Family 055 — Stability at the Gepner point and at large volume

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Family 055 · Algebraic and complex geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

Imagine assigning a complex number to every geometric object, then sorting the objects by the angle of that number. One would like every complicated object to split into a finite sequence of simpler pieces, ordered by angle. The number must also faithfully measure the size of those pieces: arbitrarily large numerical objects should not acquire charges arbitrarily close to zero.

This is the outline of a Bridgeland stability condition. It turns a derived category, a setting containing sheaves and complexes of sheaves, into a space where one can discuss stable constituents, their moduli, and changes of stability.

Family 055 asks for particular stability conditions predicted by geometry and physics. Its first manuscript claims the prescribed symmetry at the Gepner point for every smooth quintic threefold. The second claims exact large-volume charges on every smooth projective threefold with trivial canonical bundle, and separately proves uniform numerical inequalities on arbitrary smooth projective threefolds. Their subject is more specific than the existence of some stability condition.

2. Background and essential definitions

Charges, phases, and support

A coherent sheaf generalizes a vector bundle by allowing fibers of varying dimension and singular behavior. The derived category $D^b(\operatorname{Coh}X)$ also contains finite complexes of sheaves, identifying complexes with the same cohomological information. Its shift operation $[1]$ moves a complex by one degree.

The numerical Grothendieck group $K_{\mathrm{num}}(X)$ records additive classes of these objects, identifying classes that have the same Euler pairings with all others. The Euler pairing is the alternating sum of dimensions of the extension groups. A central charge is a homomorphism

$$Z:K_{\mathrm{num}}(X)\longrightarrow\mathbb C.$$

A nonzero semistable object of phase $\phi$ has $Z(E)=m e^{i\pi\phi}$ with $m>0$. Phases are real numbers, not angles modulo a full turn: shifting by $[1]$ raises the phase by one. A stability condition specifies the semistable objects and requires every object to have a finite filtration with decreasing phases, called a Harder–Narasimhan filtration.

Its support property is a bound

$$\|[E]\|\le C|Z(E)|$$

for every semistable object, using a norm on the numerical vector space. It prevents numerical complexity from becoming invisible to the charge. Local finiteness imposes a corresponding finite-length condition in short phase intervals.

Hearts and tilting

A heart is an abelian category inside the derived category where kernels, quotients, and short exact sequences can be used. Coherent sheaves form one heart. A tilt constructs another by shifting a selected class of sheaves and closing under extensions. On threefolds, two successive tilts provide the standard route to geometric stability conditions.

For a sheaf, its Chern character $\operatorname{ch}(E)$ packages rank, divisor, curve, and point data. A real divisor class $B$ gives the twisted character $\operatorname{ch}^B(E)=e^{-B}\operatorname{ch}(E)$. An ample class $H$ specifies a polarization, and $t>0$ scales it. In the large-volume problem, $t$ becomes large while the geometry of $X$ remains fixed.

3. Development history and previous partial results

Douglas’s proposal of stability for D-branes motivated the categorical viewpoint, and Bridgeland supplied its mathematical axioms and deformation theory. Bayer, Macrì and Toda developed the double-tilt approach to threefolds, identifying inequalities involving the third Chern character as a central obstacle. Bridgeland, Bayer–Macrì–Toda.

For a quintic, Toda formulated the desired Gepner symmetry and proposed the relevant pair of tilts. The symmetry functor and its fivefold periodicity were already known. The difficult part was to construct the semistable objects with finite filtrations and support for the prescribed, generally irrational, charge. Li established the existence of numerical stability conditions on every smooth quintic in 2019. Xu subsequently proved a stronger sheaf inequality that M1 uses as an input. Thus M1’s claimed addition is the exact Gepner symmetry, not the first stability condition on a quintic. Toda’s formulation, Li’s theorem, Xu’s inequality.

For large volume, Bayer’s polynomial stability describes asymptotic phase order. A genuine Bridgeland condition at each sufficiently large finite value of $t$ requires additional control. The threefold inequalities were established in important cases such as abelian threefolds, but an uncorrected inequality at every parameter is false: Schmidt and Martinez–Schmidt produced counterexamples. Corrections depending on the geometry became part of the problem. Bayer’s polynomial stability, Schmidt’s counterexample.

The historical position changed further in 2026. Li and collaborators developed general constructions; Cheng established full numerical support, while Cheng–Feyzbakhsh identified threefold hearts and proved a weak form of the BMT conjecture. M2 explicitly builds on these results. It claims prescribed ordinary and Todd-corrected charges, a common threshold over a real parameter sector, and stronger uniform inequalities. It does not claim that general nonemptiness was still open. Cheng, Cheng–Feyzbakhsh.

4. What the new manuscripts claim

M1: exact quintic Gepner symmetry

Let $X\subset\mathbb P^4_{\mathbb C}$ be any smooth degree-five hypersurface. It has trivial canonical bundle. Define

$$\Phi=T_{\mathcal O_X}\circ(-\otimes\mathcal O_X(1)),$$

where tensoring by the hyperplane bundle acts first. The spherical twist $T_{\mathcal O_X}$ is the categorical operation obtained from the evaluation map from copies of $\mathcal O_X$ to an object. The established relation is $\Phi^5\simeq[2]$.

M1 claims a numerical stability condition $(Z,\mathcal P)$ with full support satisfying

$$\begin{aligned} Z(\Phi E)&=e^{2\pi i/5}Z(E),\\ \Phi\bigl(\mathcal P(\phi)\bigr)&=\mathcal P(\phi+2/5),\\ Z(\mathcal O_x)&=-1. \end{aligned}$$

Here $\mathcal P(\phi)$ is the category of semistable objects of phase $\phi$, and $\mathcal O_x$ is the sheaf concentrated at a point. The normalization identifies the charge with Toda’s proposed one. The theorem holds for every smooth quintic, without a general-position restriction.

M2: exact large-volume charges

Let $X$ be any smooth projective complex threefold with $K_X\simeq\mathcal O_X$. For every chosen real twist $B_*$ and ample real direction $H_*$, M2 claims neighborhoods $V,W$ and a single threshold $T$ working for all $B\in V$, $H\in W$, and $t\ge T$.

There are geometric numerical stability conditions with each of the exact charges

$$\begin{aligned} Z_{B,tH}(E)&=-\int_X e^{-B-itH}\operatorname{ch}(E),\\ Z^{\mathrm{LV}}_{B,tH}(E)&=-\int_X e^{-B-itH}\operatorname{ch}(E)\sqrt{\operatorname{td}(X)}. \end{aligned}$$

The integral selects the top-degree part of the expression. The Todd class is a characteristic class of the tangent bundle; here its square root is $1+c_2(X)/24$. “Geometric” means all point sheaves are stable of phase one.

Both constructions have support on the full numerical group. The ordinary charge has the standard double-tilt heart. The Todd-corrected charge has a separately constructed heart; the source does not identify the two hearts. No additional vanishing of $H^1(X,\mathcal O_X)$ or $H^2(X,\mathcal O_X)$ is required.

An independent inequality on arbitrary threefolds

Fix any smooth projective complex threefold, an integral ample $H$, and a rational divisor class $B_0$. Put $B=B_0+bH$, with $b\in\mathbb R$. For objects semistable at zero slope for

$$\nu_{a,b}(E)=\frac{H\operatorname{ch}_2^B(E)-a^2H^3\operatorname{ch}_0(E)/2}{H^2\operatorname{ch}_1^B(E)},$$

M2 claims constants $k\ge0$ and $R_*$, independent of the object and $b$, such that

$$\operatorname{ch}_3^B(E)\le\left(\frac{a^2}{6}+k\right)H^2\operatorname{ch}_1^B(E)$$

for every $a>0$. For $a>R_*$, the same inequality holds with $k=0$. This assertion requires finite slope, and has no canonical-bundle assumption. An all-slope quadratic version follows in a corresponding large-volume region. Its parameter is related to the physical volume by $t=\sqrt3\,a$; confusing these normalizations changes the coefficient.

5. Why the advance matters

Importance: 5/5. The claims would connect two distinguished regions of threefold stability theory with explicit constructions: a highly symmetric point on every quintic and prescribed large-volume charges on all threefolds with trivial canonical bundle. They also address a uniform numerical inequality on arbitrary polarized threefolds.

Exactness matters. A nearby charge, or a charge containing an extra curve-class correction, need not represent the same geometric or physical specification. Likewise, a condition that controls only four intersection numbers may miss numerical directions when the Picard rank is greater than one. Full support and one threshold over an open parameter sector substantially strengthen a collection of isolated existence statements.

The Gepner theorem would resolve a specific longstanding symmetry conjecture. The large-volume theorem would offer a consistent starting region for studying stable complexes and their variation. Neither manuscript claims that all paths between these regions or all resulting moduli spaces have thereby been determined.

6. The main idea

Build support before completing the slicing

M1 begins with Toda’s two tilts, but varies the second cut in two independent numerical directions. Xu’s inequality supplies room for these perturbations. Combining the perturbed hearts with the symmetry $\Phi$ produces a periodic grid of categories whose charges lie in narrow angular sectors.

Testing an object against all perturbations controls its rank and first Chern coordinate. Together with the complex charge, these control the remaining two coordinates of the quintic’s rank-four numerical lattice. This yields support before the final semistable filtration is constructed.

The distinction between a lattice and its image in $\mathbb C$ is crucial. The charge image need not be discrete. The full numerical lattice is discrete, however, so a bound on numerical classes gives finite refinement. The source then merges shared boundary phases and establishes the slicing and local finiteness.

Preserve all numerical directions at large volume

M2 embeds the threefold in a product of projective spaces using sufficiently many line bundles. This retains the full numerical class rather than just the degrees along one polarization. It constructs ambient stability conditions with a support estimate after rescaling each Chern-character coordinate by the appropriate power of $t$.

In those scaled coordinates, fixed higher-degree corrections become small at large volume. Deformation and restriction can therefore produce the exact desired charge, with the unwanted Todd factors canceled beforehand. Surface restrictions identify the ordinary charge’s heart as the standard double tilt; the boundary at zero slope receives a separate argument.

The numerical inequality follows by another route. Restriction to surfaces and cohomological estimates give a rank-uniform error bound at one fixed volume. Reduction to positive characteristic and Frobenius pullback remove object-dependent errors in a limit. Moving along stability walls then yields the corrected all-volume inequality and an uncorrected large-volume tail. The manuscript treats this argument as independent of the categorical construction.

7. Scope, limitations, and open questions

The Gepner statement concerns smooth quintic threefolds, not arbitrary Calabi–Yau varieties or arbitrary hypersurface degrees. Its precise functor order, phase shift, and point-charge normalization belong to the theorem.

The prescribed-charge theorem requires trivial $K_X$, whereas the separate inequality allows arbitrary smooth projective threefolds. Its common threshold is local in the real twist and ample-direction parameters; it is not a single bound for the entire ample cone of every variety.

Known counterexamples to the uncorrected BMT inequality remain relevant. M2 allows a correction at small volume and removes it only past a threshold. Tensoring known counterexamples can move the twist without moving them into arbitrarily large volume, so twist uniformity and volume uniformity are different issues.

Natural further questions concern the global stability manifold, paths connecting distinguished regions, wall-crossing of particular classes, and effective thresholds. Existence with a specified charge does not automatically classify stable objects or prove every predicted enumerative correspondence.

8. Sources and further reading

M1’s Theorem 1.1 states the normalized Gepner claim; Sections 3–4 explain the two-parameter construction and the support-first refinement. M2’s Theorems 1.1–1.2 separate the prescribed-charge and numerical statements; Sections 2–3 and 4–7 develop their independent arguments, and Section 8 compares the known counterexamples.

The primary historical links above give the successive stages: Bridgeland’s axioms, Toda’s conjecture, Li’s quintic existence theorem, Xu’s sheaf bound, and the 2026 constructions of Cheng and Cheng–Feyzbakhsh. For the limitations of uncorrected inequalities, see also Martinez–Schmidt.

Source manuscripts

• M1. OpenAI, A Gepner stability condition on every smooth quintic threefold (September 24, 2026).
• M2. OpenAI, Prescribed large-volume charges on threefolds with trivial canonical bundle (September 24, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.