Table Of ContentLecture Notes in Mathematics 2095
Daniele Angella
Cohomological
Aspects in
Complex
Non-Kähler
Geometry
Lecture Notes in Mathematics 2095
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
AdvisoryBoard:
CamilloDeLellis(Zurich)
MarioDiBernardo(Bristol)
AlessioFigalli(Pisa/Austin)
DavarKhoshnevisan(SaltLakeCity)
IoannisKontoyiannis(Athens)
GaborLugosi(Barcelona)
MarkPodolskij(Heidelberg)
SylviaSerfaty(ParisandNY)
CatharinaStroppel(Bonn)
AnnaWienhard(Heidelberg)
Forfurthervolumes:
http://www.springer.com/series/304
Daniele Angella
Cohomological Aspects
in Complex Non-KaRhler
Geometry
123
DanieleAngella
DipartimentodiMatematica
UniversitaJdiPisa
Pisa,Italy
ISBN978-3-319-02440-0 ISBN978-3-319-02441-7(eBook)
DOI10.1007/978-3-319-02441-7
SpringerChamHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2013952956
MathematicsSubjectClassification(2010):32Q99,53C55,32Q60,32C35,57T15,32G05,32G07,
53D05,53D18
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In ricordodimiononno,
il MaestroGiovanni
Introduction
ByaremarkableresultbyW.L.Chow,[Cho49,TheoremV],seealso(cid:2)[Ser56],pr(cid:3)oı-
jectivemanifolds(thatis,compactcomplexsubmanifoldsofCPn WD CnC1nf0g
.Cnf0g/, for n 2 N) are in fact algebraic (that is, they can be described as
the zero set of finitely many homogeneous holomorphic polynomials). One is
henceinterestedinrelaxingtheprojectivecondition,lookingforspecialproperties
on compact manifolds sharing a weaker structure than projective manifolds. For
example, a large amount of developed analytic techniques allows to prove strong
cohomologicalpropertiesforcompactKählermanifolds(thatis,compactcomplex
manifoldsendowedwith a Kähler metric, namely,a Hermitian metric admitting a
localpotentialfunction),[SvD30,Käh33],seealso[Wei58],whichare,inacertain
sense,the “analytic-versus-algebraic”,[Cho49, TheoremV], orthe“R-versus-Q”,
[Kod54, Theorem 4], version of projective manifolds. On the one side, there is
the class of Kähler manifolds, which are in fact endowed with three different
structures, interacting each other: a complex structure, a symplectic structure, and
a metric structure; it is the strong linking between them that allows to develop
many analytic tools and hence to derive the very special properties of Kähler
manifolds. On the other side (the Dark Side...), in order to further investigate
anyof such propertiesandto understandwhatof these three structuresis actually
involvedandrequired,itisnaturaltolookformanifoldsnotadmittinganyKähler
structure: a large amount of interesting non-Kähler manifolds has been provided
since[Thu76].Inotherwords,oneisledtostudycomplex,symplectic,andmetric
contributionsseparately,possiblyweakeningeithertheinteractionsbetweenthem,
or one of these structures. For example, by relaxing the metric condition, one
could ask what properties of a compact complex manifold can be deduced by
the existence of special Hermitian metrics defined by conditions similar to, but
weaker than, the defining condition of the Kähler metrics (for example, metrics
being balanced in the sense of M. L. Michelsohn [Mic82], pluriclosed [Bis89],
astheno-Kähler[JY93,JY94],Gauduchon[Gau77],strongly-Gauduchon[Pop13]);
byrelaxingthecomplexstructure,oneisledtostudypropertiesofalmost-complex
manifold,possiblyendowedwithcompatiblesymplecticstructures.
vii
viii Introduction
In these notes, we are concerned with summarizing some recent results on the
cohomologicalpropertiesofcompactcomplexmanifoldsendowedwithnoKähler
structure. Cohomological aspects of manifolds endowed with almost-complex
structures, or with other special structures (such as, for example, symplectic,
generalized-complex,...)arealsoconsidered.
WerecallthatacomplexmanifoldX isendowedwithanaturalalmost-complex
structure, that is, an endomorphism J 2 End.TX/ of the tangent bundle of X
such that J2 D (cid:2)id , which actually satisfies a further integrability condition,
TX
[NN57,Theorem1.1].Byconsideringthedecompositionintoeigen-spaces,justthe
datumofthealmost-complexstructureyieldsasplittingofthecomplexifiedtangent
bundle,namely,
TX˝C D T1;0X ˚T0;1X ;
and hence it induces also a splitting of the bundle of complex differential forms,
namely,
M
^(cid:2)X ˝RC D ^p;qX :
pCqD(cid:2)
Furthermore,onacomplexmanifold,theintegrabilityconditionofsuchanalmost-
complex(cid:4)structureyiel(cid:5)dsafurtherstructureon^(cid:2);(cid:2)X,namely,astructureofdouble
complex ^(cid:2);(cid:2)X; @; @ ,where@and@arethecomponentsoftheC-linearextension
oftheexteriordifferentiald.
Hence,onacomplexmanifoldX,onecanconsiderboththedeRhamcohomo-
logy
kerd
H(cid:2) .XIC/ WD
dR imd
andtheDolbeaultcohomology
ker@
H(cid:2);(cid:2).X/ WD I
@ im@
wheneverX is compact,the Hodgetheoryassuresthattheyhave finite dimension
as C-vector spaces. On a compact complex manifold, in general, no natural map
betweenH(cid:2);(cid:2).X/andH(cid:2) .XIC/exists;ontheotherhand,thestructureofdouble
@(cid:4) d(cid:5)R
complexof ^(cid:2);(cid:2)X; @; @ givesrisetoaspectralsequence
E(cid:2);(cid:2) ' H(cid:2);(cid:2).X/ ) H(cid:2) .XIC/;
1 @ dR
Introduction ix
fromwhichonegetstheFrölicherinequality,[Frö55,Theorem2]:foreveryk 2N,
X
dimCHdkR.XIC/ (cid:3) dimCH@p;q.X/:
pCqDk
On a complex manifold, a “bridge” between the Dolbeault and the de Rham
cohomologyisprovided,inasense,bytheBott-Cherncohomology,
ker@\ker@
H(cid:2);(cid:2).X/ WD ;
BC
im@@
andtheAepplicohomology,
ker@@
H(cid:2);(cid:2).X/ WD :
A
im@Cim@
Infact,theidentityinducesthemapsof(bi-)gradedC-vectorspaces
H(cid:2);(cid:2).X/
(cid:3)(cid:3)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2) BC(cid:2)(cid:2) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:4)(cid:4)
H(cid:2);(cid:2).X/ H(cid:2) .XIC/ H(cid:2);(cid:2).X/
@ (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:4)(cid:4) dR (cid:2)(cid:2) (cid:3)(cid:3)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2) @
H(cid:2);(cid:2).X/
A
which,ingeneral,areneitherinjectivenorsurjective.
These cohomology groups have been introduced by R. Bott and S. S. Chern
in [BC65], and by A. Aeppli in [Aep65], and studied by several authors in
different contexts: among others, B. Bigolin [Big69,Big70], A. Andreotti and
F. Norguet[AN71], J. Varouchas[Var86], M. Abate [Aba88], L. Alessandriniand
G.Bassanelli[AB96],S.Ofman[Ofm85a,Ofm85b,Ofm88],S.Boucksom[Bou04],
J.-P.Demailly[Dem12],M. Schweitzer[Sch07],L.Lussardi[Lus10], R. Kooistra
[Koo11],J.-M.Bismut[Bis11b,Bis11a],L.-S.Tseng,andS.-T.Yau[TY11].
We recall that whenever X is compact, the Hodge theory can be performed
also for Bott-Chern and Aeppli cohomologies, [Sch07, §2], yielding their finite-
dimensionality;moreprecisely,onehasthat,onacompactcomplexmanifoldX of
complexdimensionnendowedwithaHermitianmetric,
H(cid:2);(cid:2).X/ ' (cid:2)Q and H(cid:2);(cid:2).X/ ' (cid:2)Q ;
BC BC A A
where(cid:2)Q and(cid:2)Q are4thorderself-adjointellipticdifferentialoperators;further-
BC A
more, the Hodge-(cid:4)-operatorassociated to any Hermitian metric on X induces an
isomorphismHp;q.X/'Hn(cid:3)q;n(cid:3)p.X/,foreveryp;q 2N.
BC A
