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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@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/}
}

TY  - JOUR
AU  - Indranil Biswas
AU  - Kingshook Biswas
TI  - A Torelli type theorem for exp-algebraic curves
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2020
SP  - 357
EP  - 370
VL  - 29
IS  - 2
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1634/
DO  - 10.5802/afst.1634
LA  - en
ID  - AFST_2020_6_29_2_357_0
ER  -

%0 Journal Article
%A Indranil Biswas
%A Kingshook Biswas
%T A Torelli type theorem for exp-algebraic curves
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2020
%P 357-370
%V 29
%N 2
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1634/
%R 10.5802/afst.1634
%G en
%F AFST_2020_6_29_2_357_0

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/

Bibliographie
Cité par

[1] Naum I. Ahiezer Continuous analogues of orthogonal polynomials on a system of intervals, Dokl. Akad. Nauk SSSR, Volume 141 (1961), pp. 263-266 | MR

[2] Henry F. Baker Note on the foregoing paper “commutative ordinary differential operators” by J. L. Burchnall and T. W. Chaundy, Proc. Royal Soc. London, Volume A118 (1928), pp. 584-593 | Zbl

[3] Kingshook Biswas Algebraic deRham cohomology of log-riemann surfaces of finite type (2016) (https://arxiv.org/abs/1602.08219)

[4] Kingshook Biswas; Ricardo Perez-Marco Uniformization of higher genus finite type log-riemann surfaces (2013) (https://arxiv.org/abs/1305.2339)

[5] Kingshook Biswas; Ricardo Perez-Marco Log-Riemann surfaces, Caratheodory convergence and Euler’s formula, Geometry, groups and dynamics (Contemporary Mathematics), Volume 639, American Mathematical Society, 2015, pp. 197-203 | MR | Zbl

[6] Kingshook Biswas; Ricardo Perez-Marco Log-Riemann surfaces, Carathéodory convergence and Euler’s formula, Geometry, groups and dynamics. ICTS program: groups, geometry and dynamics, Almora, India, December 3–16, 2012 (Contemporary Mathematics), Volume 639, American Mathematical Society, 2015, pp. 197-203 | Zbl

[7] Kingshook Biswas; Ricardo Perez-Marco Uniformization of simply connected finite type log-Riemann surfaces, Geometry, groups and dynamics (Contemporary Mathematics), Volume 639, American Mathematical Society, 2015, pp. 205-216 | MR | Zbl

[8] Pascual Cutillas Ripoll Construction of certain function fields associated with a compact Riemann surface, Am. J. Math., Volume 106 (1984) no. 6, pp. 1423-1450 | MR | Zbl

[9] Pascual Cutillas Ripoll On ramification divisors of functions in a punctured compact Riemann surface, Publ. Mat., Barc., Volume 33 (1989) no. 1, pp. 163-171 | DOI | MR | Zbl

[10] Pascual Cutillas Ripoll On the immersions between certain function fields on punctured compact Riemann surfaces, Arch. Math., Volume 54 (1990) no. 3, pp. 304-306 | MR | Zbl

[11] Boris A. Dubrovin Theta functions and non-linear equations, Usp. Mat. Nauk, Volume 36 (1981) no. 6, pp. 11-80 | Zbl

[12] Boris A. Dubrovin; Igor M. Krichever; Sergei P. Novikov Integrable systems. I, Current problems in mathematics. Fundamental directions, Vol. 4 (Itogi Nauki i Tekhniki), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., 1985, p. 179-284, 291 | Zbl

[13] S. Y. Gusman The uniform approximation of continuous functions on compact riemann surfaces, Izv. Vyssh. Uchebn. Zaved., Mat., Volume 5 (1960), pp. 43-51 | MR | Zbl

[14] Igor M. Krichever An algebraic-geometrical construction of the zakharov-shabat equation and their periodic solutions, Dokl. Akad. Nauk SSSR, Volume 227 (1976), pp. 291-294

[15] Igor M. Krichever Integration of nonlinear equations by the methods of algebraic geometry, Funct. Anal. Appl., Volume 11 (1977) no. 1, pp. 12-26 | DOI | MR | Zbl

[16] Igor M. Krichever Methods of algebraic geometry in the theory of nonlinear equations, Usp. Mat. Nauk, Volume 32 (1977) no. 6, pp. 185-213 | MR | Zbl

[17] Igor M. Krichever; Sergei P. Novikov Holomorphic bundles over algebraic curves and non-linear equations, Usp. Mat. Nauk, Volume 35 (1980) no. 6, pp. 47-68

[18] Masahiko Taniguchi Explicit representation of structurally finite entire functions, Proc. Japan Acad., Ser. A, Volume 77 (2001) no. 4, pp. 68-70 | DOI | MR | Zbl

[19] Masahiko Taniguchi Synthetic deformation space of an entire function, Value distribution theory and complex dynamics (Hong Kong, 2000) (Contemporary Mathematics), Volume 303, American Mathematical Society, 2002, pp. 107-136 | DOI | MR | Zbl

Cité par Sources :