gerbe.
0 The 2-gerbe will be constructed within the framework of bundle gerbes, intro-
. duced in [Mur96] and further studied in [MS00, Mur10, Ste04].
1
0 To set this work apart from related results, we now briefly review the existing
6 literature. In [BM96, BM94, Bry08] tautological gerbes are defined as Dixmier-
1 Douady sheaves of groupoids, but the constructions involve infinite-dimensional
:
v spaces. This makes them less useful from the point of view of string topology. We
i shall construct instead a very explicit and finite-dimensional model. Moreover, we
X
have smooth gerbes over compact, simple, simply-connected Lie groups [Mei03],
r
a certain quotients therefore [GR04a], for SU(n) [GR02], and complex reductive Lie
groups [Bry]. Of course, GL(n,C) is of none of these types. For compact semi-
simple Lie groups G, a smooth multiplicative structure on the tautological gerbe
on G was constructed by cohomologicalarguments in [CJM+05] and [Wal10].
We now state our main results and give an overview of the present paper.
We begin in Section 1 by developing further the theory of holomorphic bundle
gerbes started in [MS03, Section 7]. In 1.2 we study the relationship to smooth
gerbes and present an analogue of the Koszul-Malgrange Theorem [KM58]. For
holomorphic line bundles, the Atiyah class is the obstruction against holomorphic
connections. In1.4wedemonstratehowthisideaextendstogerbes. Therestofthe
section reviews well-known terminology for smooth gerbes [Mur96, MS00, Mur10,
Wal07] in our holomorphic context.
1
2 MARKUSUPMEIER
Our first theorem gives a very explicit and finite-dimensional construction of a
holomorphicgerbeG thatiscanonicalinthesensethatitbelongstothegenerator
can
ofH2(GL(n,C);O×)=Z. TheconstructionofGcan inSection2.1isanextensionof
thework[MS08]forthesmoothLiegroupU(n)tothenon-compactgroupGL(n,C)
and the holomorphic category. In Section 2.2 we then prove:
Theorem 1. G = (π: Y → GL (C),L,m) defines a holomorphic gerbe. Its
can n
Dixmier-Douady class is the generator of H2(GL(n,C),O×).
The complex Lie group GL(n,C) is a Stein group. Using cohomological argu-
ments developed in Section 3 for Stein manifolds, we prove in Section 4 that the
existence of a multiplicative structure is a purely topological problem:
Theorem 2. Let G be a Stein group and let G be a holomorphic gerbe on G with
holomorphic connection. ThenGadmits aholomorphic multiplicativestructurewith
connectionpreciselywhenthe topologicalDixmier-DouadyclassDD(G)∈H3(G;Z)
is in the image of the transgression map H4(BG;Z)→H3(G;Z).
In particular, every holomorphic gerbe with connection on GL(n,C) admits a
multiplicative structure (Corollary 23). Following the proof, we also give a more
explicitdescription. ItisspecialtoourtreatmentthatweuseSteinspacestostudy
the Be˘ılinson-Chernclasses,instead of Deligne’s theory of mixed Hodge structures
[Del74] used in [Be˘ı84, EV88]. Sections 5.1 and 5.2 introduce a notion of 2-gerbe,
which is slightly weaker than that in [Ste04], and their Dixmier-Douady class. For
each holomorphic vector bundles E → X we present in Section 5.3 a construction
for a 2-gerbe G(E). We show:
Theorem 3. The associated 2-gerbe has the following properties:
(1) G(f∗E)∼=f∗G(E) (functorial)
(2) The topological Dixmier-Douady class is c (E).
2
(3) For X algebraic, DD(G(E))=cB(E) is the Be˘ılinson-Chern class of E.
2
Applied to E =T∗X we obtain the canonical 2-gerbe of a complex manifold X.
Notation. For iterated fiber products we use the notation Y[i] =Y × ···× Y
X X
and similarly for maps over X. By pr we mean the projections onto the in-
ijk···
dicated factors. The sheaf of holomorphic k-forms is Ωk and O∗ is the sheaf of
nowherevanishing holomorphicfunctions. Unless statedotherwise,all bundles and
bundle maps below are assumed to be holomorphic.
Acknowledgements. I would like to thank JoelFine for innumerable discussions
on the results presented here.
1. Holomorphic Bundle Gerbes and Connections
In this section we review well-established concepts for smooth gerbes in the
holomorphic context, see [MS03]. Also some new results are developed, such a
Koszul-MalgrangeTheoremfor gerbesin1.2orageneralizationofthe Atiyahclass
in1.4. NotethatBrylinski’sholomorphicgerbes[Bry08],sheavesofgroupoidswith
band O×, are equivalent to the holomorphic bundle gerbes considered here.
THE CANONICAL 2-GERBE OF A COMPLEX MANIFOLD 3
1.1. Holomorphic Gerbes. Ordinarygerbes are geometricrepresentativesofco-
homology classes H2(X,C∗)∼=H3(X;Z). In the simplest approach,one uses Čech
cocycles for anopen coverof X, leading to Hitchin-Chatterjee gerbes [Cha98]. Be-
low,weshalldefineholomorphicgerbesoncomplexmanifoldsX. Theseobjectsare
designedtorepresentcohomologyclassesinH2(X,O∗ ). Wewilladoptthe ‘bundle
X
gerbe’ approach from [Mur96] whose main advantage is the use of descent theory
to avoid artificial choices of an open cover.
Definition 4. A holomorphic gerbe G on X is a triple (π,L,m) consisting of a
holomorphicsubmersionπ: Y →X,holomorphiclinebundleL→Y[2],andbundle
isomorphismm: pr∗ L⊗pr∗ L→pr∗ L. DenotingbyL thefibersofL,weget
12 23 13 y1,y2
maps m: L ⊗L →L , called multiplication, which are required to be
y1,y2 y2,y3 y1,y3
associative (see [Mur10, Definition 4.1]).
Definition 5. A connection on G=(Y,L,m) is a holomorphic connection on L
(see [Huy05, Definition 4.2.17])preservedby the multiplication m. A curving on a
gerbe with connection is a form f ∈Ω2(Y) related to the curvature of L by
pr∗f −pr∗f =R(∇L). (1)
2 1
The 3-curvature is then the unique ρ=R(G,∇L,f)∈Ω3(X) with π∗ρ=df.
There is also the weaker notion of compatible connection on G. It consists of
a connection on L only compatible with the holomorphic structure (see [Huy05]).
The curving may then be a form of type (2,0)+(1,1).
1.2. Relationship to Smooth Gerbes. Holomorphic gerbes with holomorphic
connectionsmay equivalently be describedby connectivedata with certainproper-
ties. ThereisanobviousvariantofDefinition4inthecategoryofsmoothmanifolds.
Theorem 6. Let G=(π: Y →X,L,m) be a smooth gerbe whose projection π is a
holomorphic submersion. Then the following are equivalent:
(1) G is a holomorphic gerbe with holomorphic connection and curving f.
(2) Gis asmooth gerbewith smoothconnection and curvingf of type(2,0)and
3-curvature ρ of type (3,0).
Proof. Itisobviousthataholomorphicconnectionhasthe propertiesstatedin(2).
Conversely,the(0,1)-part(∇L)(0,1) ofthesmoothconnectiondeterminesaCauchy-
RiemannoperatoronL. Byequation(1),thecurvatureoftheconnectionisoftype
(2,0)which showsin particularthat (∇L)(0,1)◦(∇L)(0,1) =0 is flat. Thereforethe
Theorem of Koszul-Malgrange [KM58] shows that this Cauchy-Riemann operator
defines a holomorphic structure on L. The multiplication m preserves (the (0,1)-
part of) the connection and is therefore holomorphic. Since the 3-curvature ρ of
type (3,0) satisfies π∗ρ=df, we have ∂¯f =0, so f is holomorphic. (cid:3)
Asimilarstatementappliesto holomorphicgerbeswithconnection,butwithout
curving (the hypothesis is then that R(∇L) is of type (2,0)). Moreover, a com-
patible connection with curving on G corresponds to a smooth connection whose
curving has type (2,0)+(1,1).
4 MARKUSUPMEIER
1.3. Class in Deligne Cohomology. Recall that the Deligne complex Z(p) is
D
the following complex of sheaves (see [Bry08, Section 1.5])
Z(p) : 0 O∗ dlog Ω1 d ··· d Ωp−1.
D
TheDeligne cohomology groupsHq(X,Z(p) )aredefinedasthehypercohomology
D
groups of this complex. For example, H2(X,Z(2) ) is the group of holomorphic
D
line bundles on X with holomorphic connection, see [Del91]. We now extend this
classification to holomorphic gerbes G, leading to the Dixmier-Douady class.
Lemma 7 ([Lur09, Lemma 7.2.3.5]). Let U = {U } be an open covering of a
α
paracompact space X, k ∈ N. Given an open cover Vα0···αk of each (k +1)-fold
intersection Uα0···αk, there exists a refinement {Wβ} of U with the property that
each (k+1)-fold intersection Wβ0···βk is a subset of some element of Vα0···αk.
Corollary 8. Let G = (π,L,m) be a holomorphic gerbe. Then we find arbitrarily
fine open covers {U } which admit holomorphic sections s : U →Y of π and
α α α
holomorphic trivializations σ of (s ,s )∗L.
αβ α β
For such an open cover we define g ∈O∗(U ) by the equation
αβγ αβγ
m(σ ,σ )=g ·σ .
αβ βγ αβγ αγ
Associativity of m implies that (g ) is closed. Different trivializations give coho-
αβγ
mologous cocycles. Taking the limit over a cofinal sequence of covers, we define:
Definition 9. The Dixmier-Douady class DD(G)∈Hˇ3(X,Z(1) ) ofthe holomor-
D
phic gerbe G on X is represented by the Čech cocycle (g ).
αβγ
The exponential map to H3(X;Z) maps this class to the topological Dixmier-
Douady class of G, defined for example in [Mur96, Mur10].
If G is equipped with a connection, this class may be refined to H3(X,Z(2) ).
D
For this we write the connection on (s ,s )∗L as d+A . Since m preserves
α β αβ
the connection we have dlog(g ) = A −A + A . Hence (g ,A ) ∈
αβγ βγ αγ αβ αβγ αβ
Cˇ3({U },Z(2) ) is a Čech cocycle. If, in addition, we suppose G to be equipped
α D
with a curving, then f −f = dA for f = s∗f, by (1) and so (g ,A ,f )
β α αβ α α αβγ αβ α
defines a class in H3(X,Z(3) ).
D
1.4. Atiyah Class. Contrasting the smooth case, holomorphic connections and
curvings cannot alwaysbe found. This is measuredby the following generalization
of the Atiyah class A(L) for holomorphic line bundles. The short exact sequence
of sheaves 0→Ωp[−p−1]→Z(p+1) →Z(p) →0 induces the following exact
D D
sequences,whosecoboundarymaps maybe regardedasgeneralizedAtiyahclasses:
H3(X,Z(2) ) H3(X,Z(1) ) B H2(X,Ω1)
D D
H3(X,Z(3) ) H3(X,Z(2) ) C H1(X,Ω2)
D D
Proposition 10. The images of the classes B(G) and DD(G) in H3(X;C) agree.
THE CANONICAL 2-GERBE OF A COMPLEX MANIFOLD 5
Proof. From the definition of the codifferential, one sees that B maps the Čech
cocycleg ∈O∗(U ) to the classof dlogg in H2(A1). Therefore the image
αβγ αβγ αβγ cl
of dlogg in H3(X;C) is the image of the Dixmier-Douady class δ(g ). (cid:3)
αβγ αβγ
Corollary 11. For a compact Kähler manifold X the topological Dixmier-Douady
class DD(G)∈H3(X;Z) of a gerbe with holomorphic connection is torsion.
1.5. Morphismsof Gerbes. Thereisanaivenotion(whichwedonotpresent)of
isomorphismwithwhichtwogerbesmayhavethesameDixmier-Douadyclasswith-
outbeing isomorphic. Whenwishing toemphasisethis, wealsocallthe morphisms
below stable isomorphisms, in accordance with usual terminology.
Definition 12. Let G=(Y,L,m), G′ =(Y′,L′,m′) be holomorphic gerbes on X.
A morphism F from G to G′ is a triple ((ς,ς′),R,φ) consisting of a submersion
(ς,ς′): Z →Y × Y′, line bundle R→Z, and bundle isomorphism
X
φ: (ς[2])∗L⊗pr∗R→pr∗R⊗(ς′[2])∗L′.
2 1
HenceφgivesmapsL ⊗R →R ⊗L′ for(z ,z )∈Z[2],whereς(z )=y ,
y1,y2 z2 z1 y1′,y2′ 1 2 k k
ς′(z )=y′. For (z ,z ,z )∈Z[3] we require commutative diagrams:
k k 1 2 3
L ⊗L ⊗R id⊗φ L ⊗R ⊗L′ φ⊗id R ⊗L′ ⊗L′
y1,y2 y2,y3 z3 y1,y2 z2 y2′,y3′ z1 y1′,y2′ y2′,y3′
m⊗id id⊗m′
L ⊗R φ R ⊗L′
y1,y3 z3 z1 y1′,y3′
The composite of the morphism(Z,R,φ): G→G′ with (Z′,R′,φ′): G′ →G′′ has
the submersion Z×Y′ Z′ →Y ×X Y′′ and line bundle R⊗R′.
Definition13. Aconnection onamorphismF =(Z,R,φ): G→G′ ofgerbeswith
connections is a holomorphic connection on R making φ connection-preserving.
Proposition 14. Given a morphisms F: G → G′, the Dixmier-Douady classes
DD(G) = DD(G′) in H3(X;Z(1) ) agree. If the gerbes and the morphism F have
D
connections, the classes in H3(X;Z(2) ) agree.
D
Anytwomorphisms(R,φ),(R˜,φ˜): G→G′ differbyalinebundleonthe baseX:
the maps φ and φ˜ define descent data for the line bundle R ⊗R˜∗ on Z. Hence it
z z
is the pullback of a bundle on X. Similarly with connections.
1.6. Transformations. We now come to the 2-morphisms of our bicategory.
Definition 15. Let F = (Z,R,φ) and F˜ = (Z˜,R˜,φ˜) be morphisms from G =
(Y,L,m) to G′ =(Y′,L′,m′). A transformation α: F ⇒F˜ consists of
(1) A submersion (κ,κ˜): W →Z×Y×Y′ Z˜
(2) A bundle isomorphism ψ: κ∗R→κ˜∗R˜ over W.
6 MARKUSUPMEIER
So ψ induces maps ψ : R → R˜ , for (κ,κ˜)(w) = (z,z˜). For (w ,w )∈W[2] let
w z z˜ 1 2
(κ,κ˜)(w )=(z ,z˜ ), (ς,ς′)(z )=(y ,y′). We require the commutativity of
k k k k k k
L ⊗R φ R ⊗L′
y1,y2 z2 z1 y1′,y2′
id⊗ψ ψ⊗id
L ⊗R˜ R˜ ⊗L′
y1,y2 z˜2 φ z˜1 y1′,y2′
IfthemorphismsF,F˜ areequippedwithconnections,thetransformationαiscom-
patible with the connections if ψ preserves connections.
Two transformations are identified if they coincide on a pullback, see [Wal07].
This gives a bicategory of gerbes.
1.7. Further Operations. LetG=(Y,L,m)beagerbeonX andletf: X′ →X
be a holomorphic map. The pullback gerbe f∗G is given by Y′ =Y × X′ →X′
X
and the pullback line bundle of L along (Y′)[2] →Y[2].
The tensor product G⊗G′ of two gerbes G = (Y,L,m), G′ = (Y′,L′,m′) on X
has the submersion Y × Y′ →X and the exterior product line bundle L⊗L′.
X
Similarly, we have a tensor product of morphisms. For more details, see [Wal07].
TheDixmier-Douadyclassiscompatiblewithpullbackandadditivewithrespect
totensorproducts. Equippingf∗GandG⊗G′ withthetensorproductandpullback
connections, this also holds for the 3-curvature.
2. The Canonical Gerbe on GL (C)
n
In this section, we give the details of the construction for the canonical gerbe
Gcan onGL(n,C). Theorem1assertsthatGcan isarepresentativeforthe generator
of H2(GL(n,C),O×). Its proof is given in 2.2.
2.1. Construction of Gcan. The sector of radii 0 ≤ r < R ≤ +∞ between z =
1
r1eiϕ1 and z2 =r2eiϕ2, with 0<ϕ2−ϕ1 ≤2π and r1,r2 >0, is the set
Sr,R(z ,z )= seiϕ |r ≤s≤R,ϕ ≤ϕ≤ϕ .
1 2 1 2
(cid:8) (cid:9)
We write S(z ,z ) if r=0,R=+∞.
1 2
RecallthatifχA(X)=det(A−X·id)= λ(X −λ)nλ isthecharacteristicpoly-
nomialof a matrix A, then Cn may be decomQposedinto the n -dimensionalgener-
λ
alized eigenspaces Vλ(A)=ker(A−λ·id)nλ. For a subset S ⊂C let
V (A)= {V (A)|κ∈S}.
S κ
M
For λ,µ∈C and A∈GL (C) we introduce the notation
n
A(λ→µ)=Λtop V (A) .
S(λ,µ)
(cid:0) (cid:1)
By an eigendirection we mean a ray R λ containing an eigenvalue of A. By
>0
comparing sectors, the following is easy to show:
Lemma 16. If µ and at least one of {λ,κ} are not eigendirections of A we have
A(λ→µ)⊗A(µ→κ)=A(λ→κ). (2)
THE CANONICAL 2-GERBE OF A COMPLEX MANIFOLD 7
Moreover, for λ not an eigendirection we have
A(λ→λ)=C, (3)
A(λ→µ)∼=A(µ→λ)∗. (4)
We now define the projection π: Y →GL (C) of the gerbe Gcan. Let
n
Y = (A,z)∈GL (C)×C× ∀r >0:det(A−rz·id)6=0 .
n
(cid:8) (cid:12) (cid:9)
Using that all eigenvalues are bounded(cid:12) by the operator norm of a matrix, one
shows the set (A,λ)∈GL (C)×S1 the ray R λ contains an eigenvalue of A
n >0
ofeigendirectio(cid:8)nstobeclosed. Itfollow(cid:12)sthatY isanopensubsetofGL (C)×C×(cid:9),
(cid:12) n
soitisacomplexmanifold. Clearly,theprojectionπ: Y →GL (C)isasubmersion.
n
The line bundle L→Y[2] of the gerbe Gcan is given by the determinant lines
L =A(λ→µ).
(A,λ,µ)
In particular, L depends only on R λ and R µ. The multiplication m in
(A,λ,µ) >0 >0
thegerbeisgivenby(2). Itisassociativesincetheidentifications(2)areassociative.
2.2. Proof of Theorem 1. We prove that L is holomorphic near each point
(A ,λ ,µ ). Suppose first that λ 6= µ . Pick real numbers 0 < r < R so that
0 0 0 0 0
the spectrum of A is contained in the annulus of radii r,R. Since the spectrum of
0
A is finite we find disjoint sectors S =Sr,R(µ−,µ+), S =Sr,R(λ−,λ+) with
0 µ0 λ0
non-emptyinteriorandcontainingµ ,λ respectively,butnoeigenvaluesofA . Let
0 0 0
O=Sr,R(λ+,µ−), O′ =Sr,R(µ+,λ−) be the complementary sectors.
O′ Sµ0
µ O
0
µ+ µ−
λ+
λ0 Sλ0
λ−
Considerthelowersemi-continuousfunctionϕ(A)=sup|det(A−λE )|onGL (C),
n n
the supremum ranging over λ ∈ ∂O∪∂O′ ∪∂S ∪∂S . As ϕ(A ) > 0 we find
µ0 λ0 0
a connected neighbourhood U ⊂{A|ϕ(A)>ϕ(A )/2} of A . The zero counting
0 0
integral is a holomorphic Z-valued function of A and therefore constant on U:
1 χ′ (λ)
A dλ (5)
2πiI χ (λ)
A
Applied to the boundary paths of O,O′,S ,S we see that A has no eigenvalues
µ0 λ0
in S or S and also that all eigenvalues remain within O∪O′.
λ0 µ0
It follows that for (A,λ,µ) ∈ U ×S ×S the line L may be identified
λ0 µ0 (A,λ,µ)
with the top exterior power of V (A). It suffices now to prove that V → U is a
O O
holomorphic sub vector bundle of the trivial bundle. Fix A ∈U.
1
8 MARKUSUPMEIER
We have holomorphic maps e,e′ on O ∪ O′ defined by e|O = 1, e|O′ = 0
and e′ =1−e. Holomorphic functional calculus gives us spectral idempotents
e ,e′ ∈M (C) depending holomorphically on A ∈ U. It is well-known that e
A A n A
ande′ arethecomplex-linearprojectionmapsontothegeneralizedeigenspacesV
A O
andVO′. Weregarde(A)=eAasabundleendomorphismofU×Cn. Thesubspaces
kere and ime are complementary in Cn. It follows that the holomorphic map
A1 A1
ψ =e⊕incl: (U ×ime )⊕(U ×kere )→U ×Cn
A1 A1
is the identity on the fiber over A . The map ψ remains invertible on fibers close
1
to A , so by restricting we get a holomorphic isomorphism from the trivial bundle
1
U ×ime to ψ(U ×ime ). Applying (5) to ∂O shows that the fiber dimensions
A1 A1
in ψ(U ×imeA1)⊂im(e) coincide, so we have equality. That is, ψ|U×imA1 is a
trivialization of im(e)=V near A .
O 1
If λ = µ we proceed similarly: choose a small sector S around λ containing
0 0 0
noeigenvaluesofA . ApplyingtheaboveargumentgivesaneighbourhoodU ofA
0 0
sothatno A∈U hasaneigenvaluein S. Hence L|U×S×S is the trivialline bundle.
We now show that m is holomorphic in a neighbourhoodof (A ,λ ,µ ,κ ). For
0 0 0 0
thiswefirstsupposethatthedirectionsR λ ,R µ ,R κ arepairwisedifferent.
>0 0 >0 0 >0 0
Choose disjoint sectors λ ∈ S = Sr,R(λ−,λ+), µ ∈ S = Sr,R(µ−,µ+), κ ∈
0 λ0 0 µ0 0
S = Sr,R(κ−,κ+) with non-empty interior and containing no eigenvalues of A .
κ0 0
Letting O = Sr,R(λ+,µ−), O′ = Sr,R(µ+,κ−), and O′′ = Sr,R(κ+,λ−) we find as
above a neighbourhood U of A in which
0
L(A,λ,µ) =Λtop(VO), L(A,µ,κ) =Λtop(VO′), L(A,λ,κ) =Λtop(VO′′)
whenever (A,λ,µ,κ) ∈ U ×S ×S ×S . There are two cases: arg(κ /µ )+
λ0 µ0 κ0 0 0
arg(µ /λ ) < 2π and > 2π. Depending on the case, the map m is given on this
0 0
neighbourhoodbythetopexteriorpowerofeithertheadditionmapVO⊕VO′ →VO′′
or of VO ⊕VO′ →VC⊕VO′′ →VO′′, both of which are holomorphic. The cases
where some of the directions λ ,µ ,κ agree are even simpler.
0 0 0
The restriction of G to U(n) is the tautological smooth gerbe of Murray-
can
Stevenson [MS08]. Therefore the Dixmier-Douady class of G is the generator
can
of H3(GL (C);Z), by functoriality. This will be impoved in Corollary 19 below,
n
completing the proof of Theorem 1.
3. Cohomological Theory on Stein Manifolds
In this section we collect a number of facts for the Deligne cohomology of Stein
manifolds. These will be needed in the proof of Theorem 2 in the next section.
3.1. Stein manifolds. We recall that a domain U ⊂Cn is a Stein manifold if for
everyboundarypointz ∈∂U thereexistsaholomorphicfunctiononU thatcannot
be continued analytically across z (see [GR04b]).
Definition 17. AcomplexLiegroupGisaStein group iftheunderlyingmanifold
is a Stein manifold (see [GR04b, p. 136]).
GL(n,C) and any closed complex subgroup of GL(n,C) is a Stein group. Any
semi-simple connected or simply-connected solvable complex Lie group is Stein.
THE CANONICAL 2-GERBE OF A COMPLEX MANIFOLD 9
Proposition 18. Let X be a contractible Stein manifold (for example, a poly-
cylinder) and let G,G′ be gerbes on X. Then any two stable isomorphisms are
isomorphic, meaning that we find a transformation between them.
Proof. Two stable morphisms differ by a line bundle on X, which in our case is a
holomorphic line bundle L. But L is holomorphically trivial, by the Grauert-Oka
Theorem [Gra58], and a trivialization defines a transformation. (cid:3)
3.2. Exponential Sequence. The long exact sequence induced by
0→Ω•<p[−1]→Z(p) →Z→0
X D
with the fact H∗(Ω<p)=H∗(X;C) for ∗<p−1 shows that Hq(Z(p) ) sits inside
D
a Bockstein sequence. By the Five Lemma we get
Hq(X;Z(p)D)∼=Hq−1(X;C/Z), 0<q <p−1 (6)
The case q ≥p is more difficult and leads to the exponential sequence
→Hq−1(X;Z)→Hq−1(Ω<p)→Hq(Z(p) )→Hq(X;Z)→Hq(Ω<p)→
D
If X is a Stein manifold and q >p>0 then Hq(Ω<p)=0, by Cartan’s TheoremB
[Car53, p. 51]. Putting this into the exponential sequence gives
Hq(Z(p)D)∼=Hq(X;Z), q >p. (7)
Corollary 19. The class DD(G ) of the canonical gerbe is the generator of
can
H3(GL(n,C),Z(1)D)∼=H3(GL(n,C),Z)=Z.
Since H3(GL(n,C),Z(2)D) ∼= H3(GL(n,C),Z(1)D) there is a unique holomor-
phic connection on G , up to stable isomorphisms with connection. This connec-
can
tion may also be constructed explicitly by C-linear projection Cn → V (A) onto
O
the eigenspace bundles.
4. Multiplicative Structure
The existence of a multiplicative structure depends only on the stable isomor-
phism class of the gerbe and is therefore a cohomologicalproblem. After the proof
of existence, we describe the multiplicative structure more explicitly.
Definition 20. Let G be complex Lie group with product µ. A multiplicative
holomorphic gerbe on G consists of a holomorphic gerbe G on G, a morphism
M: pr∗G⊗pr∗G → µ∗G, and a transformation α: M ◦(M ⊗id) ⇒ M ◦(id⊗M).
1 2
The transformation α should fit into the usual coherence pentagon [Wal10, p. 47].
A connection ona multiplicative gerbe consists ofconnections on G and M so that
α is compatible with the connections.
Ingeneral,thereareobstructionstofindingamultiplicativestructureonagiven
gerbe G. For Stein groups,Theorem2, which we shall provein this section, asserts
that this obstruction against a holomorphic multiplicative structure reduces to a
topological problem.
Recall from [Del74] that a simplicial space is a functor X: ∆op → Top. For
example, the constant simplicial space has X = X for a fixed space X and all
n
facesanddegeneraciesaretheidentity. Next,foratopologicalgroupGweletBG•
denote the nerve of G with n-simplices BG = Gn. One finds a simplicial map
n
EG• →BG• whose fibers are the constant simplicial spaces G, see [Seg68].
10 MARKUSUPMEIER
The cohomology of simplicial space X• with coefficients in a sheaf F is the co-
homology of the total complex of the double Čech complex Cpq = Cˇq(X ,F) (one
p
may also use injective resolutions). Reduced cohomology is defined as usual way
bypassingtoquotientcomplexes. Onconstantsimplicialspacesonerecoversusual
sheaf cohomology. Similarly, for a complex of sheaves F∗ one defines H(X•,F∗)
using the Čech hypercomplex [Bry08, p. 28]. From [Seg68] we take that the sim-
plicial cohomology H(BG•;Z) computes the cohomology of the classifying space
BG=|BG•| of a Lie group G.
For a double complex Fp denotes the p-th verticalfiltrationCiq,i≥p. We have
a short exact sequence of Čech cochain complexes
0→F2 →Cˇ∗(BG•,pt;F)−→τ Cˇ∗(BG•,pt;F) →0
F2
Identifying the rightmost term with Cˇ∗−1(G,F), the map τ simply collapses all
but the second column of the double complex. In cohomologyτ induces the trans-
gressionmap, see [Wal10, Lemma 2.9]. Using Lemma 7, a gerbe with holomorphic
connection may be described, up to isomorphism, by a cocycle in Cˇ3(G,Z(2) ).
D
The data for the morphism M and transformation α in Definition 20 corresponds
exactly to an extension to a cocycle of the double complex Cˇ4(BG•,pt;Z(2)D).
To prove Theorem 2 it remains therefore only to show:
Lemma 21. The map H∗(BG•,Z(2)D)→H∗(BG;Z) is an isomorphism (∗>0).
Proof. By the long exact sequence induced by 0 →Ω<2[−1]→ Z(2) →Z→ 0 it
D
suffices to show H∗(BG•,Ω<2)=0. To do this, we consider the spectral sequence
for simplicial spaces E1pq =Hq(BGp,Ω<2)⇒Hp+q(BG•;Ω<2).
From Cartan’s Theorem B [Car53, p. 51] and the sheaf hypercohomology spec-
tral sequence we deduce for any Stein manifold X that the space H∗(X,Ω<2) is
H0(X;C) for ∗=0, H1(X;C) for ∗=1, and zero else.
It follows that Epq =0 unless q = 0,1. The first row Ep0 = H0(Gp;C) is given
1 1
by C −→0 C −→1 C → ··· and in particular exact apart from the first term. The
second row Ep1 reads 0→H1(G;C)→H1(G2;C)→H1(G3;C)→···, where the
1
maps are (−1)id∗ induced by the face maps d of the nerve. Since we are in
i i i
degreeoneP, µ∗: H1(G;C)→H1(G2;C)equalspr∗+pr∗,fromwhichitfollowsthat
1 2
the row Ep1 is exact. Hence Epq =0 for p+q >0, whence the result. (cid:3)
1 2
Definition 22. A choice of preimage of DD(G) ∈ H3(G;Z(2) ) is the multiplica-
D
tive class λ(G) ∈ H4(BG•;Z(2)D) of the multiplicative gerbe with holomorphic
connection (see [Wal10] in the smooth category).
From the proof above, it is clear that the multiplicative class determines the
multiplicative structure, up to (multiplicative) isomorphism, see [Wal10].
Corollary 23. For the group G=GL(n,C) every holomorphic gerbe with connec-
tion admits a multiplicative structure.
Proof. From Lemma 21 we deduce
H4(BGL(n,C)•,Z(2)D)∼=H4(BGL(n,C);Z)=Zc21⊕Zc2.