V Further developments

Arnaud Beauville

Description

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

Sujets

• Beauville
• Universidad de Niza Sophia Antipolis

Informations

• Publié par: pefav
• Nombre de lectures: 21
• Langue: English

Extrait

V.

Further developments

Arna

Arnaud Beauville

u

Universit´edeNice

March 28, 2008

dBeauvilleV.uFrtheredevlopments

Poropogomolovsition(B X )yk epyh reV-stib su,Aalb-¨arkerhl)Qps( ∗ XoaHfegrbQ). T 2 (XdbyHanne nioPsefisitasAneh ∗ (;Htyliuaedr´ca QX(wJ/)Jhti r xh= Q)X⊕A=A A= ⊥ ; H 2 S ∗ =)i0 oCorllra S y H p +1 |x∈H 2 (XQ)q(x evitcejn.r≤prof→HQ)X 2 ()iQ(X 2p

developments

Further

V.

Beauville

Arnaud

Cohomology

2 ( p HQ)X 2p →HQ(Xjni)itceofevr≤pr oCorllra S y

A=S ∗ H 2 (XQ)/J with J=hx r+1 |x∈H 2 (XQ)q(x) = 0i

ThenAsatisfiesPoincar´eduality;H ∗ (XQ) =A⊕A ⊥ ;

Xhyperk¨ahler,Asub-algebraofH ∗ (XQ)spanned by H 2 (XQ).

Proposition (Bogomolov-Verbitsky)

stnemlielaevuuaBdA . nreloprdevrtheV.Fu logoyChomo

Cohomology

Proposition (Bogomolov-Verbitsky) Xhyperka¨hler,Asub-algebraofH ∗ (XQ)spanned by H 2 (XQ).

ThenAsatisfiesPoincar´eduality;H ∗ (XQ) =A⊕A ⊥ ;

A=S ∗ H 2 (XQ)/J with J=hx r+1 |x∈H 2 (XQ)q(x) = 0i

Corollary S p H 2 (XQ)→H 2p (XQ)injective for p≤r .

ArnaudBeauville.VFurtherdevelopments

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of.

developments

Further

V.

Beauville

Arnaud

input:

Pro

Geometric

.nard 1 o0Jtpsmafaceen,h H ∗ SnehT.)QX( ∗ H→ 2r )6hλ(B Repr =0. 3 asλstcrow,tiB:A α r 0⇒=0 +1 ω∈ ; ∃(XH 2 q=)α( =H 2 (XQ),B=S ∗ H/J Q,) 2 ω6 r 0= 2 . uPHt