# Exact sequence of Quillen metrics

Let $M$ be a compact Riemann surface equipped with its Arakelov metric. Let $\xi$ be a holomorphic line bundle on $M$ equipped with an admissble metric. For any point $P\in M$ we get an admissible metric on $\xi\otimes \mathcal{O}(-P)$ by multiplying the given metric on $\xi$ witht the caonical metric given by Green function on $\mathcal{O}(-P)$. From these data we get Quillen metrics on the line bundle $$\det(\overline{\partial}_{\xi}), \det(\overline{\partial}_{\xi\otimes\mathcal{O}(-P)})$$ From the exact sequence $$0\rightarrow \xi\otimes \mathcal{O}(-P)\rightarrow \xi\rightarrow \xi|_{P}\rightarrow 0$$ up to a multiplicative constant depending on $M$ and $\deg(\xi)$, Bost claim that we have an isometry of one dimensional vector spaces $$(\det(\overline{\partial}_{\xi\otimes\mathcal{O}(-P)}))^{-1}\cong \det(\overline{\partial}_{\xi})^{-1}\otimes \xi_{P}$$ (there may be a typo with the sign of $\det$)

However I have never seen any proof of this important fact from literature. Neither do I know the specifics of the constant involved. Does anyone know the rough idea of the proof?

Notice that the constant at here is very important. Otherwise we may "go down" to the trivial line bundle and decompose the determinant line bundle as the tensor product of skyscraper sheaves, which seems totally absurd because it involves analytic torsion. I am interested in this theorem because if $\deg(\xi)$ is large, this implies $h^{0}(X,\mathcal{O}(P))$ should be approximated by the Green functions $G(P)$. But I am not sure how to prove the statement rigorously (is some bosonization formula needed here?).

The main paper Bost referenced is carefully written, but somehow only gave a one line proof (end of page 544): " The third assertion is a consequence ofthe definition of the holomorphic structure on the determinant line bundle. ", which I am unable to follow. Since the same exact sequence showed up in Faltings' paper (page 395, Theorem 1), this provided a hint that Quillen's metric and Faltings' metric must differ by a constant only dependent on $X$.