ALGEBRAIC CAPACITIES AS TROPICAL POLYNOMIALS OVER THE REDUCED c₁-POSITIVE SYMPLECTIC CONE

In a series of work by Wormleighton [Selecta Math. (N.S.) 28 (2022), p 9], Wormleighton [J. Symplectic Geom. 19 (2021), pp. 475-506], and Chaidez and Wormleighton [ECH embedding obstructions for rational surfaces, arXiv:2008.10125, 2020], algebraic capacities were introduced in an algebraic manner for polarized algebraic surfaces and applied to the symplectic embedding problems. In this paper, we give a reformulation of algebraic capacities in terms of only a tamed pair of symplectic form and almost complex structure. We show that they actually only depend on the cohomology class of the symplectic form for a rational manifold. Since it is not known that any symplectic form on a rational manifold is Kähler, this novel formulation potentially is more general on a rational manifold. Additionally, for manifolds with b⁺₂ = 1, we derive asymptotic results that are parallel to the context of ECH (Embedded Contact Homology) and algebraic settings. When assuming c1 · [ω] > 0 on rational manifolds, we further introduce a sequence of tropical polynomials which will succinctly describe those capacities viewed as functions over the domain parametrizing such symplectic forms. As an application, we give a purely symplectic proof of the correspondence between algebraic capacities and ECH capacities for smooth toric surfaces.