Surjectivity of the Haiden–Katzarkov–Kontsevich flat-surfaces-to-stability-conditions map: the unresolved case of poles of order at least two

Haiden, Katzarkov and Kontsevich (2017) constructed, for every graded surface $X$ of finite type, a canonical map from the moduli space $\mathcal{M}(X)$ of marked flat structures (equivalently, framed meromorphic quadratic differentials with prescribed singularity type) to the space $\operatorname{Stab}(\mathcal{F}(X))$ of Bridgeland stability conditions on the partially wrapped Fukaya category $\mathcal{F}(X)$, proved it injective with open and closed image, and asked whether the image is all of $\operatorname{Stab}(\mathcal{F}(X))$. The affirmative is known for all 'fully stopped' surfaces — those without unmarked boundary circles, i.e. quadratic differentials with no pole of order at least two — by the authors' disk, annulus and elliptic computations and by Takeda's cutting-and-gluing theorem for relative stability conditions (2022). The unresolved core is the complementary class: flat surfaces with at least one pole of order at least two, where the Fukaya category is defined as a subcategory of a wrapped category excluding infinite-length objects and its Grothendieck group is typically infinitely generated. The problem is to decide, for that class, whether every Bridgeland stability condition is induced by a flat structure, i.e. whether every connected component of $\operatorname{Stab}(\mathcal{F}(X))$ is contained in the image of the HKK map.

Background

A Bridgeland stability condition on a triangulated category $\mathcal{D}$ (Bridgeland 2007) is the data of a group homomorphism $Z:K_0(\mathcal{D})\to\mathbb{C}$ (the central charge, defined on the Grothendieck group $K_0$) together with a slicing — a family of full subcategories of semistable objects of phase $\phi\in\mathbb{R}$ — subject to axioms modeled on slope stability of vector bundles: $Z$ sends a nonzero semistable object of phase $\phi$ to the ray $\mathbb{R}_{>0}e^{i\pi\phi}$, the shift functor rotates phases by one, there are no nonzero morphisms from larger to smaller phase, and every object admits a finite Harder–Narasimhan filtration with semistable subquotients of strictly decreasing phase. Bridgeland showed that the set $\operatorname{Stab}(\mathcal{D})$ of stability conditions satisfying a support property (a boundedness condition for $Z$ on a lattice in $K_0\otimes\mathbb{R}$) is a complex manifold on which $Z$ acts as a local homeomorphism; its connected-component structure is a basic invariant of $\mathcal{D}$.

HKK study these spaces on the symplectic, or A-side, of mirror symmetry. A graded marked surface $(S,M,\eta)$ is a compact oriented surface with boundary whose boundary is decomposed into marked intervals (the set $M$) and unmarked intervals, together with a grading, that is, a line field $\eta$ making intersection numbers of curves integer-valued; each boundary circle is either contained entirely in $M$, or alternates between marked and unmarked intervals, or — a case HKK introduce precisely for the problem at issue — is entirely unmarked. The partially wrapped Fukaya category $\mathcal{F}(S,M)$ is the topological, purely combinatorial, model of the Fukaya category of $S$ (the category of Lagrangian submanifolds of a symplectic surface, with morphisms given by intersection theory — the basic object of the symplectic side of mirror symmetry): it is generated by arcs with endpoints on $M$, and morphism spaces are computed by wrapping arcs along the boundary around marked intervals; when $M=\partial S$ this is the wrapped Fukaya category, and for a full arc system the category is equivalent to the derived category of a gentle algebra (a quiver algebra whose relations are paths of length two). For marked surfaces with unmarked boundary circles HKK define $\mathcal{F}(S,M)$ as the full subcategory of the category $\mathcal{F}(S,M')$ of the fully marked surface (where $M'$ adds all unmarked circles to $M$) consisting of objects carried by immersed curves avoiding the added circles — a device that removes the objects of infinite flat length created by higher-order poles.

The other side of the dictionary is flat geometry. A flat surface (half-translation surface) is a surface glued from Euclidean polygons along parallel edges; equivalently, a Riemann surface $C$ carrying a meromorphic quadratic differential $\varphi$ (a section of the square of the canonical bundle), whose norm defines the flat metric. A zero of order $n$ gives a cone point of angle $(n+2)\pi$, a simple pole gives angle $\pi$, while a pole of order at least two lies at infinite distance and the corresponding end has infinite area (a double pole with closed trajectories is a semi-infinite flat cylinder); exponential singularities, locally $\exp(f(z))g(z)dz^2$ with $f$ meromorphic, are transcendental infinite-angle singularities. Geodesics for the flat metric connecting singularities are saddle connections. HKK fix the topological type by taking $\mathcal{M}(X)$, the moduli space of marked flat structures on a graded surface $X$ of finite type: flat structures together with an identification of the underlying graded surface, up to isotopy.

HKK's two main results connect these worlds. First, they classify indecomposable objects of $\mathcal{F}(S)$: up to shift, every indecomposable is an immersed curve with an indecomposable local system (a locally constant sheaf of modules over a coefficient ring), a tame classification analogous to the classification of indecomposable representations of Dynkin quivers (quivers whose underlying graph is a Dynkin diagram, for which the indecomposables are finite in number and explicitly classified). Second, to a flat structure they associate the period homomorphism sending a curve class to the pairing of its homology class in $H_1(X,\partial X;\mathbb{Z}_\tau)$ (homology with coefficients twisted by the grading) with the square root of $\varphi$, and prove the Main Theorem: the resulting map $\mathcal{M}(X)\to\operatorname{Stab}(\mathcal{F}(X))$ is injective and its image is open and closed, hence a union of connected components of the stability manifold. The introduction states: 'For now it remains an open question whether the image is in fact all of $\operatorname{Stab}(\mathcal{F}(S))$.'

Surjectivity is known in cases with no unmarked boundary circles. HKK prove it directly for three families: the disk with $n+1$ marked boundary intervals (of Dynkin type $A_n$, the representation-theoretic type of the $A_n$ quiver, where $\mathcal{M}=\operatorname{Stab}\cong\mathbb{C}^n$), the annulus with exponential-type singularities $\exp(z^p+z^{-q})dz^2$ (of affine, i.e. extended-Dynkin, type, $\cong\mathbb{C}^{n+1}$), and elliptic curves (via homological mirror symmetry — the conjectural equivalence between the Fukaya category of a symplectic manifold and the derived category of coherent sheaves on a mirror algebraic variety — together with Bridgeland's computation of the stability space of the derived category of an elliptic curve, which identifies it with the universal cover of $\mathrm{GL}^+(2,\mathbb{R})$). Takeda (2022) then settled all fully stopped surfaces — those in which every boundary circle carries at least one marked interval — by introducing relative stability conditions (stability data compatible with a decomposition of the surface) with functorial cutting and gluing, using Bridgeland's generalized metric on $\operatorname{Stab}$ to show that cutting never collapses distinct connected components, and reducing to three base cases: the disk and annulus (from HKK) and the punctured torus (new). His Theorem 2: every Bridgeland stability condition on the Fukaya category of a fully stopped surface is induced by a flat structure, i.e. the HKK map is a global homeomorphism in that case.

Fully stopped excludes exactly the unmarked boundary circles, which in HKK's dictionary are precisely the ends produced by poles of order at least two. The open core is therefore the class of graded surfaces of finite type with unmarked boundary circles, where two structural obstacles appear: HKK's Proposition 5.1 notes that $K_0(\mathcal{F}(S,M))$ is often infinitely generated, so the usual finite-rank support-property and deformation arguments do not directly apply, and the objects excluded near the unmarked ends are exactly the wrapping (band-type) objects around the pole cylinders — for which HKK's Harder–Narasimhan argument explicitly assumes the absence of second-order poles. For context, Bridgeland and Smith had earlier identified moduli of quadratic differentials with unions of components of stability spaces of certain Calabi–Yau-3 categories (triangulated categories modeled on derived categories of Calabi–Yau threefolds, i.e. whose Serre functor acts as the triple suspension) built from quivers with potential (a quiver — a directed graph — together with a formal linear combination of oriented cycles in its path algebra); HKK note that the relation between that category and the wrapped Fukaya category 'is still to be clarified' and suspect that in the case of double poles an additional deformation is necessary — the same singularity regime as the surjectivity question.

Problem Statement

Let $X$ be a graded surface of finite type, and let $\mathcal{M}(X)$ be the moduli space of marked flat structures on $X$ — equivalently, of framed meromorphic quadratic differentials with the prescribed singularity type (zeros, simple poles, poles of order at least two, and exponential singularities) — where 'marked' means that the identification of the underlying graded surface with $X$ is fixed up to isotopy. Let $\operatorname{Stab}(\mathcal{F}(X))$ be the space of Bridgeland stability conditions on the partially wrapped Fukaya category $\mathcal{F}(X)$ of $X$. In HKK's construction, a flat structure with at least one pole of order at least two has, at each such pole, an end of infinite area; one removes a punctured open disk around each of these poles, and each removed disk leaves behind a boundary circle that is declared unmarked. Writing $S$ for the resulting compact core, $M$ for the set of its marked boundary intervals, and $M'$ for the marking obtained by also marking the unmarked circles, $\mathcal{F}(X)=\mathcal{F}(S,M)$ is defined as the full subcategory of the wrapped Fukaya category $\mathcal{F}(S,M')$ consisting of objects carried by immersed curves avoiding the unmarked circles; the presence of an unmarked boundary circle is thus equivalent to the presence of a pole of order at least two, while flat structures with no such pole (zeros, simple poles and exponential singularities only) have no unmarked boundary circles.

Haiden–Katzarkov–Kontsevich constructed a canonical map

$$\mathcal{M}(X)\longrightarrow\operatorname{Stab}(\mathcal{F}(X))$$

and proved it injective with open and closed image, hence a union of connected components of the stability manifold. The problem: decide whether this map is surjective — that is, whether every Bridgeland stability condition on $\mathcal{F}(X)$ is induced by a marked flat structure on $X$ (such conditions are the HKK stability conditions) — for every graded surface of finite type; in the negative direction, exhibit a graded surface $X$ of finite type and a connected component of $\operatorname{Stab}(\mathcal{F}(X))$ containing no HKK stability condition.

Surjectivity is known for surfaces without unmarked boundary circles — quadratic differentials with no pole of order at least two — by HKK's disk, annulus and elliptic computations and by Takeda's theorem for all fully stopped surfaces. The unresolved case is exactly the presence of unmarked boundary circles — flat structures with at least one pole of order at least two, including double poles — where $K_0(\mathcal{F}(X))$ is typically infinitely generated and the category is a proper subcategory of a wrapped category. The question concerns the stated class in its source generality (arbitrary genus and arbitrary configuration of the allowed singularity types), not only special families.

References

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