A comparison of positivity in complex and tropical toric geometry

Abstract

Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the cone of invariant closed positive currents on the complex toric variety with closed positive currents on the tropicalization. In a subsequent paper, this correspondence will be used to develop a Bedford–Taylor theory of plurisubharmonic functions on the tropicalization.

Monge–Ampère measures for toric metrics on abelian varieties

“…in[8, Remarks 7.1.2, 8.1.3]. Since ∥ • ∥ H is a model metric, we deduce from(23) The line bundle L is ample if and only if H is ample and b is positive definite, see[7, Theorem 6.13].We conclude that the assumptions in 3.5 are satisfied.…”

“…They have similar properties as the complex (p, q)-forms. More generally, by restriction we get a bigraded differential sheaf A f j α j ∧ Jα j for smooth non-negative functions f j and smooth (p, 0)-forms α j on S. Again, positive forms are obtained from positive forms on N R and the latter are studied in [8]. In particular, we deduce that positive Lagerberg forms on S are closed under products.…”

Dynamical Tropicalisation

“…Given an algebraic subvariety Z ⊆ (C * ) n , the set Log(Z ) is called the amoeba of Z . By Bergman's theorem [9] there exists a close subset of R n such that…”