Sunday, 3 December 2006

complex geometry - Is there a "simple commutation" relation between DD′′ and deltadelta, with DD′′ the (0,1) part of the chern connection of a vector bundle and deltadelta the adjoint of the (1,0) part?

Yes, the relation is that they anti-commute.



Let's see this very briefly (you can find it in almost all books on Kähler geometry and Hodge theory).



We want to compute [DE,deltaE][D′′E,deltaE], where [bullet,bullet][bullet,bullet] is the graded commutator and let me call omegaomega (instead of gg) the Kähler form. Then one has that
[Lambdaomega,DE]=ideltaE,[Lambdaomega,D′′E]=ideltaE,
so that
[DE,deltaE]=i[DE,[Lambdaomega,DE]],[D′′E,deltaE]=i[D′′E,[Lambdaomega,D′′E]],
where Lambdaomega=1Lomega is the formal adjoint of the operator of degree (1,1) given by Lomegabullet=omegawedgebullet.



Now, the (graded) Jacobi identity gives
[DE,[Lambdaomega,DE]]+[DE,underbrace[DE,Lambdaomega]=[Lambdaomega,DE]]+[Lambdaomega,underbrace[DE,DE]=0]=0,
thus 2[DE,[Lambdaomega,DE]]=0 and
DEdeltaE+deltaEDE=[DE,deltaE]=0.

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