Remove the rationality assumption for infinite ECH capacities

Establish that the ECH capacities of every closed symplectic 4-manifold are infinite without assuming that the symplectic form is rational, including the case of an irrational-area-ratio symplectic form on S^2 times S^2.

Background

The paper proves that all ECH capacities are infinite for closed rational symplectic 4-manifolds. The proof uses Donaldson divisors, which require an integral or rational cohomology class after rescaling. The authors indicate that they expect both the general infiniteness theorem and its stronger version under the condition c_1(TX,Ω)·[Ω]≤0 to remain valid without rationality.

The paper identifies a symplectic form on S2×S2 for which the ratio of the areas of the two sphere factors is irrational as the simplest example whose ECH capacities remain unresolved. Resolving this case would provide a basic test of whether the rationality hypothesis is merely technical.

References

We expect \Cref{thm:closed_inf,thm:posinf} to hold without the rationality assumption. Note that Cor.! 4.13 gives many closed irrational symplectic 4-manifolds whose ECH capacities are infinite. The simplest example of a closed symplectic 4-manifold where the ECH capacities remain unknown is a symplectic form on $S2\times S2$ where the ratio of the areas of the $S2$ factors is irrational.

— The ECH and alternative ECH capacities of closed symplectic 4-manifolds  (2608.19042 - Beiner, 19 Aug 2026) in Section 1, subsection “Results on the ECH capacities”