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18 January 2022
 
  » arxiv » 1606.6449

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A Torelli type theorem for exp-algebraic curves
Indranil Biswas ; Kingshook Biswas ;
Date 21 Jun 2016
AbstractAn 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 $H^1_{dR}(S, HH) := Omega^0_{HH}/dOO_{HH}$.
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 $(L_{HH},, K_{HH} subset W^*_{HH})$.
Source arXiv, 1606.6449
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