Question I'm given a Laplacian $\Delta_n=-4y^2 \cdot \frac{\partial^2}{\partial\bar{z} \partial z} + 4 iny \cdot \frac{\partial}{\partial\bar{z}}$, and I want it to be the Laplace operator associated to a Cauchy-Riemann operator $\bar\partial:\Omega^{0,0}(E) \rightarrow \Omega^{0,1}(E)$, which is a differential operator of the form $\bar\partial=(\partial_{\bar{z}} + \alpha(z))d\bar{z}$. How to find a suitable Cauchy-Riemann operator?
Here $E$ is a smooth vector bundle over a compact Riemann surface $X$, and $\alpha(z)$ is a smooth function.
Since the problem is dependent on the metric we have on $X$, we have to specify it. Let $X$ be the quotient of the upper half plane $\mathbb{H}$ by a cocompact Fuchsian subgroup of the first kind, and the metric is the quotient of the hyperbolic metric on $\mathbb{H}$ by the action of the group.
How to associate a Generalized Laplacian to a Cauchy-Riemann operator
The first step is to define an $L^2$-scalar product in $\Omega^{0,0}(E)$ and $\Omega^{0,1}(E)$. The hyperbolic metric on $X$ corresponds to an Hermitian metric $h$ on the holomorphic tangent space of $X$ , i.e. the subbundle $T^{1,0}(X)$ of the complex tangent bundle of $X$. This metric induces a metric on the dual of $T^{1,0}(X)$ , i.e. differential forms of type $(1,0)$, and, by complex conjugation, on forms of type $(0,1)$. By taking the exterior powers of this metric, and by tensoring with the Hermitian metric on $E$ we get a pointwise scalar Hermitian product $(s(x),t(x))$ for two sections of $\Omega^{0,i}(E)=\Omega^{0,i}(X) \otimes_{C^\infty(X)} \Gamma(X,E)$. On the other hand, let $\omega_0$ be the normalized Kähler form:
$\omega_0=\frac{i}{2\pi}h(\frac{\partial}{\partial z}, \frac{\partial}{\partial z}) dz d\bar{z}$
Then the $L^2$ scalar product of two sections $s,t \in \Omega^{0,i}(E)$ is defined by: $(s,t)_{L^2}=\int_X(s(x),t(x))\omega_0$.
The second step consists in defining the operator $\bar{\partial}^*$ in such a way that it is the $L^2$-adjoint of $\bar{\partial}$. So $\bar\partial^*:\Omega^{0,1}(E) \rightarrow \Omega^{0,0}(E)$ and $(s,\bar\partial^* t)_{L^2}=(\bar\partial s, t)_{L^2}$. Then the Laplace operator is defined by: $\Delta_n=\bar\partial^*\bar\partial:\Omega^{0,0}(E) \rightarrow \Omega^{0,0}(E)$.
- This blockquote is a, more or less faithfully, quote from "Lectures on Arakelov Geometry" of Soulé, Abramovich, Burnol, Kramer. Chapter V, §2.2.*
My Approach My idea to find the right Cauchy-Riemann operator is to compute in full generality it's $L^2$-adjoint and find a general expression for the induced Laplacian.
My Computations As Willie kindly pointed out I was working on the wrong definition of Laplacian, so all my previous computations were useless. I will post some progress here as soon as I will have them.
Thank you very much!