Lefschetz section theorems for tropical hypersurfaces

Charles Arnal, Arthur Renaudineau, Kris Shaw

Abstract:

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 Rn are torsion free. We prove a relationship between the
coefficients of the χy 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.