Cycles, cocycles, and duality on tropical manifolds

Gross, Andreas; Shokrieh, Farbod

doi:10.1090/proc/15468

Cycles, cocycles, and duality on tropical manifolds
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by Andreas Gross and Farbod Shokrieh

Proc. Amer. Math. Soc. 149 (2021), 2429-2444

DOI: https://doi.org/10.1090/proc/15468

Published electronically: March 26, 2021

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Abstract:

We prove a Poincaré duality for the Chow rings of smooth fans whose support are tropical linear spaces. As a consequence, we show that cycles and cocycles on tropical manifolds are Poincaré dual to each other. This allows us to define pull-backs of tropical cycles along arbitrary morphisms with smooth target.

References

Federico Ardila, Graham Denham, and June Huh, Lagrangian geometry of matroids, 2020. arXiv:2004.13116.
Karim Adiprasito, Combinatorial lefschetz theorems beyond positivity, 2018. arXiv:1812.10454.
Karim Adiprasito, June Huh, and Eric Katz, Hodge theory for combinatorial geometries, Ann. of Math. (2) 188 (2018), no. 2, 381–452. MR 3862944, DOI 10.4007/annals.2018.188.2.1
Federico Ardila and Caroline J. Klivans, The Bergman complex of a matroid and phylogenetic trees, J. Combin. Theory Ser. B 96 (2006), no. 1, 38–49. MR 2185977, DOI 10.1016/j.jctb.2005.06.004
Omid Amini and Matthieu Piquerez, Hodge theory for tropical varieties, 2020. arXiv:2007.07826.
Lars Allermann and Johannes Rau, First steps in tropical intersection theory, Math. Z. 264 (2010), no. 3, 633–670. MR 2591823, DOI 10.1007/s00209-009-0483-1
Spencer Backman, Christopher Eur, and Connor Simpson, Simplicial generation of Chow rings of matroids, 2019. arXiv:1905.07114.
Michel Brion, Piecewise polynomial functions, convex polytopes and enumerative geometry, Parameter spaces (Warsaw, 1994) Banach Center Publ., vol. 36, Polish Acad. Sci. Inst. Math., Warsaw, 1996, pp. 25–44. MR 1481477
David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322, DOI 10.1090/gsm/124
W. Fulton, R. MacPherson, F. Sottile, and B. Sturmfels, Intersection theory on spherical varieties, J. Algebraic Geom. 4 (1995), no. 1, 181–193. MR 1299008
Georges François and Johannes Rau, The diagonal of tropical matroid varieties and cycle intersections, Collect. Math. 64 (2013), no. 2, 185–210. MR 3041763, DOI 10.1007/s13348-012-0072-1
Georges Francois, Cocycles on tropical varieties via piecewise polynomials, Proc. Amer. Math. Soc. 141 (2013), no. 2, 481–497. MR 2996952, DOI 10.1090/S0002-9939-2012-11359-0
Eva Maria Feichtner and Bernd Sturmfels, Matroid polytopes, nested sets and Bergman fans, Port. Math. (N.S.) 62 (2005), no. 4, 437–468. MR 2191630
William Fulton and Bernd Sturmfels, Intersection theory on toric varieties, Topology 36 (1997), no. 2, 335–353. MR 1415592, DOI 10.1016/0040-9383(96)00016-X
William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
Andreas Gross and Farbod Shokrieh, A sheaf-theoretic approach to tropical homology, 2019. arXiv:1906.09245.
Philipp Jell, Johannes Rau, and Kristin Shaw, Lefschetz $(1,1)$-theorem in tropical geometry, Épijournal Géom. Algébrique 2 (2018), Art. 11, 27 (English, with English and French summaries). MR 3894860, DOI 10.46298/epiga.2018.volume2.4126
Philipp Jell, Kristin Shaw, and Jascha Smacka, Superforms, tropical cohomology, and Poincaré duality, Adv. Geom. 19 (2019), no. 1, 101–130. MR 3903579, DOI 10.1515/advgeom-2018-0006
Eric Katz, A tropical toolkit, Expo. Math. 27 (2009), no. 1, 1–36. MR 2503041, DOI 10.1016/j.exmath.2008.04.003
Eric Katz, Tropical intersection theory from toric varieties, Collect. Math. 63 (2012), no. 1, 29–44. MR 2887109, DOI 10.1007/s13348-010-0014-8
Eric Katz and Sam Payne, Piecewise polynomials, Minkowski weights, and localization on toric varieties, Algebra Number Theory 2 (2008), no. 2, 135–155. MR 2377366, DOI 10.2140/ant.2008.2.135
Diane Maclagan and Bernd Sturmfels, Introduction to tropical geometry, Graduate Studies in Mathematics, vol. 161, American Mathematical Society, Providence, RI, 2015. MR 3287221, DOI 10.1090/gsm/161
James Oxley, Matroid theory, 2nd ed., Oxford Graduate Texts in Mathematics, vol. 21, Oxford University Press, Oxford, 2011. MR 2849819, DOI 10.1093/acprof:oso/9780198566946.001.0001
Dan Petersen, Poincaré duality of wonderful compactifications and tautological rings, Int. Math. Res. Not. IMRN 17 (2016), 5187–5201. MR 3556436, DOI 10.1093/imrn/rnv296
Johannes Rau, Intersections on tropical moduli spaces, Rocky Mountain J. Math. 46 (2016), no. 2, 581–662. MR 3529085, DOI 10.1216/RMJ-2016-46-2-581
Kristin M. Shaw, A tropical intersection product in matroidal fans, SIAM J. Discrete Math. 27 (2013), no. 1, 459–491. MR 3032930, DOI 10.1137/110850141
Bernd Sturmfels and Jenia Tevelev, Elimination theory for tropical varieties, Math. Res. Lett. 15 (2008), no. 3, 543–562. MR 2407231, DOI 10.4310/MRL.2008.v15.n3.a14
Jenia Tevelev, Compactifications of subvarieties of tori, Amer. J. Math. 129 (2007), no. 4, 1087–1104. MR 2343384, DOI 10.1353/ajm.2007.0029
Jarosław Włodarczyk, Decomposition of birational toric maps in blow-ups & blow-downs, Trans. Amer. Math. Soc. 349 (1997), no. 1, 373–411. MR 1370654, DOI 10.1090/S0002-9947-97-01701-7