Tropical (1, 1)-homology for floor decomposed surfaces

TL;DR

This paper computes the (1, 1)-homology and intersection form of non-singular floor decomposed tropical surfaces, advancing understanding of tropical analogues of Hodge theory in algebraic geometry.

Contribution

It provides explicit calculations of (1, 1)-homology groups and intersection pairings for a class of tropical surfaces, filling a gap in tropical Hodge theory.

Findings

Explicit (1, 1)-homology groups computed

Intersection form on tropical surfaces determined

Advances understanding of tropical Hodge structures

Abstract

The tropical (p, q)-homology groups of Itenberg, Katzarkov, Mikhalkin and Zharkov are the tropical analogues of the Hodge decomposition of the cohomology of complex algebraic varieties. There is a well-defined intersection pairing on tropical (1, 1)-classes of a compact non-singular tropical surface. Here we compute directly the (1, 1)-homology of a non-singular floor decomposed tropical surface in tropical projective space, along with the intersection form.