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Article Dans Une Revue Journal of Geometry and Physics Année : 2023

Parabolic opers and differential operators

Indranil Biswas
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Niels Borne
Christian Pauly
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Résumé

Parabolic ${\rm SL}(r,\mathbb C)$--opers were defined and investigated in \cite{BDP} in the set-up of vector bundles on curves with a parabolic structure over a divisor. Here we introduce and study holomorphic differential operators between parabolic vector bundles over curves. We consider the parabolic ${\rm SL}(r,\mathbb C)$--opers on a Riemann surface $X$ with given singular divisor $S \,\subset\, X$ and with fixed parabolic weights satisfying the condition that all parabolic weights at any $x_i\, \in\, S$ are integral multiples of $\frac{1}{2N_i+1}$, where $N_i\,>\,1$ are fixed integers. We prove that this space of opers is canonically identified with the affine space of holomorphic differential operators of order $r$ between two natural parabolic line bundles on $X$ (depending only on the divisor $S$ and the weights $N_i$) satisfying the conditions that the principal symbol of the differential operators is the constant function $1$ and the sub-principal symbol vanishes identically. The vanishing of the sub-principal symbol ensures that the logarithmic connection on the rank $r$ bundle is actually a logarithmic ${\rm SL}(r,\mathbb C)$--connection.

Dates et versions

hal-04396497 , version 1 (16-01-2024)

Identifiants

Citer

Indranil Biswas, Niels Borne, Sorin Dumitrescu, Sebastian Heller, Christian Pauly. Parabolic opers and differential operators. Journal of Geometry and Physics, 2023, 187, pp.104791. ⟨10.1016/j.geomphys.2023.104791⟩. ⟨hal-04396497⟩
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