Tropical Abelian Varieties and Lefschetz Hyperplane Theorem

In a recent work, Adiprasito and Björner established a tropical Lefschetz hyperplane theorem for projective tropical varieties. Building on this, we exhibit a Lefschetz hyperplane theorem for principally polarized tropical abelian varieties (PPTAV). These results combine stratified Morse theory, Voronoi decompositions, and Adiprasito and Bjorner’s Lefschetz theorem for matroidal fans. Furthermore, there are applications to the topology of complete intersections in PPTAV’s.