Is there a "simple commutation" relation between $D^{''}$ and $\delta^{'}$, with $D^{''}$ the (0,1) part of the chern connection of a vector bundle and $\delta^{'}$ the adjoint of the (1,0) part?

Question

Hi, as the title says i'm wondering if there's a "simple" and known commutation relation between the following two differential operators. Let $E$ be a holomorphic vector bundle over a compact kahler manifold $X$, fix a Hermitian metric $h$ on $E$ and a kahler metric $g$ on $X$. Denote with $D=D^{'}+D^{''}$ the associated Chern connection and its (1,0) and (0,1) parts. Denote with $\delta=\delta^{'}+\delta^{''}$ the adjoint respect to $h$ of $D$. So my question is what is $[D^{''},\delta^{'}]$? Is there a relation like kahler relations?

Thank you in advance.

Answer 1

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 $[D''_E,\delta'_E]$, where $[\bullet,\bullet]$ is the graded commutator and let me call $\omega$ (instead of $g$) the Kähler form. Then one has that $$ [\Lambda_\omega,D''_E]=-i\delta'_E, $$ so that $$ [D''_E,\delta'_E]=i[D''_E,[\Lambda_\omega,D''_E]], $$ where $\Lambda_\omega=*^{-1}L_\omega*$ is the formal adjoint of the operator of degree $(1,1)$ given by $L_\omega\bullet=\omega\wedge\bullet$.

Now, the (graded) Jacobi identity gives $$ -[D''_E,[\Lambda_\omega,D''_E]]+[D''_E,\underbrace{[D''_E,\Lambda_\omega]}_{=-[\Lambda_\omega,D''_E]}]+[\Lambda_\omega,\underbrace{[D''_E,D''_E]}_{=0}]=0, $$ thus $-2[D''_E,[\Lambda_\omega,D''_E]]=0$ and $$ D''_E\delta'_E+\delta'_ED''_E=[D''_E,\delta'_E]=0. $$