[Paper Review] Tropical contractions to integral affine manifolds with singularities

Yuto Yamamoto|arXiv (Cornell University)|May 21, 2021

Algebraic Geometry and Number Theory24 references4 citations

TL;DR

This paper constructs tropical contractions—maps from tropical varieties to integral affine manifolds with singularities (IAMS)—that preserve integral affine structures and tropical cohomology. It shows that the IAMS arising as the dual intersection complex of a toric degeneration is embedded in the tropical variety via a stable intersection of hypersurfaces, and proves that the contraction map preserves the eigenwave class and radiance obstruction, generalizing tropical modifications and providing local models for key singularities in mirror symmetry.

ABSTRACT

We consider a toric degeneration of Calabi--Yau complete intersections of Batyrev--Borisov in the Gross--Siebert program. One can associate two types of tropical spaces with it. One is a tropical variety obtained by tropicalization. The other one is an integral affine manifold with singularities, which arises as the dual intersection complex of the toric degeneration. In this article, we show that the latter is contained in the former as a subset, and construct an integral affine contraction map from the former to the latter. We also show that the contraction preserves tropical cohomology groups, and sends the eigenwave to the radiance obstruction.

Motivation & Objective

• To establish a geometric bridge between tropical varieties and integral affine manifolds with singularities (IAMS), central to the Gross–Siebert program and non-archimedean mirror symmetry.
• To construct a contraction map from a tropical variety (defined as a stable intersection of tropical hypersurfaces) to an IAMS, which arises as the dual intersection complex of a toric degeneration of Calabi–Yau complete intersections.
• To show that this contraction preserves tropical cohomology groups and maps the eigenwave class to the radiance obstruction, linking geometric and cohomological invariants.
• To provide local models of such contractions that realize key singularities in topological and tropical mirror symmetry, such as focus-focus points and tropical nodes.
• To generalize Mikhael's tropical modifications by introducing a broader class of morphisms—tropical contractions—that are locally isomorphic to these models.

Proposed method

• The construction uses a lattice polytope configuration in dual spaces M′ and N′, with associated cones C in NR generated by products of dual polytopes and standard basis vectors.
• Tropical hypersurfaces are defined via min-tropical polynomials fi(n) = min_{m∈Ai} ⟨m,n⟩, where Ai includes 0 and lattice points in ∆i × {−e*i}.
• The tropical variety is realized as the stable intersection X(f1,…,fr)◦ of these hypersurfaces, which is a weighted balanced polyhedral complex in NR.
• A subset B ⊂ NR is defined as the locus where fi(n) = 0 for some non-zero m∈Ai, and equipped with an integral affine structure via projections to tropical torus orbits.
• A contraction map δ: V → U is constructed locally, where V is a tropical manifold (locally isomorphic to a Bergman fan), and U ⊂ B is a neighborhood in the IAMS.
• The map is shown to be a morphism of tropical spaces, preserving integral affine structures and satisfying conditions for cohomological invariance.

Experimental results

Research questions

• RQ1 How can one construct a morphism from a tropical variety to an integral affine manifold with singularities that preserves the underlying affine structure?
• RQ2 Does the tropical contraction map preserve tropical cohomology groups and the eigenwave class?
• RQ3 Can the contraction map realize known singularities such as focus-focus points and tropical nodes as local models?
• RQ4 Under what conditions on the IAMS does the contraction become locally representable as a tropical modification or a very good local model?
• RQ5 Is the dual intersection complex of a toric degeneration of Calabi–Yau complete intersections naturally embedded in the tropical variety via stable intersection?

Key findings

• The IAMS B arising as the dual intersection complex of a toric degeneration is embedded as a subset of the tropical variety X(f1,…,fr) via the stable intersection construction.
• A tropical contraction map δ: V → U is constructed from a tropical manifold V to a neighborhood U ⊂ B in the IAMS, preserving integral affine structures.
• The contraction map preserves tropical cohomology groups, ensuring that cohomological invariants are respected under the morphism.
• The map sends the eigenwave class to the radiance obstruction, linking a geometric invariant of the tropical variety to a characteristic class of the IAMS.
• When the IAMS is quasi-simple (e.g., simple in the Gross–Siebert sense), the contraction is locally isomorphic to a local model, and when very simple, the local model is 'very good'—a necessary condition for further applications.
• The construction generalizes Mikhael's tropical modifications, showing that they are a special case of tropical contractions.