A Torelli type theorem for exp-algebraic curves

Indranil Biswas; Kingshook Biswas

doi:10.5802/afst.1634

Indranil Biswas ¹ ; Kingshook Biswas ²

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005 (India)
² Indian Statistical Institute, Stat-Math Unit, 203, Barrackpore Trunk Road, Baranagar, Kolkata, 700108 (India)

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 2, pp. 357-370.

Résumé
Abstract

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 $ℋ = {[h_{1}], \dots, [h_{n}]}$ , avec des pôles d’ordre $d_{1}, \dots, d_{n} \geq 1$ aux points $p_{1}, \dots, p_{n}$ . Cette donnée détermine un espace de fonctions $𝒪_{ℋ}$ (respectivement, un espace de $1$ -formes $Ω_{ℋ}^{0}$ ) holomorphes sur la surface épointée $S^{'} = S - {p_{1}, \dots, p_{n}}$ avec des singularités exponentielles aux points $p_{1}, \dots, p_{n}$ de type $[h_{1}], \dots, [h_{n}]$ , i.e., au voisinage du point $p_{i}$ toute $f \in 𝒪_{ℋ}$ est de la forme $f = g e^{h_{i}}$ pour un germe de fonction méromorphe $g$ (respectivement toute forme $ω \in Ω_{ℋ}^{0}$ est de la forme $ω = α e^{h_{i}}$ pour un germe de $1$ -forme méromorphe $α$ .

Pour toute $ω \in Ω_{ℋ}^{0}$ la complétion de $S^{'}$ par rapport à la métrique plate $| ω |$ donne un espace $S^{*} = S^{'} \cup ℛ$ obtenu en ajoutant un ensemble fini $ℛ$ de $\sum_{i} d_{i}$ points. Il est connu que l’intégration le long des courbes fournit un accouplement non dégénéré sur l’homologie relative $H_{1} (S^{*}, ℛ; ℂ)$ où le groupe de cohomologie de de Rham est défini par $H_{d R}^{1} (S, ℋ) : = Ω_{ℋ}^{0} / d 𝒪_{ℋ}$ .

Il existe un fibré en droites $L_{ℋ}$ de degré zéro associé à toute courbe exp-algébrique, avec un isomorphisme naturel entre $Ω_{ℋ}^{0}$ et l’espace $W_{ℋ}$ des $1$ -formes méromorphes à valeurs dans $L_{ℋ}$ , holomorphes sur $S^{'}$ et tel que $H_{1} (S^{*}, ℛ; ℂ)$ s’envoie sur un sous-espace $K_{ℋ} \subset W_{ℋ}^{*}$ . Nous montrons que la courbe exp-algébrique $(S, ℋ)$ est déterminée de façon univoque par la paire $(L_{ℋ}, K_{ℋ} \subset W_{ℋ}^{*})$ .

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, $ℋ = {[h_{1}], \dots, [h_{n}]}$ , with poles of orders $d_{1}, \dots, d_{n} \geq 1$ at points $p_{1}, \dots, p_{n}$ . This data determines a space of functions $𝒪_{ℋ}$ (respectively, a space of $1$ -forms $Ω_{ℋ}^{0}$ ) holomorphic on the punctured surface $S^{'} = S - {p_{1}, \dots, p_{n}}$ with exponential singularities at the points $p_{1}, \dots, p_{n}$ of types $[h_{1}], \dots, [h_{n}]$ , i.e., near $p_{i}$ any $f \in 𝒪_{ℋ}$ is of the form $f = g e^{h_{i}}$ for some germ of meromorphic function $g$ (respectively, any $ω \in Ω_{ℋ}^{0}$ is of the form $ω = α e^{h_{i}}$ for some germ of meromorphic $1$ -form).

For any $ω \in Ω_{ℋ}^{0}$ the completion of $S^{'}$ with respect to the flat metric $| ω |$ gives a space $S^{*} = S^{'} \cup ℛ$ obtained by adding a finite set $ℛ$ of $\sum_{i} d_{i}$ points, and it is known that integration along curves produces a nondegenerate pairing of the relative homology $H_{1} (S^{*}, ℛ; ℂ)$ with the de Rham cohomology group defined by $H_{d R}^{1} (S, ℋ) : = Ω_{ℋ}^{0} / d 𝒪_{ℋ}$ .

There is a degree zero line bundle $L_{ℋ}$ associated to an exp-algebraic curve, with a natural isomorphism between $Ω_{ℋ}^{0}$ and the space $W_{ℋ}$ of meromorphic $L_{ℋ}$ -valued $1$ -forms which are holomorphic on $S^{'}$ , so that $H_{1} (S^{*}, ℛ; ℂ)$ maps to a subspace $K_{ℋ} \subset W_{ℋ}^{*}$ . We show that the exp-algebraic curve $(S, ℋ)$ is determined uniquely by the pair $(L_{ℋ}, K_{ℋ} \subset W_{ℋ}^{*})$ .

Reçu le : 2016-06-26
Accepté le : 2018-05-14
Publié le : 2020-08-28

DOI : 10.5802/afst.1634

Classification : 30F30, 34M03

Affiliations des auteurs :

Indranil Biswas ¹ ; Kingshook Biswas ²

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005 (India)
² Indian Statistical Institute, Stat-Math Unit, 203, Barrackpore Trunk Road, Baranagar, Kolkata, 700108 (India)

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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},
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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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