Lifts of cycles in tropical hypersurfaces and the Gamma conjecture

TL;DR

The paper develops a bridge between tropical and classical geometry for complex hypersurfaces in toric varieties by constructing lifts of tropical $(p,q)$-cycles with $p+q=d$ and analyzing their topological intersections and period integrals. It introduces lifts built from Minkowski weights on unimodular fans, derives explicit tropical intersection formulas, and establishes period asymptotics in the tropical limit that mirror the mirror Gamma conjecture. By translating tropical data into $d$-cycles via torus fibrations and a Kodaira–Spencer–type residue calculus, the work provides a framework to compute intersection numbers of lifts from tropical information and links these to gamma-classes on toric varieties. In special Calabi–Yau settings, the results specialize to Gamma-conjecture type period identities, and the final theorem equates lift intersections with tropical intersections modulo a sign, enabling combinatorial computation of classical invariants from tropical geometry.

Abstract

For a complex hypersurface of dimension $d \geq 1$ in a toric variety, we construct lifts of tropical $(p, q)$-cycles with $p+q=d$ in the associated tropical hypersurface. The tropical cycles we consider are described by Minkowski weights, and their lifts are realized as topological cycles admitting a torus fibration structure over the tropical cycles. The intersection numbers of these lifted cycles are computed in terms of tropical intersection theory. We further derive the asymptotic formulas for the period integrals of the lifts in the tropical limit, which are closely related to the mirror symmetric Gamma conjecture. Throughout the paper, we assume that the tropicalization is dual to a unimodular triangulation of the Newton polytope.

Figures (3)

• Figure 1: The tropical cycle $c(a_0)$ in the tropical hypersurface $X(\operatorname{trop}(f))$ and its self-intersection
• Figure 2: The tropical cycles $c(a_1), c(a_2)$
• Figure 3: The decomposition of $\partial \nabla_{w=0}$ and the fan $\check{\Sigma}$

Theorems & Definitions (38)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Remark 2.1
• Remark 2.2
• Theorem 2.3
• Example 3.1