Subleading asymptotics of alternative ECH capacities

Prove or disprove the conjectured asymptotic formula for the error terms e_k^Alt of closed symplectic 4-manifolds with b_2^+=1: liminf e_k^Alt=−(1/2)[Ω]·c_1(TX,Ω) and limsup e_k^Alt=−(1/2)[Ω]·c_1(TX,Ω)+ρ([Ω]), with convergence if and only if (X,Ω) is irrational.

Background

The paper proves that the alternative ECH capacities of closed symplectic 4-manifolds with b_2+=1 obey a Weyl law with a uniformly bounded error term. The next unresolved issue is to identify the precise lower and upper accumulation values of that error term.

The conjecture introduces a pseudonorm ρ on H2(X;R), distinguishing rational primitive classes from irrational classes. The paper verifies the proposed behavior in several elementary examples and proves partial inequalities, including equality for the liminf in the monotone or first-Chern-proportional case.

References

We now make a conjecture about the asymptotics of $e_kAlt$ for closed 4-manifolds, which we hope can motivate further study of the general subleading asymptotics of ECH-type spectral invariants.

— The ECH and alternative ECH capacities of closed symplectic 4-manifolds  (2608.19042 - Beiner, 19 Aug 2026) in Conjecture 1.10, labeled conj:ek, subsection “More on the asymptotics of e_k^Alt”