Lefschetz section theorems for tropical hypersurfaces

Arnal, Charles; Renaudineau, Arthur; Shaw, Kris

Annales Henri Lebesgue, Tome 4 (2021), pp. 1347-1387.

Nous démontrons des analogues au théorème de la section hyperplane de Lefschetz pour l’homologie tropicale entière d’hypersurfaces tropicales dans des variétés toriques tropicales. Nous en déduisons que les groupes d’homologie des hypersurfaces tropicales non-singulières compactes (ou contenues dans ${ℝ}^n$) sont sans torsion. Nous en déduisons également une relation entre les coefficients du genre ${\chi }_y$ des hypersurfaces complexes dans les variétés toriques et les caractéristiques d’Euler des complexes de chaînes cellulaires tropicales des hypersurfaces tropicales. Il s’ensuit que les groupes d’homologies tropicales à coefficient entier ont pour rang les nombres de Hodges d’hypersurfaces compactes non-singulières dans des variétés toriques complexes. Finalement pour les hypersurfaces tropicales dans certaines variétés toriques affines, nous relions les rangs de leurs groupes d’homologie tropicale aux nombres de Hodge–Deligne des hypersurfaces complexes correspondantes.

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 ${ℝ}^n$ 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.

Classification : 14T05, 58A14, 32S60
Mots clés : tropical geometry, tropical homology, Lefschetz section theorems, Hodge theory

[AB14] Adiprasito, Karim Alexander; Björner, Anders Filtered geometric lattices and Lefschetz Section Theorems over the tropical semiring (2014) (https://arxiv.org/abs/1401.7301)

[BIMS15] Brugallé, Erwan; Itenberg, Ilia; Mikhalkin, Grigory; Shaw, Kristin 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] Curry, Justin Michael Sheaves, cosheaves and applications, ProQuest LLC, 2014 Thesis (Ph.D.)–University of Pennsylvania, USA https://www.proquest.com/docview/1553207954 | MR

[DK86] Danilov, Vladimir I.; Khovanskiĭ, Askold G. 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] Fulton, William 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] Gross, Andreas; Shokrieh, Farbod Sheaf-theoretic approach to tropical homology (2019) (https://arxiv.org/abs/1906.09245)

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

[IKMZ19] Itenberg, Ilia; Katzarkov, Ludmil; Mikhalkin, Grigory; Zharkov, Ilia Tropical homology, Math. Ann., Volume 374 (2019) no. 1-2, pp. 963-1006 | DOI | MR | Zbl

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

[JRS18] Jell, Philipp; Rau, Johannes; Shaw, Kristin Lefschetz $(1,1)$-theorem in tropical geometry, Épijournal de Géom. Algébr., EPIGA, Volume 2 (2018), 11 | MR | Zbl

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

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

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

[KS17] Kastner, Lars; Shaw, Anna-Lena Kristinand Winz 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] de Cataldo, Mark Andrea; Migliorini, Luca; Mustaţă, Mircea Combinatorics and topology of proper toric maps, J. Reine Angew. Math., Volume 2018 (2018) no. 744, pp. 133-163 | MR | Zbl

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

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

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

[MZ14] Mikhalkin, Grigory; Zharkov, Ilia 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] Osserman, Brian; Rabinoff, Joseph 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] Payne, Sam Analytification is the limit of all tropicalizations, Math. Res. Lett., Volume 16 (2009) no. 2-3, pp. 543-556 | DOI | MR | Zbl

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

[Sha93] Shapiro, Boris Z. 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] Shepard, Allen Dudley 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] Wisniewski, Jarosław A. 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] Zharkov, Ilia The Orlik–Solomon algebra and the Bergman fan of a matroid, J. Gökova Geom. Topol. GGT, Volume 7 (2013), pp. 25-31 | MR | Zbl