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a choice of nonconstant meromorphic function @xmath28 on a compact riemann surface @xmath0 realizes @xmath0 as a finite sheeted branched covering of the riemann sphere @xmath29 . _ log - riemann surfaces of finite type _ are certain branched coverings , in a generalized sense , of @xmath30 by a punctured compact riemann surface , namely , which are given by certain transcendental functions of infinite degree . formally a log - riemann surface consists of a riemann surface together with a local holomorphic diffeomorphism @xmath31 from the surface to @xmath30 such that the set of points @xmath19 added to the surface , in the completion @xmath18 with respect to the path - metric induced by the flat metric @xmath32 , is discrete . log - riemann surfaces were defined and studied in @xcite ( see also @xcite ) , where it was shown that the map @xmath31 restricted to any small enough punctured metric neighbourhood of a point @xmath33 in @xmath19 gives a covering of a punctured disc in @xmath30 , and is thus equivalent to either @xmath34 restricted to a punctured disc @xmath35 ( in which case we say @xmath33 is a ramification point of order @xmath1 ) or to @xmath36 restricted to a half - plane @xmath37 ( in which case we say @xmath33 is a ramification point of infinite order ) .    a log - riemann surface is said to be of finite type if it has finitely many ramification points and finitely generated fundamental group . we will only consider those for which the set of infinite order ramification points is nonempty ( otherwise the map @xmath31 has finite degree and is given by a meromorphic function on a compact riemann surface ) . in @xcite , @xcite , uniformization theorems were proved for log - riemann surfaces of finite type , which imply that a log - riemann surface of finite type is given by a pair @xmath38 , where @xmath0 is a compact riemann surface , and @xmath31 is a meromorphic function on the punctured surface @xmath16 such that the differential @xmath39 has essential singularities at the punctures of a specific type , namely _ exponential singularities_.    given a germ of meromorphic function @xmath40 at a point @xmath41 of a riemann surface , a function @xmath42 with an isolated singularity at @xmath41 is said to have an exponential singularity of type @xmath40 at @xmath41 if locally @xmath43 for some germ of meromorphic function @xmath13 at @xmath41 , while a 1-form @xmath44 is said to have an exponential singularity of type @xmath40 at @xmath41 if locally @xmath45 for some germ of meromorphic 1-form @xmath46 at @xmath41 . note that the spaces of germs of functions and 1-forms with exponential singularity of type @xmath40 at @xmath41 only depend on the equivalence class @xmath47 $ ] in the space @xmath48 of germs of meromorphic functions at @xmath41 modulo germs of holomorphic functions at @xmath41 . thus the uniformization theorems of @xcite , @xcite give us @xmath1 germs of meromorphic functions @xmath49 at the punctures @xmath50 , with poles of orders @xmath3 say , such that near a puncture @xmath51 the map @xmath31 is of the form @xmath52 , where @xmath53 is a germ of meromorphic function near @xmath51 and @xmath28 a local coordinate near @xmath51 . the punctures correspond to ends of the log - riemann surface , where at each puncture @xmath51 , @xmath54 infinite order ramification points are added in the metric completion , so that the total number of infinite order ramification points is @xmath55 . the @xmath54 infinite order ramification points added at a puncture @xmath51 correspond to the @xmath54 directions of approach to the puncture along which @xmath56 so that @xmath57 decays exponentially and @xmath58 converges . in the case of genus zero with one puncture for example , which is considered in @xcite , @xmath31 must have the form @xmath59 where @xmath60 is a rational function and @xmath61 is a polynomial of degree equal to the number of infinite order ramification points .    in @xcite , certain spaces of functions and @xmath6-forms on a log - riemann surface @xmath62 of finite type were defined , giving rise to a derham cohomology group @xmath63 . the integrals of the @xmath6-forms considered along curves in @xmath62 joining the infinite ramification points converge , giving rise to a pairing between @xmath63 and @xmath21 , which was shown to be nondegenerate ( @xcite ) . the spaces of functions and @xmath6-forms defined were observed to depend only on the types @xmath49 of the exponential singularities , and so a notion less rigid than that of a log - riemann surface was defined , namely the notion of an _ exp - algebraic curve _ , which consists of a compact riemann surface @xmath0 together with @xmath1 equivalence classes of germs of meromorphic functions modulo germs of holomorphic functions , @xmath2 , \cdots , [ h_n ] \}$ ] , with poles of orders @xmath3 at points @xmath4 . the relevant spaces of functions and @xmath6-forms with exponential singularities at @xmath4 of types @xmath9 , \cdots , [ h_n]$ ] can then be defined as follows : @xmath64 , \cdots , [ h_n ] \ } \\ { { \mathcal o}}_{{{\mathcal h } } } : = & \ { f \in { { \mathcal m}}_{{{\mathcal h } } } \,\mid\ , f \hbox { holomorphic on } s ' \ } \\ \omega_{{{\mathcal h } } }   : = & \ { \omega \,\mid\ , \omega \hbox { meromorphic 1-form on } s ' \hbox { with exponential singularities}\\ & \hbox {    of types } [ h_1 ] , \cdots , [ h_n ] \ } \\ \omega^0_{{{\mathcal h } } } : = & \ { \omega \in \omega_{{{\mathcal h } } } \,\mid\ , \omega \hbox { holomorphic on } s ' \}.\end{aligned}\ ] ]    for @xmath65 ( respectively , @xmath66 ) we can define a divisor @xmath67 ( respectively , @xmath68 ) by @xmath69 if @xmath70 and @xmath71 if @xmath72 , where @xmath13 is a germ of meromorphic function at @xmath10 such that @xmath12 ( respectively , @xmath73 if @xmath70 and @xmath74 if @xmath72 , where @xmath46 is a germ of meromorphic @xmath6-form at @xmath10 such that @xmath15 ) .    in @xcite it is shown how to naturally associate to an exp - algebraic curve @xmath75 a degree zero line bundle @xmath76 together with a meromorphic connection @xmath77 with poles at @xmath4 . the connection @xmath6-form of @xmath78 near @xmath10 is given ( with respect to an appropriate local trivialization ) by @xmath79 , so that the pair @xmath80 determines the exp - algebraic curve @xmath81 . there are naturally defined isomorphisms between the space of meromorphic sections of @xmath23 ( respectively , the space of meromorphic @xmath23-valued @xmath6-forms ) and @xmath82 ( respectively , @xmath83 ) , such that a meromorphic section @xmath84 of @xmath23 ( respectively , a meromorphic @xmath23-valued @xmath6-form @xmath46 ) maps to an @xmath65 with the same divisor as @xmath84 ( respectively , an @xmath66 with the same divisor as @xmath46 ) .    in particular the space @xmath24 of meromorphic @xmath23-valued @xmath6-forms which are holomorphic on @xmath16 is naturally isomorphic to the space @xmath7 . fixing an @xmath11 inducing a log - riemann surface structure on @xmath0 , with completion @xmath18 , the @xmath6-forms in @xmath7 can be integrated along curves in @xmath21 , giving a map @xmath85 let @xmath25 denote the image of @xmath21 in @xmath86 . then our torelli - type theorem for exp - algebraic curves states that the pair @xmath87 determines the exp - algebraic curve @xmath88 :    [ mainthm ] let @xmath89 be two exp - algebraic curves with the same underlying riemann surface @xmath0 , and the same set of punctures @xmath4 . suppose that @xmath90 is nontrivial , that the line bundles @xmath91 are isomorphic and that the induced isomorphism @xmath92 maps @xmath93 to @xmath94 . then @xmath95 . finally , we mention briefly some appearances of functions with exponential singularities in the literature . certain functions with exponential singularities , namely the @xmath1-point _ baker - akhiezer functions _ ( @xcite , @xcite ) , have been used in the algebro - geometric integration of integrable systems ( see , for example , @xcite , @xcite and the surveys @xcite , @xcite , @xcite , @xcite ) . given a divisor @xmath96 on @xmath16 , an @xmath1-point baker - akhiezer function ( with respect to the data @xmath97 ) is a function @xmath42 in the space @xmath98 satisfying the additional properties that the divisor @xmath99 of zeroes and poles of @xmath42 on @xmath16 satisfies @xmath100 , and that @xmath101 is holomorphic at @xmath51 for all @xmath102 . for @xmath96 a non - special divisor of degree at least @xmath13 , the space of such baker - akhiezer functions is known to have dimension @xmath103 . functions and differentials with exponential singularities on compact riemann surfaces have also been studied by cutillas ripoll ( @xcite , @xcite , @xcite ) , where they arise naturally in the solution of the _ weierstrass problem _ of realizing arbitrary divisors on compact riemann surfaces , and by taniguchi ( @xcite , @xcite ) , where entire functions satisfying certain topological conditions ( called  structural finiteness " ) are shown to be precisely those entire functions whose derivatives have an exponential singularity at @xmath104 , namely functions of the form @xmath105 , where @xmath106 are polynomials . * acknowledgements . * this work grew out of a visit of the second author to tifr , mumbai . he would like to thank tifr for its hospitality . we recall some basic definitions and facts from @xcite , @xcite , @xcite . a log - riemann surface is a pair @xmath107 , where @xmath0 is a riemann surface and @xmath108 is a local holomorphic diffeomorphism such that the set of points @xmath19 added to @xmath0 in the completion @xmath109 with respect to the path metric induced by the flat metric @xmath32 is discrete . the map @xmath31 extends to the metric completion @xmath62 as a @xmath6-lipschitz map . in @xcite it is shown that the map @xmath31 restricted to a sufficiently small punctured metric neighbourhood @xmath110 of a ramification point is a covering of a punctured disc @xmath111 in @xmath30 , and so has a well - defined degree @xmath112 , called the order of the ramification point ( we assume that the order is always at least @xmath113 , since order one points can always be added to @xmath0 and @xmath31 extended to these points in order to obtain a log - riemann surface ) . a log - riemann surface is of finite type if it has finitely many ramification points and finitely generated fundamental group . for example , the log - riemann surface given by @xmath114 is of finite type ( with the metric @xmath32 it is isometric to the riemann surface of the logarithm , which is simply connected , with a single ramification point of infinite order ) , as is the log - riemann surface given by the gaussian integral @xmath115 , which has two ramification points , both of infinite order , as in the figure below :    log - riemann surface of the gaussian integral    in @xcite , it is shown that a log - riemann surface of finite type ( which has at least one infinite order ramification point ) is of the form @xmath116 , where @xmath16 is a punctured compact riemann surface @xmath8 and @xmath31 is meromorphic on @xmath16 and @xmath39 has exponential singularities at the punctures @xmath4 . let @xmath49 be the types of the exponential singularities of @xmath39 at the punctures @xmath4 . as described in @xcite , each puncture @xmath51 corresponds to an end of the log - riemann surface where @xmath54 infinite order ramification points are added , @xmath54 being the order of the pole of @xmath117 at @xmath51 . let @xmath33 be an infinite order ramification point associated to a puncture @xmath51 . an @xmath118-ball @xmath119 around @xmath33 is isometric to the @xmath118-ball around the infinite order ramification point of the riemann surface of the logarithm ( given by cutting and pasting infinitely many discs together ) , and there is an argument function @xmath120 defined on the punctured ball @xmath121 . while the function @xmath31 , which is of the form @xmath122 in a punctured neighbourhood of @xmath51 for some meromorphic @xmath6-form @xmath123 , extends continuously to @xmath33 for the metric topology on @xmath62 , in general functions of the form @xmath124 ( where @xmath46 is a @xmath6-form meromorphic near @xmath51 ) do * not * extend continuously to @xmath33 for the metric topology ( @xcite ) . limits of these functions in sectors @xmath125 do exist however and are independent of the sector ; we say that the function is _ stolz continuous _ at points of @xmath19 . define spaces of functions and @xmath6-forms on @xmath62 : @xmath126    we remark that these are simply the spaces @xmath127 defined in the introduction , where @xmath2 , \cdots , [ h_n ] \}$ ] . functions in @xmath98 are stolz continuous at points of @xmath19 taking the value @xmath128 there . the integrals of @xmath6-forms @xmath44 in @xmath129 over curves @xmath130 \to s^*$ ] joining points @xmath131 of @xmath19 converge if @xmath132 is disjoint from the poles of @xmath44 and tends to these points through sectors @xmath133 ( since any primitive of @xmath44 on a sector is stolz continuous ) . the definitions of the above spaces only depend on the types @xmath134 \in { { \mathcal m}}_{p_i}/{{\mathcal o}}_{p_i } \}$ ] of the exponential singularities of the @xmath6-form @xmath39 , which do not change if @xmath39 is multiplied by a meromorphic function . it is natural to define then a structure less rigid than that of a log - riemann surface of finite type .    given a punctured compact riemann surface @xmath8 , two meromorphic functions @xmath135 on @xmath16 inducing log - riemann surface structures of finite type are considered equivalent if @xmath136 is meromorphic on the compact surface @xmath0 . an exp - algebraic curve is an equivalence class of such log - riemann surface structures of finite type . it follows from the uniformization theorem of @xcite that an exp - algebraic curve is given by the data of a punctured compact riemann surface and @xmath1 ( equivalence classes of ) germs of meromorphic functions @xmath137 \in { { \mathcal m}}_{p_i}/{{\mathcal o}}_{p_i } \}$ ] with poles at the punctures . we can associate a topological space @xmath138 to an exp - algebraic curve , given as a set by @xmath139 , where @xmath19 is the set of infinite ramification points added with respect to any map @xmath31 in the equivalence class of log - riemann surfaces of finite type , and the topology is the weakest topology such that all maps @xmath140 in the equivalence class extend continuously to @xmath138 .    finally , for a meromorphic function @xmath42 on @xmath16 ( respectively , meromorphic @xmath6-form @xmath44 on @xmath16 ) with exponential singularities of types @xmath49 at points @xmath4 we can define a divisor @xmath67 ( respectively , @xmath68 ) by @xmath69 if @xmath70 and @xmath71 if @xmath72 , where @xmath13 is a germ of meromorphic function at @xmath10 such that @xmath12 ( respectively , @xmath73 if @xmath70 and @xmath74 if @xmath72 , where @xmath46 is a germ of meromorphic @xmath6-form at @xmath10 such that @xmath15 ) . note that the divisor @xmath99 can also be defined by @xmath141 , so it follows from the residue theorem applied to the meromorphic @xmath6-form @xmath142 that the divisor @xmath99 has degree zero . let @xmath143 be an exp - algebraic curve , where @xmath0 is a compact riemann surface of genus @xmath13 and @xmath49 are germs of meromorphic functions at points @xmath4 . let @xmath144 be the space of holomorphic @xmath6-forms on @xmath0 . the data @xmath145 defines a degree zero line bundle @xmath76 together with a transcendental section @xmath146 of this line bundle which is non - zero on the punctured surface @xmath16 as follows :    solving the mittag - leffler problem locally for the distribution @xmath147 gives meromorphic functions on an open cover such that the differences are holomorphic on intersections , and hence gives an element of @xmath148 . under the exponential this gives a degree zero line bundle as an element of @xmath149 . explicitly this is constructed as follows :    let @xmath150 be pairwise disjoint coordinate disks around the punctures @xmath4 and let @xmath151 be an open subset of @xmath16 intersecting each disk @xmath152 in an annulus @xmath153 around @xmath10 such that @xmath154 is an open cover of @xmath0 . define a line bundle @xmath76 by taking the functions @xmath155 to be the transition functions for the line bundle on the intersections @xmath156 . define a holomorphic non - vanishing section of @xmath76 on @xmath16 by :    @xmath157    define a connection @xmath77 on @xmath76 by declaring that @xmath158 . then for any holomorphic section @xmath84 on @xmath151 , @xmath159 for some holomorphic function @xmath42 , and @xmath160 , so @xmath77 is holomorphic on @xmath151 . on each disk @xmath152 , letting @xmath161 be the section which is constant equal to @xmath6 on @xmath152 ( with respect to the trivialization on @xmath152 ) , for any holomorphic section @xmath84 on @xmath152 , @xmath162 for some holomorphic function @xmath42 , and @xmath163 , so @xmath164 thus the connection @xmath6-form of @xmath77 with respect to @xmath161 is given by @xmath79 , so @xmath77 is meromorphic on @xmath152 with a single pole at @xmath10 of order @xmath165 . let @xmath166 be the unique section of the dual bundle @xmath167 on @xmath16 such that @xmath168 on @xmath16 . then for any non - zero meromorphic section @xmath84 of @xmath23 , the function @xmath169 is meromorphic on @xmath16 with exponential singularities at @xmath4 of types @xmath49 , and the divisors of @xmath84 and @xmath42 coincide . thus the line bundle @xmath76 has degree zero . in summary we have :    [ existence ] for any log - riemann surface of finite type @xmath62 , the line bundle @xmath76 has degree zero and the maps @xmath170 ( respectively , @xmath171 ) are mutually inverse isomorphisms between the spaces of meromorphic sections of @xmath23 and @xmath98 ( respectively , the spaces of meromorphic @xmath23-valued @xmath6-forms and @xmath172 ) preserving divisors .    in particular the vector spaces @xmath173 are non - zero . since the isomorphisms above preserve divisors , the spaces @xmath174 correspond to the spaces of meromorphic sections of @xmath23 and meromorphic @xmath23-valued @xmath6-forms which are holomorphic on @xmath16 , both of which are non - empty . the correspondence @xmath175 gives a one - to - one correspondence between exp - algebraic structures on @xmath0 and degree zero line bundles on @xmath0 with meromorphic connections with all poles of order at least two , zero residues , and trivial monodromy . since the connection @xmath6-form of @xmath78 is given by @xmath79 on @xmath152 , all residues of @xmath78 are equal to zero , while the monodromy of @xmath78 is trivial since @xmath176 is a single - valued horizontal section . conversely , given such a meromorphic connection @xmath177 on a degree zero line bundle @xmath178 , if @xmath4 are the poles of @xmath177 and @xmath179 are the connection @xmath6-forms of @xmath177 with respect to trivializations near @xmath4 , then each @xmath180 has zero residue at @xmath10 and pole order at least two , hence there exist meromorphic germs @xmath49 near @xmath4 such that @xmath181 . we obtain an exp - algebraic curve @xmath182 . it is clear for an exp - algebraic curve @xmath26 that @xmath183 , so the correspondences are inverses of each other .    finally we remark that by serre duality , the degree zero line bundle @xmath76 , given as an element of @xmath148 , can also be described as an element of @xmath184 using residues , as the linear functional @xmath185 we proceed to the proof of theorem [ mainthm ] . we will need the following theorems from @xcite and @xcite :      [ approxn ] let @xmath0 be a compact riemann surface and @xmath188 a closed subset such that @xmath189 has finitely many connected components @xmath190 , and for each @xmath191 let @xmath192 be a point of @xmath193 . then any continuous function @xmath42 on @xmath194 which is holomorphic in the interior of @xmath194 can be uniformly approximated on @xmath194 by functions meromorphic on @xmath0 with poles only in the set @xmath195 . let @xmath196 be two exp - algebraic curves with the same underlying riemann surface @xmath0 and the same set of punctures @xmath4 , and suppose the hypothesis of theorem 1.1 holds , namely the line bundles @xmath91 are isomorphic and the induced isomorphism @xmath197 maps @xmath93 to @xmath94 . since the spaces @xmath198 are isomorphic to the spaces of meromorphic sections of @xmath199 and @xmath200 respectively , there is an induced isomorphism @xmath201 which preserves divisors . we fix non - zero functions @xmath202 which correspond to each other under this isomorphism , and let @xmath203 denote the completions induced by the corresponding log - riemann surface structures . we also fix a meromorphic @xmath6-form @xmath204 on @xmath0 . then the divisor preserving isomorphisms @xmath201 and @xmath205 induced by the isomorphism @xmath206 can be expressed as @xmath207 respectively , where @xmath13 varies over all meromorphic functions on @xmath0 . the hypothesis of theorem [ mainthm ] implies that for any @xmath212 , there is a @xmath213 such that @xmath214 for all meromorphic functions @xmath13 on @xmath0 such that @xmath215 is holomorphic on @xmath16 for @xmath216 . if @xmath208 is a meromorphic function on @xmath0 such that @xmath217 is holomorphic on @xmath16 , then @xmath218 for some meromorphic function @xmath13 on @xmath0 such that @xmath219 is holomorphic on @xmath16 . since the isomorphism @xmath220 is divisor preserving , we have that @xmath221 is also holomorphic on @xmath16 , so for any @xmath212 there is a @xmath222 such that @xmath223 since this is true for all @xmath212 , it follows from theorem [ pairing ] that @xmath224 , so there exists a meromorphic function @xmath225 on @xmath0 such that @xmath226 is holomorphic on @xmath16 and @xmath227 . in this case there exists a closed curve @xmath132 disjoint from the punctures @xmath4 and the poles and zeroes of @xmath235 such that @xmath236 is connected . fix a non - zero meromorphic function @xmath237 on @xmath0 such that @xmath238 ( and hence also @xmath239 ) is holomorphic on @xmath16 . if the meromorphic @xmath6-form @xmath240 ( which is holomorphic outside the punctures and the zeroes and poles of @xmath235 ) is not identically zero , then we can choose a continuous function @xmath241 on @xmath132 such that @xmath242 ( since the @xmath6-form @xmath243 is holomorphic and not identically zero on @xmath132 ) . by theorem [ approxn ] , since @xmath236 is connected and contains @xmath244 , we can choose a meromorphic function @xmath245 on @xmath0 which is holomorphic on @xmath246 and uniformly close enough to @xmath241 on @xmath132 such that @xmath247 . letting @xmath248 , we have that @xmath217 is holomorphic on @xmath16 and @xmath249 , contradicting lemma [ exact ] . it follows that @xmath250 , from which it follows easily that @xmath95 .      in this case @xmath251 and we may assume @xmath252 . fix a non - zero polynomial @xmath61 such that @xmath253 are holomorphic on @xmath16 . then by lemma [ exact ] , for all @xmath254 , taking @xmath255 we have @xmath256 from which it follows that the laurent series of @xmath257 around @xmath258 vanishes identically , hence @xmath259 and @xmath95 . in this case @xmath251 and we may assume the single puncture @xmath260 , and that the functions @xmath235 are of the form @xmath261 for some polynomials @xmath262 . in this case it follows from the main theorem of @xcite that the dimension of @xmath263 equals @xmath264 . since @xmath93 and @xmath94 are isomorphic by hypothesis , it follows that @xmath265 say , where @xmath266 since @xmath267 is non - trivial . let @xmath268 . let @xmath269 be a basis for @xmath267 as described in section 4 of @xcite , each @xmath270 being a curve joining a pair of ramification points @xmath271 , where @xmath272 . by hypothesis , for each curve @xmath273 there is a @xmath274 such that @xmath275 for all polynomials @xmath276 . consider the @xmath277 matrix @xmath278 it follows from theorem iii.1.5.1 of @xcite that the @xmath279 @xmath6-forms @xmath280 span @xmath281 , and hence form a basis for @xmath281 . since by theorem [ pairing ] the pairing with @xmath267 is nondegenerate , it follows that the @xmath282 submatrix formed by the first @xmath279 columns of @xmath283 is nonsingular , thus @xmath283 has rank @xmath279 . on the other hand , since @xmath284 , it follows that @xmath285 hence there is a scalar @xmath286 such that @xmath287 , so @xmath288 . it follows from lemma [ exact ] that for any polynomial @xmath276 the @xmath6-form @xmath289 lies in @xmath211 . thus if @xmath290 , then @xmath291 , hence @xmath292 for all polynomials @xmath276 . since all @xmath6-forms in @xmath293 are of the form @xmath294 for some polynomial @xmath61 and any @xmath295 for some polynomial @xmath276 , it follows that @xmath296 is trivial , a contradiction . thus @xmath297 , so @xmath298 and hence @xmath95 .
an exp - algebraic curve consists of a compact riemann surface @xmath0 together with @xmath1 equivalence classes of germs of meromorphic functions modulo germs of holomorphic functions , @xmath2 , \cdots , [ h_n ] \}$ ] , with poles of orders @xmath3 at points @xmath4 . 
 this data determines a space of functions @xmath5 ( respectively , a space of @xmath6-forms @xmath7 ) holomorphic on the punctured surface @xmath8 with exponential singularities at the points @xmath4 of types @xmath9 , \cdots , [ h_n]$ ] , i.e. , near @xmath10 any @xmath11 is of the form @xmath12 for some germ of meromorphic function @xmath13 ( respectively , any @xmath14 is of the form @xmath15 for some germ of meromorphic @xmath6-form ) .    for any @xmath14 
 the completion of @xmath16 with respect to the flat metric @xmath17 gives a space @xmath18 obtained by adding a finite set @xmath19 of @xmath20 points , and it is known that integration along curves produces a nondegenerate pairing of the relative homology @xmath21 with the derham cohomology group defined by @xmath22 . 
 there is a degree zero line bundle @xmath23 associated to an exp - algebraic curve , with a natural isomorphism between @xmath7 and the space @xmath24 of meromorphic @xmath23-valued @xmath6-forms which are holomorphic on @xmath16 , so that @xmath21 maps to a subspace @xmath25 . 
 we show that the exp - algebraic curve @xmath26 is determined uniquely by the pair @xmath27 . 
 [ section ] [ theorem]proposition [ theorem]lemma [ theorem]corollary    [ theorem]definition