Abstract
We study the singularities of algebraic difference equations on curves from the point of view of equivariant sheaves. We propose a definition for the formal local type of an equivariant sheaf at a point in the case of a reduced curve acted on by a group which is virtually the integers. We show that with this definition, equivariant sheaves can be glued from an “open cover”. Precisely, we show that an equivariant sheaf can be uniquely recovered from the following data: the restriction to the complement of a point, the local type at the point itself, and an isomorphism between the two over the punctured neighborhood of said point. We study symmetric elliptic difference equations (“elliptic equations” from now on) from this point of view. We consider several natural notions for an algebraic version of symmetric elliptic difference equations, i.e. symmetric elliptic difference modules (“elliptic modules”). We show that different versions are not equivalent, but we detail how they are related: all the versions embed fully faithfully into the same category of equivariant sheaves. This implies that we can use the theory for equivariant sheaves to study singularities of elliptic equations as well. One reason to study elliptic equations is that they generalize, and degenerate to, (q-)difference equations (i.e. equivariant sheaves) and differential equations (i.e. D-modules) on the projective line. We discuss this from the elliptic module point of view, which requires studying elliptic modules on singular curves. We study the relation between elliptic modules on singular curves and their normalization. We show that for modules which are flat at the singular points there is an equivalence and we give examples showing that this cannot be improved upon.
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References
Arinkin, D., Borodin, A.: Moduli spaces of \(d\)-connections and difference Painlevé equations. Duke Math. J. 134(3), 515–556 (2006). https://doi.org/10.1215/S0012-7094-06-13433-6
Beauville, A., Laszlo, Y.: Un lemme de descente. C. R. Acad. Sci. Paris Sér. I Math. 320(3), 335–340 (1995)
Deligne, P.: Équations différentielles à points singuliers réguliers. Lecture Notes in Mathematics, vol. 163. Springer-Verlag, Berlin-New York (1970)
Ferrand, D.: Conducteur, descente et pincement. Bulletin de la Société Mathématique de France 131(4), 553–585 (2003). https://doi.org/10.24033/bsmf.2455. http://www.numdam.org/item/BSMF_2003__131_4_553_0
Herradón Cueto, M.: The local information of difference equations. Math. Res. Lett. (2020). Accepted for publication. arXiv:1803.08611
Laumon, G.: Transformation de Fourier generalisee. (1996) arXiv:alg-geom/9603004
Okamoto, K.: Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé. Jpn. J. Math. (N.S.) 5(1), 1–79 (1979)
Rains, E.M.: An isomonodromy interpretation of the hypergeometric solution of the elliptic Painlevé equation (and generalizations). SIGMA Symmetry Integrability Geom. Methods Appl. 7, Paper 088, 24 (2011). https://doi.org/10.3842/SIGMA.2011.088
Sakai, H.: Rational surfaces associated with affine root systems and geometry of the Painlevé equations. Commun. Math. Phys. 220(1), 165–229 (2001). https://doi.org/10.1007/s002200100446
Stacks Project Authors, T.: Stacks Project. http://stacks.math.columbia.edu (2019)
Acknowledgements
I am very grateful to Dima Arinkin for suggesting the problem and for many useful discussions. I also wish to thank Eva Elduque for helpful conversations and comments. This work was partially supported by National Science Foundation grant DMS-1603277. This material is based upon work supported by the National Science Foundation under Grant No. 1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the spring semester of 2019.
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Partially supported by National Science Foundation grant DMS-1603277. This material is based upon work supported by the National Science Foundation under Grant No. 1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the spring semester of 2019.
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Herradón Cueto, M. The local information of equivariant sheaves and elliptic difference equations. RACSAM 116, 2 (2022). https://doi.org/10.1007/s13398-021-01153-w
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DOI: https://doi.org/10.1007/s13398-021-01153-w