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A Torelli type theorem for exp-algebraic curves
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 2, pp. 357-370.

Une courbe exp-algébrique est une surface de Riemann S munie de n classes d’équivalence de germes de fonctions méromorphes modulo les germes de fonctions holomorphes ={[h 1 ],,[h n ]}, avec des pôles d’ordre d 1 ,,d n 1 aux points p 1 ,,p n . Cette donnée détermine un espace de fonctions 𝒪 (respectivement, un espace de 1-formes Ω 0 ) holomorphes sur la surface épointée S =S-{p 1 ,,p n } avec des singularités exponentielles aux points p 1 ,,p n de type [h 1 ],,[h n ], i.e., au voisinage du point p i toute f𝒪 est de la forme f=ge h i pour un germe de fonction méromorphe g (respectivement toute forme ωΩ 0 est de la forme ω=αe h i pour un germe de 1-forme méromorphe α.

Pour toute ωΩ 0 la complétion de S par rapport à la métrique plate |ω| donne un espace S * =S obtenu en ajoutant un ensemble fini de i d i points. Il est connu que l’intégration le long des courbes fournit un accouplement non dégénéré sur l’homologie relative H 1 (S * ,;) où le groupe de cohomologie de de Rham est défini par H dR 1 (S,):=Ω 0 /d𝒪 .

Il existe un fibré en droites L de degré zéro associé à toute courbe exp-algébrique, avec un isomorphisme naturel entre Ω 0 et l’espace W des 1-formes méromorphes à valeurs dans L , holomorphes sur S et tel que H 1 (S * ,;) s’envoie sur un sous-espace K W * . Nous montrons que la courbe exp-algébrique (S,) est déterminée de façon univoque par la paire (L ,K W * ).

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, ={[h 1 ],,[h n ]}, with poles of orders d 1 ,,d n 1 at points p 1 ,,p n . This data determines a space of functions 𝒪 (respectively, a space of 1-forms Ω 0 ) holomorphic on the punctured surface S =S-{p 1 ,,p n } with exponential singularities at the points p 1 ,,p n of types [h 1 ],,[h n ], i.e., near p i any f𝒪 is of the form f=ge h i for some germ of meromorphic function g (respectively, any ωΩ 0 is of the form ω=αe h i for some germ of meromorphic 1-form).

For any ωΩ 0 the completion of S with respect to the flat metric |ω| gives a space S * =S obtained by adding a finite set of i d i points, and it is known that integration along curves produces a nondegenerate pairing of the relative homology H 1 (S * ,;) with the de Rham cohomology group defined by H dR 1 (S,):=Ω 0 /d𝒪 .

There is a degree zero line bundle L associated to an exp-algebraic curve, with a natural isomorphism between Ω 0 and the space W of meromorphic L -valued 1-forms which are holomorphic on S , so that H 1 (S * ,;) maps to a subspace K W * . We show that the exp-algebraic curve (S,) is determined uniquely by the pair (L ,K W * ).

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DOI : https://doi.org/10.5802/afst.1634
Classification : 30F30,  34M03
@article{AFST_2020_6_29_2_357_0,
     author = {Indranil Biswas and Kingshook Biswas},
     title = {A Torelli type theorem for exp-algebraic curves},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {357--370},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 29},
     number = {2},
     year = {2020},
     doi = {10.5802/afst.1634},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1634/}
}
Indranil Biswas; Kingshook Biswas. A Torelli type theorem for exp-algebraic curves. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 2, pp. 357-370. doi : 10.5802/afst.1634. https://afst.centre-mersenne.org/articles/10.5802/afst.1634/

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