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: <ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://arxiv.org/abs/2008.10125">2008.10125</ext-link> , 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b 2 Superscript plus Baseline equals 1"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>b</mml:mi> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">b_2^+=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we derive asymptotic results that are parallel to the context of ECH (Embedded Contact Homology) and algebraic settings. When assuming <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 1 dot left-bracket omega right-bracket greater-than 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> ⋅ </mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mi> ω </mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c_1\cdot [\omega ]&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.