Let $C_5:=\{{(z_1 \dots, z_5) \in (\mathbb{C})^5 | z_i \neq z_j \forall i\neq j }\}$ be the configuration space of five distinct ordered points in $\mathbb{C}$. Arnold showed that the holomorphic one forms $w_{i,j}:=d log (z_i - z_j)$ satisfy the relation $w_{i,j}\wedge w_{j,k}+ w_{j,k}\wedge w_{k,i}+w_{k,i}\wedge w_{i,j}=0$.This relation is necessary for proving the flatness of KZ connection.

Let $\mathcal{A}_2[2]$ be the space of principally polarized abelian surfaces with level 2 structure. Since $C_5$ parametrizes genus 2 curves with an ordering on their Weirstrass points the Torelli map gives an injection $T:C_5/G \hookrightarrow \mathcal{A}_2[2]$ whose image is the complement of the divisor parametrizing Abelian surfaces which are products of two elliptic curves. Here $G$ is the 2-dimensional group of affine transformations of $\mathbb{C}$.

Is it known which theta functions (perhaps with characteristics) on $\mathcal{A}_2[2]$ pull back to the functions $(z_i -z_j)$ on $C_5$ by the Torelli map $T$? Logarithmic derivative of such theta functions will correspond to the differential forms $w_{i, j}$ on $C_5$ and should satisfy Arnold's relation. Moreover, such an identification will allow to write the KZ connection in terms of logarithmic derivatives of theta functions.

Any comments or references are more than welcome.

  • $\begingroup$ Thanks for the comment. Indeed, I meant which theta functions pull back to functions $(z_i-z_j)$ on $C_5$, then the logarithmic derivatives of such theta functions would correspond to the differential forms $d log (z_i- z_j)$ and should satisfy the Arnold relation. I'll make the edit in my question. $\endgroup$ Apr 23, 2018 at 15:43
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    $\begingroup$ Thomae's formula (Mumford, Tata Lectures on Theta II) relates the cross-ratios of the branch points $z_1,...,z_5,\infty$ to fourth powers of the theta-nulls. However, anything pulled back from $\mathcal A_2[2]$ will be $G$-invariant, so finding the individual $z_i-z_j$ seems to me to be excluded. $\endgroup$
    – inkspot
    Apr 23, 2018 at 17:40
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    $\begingroup$ @inkspot Very nice! A remark is that taking dlog of a cross-ratio gives a sum of four "Arnold forms" $w_{ij}$. Moreover the dlog's of cross-ratios can be used to give an alternative presentation of the cohomology ring of $M_{0,n} = C_{n-1}/G$, analogous to Arnolds presentation of the cohomology ring of $C_n$. See Section 2 of my joint paper with Johan Alm arxiv.org/abs/1509.09274 $\endgroup$ Apr 24, 2018 at 8:50


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