Geodesic
Lefschetz (1,1)-theorem in tropical geometry
Épijournal de Géométrie Algébrique, Tome 2 (2018)

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For a tropical manifold of dimension n we show that the tropical homology classes of degree (n-1, n-1) which arise as fundamental classes of tropical cycles are precisely those in the kernel of the eigenwave map. To prove this we establish a tropical version of the Lefschetz (1, 1)-theorem for rational polyhedral spaces that relates tropical line bundles to the kernel of the wave homomorphism on cohomology. Our result for tropical manifolds then follows by combining this with Poincaré duality for integral tropical homology. 
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     author = {Jell, Philipp and Rau, Johannes and Shaw, Kristin},
     title = {Lefschetz (1,1)-theorem in tropical geometry},
     journal = {\'Epijournal de G\'eom\'etrie Alg\'ebrique},
     publisher = {mathdoc},
     volume = {2},
     year = {2018},
     doi = {10.46298/epiga.2018.volume2.4126},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/epiga.2018.volume2.4126/}
}
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Jell, Philipp; Rau, Johannes; Shaw, Kristin. Lefschetz (1,1)-theorem in tropical geometry. Épijournal de Géométrie Algébrique, Tome 2 (2018). doi: 10.46298/epiga.2018.volume2.4126

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