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A Torelli type theorem for expalgebraic curves  Indranil Biswas
; Kingshook Biswas
;  Date: 
21 Jun 2016  Abstract:  An expalgebraic 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 expalgebraic
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 expalgebraic 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 


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