Question

Given an $N$-dimensional Riemannian manifold $M$, with associated Hodge $\ast$-mapping $\ast$, we have the chain complex $$ \Omega^{0} {\buildrel {\text d}^\ast \over \longleftarrow} \Omega^{N} {\buildrel {\text d}^\ast \over \longleftarrow}\cdots \Omega^{N}
$$ For a Kahler manifold $M$ of complex dimension $N$, with associated Hodge $\ast$-mapping $\ast$, we have a second chain complex $$ \Omega^{(0,0)} {\buildrel \overline{\partial}^\ast \over \longleftarrow} \Omega^{(0,1)} {\buildrel \overline{\partial}^\ast \over \longleftarrow} \cdots \Omega^{(0,N)}. $$ What I would like to know is for which other Hermitian manifolds does this complex exist? I suppose I'm asking what condition on the Hermitian metric will give a Hodge $\ast$-map for which $\overline{\partial}^\ast(\Omega^{(0,k)}) \subset \Omega^{(0,k-1)}$ and $\overline{\partial}^2=0$.

Answer 1

Hodge $*$ and $\overline{\partial}^*$ are defined and have the named properties on any complex manifold with respect to any Hermitian metric. No Kahlerness, nor compactness, is necessary. Note that Voisin's book presents them in the chapter before she defines the Kahler condition.

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John McCarthy points out that the definition of $*$ seems to vary across textbooks. Looking at the $3$ books I have available:

Voisin (Section 5.1) defines $*$ to be a $\mathbb{C}$-linear map, which takes $(p,q)$-forms to $(n-q, n-p)$ forms, and for which $\overline{\partial}^* = - * \partial *$. This is the convention I have been following.

Wells (Secton V.2) defines $*$ in the same way as Voisin, but then also defines $\overline{*}$ by $\overline{*} \phi = * \overline{\phi}$, and basically doesn't mention $*$ again. He writes $\overline{\partial}^* = - \overline{*} \overline{\partial} \overline{*}$. This defines the same map as Voisin does, but generalizes more nicely to vector bundles. This is a nice expository choice; I probably would have done better to follow this in my course.

Griffiths and Harris (Section 0.6) define $*$ to be a $\mathbb{C}$-antilinear map which takes $(p,q)$-forms to $(n-p, n-q)$ forms. (See a little earlier on p. 82.) This suggests that their $*$ is Wells' $\overline{*}$. I am not sure that they are exactly the same, though. Looking at $\mathbb{C}$, with the standard Kahler form, I think Wells' $\overline{*} (dz)$ should be his $* d\overline{z}$ which I think is $dy + i dx = i d \overline{z}$. That is off by $i$ from Griffiths and Harris' formula, I'm not sure why.

In any case, it appears that G and H write $*$ for a map which has the complex conjugate built in, where the others do not. They all seem to agree as to what $\overline{\partial}^*$ means.