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

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, $ℋ=\left\{\left[{h}_{1}\right],\cdots ,\left[{h}_{n}\right]\right\}$, with poles of orders ${d}_{1},\cdots ,{d}_{n}\ge 1$ at points ${p}_{1},\cdots ,{p}_{n}$. This data determines a space of functions ${𝒪}_{ℋ}$ (respectively, a space of $1$-forms ${\Omega }_{ℋ}^{0}$) holomorphic on the punctured surface ${S}^{\prime }=S-\left\{{p}_{1},\cdots ,{p}_{n}\right\}$ with exponential singularities at the points ${p}_{1},\cdots ,{p}_{n}$ of types $\left[{h}_{1}\right],\cdots ,\left[{h}_{n}\right]$, 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 $\omega \in {\Omega }_{ℋ}^{0}$ is of the form $\omega =\alpha {e}^{{h}_{i}}$ for some germ of meromorphic $1$-form).

For any $\omega \in {\Omega }_{ℋ}^{0}$ the completion of ${S}^{\prime }$ with respect to the flat metric $|\omega |$ gives a space ${S}^{*}={S}^{\prime }\cup ℛ$ 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 ${H}_{1}\left({S}^{*},ℛ;ℂ\right)$ with the de Rham cohomology group defined by ${H}_{\mathrm{d}R}^{1}\left(S,ℋ\right):={\Omega }_{ℋ}^{0}/d{𝒪}_{ℋ}$.

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

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 $ℋ=\left\{\left[{h}_{1}\right],\cdots ,\left[{h}_{n}\right]\right\}$, avec des pôles d’ordre ${d}_{1},\cdots ,{d}_{n}\ge 1$ aux points ${p}_{1},\cdots ,{p}_{n}$. Cette donnée détermine un espace de fonctions ${𝒪}_{ℋ}$ (respectivement, un espace de $1$-formes ${\Omega }_{ℋ}^{0}$) holomorphes sur la surface épointée ${S}^{\prime }=S-\left\{{p}_{1},\cdots ,{p}_{n}\right\}$ avec des singularités exponentielles aux points ${p}_{1},\cdots ,{p}_{n}$ de type $\left[{h}_{1}\right],\cdots ,\left[{h}_{n}\right]$, 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 $\omega \in {\Omega }_{ℋ}^{0}$ est de la forme $\omega =\alpha {e}^{{h}_{i}}$ pour un germe de $1$-forme méromorphe $\alpha$.

Pour toute $\omega \in {\Omega }_{ℋ}^{0}$ la complétion de ${S}^{\prime }$ par rapport à la métrique plate $|\omega |$ donne un espace ${S}^{*}={S}^{\prime }\cup ℛ$ 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 ${H}_{1}\left({S}^{*},ℛ;ℂ\right)$ où le groupe de cohomologie de de Rham est défini par ${H}_{\mathrm{d}R}^{1}\left(S,ℋ\right):={\Omega }_{ℋ}^{0}/d{𝒪}_{ℋ}$.

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

Accepted:
Published online:
DOI: 10.5802/afst.1634
Classification: 30F30, 34M03
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)
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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, Serie 6, Volume 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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