### Metadata

### Abstract

We establish variants of the Lefschetz section theorem for the integral tropical homology groups of tropical hypersurfaces of tropical toric varieties. It follows from these theorems that the integral tropical homology groups of non-singular tropical hypersurfaces which are compact or contained in ${\mathbb{R}}^{n}$ are torsion free. We prove a relationship between the coefficients of the ${\chi}_{y}$ genera of complex hypersurfaces in toric varieties and Euler characteristics of the integral tropical cellular chain complexes of their tropical counterparts. It follows that the integral tropical homology groups give the Hodge numbers of compact non-singular hypersurfaces of complex toric varieties. Finally for tropical hypersurfaces in certain affine toric varieties, we relate the ranks of their tropical homology groups to the Hodge–Deligne numbers of their complex counterparts.

### References

[AB14] Filtered geometric lattices and Lefschetz Section Theorems over the tropical semiring (2014) (https://arxiv.org/abs/1401.7301)

[BIMS15] Brief introduction to tropical geometry, Proceedings of the Gökova Geometry-Topology Conference 2014, International Press; Gökova: Gökova Geometry-Topology Conferences (GGT) (2015), pp. 1-75 | MR | Zbl

[Cur14] Sheaves, cosheaves and applications, ProQuest LLC, 2014 Thesis (Ph.D.)–University of Pennsylvania, USA https://www.proquest.com/docview/1553207954 | MR

[DK86] Newton polyhedra and an algorithm for calculating Hodge–Deligne numbers, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 50 (1986) no. 5, pp. 925-945 | MR

[Ful93] Introduction to toric varieties. The 1989 William H. Roever Lectures in Geometry, Annals of Mathematics Studies, 131, Princeton University Press, 1993 | DOI | MR | Zbl

[GS19] Sheaf-theoretic approach to tropical homology (2019) (https://arxiv.org/abs/1906.09245)

[Hat02] Algebraic topology, Cambridge University Press, 2002 | MR | Zbl

[IKMZ19] Tropical homology, Math. Ann., Volume 374 (2019) no. 1-2, pp. 963-1006 | DOI | MR | Zbl

[Ite17] Tropical homology and Betti numbers of real algebraic varieties (2017) (https://web.ma.utexas.edu/users/sampayne/pdf/Itenberg-Simons2017.pdf)

[JRS18] Lefschetz $(1,1)$-theorem in tropical geometry, Épijournal de Géom. Algébr., EPIGA, Volume 2 (2018), 11 | MR | Zbl

[JSS19] Superforms, tropical cohomology, and Poincaré duality, Adv. Geom., Volume 19 (2019) no. 1, pp. 101-130 | DOI | MR | Zbl

[Kho77] Newton polyhedra, and toroidal varieties, Funkts. Anal. Prilozh., Volume 11 (1977) no. 4, p. 56-64, 96 | MR

[KS16] Tropical geometry, the motivic nearby fiber, and limit mixed Hodge numbers of hypersurfaces, Res. Math. Sci., Volume 3 (2016), 10 | MR | Zbl

[KS17] Cellular sheaf cohomology of polymake, Combinatorial algebraic geometry. Selected papers from the 2016 apprenticeship program, Ottawa, Canada, July–December 2016 (Fields Institute Communications), Volume 80, The Fields Institute for Research in the Mathematical Sciences, Toronto; Springer, 2017, pp. 369-385 | MR | Zbl

[MM18] Combinatorics and topology of proper toric maps, J. Reine Angew. Math., Volume 2018 (2018) no. 744, pp. 133-163 | MR | Zbl

[MR18] Tropical geometry, 2018 (https://math.uniandes.edu.co/~j.rau/downloads/main.pdf)

[MS15] Introduction to tropical geometry, Graduate Studies in Mathematics, 161, American Mathematical Society, 2015 | MR | Zbl

[Mus04] Lecture notes on toric varieties, 2004 (http://www-personal.umich.edu/~mmustata/toric_var.html)

[MZ14] Tropical eigenwave and intermediate Jacobians, Homological mirror symmetry and tropical geometry. Based on the workshop on mirror symmetry and tropical geometry, Cetraro, Italy, July 2–8, 2011 (Lecture Notes of the Unione Matematica Italiana), Volume 15, Springer, 2014, pp. 309-349 | DOI | MR | Zbl

[OR13] Lifting nonproper tropical intersections, Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011 (Contemporary Mathematics), Volume 605, American Mathematical Society (2013), pp. 15-44 | DOI | MR | Zbl

[Pay09] Analytification is the limit of all tropicalizations, Math. Res. Lett., Volume 16 (2009) no. 2-3, pp. 543-556 | DOI | MR | Zbl

[RS18] Bounding the Betti numbers of real hypersurfaces near the tropical limit (2018) (https://arxiv.org/abs/1805.02030)

[Sha93] The mixed Hodge structure of the complement to an arbitrary arrangement of affine complex hyperplanes is pure, Proc. Am. Math. Soc., Volume 117 (1993) no. 4, pp. 931-933 | DOI | MR | Zbl

[She85] A cellular description of the derived category of a stratified space, Ph. D. Thesis, Brown University, Michigan USA (1985) (published on ProQuest LLC, https://www.proquest.com/openview/ca196f7bbe67f464b8da5c5930e20635/1?pq-origsite=gscholar&cbl=18750&diss=y) | MR

[Wis02] Toric Mori theory and Fano manifolds, Geometry of toric varieties (Séminaires et Congrès), Volume 6, Société Mathématique de France, 2002, pp. 249-272 | MR | Zbl

[Zha13] The Orlik–Solomon algebra and the Bergman fan of a matroid, J. Gökova Geom. Topol. GGT, Volume 7 (2013), pp. 25-31 | MR | Zbl