A Torelli type theorem for exp-algebraic curves

TL;DR

This paper establishes a Torelli-type theorem showing that exp-algebraic curves are uniquely determined by associated line bundles and subspaces of meromorphic forms, extending classical results to a broader class of Riemann surfaces with exponential singularities.

Contribution

It introduces a Torelli-type theorem for exp-algebraic curves, linking their geometric data to algebraic invariants, which was not previously known.

Findings

Exp-algebraic curves are uniquely determined by their line bundle and subspace of meromorphic forms.

The pairing between relative homology and de Rham cohomology is nondegenerate.

A natural isomorphism exists between certain spaces of forms, enabling the reconstruction of the curve.

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 $d_1, \cdots, d_n \geq 1$ at points $p_1, \cdots, p_n$. This data determines a space of functions $\OO_{\HH}$ (respectively, a space of $1$-forms $\Omega^0_{\HH}$) holomorphic on the punctured surface $S' = S - \{p_1, \cdots, p_n\}$ 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 $f = ge^{h_i}$ for some germ of meromorphic function $g$ (respectively, any $\omega \in \Omega^0_{\HH}$ is of the form $\omega = \alpha e^{h_i}$ for some germ of meromorphic $1$-form). For any $\omega \in \Omega^0_{\HH}$ the completion of $S'$ 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 $H^1_{dR}(S, \HH) := \Omega^0_{\HH}/d\OO_{\HH}$. There is a degree zero line bundle $L_{\HH}$ associated to an exp-algebraic curve, with a natural isomorphism between $\Omega^0_{\HH}$ and the space $W_{\HH}$ of meromorphic $L_{\HH}$-valued $1$-forms which are holomorphic on $S'$, 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 $(L_{\HH},\, K_{\HH} \subset W^*_{\HH})$.