Lefschetz (1,1)-theorem in tropical geometry

Abstract

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.

27 pages, 6 figures, POSTpublished version with minor corrections and improvements; the published version is arxiv.org/abs/1711.07900v3

References in corpus (2)

• Combinatorial tropical surfaces
• Monodromy map for tropical Dolbeault cohomology

Cited by in corpus (9)

• A homology theory for tropical cycles on integral affine manifolds and a perfect pairing
• Lefschetz section theorems for tropical hypersurfaces
• Tautological cycles on tropical Jacobians
• Homology of tropical fans
• The tropical Poincaré-Hopf theorem
• The Ceresa period from tropical homology
• Tropical Theta Functions and Riemann-Roch Inequality for Tropical Abelian Surfaces
• Poincaré Duality, Degeneracy, and Real Lefschetz Property for T-Hypersurfaces
• Riemann-Roch inequality for smooth tropical toric surfaces