forgot password?
register here
Research articles
  search articles
  reviews guidelines
  articles index
My Pages
my alerts
  my messages
  my reviews
  my favorites
Members: 3158
Articles: 2'154'691
Articles rated: 2589

18 January 2022
  » arxiv » 1606.6449

 Article overview

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
Services Forum | Review | PDF | Favorites   
Visitor rating: did you like this article? no 1   2   3   4   5   yes

No review found.
 Did you like this article?

This article or document is ...
of broad interest:
Global appreciation:

  Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.

browser CCBot/2.0 (
» my Online CV
» Free

News, job offers and information for researchers and scientists:
home  |  contact  |  terms of use  |  sitemap
Copyright © 2005-2022 - Scimetrica