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I have a question about the Hodge Decomposition theorem. Let $(X,\omega)$ be a compact Fano Kaeler manifold, we know the Hodge theorem works very well with respect to the $\bar\partial-$Laplacian $\Box_{\bar\partial}=\bar\partial\bar\partial^*+\bar\partial^*\bar\partial$. However, on Fano manifold, we have $Ric-\omega=\sqrt{-1}\partial\bar\partial f$ for some smooth function. we can use this $f$ to modify the $L^2-$ metric on the space of $(p,q)-$form, say $<\varphi, \psi>=\int (\varphi,\psi)e^{f}$, correspondingly, we can define $\bar\partial^*_f$ and Laplacian $\Box_f=\bar\partial\bar\partial^*_f+\bar\partial^*_f\bar\partial$. So My question is if the Hodge decomposition with respect to $\Box_f$ still works? Thank you in advance.

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I don't know much about the complex geometry behind your question, but the Hodge decomposition theorem works in more general contexts such as the one you have described. Let me describe this generalization.

Let $M$ be a smooth manifold and $E_0, \dots, E_m$ be smooth vector bundles on $M$. The sequence of differential operators $$0 \longrightarrow \Gamma(E_0) \xrightarrow{~D_0~} \Gamma(E_1) \xrightarrow{~D_1~} \cdots \xrightarrow{~D_{m-1}~} \Gamma(E_m) \longrightarrow 0$$ is called an elliptic complex if

  1. The sequence is a cochain complex, i.e. $D_i \circ D_{i-1} = 0$ for $1 \leq i \leq m$.
  2. The associated symbol complex $$0 \longrightarrow (E_0)_x \xrightarrow{~\sigma_{D_0}(x, \xi)~} (E_1)_x \xrightarrow{~\sigma_{D_1}(x,\xi)~} \cdots \xrightarrow{~\sigma_{D_{m-1}}(x,\xi)~} (E_m)_x \longrightarrow 0$$ is exact for all $x \in M$, $\xi \in T_x^\ast M \setminus \{0\}$.

Now let each $E_i$ be equipped with an inner product and let $$D_i^\ast: \Gamma(E_{i+1}) \longrightarrow \Gamma(E_i)$$ be the formal adjoint of $D_i$ with respect to this inner product, for $1 \leq i \leq m$. Define also $$\Delta_i = D_{i-1} \circ D_{i-1}^\ast + D_i^\ast \circ D_i.$$ Then the following generalization of the Hodge theorem holds.

Theorem. Let $M$ be a compact, oriented manifold and suppose $(E_\ast, D_\ast)$ is an elliptic complex over $M$. Then the following hold.

  1. There is an orthogonal direct sum decomposition $$\Gamma(E_i) = \ker(\Delta_i) \oplus \mathrm{Im}(D_{i-1}) \oplus \mathrm{Im}(D_i^\ast).$$
  2. $H^i(E_\ast, D_\ast) \cong \ker(\Delta_i)$.
  3. $H^i(E_\ast, D_\ast)$ is finite-dimensional for all $i$.

In your case, you are still considering the usual Dolbeault complex $$0 \to \Omega^{p,0}(X) \xrightarrow{~\bar{\partial}~} \Omega^{p,1}(X) \xrightarrow{~\bar{\partial~}} \cdots,$$ which is elliptic, and instead considering a different inner product on your bundles. The above generalized Hodge theorem tells you that you still get the expected Hodge decomposition $$\Omega^{p,q}(X) = \ker(\square_f) \oplus \mathrm{Im}(\bar{\partial}) \oplus \mathrm{Im}(\bar{\partial}_f^\ast).$$

For the full details of the generalized Hodge theorem, you can see Chapter IV of Wells's Differential Analysis on Complex Manifolds.

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