Let $E\rightarrow X$ be a line bundle over a compact Riemann surface. Let $\Delta_{E}$ be the Dolbeault Laplacian $\Delta_{\overline{\partial}}$ extended to sections of $E$. Let $g_1, g_2$ be two metrics on $X$.
Q: Can we give an upper bound for $\det(\Delta_{E})$ independent of $g_1, g_2$ for fixed $X$ and $E$?
For some motivation of this, note that from Weyl asymptotics we immediately have $\lambda_{j}\rightarrow A(E, g_{i})j$ for $j$ large enough, where $A(E,g_{i})$ is some constant depending on $g_{i}$. However, if I am not mistaken, Weyl asymptotics does not give us effective control of the growth of eigenvalues. For two flat line bundles $E, E'$ induced from representation of $\pi_{1}(X)$, Ray-Singer proved the ratio of two analytic torsions $\det(\Delta_{E})/\det(\Delta_{E'})$ is independent of the metic on $E, E'$. But it is unclear how to extend their work to general line bundles (using automorphic factors?). Later, Soule proved in his paper that when $E$ is a flat unitary bundle, $\Delta_{E}<C$ for some constant $C$. There is some follow up work by Gillet and Soule in their paper, basically proving the upper bound exists for $X\cong \mathbb{CP}^{1}$ and when the sum of the betti numbers is less than $2$.
