Stokes Geometry and Presentation Complexity: Four Applications

Definition 2.1 (Presentation lower-bound datum).

Let \(\mathcal C\) be a class of objects with presentation cost \(\mathbf C_{\mathcal C}\). A presentation lower-bound datum consists of an isomorphism-invariant observable

a cost \(\mathbf C_I\) on values of \(I\), and a nondecreasing overhead function \(F_I\) such that every complete presentation of \(X\in\mathcal C_0\) determines \(I(X)\) with cost at most \(F_I\) of the presentation cost.

Theorem 2.2 (Observable lower bound).

For every presentation lower-bound datum and every \(X\in\mathcal C_0\),

In the affine-linear case \(F_I(t)=at+b\), this gives

Definition 3.1 (Turning cost).

Fix a presentation model for closed subvarieties or analytic subsets of \(D\). The turning cost of \(M\) is

One may also use a resolution cost

where \(\pi:Y\to X\) ranges over admissible modifications for which \(\pi^\dagger M\) has good formal structure.

Theorem 3.2 (Turning lower bound).

Suppose every complete global presentation of \(M\) determines \(\operatorname{Turn}(M)\) with overhead \(F_{\operatorname{turn}}\). Then

Theorem 3.3 (Resolution lower bound).

Suppose every complete global Stokes presentation determines an admissible good-formal-structure modification with overhead \(F_{\operatorname{res}}\). Then

Remark 3.4.

Teyssier's criterion identifies the good formal structure locus with a solution-theoretic local-system locus for the restrictions of the solution complexes of \(M\) and \(\operatorname{End}M\). In classes where this criterion applies, the same lower bound can be read on the solution side: a presentation of enough irregular Riemann-Hilbert data must detect where those restrictions fail to be local systems. The classical theorem identifies the obstruction; the presentation-theoretic layer turns it into a cost lower bound.

Assumption 4.1 (Tame Morse direct-image class).

We work in a tame Morse class of exponential direct images for which the relevant irregular object \(M\), for instance a direct image of \(\mathcal O_X e^g\), has formal exponential factors determined by the critical values \(c_i\), and Stokes matrices determined by the corresponding Picard-Lefschetz intersection data.

Definition 4.2 (Critical Stokes data).

Let

Let \(\Theta_{\operatorname{crit}}\) be the Stokes-direction configuration determined by the differences in \(\Delta C\), and let \(S_{\operatorname{PL}}\) denote the Picard-Lefschetz/Stokes matrices in the chosen tame class. Define

Theorem 4.3 (Direct-image lower bound).

Let \(M\) be an exponential direct-image object satisfying the tame Morse assumption. Suppose every complete presentation of \(M\) determines the critical Stokes data

with overhead \(F_{\operatorname{critSt}}\). Then

Example 4.4 (Two critical values).

If there are two critical values \(c_1,c_2\), then the exponential separation is \(c_1-c_2\). In a one-parameter asymptotic coordinate, the Stokes directions are governed by

If the relevant Picard-Lefschetz jump is represented in a chosen basis by

then any complete presentation in a cost model charging for the separation and the jump coefficient satisfies a lower bound of the form

with constants determined by the extraction overhead and the chosen encodings.

Definition 5.1 (Bounded global Stokes class).

Let \(B\) be the boundary of the real oriented blow-up along the pole divisor. A bounded global Stokes class is a class of objects whose global Stokes data are constructible with respect to cell decompositions of \(B\) having at most \(N_0\) cells, with Stokes cocycles represented on covers of bounded combinatorial type and with all local Stokes formats drawn from a bounded list.

Definition 5.2 (Stokes-determining curve family).

A finite family of holomorphic curves \(\mathcal K=\{C_1,\ldots,C_N\}\) is Stokes-determining for a bounded global Stokes class if the restricted Stokes presentations of \(M|_{C_i}\) determine the global Stokes data of \(M\) up to the equivalence relation used in the bounded class.

Theorem 5.3 (Finite curve determination under incidence hypotheses).

Let \(\mathcal F\) be a bounded global Stokes class. Suppose a finite family of curves has boundary maps meeting every cell of the Stokes cell decomposition and every incidence needed to determine the chosen Cech representative of the Stokes gluing class. Then the restricted Stokes data on these curves determine the global Stokes presentation for every \(M\in\mathcal F\).

Theorem 5.4 (Missing-cell obstruction).

In the same bounded setting, suppose a finite family of curves fails to meet a cell or incidence on which the global Stokes data are allowed to vary within the class. Then the family is not Stokes-determining.

Definition 6.1 (Bounded-presentation locus).

Fix a wild character variety

with a finite Stokes coordinate atlas. For a point \(x\), let \(\mathbf C_{\operatorname{St}}^\ast(x)\) be the minimal cost of a Stokes presentation of \(x\) in the chosen atlas. Define

Theorem 6.2 (Constructibility in fixed coordinate models).

Assume that, in each chart of the chosen finite Stokes coordinate atlas, the condition \(\mathbf C_{\operatorname{St}}(P_x)\le B\) is a finite Boolean combination of algebraic equations, inequations, rank bounds, support conditions, and global relation equations. Then

is constructible.

Theorem 6.3 (Generic dimension lower bound).

Let a component of a wild character variety have effective Stokes-parameter dimension \(d_{\operatorname{eff}}\) after imposing the global relation and quotienting by the generic gauge action. In any coordinate presentation model that charges at least linearly for independent scalar parameters, a generic point of that component has presentation complexity at least

Theorem 6.4 (Active-root lower bound).

Let \(G\) be reductive and \(Q\) an irregular type. Let

be the active-root set. On any class where the active-root data are extracted with overhead \(F_{\operatorname{act}}\),

If the active-root cost dominates cardinality, this gives a lower bound linear in \(\#R_{\operatorname{act}}(Q)\).