  ECH capacities are rich obstructions to symplectic embeddings in 4-dimensions
that have also been seen to arise in the context of algebraic positivity for
(possibly singular) projective surfaces. We extend this connection to relate
general convex toric domains on the symplectic side with towers of polarised
toric surfaces on the algebraic side, and then use this perspective to show
that the sub-leading asymptotics of ECH capacities for all convex and concave
toric domains are $O(1)$. We obtain sufficient criteria for when the
sub-leading asymptotics converge in this context, generalising results of
Hutchings and of the author, and derive new obstructions to embeddings between
toric domains of the same volume. We also propose two invariants to more
precisely describe when convergence occurs in the toric case. Our methods are
largely non-toric in nature, and apply more widely to towers of polarised
Looijenga pairs.
