Jell, Philipp and Rau, Johannes and Shaw, Kristin - Lefschetz (1,1)-theorem in tropical geometry

epiga:4126 - Épijournal de Géométrie Algébrique, November 19, 2018, Volume 2
Lefschetz (1,1)-theorem in tropical geometry

Authors: Jell, Philipp and Rau, Johannes and Shaw, Kristin

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.

Volume: Volume 2
Published on: November 19, 2018
Submitted on: December 5, 2017
Keywords: Mathematics - Algebraic Geometry,14T05 (Primary), 52B40, 55N35, 14C25, 14C22 (Secondary)


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