Validity of a genus-refined capacity and its computation formula

Determine whether the proposed genus-refined capacities Tilde{g}_k^{ℓ,h}(X_Ω) on four-dimensional convex toric domains admit the computation formula Tilde{g}_k^{ℓ,h}(X_Ω) = min_{P ∈ P_{k,ℓ,h}} Σ_s ||(i_s, j_s)||_Ω^*, where P_{k,ℓ,h} consists of multisets P = {(i_s, j_s)} with an integer g ≤ h satisfying the index constraint Σ_s(i_s + j_s) + q + g − 1 = k, the weak permissibility condition (if q ≥ 2 then (i_1,…,i_q) ≠ (0,…,0) and (j_1,…,j_q) ≠ (0,…,0)), and q ≤ ℓ. Specifically, ascertain that the relative adjunction constraint is enforced by these criteria and establish (or refute) that for all such admissible data the required J-holomorphic curves exist so that the minimum equals the capacity.

Background

The paper introduces refined elementary capacities Tilde{g}_k^ℓ that constrain the number of positive ends and establishes a closed computation formula for four-dimensional convex toric domains, yielding new embedding obstructions that sometimes surpass those from Gutt–Hutchings and McDuff–Siegel capacities.

As a further refinement, the authors propose capacities Tilde{g}_k^{ℓ,h} that also bound the genus. They present a naive computation formula for these capacities by minimizing the dual-norm sum over (k,ℓ,h)-admissible words, where admissibility augments the existing (k,ℓ)-criteria with an extra genus parameter g ≤ h in the index relation.

However, the authors explicitly note uncertainty about the correctness of this naive formula. They state that one must verify that the relative adjunction constraint is captured by these criteria and that the relevant J-holomorphic curves actually exist under these conditions, acknowledging that curves permitted by index and adjunction may still fail to exist in practice.