Given any compact Riemann surface C, there is a canonical meromorphic 2–form $\hat{\eta }$ on $C\times C$, with pole of order two on the diagonal $\mathrm{\Delta }\subset C\times C$, constructed in [4]. This meromorphic 2–form $\hat{\eta }$ produces a canonical projective structure on C. On the other hand the uniformization theorem provides another canonical projective structure on any compact Riemann surface C. We prove that these two projective structures differ in general. This is done by comparing the $(0,1)$–component of the differential of the corresponding sections of the moduli space of projective structures over the moduli space of curves. The $(0,1)$–component of the differential of the section corresponding to the projective structure given by the uniformization theorem was computed by Zograf and Takhtadzhyan in [16] as the Weil–Petersson Kähler form ${\omega }_{wp}$ on the moduli space of curves. We prove that the $(0,1)$–component of the differential of the section of the moduli space of projective structures corresponding to $\hat{\eta }$ is the pullback of a nonzero constant scalar multiple of the Siegel form, on the moduli space of principally polarized abelian varieties, by the Torelli map.

给定任何紧致的黎曼曲面C，存在一个规范的亚纯2形式$\hat{\eta }$ 上 $C\times C$，对角线的二阶极点 $\mathrm{\Delta }\subset C\times C$，在[4]中构造。亚纯2形式$\hat{\eta }$在C上产生规范的投影结构。另一方面，均匀化定理在任何紧致的黎曼曲面C上提供了另一种规范的射影结构。我们证明这两个投影结构在总体上是不同的。这是通过比较$（0，\mathrm{1个}）$–射影结构的模空间上曲线的模空间上相应截面的微分的分量。这$（0，\mathrm{1个}）$Zograf和Takhtadzhyan在[16]中以Weil-PeterssonKähler形式计算了与均匀化定理给出的投影结构相对应的截面微分的-分量 ${\omega }_{wp}$在曲线的模空间上 我们证明$（0，\mathrm{1个}）$–对应于投影结构的模空间截面的微分的分量 $\hat{\eta }$ 是通过Torelli映射在主要极化的阿贝尔变种的模空间上拉出的Siegel形式的非零常数标量倍数。