PomeProposition 17.16.2.1A5A4BBF82E3 Subgeodesics are preserved by adding an affine function of $t$, by a positive affine reparametrization of time, by finite maxima, and by convex combinations. For $h\in C^\infty (X,\mathbb R)$, replacing $\theta $ by $\theta +\mathrm{dd}^{\mathrm{c}}h$ and $u_ t$ by $u_ t-h$ also preserves them.
Proposition 17.16.21A5A4BBF82E3
Proof. Affine functions of $\operatorname{Re}z$ have zero complex Hessian, and a positive affine change of time is induced by a holomorphic affine change of the strip coordinate. Convex combinations preserve the inequality between currents, and the change $(\theta ,u_ t)\mapsto (\theta +\mathrm{dd}^{\mathrm{c}}h,u_ t-h)$ preserves the current itself. Maxima are covered by lemma 5.24.1. $\square$
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