Gargnano March

De
Publié par

Holomorphic symplectic manifolds Arnaud Beauville Universite de Nice Gargnano, March 2008 Arnaud Beauville Holomorphic symplectic manifolds

  • holomorphic symplectic

  • building bricks

  • k?1 ample

  • universite de nice


Publié le : mardi 19 juin 2012
Lecture(s) : 29
Source : math.unice.fr
Nombre de pages : 43
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Holomorphic symplectic
Arnaud Beauville
Universit´edeNice
manifolds
Gargnano, March 2008
ArnaudBaevuilleHolmorophicyspmlceitcamnioflsd
I.
Manifolds withc1
Arnaud Beauville
Universite´deNice
=
Gargnano, March 2008
ArnaudBaevuille.IMainofdlswithc1
0
=0
the
“building
bricks”
of
FenahTfilomonaitswtrhKnimaldfo)elpehT;K(sdma1mple.WewdswithKamenafiloviai;lhTaurn.AalviriKtseacehtredisnoclli0
Conjecturally, andvery algebraic geometry are:
roughly,
hc1=switfoldaMinel.IvuliBdae
lcilewWinaMdloflivu.Iel
Conjecturally, andvery roughly, the “building bricks” of algebraic geometry are:
The Fano manifolds (K1ample);
0
The manifolds withKtrivial;
The manifolds withKample.
itsw1=hcehacestKnoisedtrrnaudBearivial.A
Conjecturally, andvery roughly, the “building bricks” of algebraic geometry are:
The Fano manifolds (K1ample);
The manifolds withKtrivial;
The manifolds withKample.
We will consider the caseKtrivial.
ArnaudBaevuilleI.Manifoldswithc1=0
oCorπy(1llramostX)alian;abeloisrotXKduanrA.nllviauBeifan.MeI
T = complex torus, Y simply-connected with KY=OY.
There exists T×YXinet´etalewith
=0c1
Let X comp
LetXcompactK¨ahlerwithc1(X) = 0in H2(X,C).
Proposition
dsolthwi
Proposition LetXcompactKa¨hlerwithc1(X) = 0in H2(X,C).
There exists T×Ynite´etalewithX
T = complex torus, Y simply-connected with KY=OY.
Corollary π1(X)almost abelian; KXtorsion.
ArnaudBeauvilleI.Mainofdlswithc1=0
=0
Theorem (Decomposition theorem)
compactKa¨hlersimply-connectedwithKX X=YYi×YZ i j
X
=OX.
j
Then
i=YdeH,(0,Zc-noentc=Zsimplyfolds)ZjuaY-inamaCeribalth.(eaesoratKYofneresigareωe,ωhw=CCΩY)0(Y,3,HYmid,evitcejorpdteecnncoy-plimYs.Manifoldswithc1rAandueBuaivllIectlemaicfonis)ldrriecudeelbipmys.(thrateretheseaeronwyehegen-nedΩ2Z,0(HerevisZ)σ[C=)ZΩσerehw,]
aeBdlivuA)sduanrldfoitswI.leniMach=10
Yi= projective,Y simply-connecteddimY3, H0(Y,ΩY) =CCω, whereωis a generator of KY. (these areCalabi-Yaumanifolds)
Theorem (Decomposition theorem) XcompactKa¨hlersimply-connectedwithKX=OX. Then X=YYi×YZj i j
j=ZimZs-ylpnnocetce0H,d(Z,ΩZ)=C[σ],wheerσ0HZ(Ω,Z2i)eserwhryveeg-donen.etareneraeseht(irreetheblesduciceitmylpfilomcna
Theorem (Decomposition theorem) XcompactKa¨hlersimply-connectedwithKX=OX. Then X=YYi×YZj i j
Yi=Y simply-connected projective,dimY3, H0(Y,ΩY) =CCω, whereωis a generator of KY. (these areCalabi-Yaumanifolds)
Zj= HZ simply-connected,0(Z,ΩZ) =C[σ], where σH0(Z,Ω2Z)is everywhere non-degenerate. (these are theirreducible symplecticmanifolds)
ArnaudBaevuille.IMainofdlsiwhtc1=0
contInsyofesplicctlemprev,tsarmaxewefyArs.udnanimaldfoM.IefinauaeBllivc1=0
Remarks
:
examples etc.
oldswith
1
of
Many inPn ,
hypersurfaces
Calabi-Yau:
degree
of
+
n
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