Characterization of tropical varieties satisfying the tropical Calabi–Yau theorem

Characterize compact prek tropical varieties equipped with a strongly semipositive form for which the tropical Calabi–Yau theorem holds, proving that it holds if and only if every point of the open part is regular in codimension one.

Background

The paper studies the tropical Monge–Ampère equation on a compact prek tropical variety equipped with a strongly semipositive form. Although uniqueness is established, existence can fail in dimension at least two. The authors prove existence when every point of the open part is regular in codimension one, and prove failure under a genericity condition on a codimension-two star fan.

The conjecture proposes that regularity in codimension one is the exact geometric condition governing existence for all compactly supported measures of the prescribed total volume. Thus, the unresolved direction is to show that failure of regularity in codimension one necessarily produces failure of the tropical Calabi–Yau theorem in full generality. The paper establishes the positive direction and proves failure for a broad generic class, but does not prove the asserted equivalence.

References

Actually, we have the following conjecture. Let $(X, \omega)$ be a compact prek tropical variety equipped with a strongly semipositive form $\omega$. Then, $(X, \omega)$ verifies the tropical Calabi--Yau theorem if and only if $\CR=X_o$, i.e. every point $x \in X_o$ is regular in codimension one (see Definition \ref{def:regular_in_codim_one}).

— On solutions to tropical Monge-Ampère equations  (2610.25602 - Amini et al., 19 Oct 2026) in Conjecture \ref{conj:exact_condition_for_orthogonality_properties}, Section 10, immediately following Remark 10.11 in the section on failure of the tropical Calabi–Yau theorem

We are however only able to prove the uniqueness part assuming strong semipositivity, and do not know whether this holds in general for any semipositive form.

— On solutions to tropical Monge-Ampère equations  (2610.25602 - Amini et al., 19 Oct 2026) in Remark immediately following Theorem \ref{thm:uniqueness}, Section 7, subsection on tropical Monge–Ampère equations