Abelian tropical covers
Y. Len, M. Ulirsch, D. Zakharov, Mathematical Proceedings of the Cambridge Philosophical Society 176 (2023) 395–416.
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Author
Len, Yoav;
Ulirsch, MartinLibreCat;
Zakharov, Dmitry
Abstract
<jats:title>Abstract</jats:title><jats:p>Let<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004123000518_inline1.png"/><jats:tex-math>$\mathfrak{A}$</jats:tex-math></jats:alternatives></jats:inline-formula>be a finite abelian group. In this paper, we classify harmonic<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004123000518_inline2.png"/><jats:tex-math>$\mathfrak{A}$</jats:tex-math></jats:alternatives></jats:inline-formula>-covers of a tropical curve<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004123000518_inline3.png"/><jats:tex-math>$\Gamma$</jats:tex-math></jats:alternatives></jats:inline-formula>(which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004123000518_inline4.png"/><jats:tex-math>$\Gamma$</jats:tex-math></jats:alternatives></jats:inline-formula>. We give a realisability criterion for harmonic<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004123000518_inline5.png"/><jats:tex-math>$\mathfrak{A}$</jats:tex-math></jats:alternatives></jats:inline-formula>-covers by patching local monodromy data in an extended homology group on<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004123000518_inline6.png"/><jats:tex-math>$\Gamma$</jats:tex-math></jats:alternatives></jats:inline-formula>. As an explicit example, we work out the case<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004123000518_inline7.png"/><jats:tex-math>$\mathfrak{A}=\mathbb{Z}/p\mathbb{Z}$</jats:tex-math></jats:alternatives></jats:inline-formula>and explain how realisability for such covers is related to the nowhere-zero flow problem from graph theory.</jats:p>
Publishing Year
Journal Title
Mathematical Proceedings of the Cambridge Philosophical Society
Volume
176
Issue
2
Page
395-416
LibreCat-ID
Cite this
Len Y, Ulirsch M, Zakharov D. Abelian tropical covers. Mathematical Proceedings of the Cambridge Philosophical Society. 2023;176(2):395-416. doi:10.1017/S0305004123000518
Len, Y., Ulirsch, M., & Zakharov, D. (2023). Abelian tropical covers. Mathematical Proceedings of the Cambridge Philosophical Society, 176(2), 395–416. https://doi.org/10.1017/S0305004123000518
@article{Len_Ulirsch_Zakharov_2023, title={Abelian tropical covers}, volume={176}, DOI={10.1017/S0305004123000518}, number={2}, journal={Mathematical Proceedings of the Cambridge Philosophical Society}, publisher={Cambridge University Press (CUP)}, author={Len, Yoav and Ulirsch, Martin and Zakharov, Dmitry}, year={2023}, pages={395–416} }
Len, Yoav, Martin Ulirsch, and Dmitry Zakharov. “Abelian Tropical Covers.” Mathematical Proceedings of the Cambridge Philosophical Society 176, no. 2 (2023): 395–416. https://doi.org/10.1017/S0305004123000518.
Y. Len, M. Ulirsch, and D. Zakharov, “Abelian tropical covers,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 176, no. 2, pp. 395–416, 2023, doi: 10.1017/S0305004123000518.
Len, Yoav, et al. “Abelian Tropical Covers.” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 176, no. 2, Cambridge University Press (CUP), 2023, pp. 395–416, doi:10.1017/S0305004123000518.
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