V Further developments

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V. Further developments Arnaud Beauville Universite de Nice March 28, 2008 Arnaud Beauville V. Further developments

  • depends only

  • then s?h ?

  • bp ?

  • bogomolov-verbitsky

  • geometric input

  • universite de nice


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V.
Further developments
Arna
Arnaud Beauville
u
Universit´edeNice
March 28, 2008
dBeauvilleV.uFrtheredevlopments
Poropogomolovsition(BX)ykepyhreV-stibsu,Aalb-¨arkerhl)Qps(XoaHfegrbQ).T2(XdbyHannenioPsesitasAneh(;Htyliuaedr´caQX(wJ/)Jhtirxh=Q)XA=AA=;H2S=)i0oCorllraSyHp+1|xH2(XQ)q(xevitcejn.rprofHQ)X2()iQ(X2p
developments
Further
V.
Beauville
Arnaud
Cohomology
2(pHQ)X2pHQ(Xjni)itceofevrproCorllraSy
A=SH2(XQ)/J with J=hxr+1|xH2(XQ)q(x) = 0i
ThenAsatisesPoincar´eduality;H(XQ) =AA;
Xhyperk¨ahler,Asub-algebraofH(XQ)spanned by H2(XQ).
Proposition (Bogomolov-Verbitsky)
stnemlielaevuuaBdA.nreloprdevrtheV.FulogoyChomo
Cohomology
Proposition (Bogomolov-Verbitsky) Xhyperka¨hler,Asub-algebraofH(XQ)spanned by H2(XQ).
ThenAsatisesPoincar´eduality;H(XQ) =AA;
A=SH2(XQ)/J with J=hxr+1|xH2(XQ)q(x) = 0i
Corollary SpH2(XQ)H2p(XQ)injective for pr .
ArnaudBeauville.VFurtherdevelopments
rkmade:AioctRen.(noy)qHdneplnos6=0,cont.4IfKerλoctnaridiasn2B,rrB2p×.B.e,iinteptcefrepQ=r2BpeoryonthtatiesenersnBoGH)qfo(O
of.
developments
Further
V.
Beauville
Arnaud
input:
Pro
Geometric
.nard1o0Jtpsmafaceen,hHSnehT.)QX(H2r)6hλ(BRepr=0.3asλstcrow,tiB:Aαr0=0+1ω;(XH2q=)α(=H2(XQ),B=SH/JQ,)2ω6r0=2.uPHt