Algebraic capacities

We study invariants coming from certain optimisation problems for nef divisors on surfaces. These arise in work of the author and collaborators tying obstructions to embeddings between symplectic 4-manifolds questions positivity (possibly singular) algebraic develop general framework these prove foundational results their structure asymptotics. describe connections have embedded contact homology (ECH) Ruelle invariant geometry, min–max widths minimal hypersurfaces. use first obtain optimal bounds sub-leading asymptotics ECH capacities many toric domains.