Tropical algebraic geometry is a set of methods combining algebraic, geometric and combinatorial tools, that have broad applications in algebraic geometry, particularly in toric geometry. This paper works with tropical toric varieties – topological spaces appearing as (partial) compactifications of \(\mathbb{R}^{n+1}\) – and tropical hypersurfaces inside them, which are some particular type of polyhedral topological spaces which are also (partial) compactifications of rational polyhedral complexes inside \(\mathbb{R}^{n+1}\).
Tropical homology is part of these methods, which is a homology theory for tropical varieties. It first appeared in comparison results to the homology of a generic member of a family of complex projective varieties. The results in this paper can also be seen as comparison results to the homology of complex (and real) toric varieties through a tropical toric version of Lefschetz hyperplane section theorems, which are a collection of very important statements in complex projective algebraic geometry: these are comparison results between the homology groups of certain projective varieties and a hyperplane section of them.
The paper under discussion can be divided in three parts. First it establishes tropical variants of the Lefschetz hyperplane section theorem for the tropical toric case, more concretely, these are comparison results between the integral tropical homology groups of a tropical toric variety and a hypersurface of it. It states that under the correct hypotheses on the ambient space \(Y\), the hypersurface \(X\), and the embedding \(X\xrightarrow{i} Y\), the map \(i_*:H_q(X;\mathcal{F}_p^X)\xrightarrow{}H_q(Y;\mathcal{F}_p^Y)\) on tropical homology induced by \(i\) is an isomorphism when \(p+q<n\) and a surjection when \(p+q=n\).
Then, by studying this comparison result, it is shown that these tropical homology groups are torsion-free for a nonsingular tropical hypersurface \(X\) in a compact nonsingular tropical toric variety \(Y\). This result is a consequence of the fact that the integral tropical homology groups of a nonsingular compact tropical toric variety are torsion-free. The paper contains versions for the non-compact case too.
Finally, it gives applications towards the tropical computation of the Hodge-Deligne numbers of (torically non-degenerate) complex hypersurface \(X_{\mathbb{C}}\subset Y_{\mathbb{C}}\) using a nonsingular tropical hypersurface \(X\subset Y\) with the same Newton polytope, encoded as the coefficients of the so-called \(E\)- polynomial of \(X_{\mathbb{C}}\). Also it outlines the possibility to write some bounds on the Betti numbers of the real part of (patchworked) real algebraic hypersurfaces in terms of Hodge-Deligne numbers of the complexification of this real variety.
The inspiration for the results in this paper comes from Lefschetz section theorems, since in the compact case, a hypersurface of a tropical toric variety can be recovered as a hyperplane section after embedding the toric variety into a projective space using a very ample line bundle. So they first provide such type of theorem for the tropical toric case under consideration.
Section 3 is the technical heart of this paper. Here they prove (homological) vanishing theorems that arise from short exact sequences relating the structure cosheaf of the tropical hypersurface in question with that of its ambient tropical toric variety.
In Section 4 they show that the integral tropical homology groups of a non-singular tropical toric variety are torsion-free. The method of the proof consist in finding the cellular homology groups of the variety by computing the corresponding tropical cohomology groups, taking advantage of the fact that the tropical variety under discussion is a tropical manifold, and thus satisfies Poincaré duality for tropical homology.
Given a nonsingular tropical variety \(Y\), Section 5 shows a relation between the Betti numbers of tropical homology of a nonsingular tropical hypersurface \(X\subset Y\) and the Hodge-Deligne numbers of (torically non-degenerate) complex hypersurface \(X_{\mathbb{C}}\subset Y_{\mathbb{C}}\) sharing the same Newton polytope. The proof can be reduced to the case of very affine hypersurfaces, where it follows from direct computation of both sides.