A Torelli type theorem for exp-algebraic curves

Abstract

An exp-algebraic curve consists of a compact Riemann surface $S$ together with $n$ equivalence classes of germs of meromorphic functions modulo germs of holomorphic functions, $\HH = \{ [h_1], \cdots, [h_n] \}$, with poles of orders $d_1,\cdots ,d_n\ge 1$ at points $p_1,\cdots ,p_n$. This data determines a space of functions $\OO_{\HH}$ (respectively, a space of $1$-forms $Ω^0_{\HH}$) holomorphic on the punctured surface $S^{\prime }=S-\{p_1,\cdots ,p_n\}$ with exponential singularities at the points $p_1,\cdots ,p_n$ of types $[h_1],\cdots ,[h_n]$, i.e., near $p_i$ any $f \in \OO_{\HH}$ is of the form $f=g{e}^{h_i}$ for some germ of meromorphic function $g$ (respectively, any $ω\in Ω^0_{\HH}$ is of the form $\omega =\alpha e^{h_i}$ for some germ of meromorphic $1$-form). For any $ω\in Ω^0_{\HH}$ the completion of $S^{\prime }$ with respect to the flat metric $|\omega |$ gives a space $S^* = S' \cup \RR$ obtained by adding a finite set $\RR$ of ${\sum }_i{d}_i$ points, and it is known that integration along curves produces a nondegenerate pairing of the relative homology $H_1(S^*, \RR ; \C)$ with the deRham cohomology group defined by $H^1_{dR}(S, \HH) := Ω^0_{\HH}/d\OO_{\HH}$. There is a degree zero line bundle $L_{\HH}$ associated to an exp-algebraic curve, with a natural isomorphism between $Ω^0_{\HH}$ and the space $W_{\HH}$ of meromorphic $L_{\HH}$-valued $1$-forms which are holomorphic on $S^{\prime }$, so that $H_1(S^*, \RR ; \C)$ maps to a subspace $K_{\HH} \subset W^*_{\HH}$. We show that the exp-algebraic curve $(S, \HH)$ is determined uniquely by the pair $(L_{\HH},\, K_{\HH} \subset W^*_{\HH})$.

arXiv admin note: substantial text overlap with arXiv:1602.08219