Let $\overline{M}$ be a compact Riemann surface and let $g^{TM}$ be a metric over $\overline{M} \setminus D_M$, where $D_M \subset \overline{M}$ is a finite set of points. We suppose that $g^{TM}$ is equal to the Poincare metric over a punctured disks around the points of $D_M$. The metric $g^{TM}$ endows the twisted canonical line bundle $\omega_M(D)$ with the induced Hermitian norm $\|\cdot\|_M$ over $\overline{M} \setminus D_M$. Let $(\xi, h^{\xi})$ be a holomorphic Hermitian vector bundle over $\overline{M}$. In this article we define the analytic torsion $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ associated with $(M, g^{TM})$ and $(\xi \otimes \omega_M(D)^n, h^{\xi} \otimes \|\cdot\|_M^{2n})$ for $n \leq 0$. We prove that $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ is related to the analytic torsion of non-cusped surfaces. Then we prove the anomaly formula for the associated Quillen norm. The results of this paper will be used in the sequel to study the regularity of the Quillen norm and its asymptotics in a degenerating family of Riemann surfaces with cusps and to prove the curvature theorem. We also prove that our definition of the analytic torsion for hyperbolic surfaces is compatible with the one obtained through Selberg trace formula by Takhtajan-Zograf.

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带尖点表面的解析扭转 I：紧致微扰定理和异常公式

令 $\overline{M}$ 是一个紧凑的黎曼曲面，令 $g^{TM}$ 是 $\overline{M} \setminus D_M$ 上的一个度量，其中 $D_M \subset \overline{M}$ 是一个点的有限集。我们假设 $g^{TM}$ 等于 $D_M$ 点周围穿孔磁盘上的 Poincare 度量。度量 $g^{TM}$ 赋予扭曲的规范线丛 $\omega_M(D)$ 在 $\overline{M} \setminus D_M$ 上的诱导 Hermitian 范数 $\|\cdot\|_M$。令 $(\xi, h^{\xi})$ 是 $\overline{M}$ 上的全纯 Hermitian 向量丛。在本文中，我们定义与 $(M, g^{TM}) 相关的解析扭 $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ $ 和 $(\xi \otimes \omega_M(D)^n, h^{\xi} \otimes \|\cdot\|_M^{2n})$ 用于 $n \leq 0$。我们证明 $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ 与非尖头曲面的解析扭转有关。然后我们证明关联的 Quillen 范数的异常公式。本文的结果将用于后续研究 Quillen 范数的正则性及其在具有尖点的退化黎曼曲面族中的渐近性，并证明曲率定理。我们还证明了我们对双曲曲面解析扭的定义与 Takhtajan-Zograf 通过 Selberg 迹公式获得的定义是兼容的。