Establish the $G$-equivariant noncommutative minimal model program: quasi-convergent paths of equivariant stability conditions from the equivariant quantum differential equation

The $G$-noncommutative minimal model program ($G$-NMMP) of Wu and Zhang (arXiv:2602.20335, 2026) conjectures that to every $G$-contraction $\pi:X\to Y$ of a smooth projective variety with an action of an algebraic group $G$ one can attach canonical quasi-convergent paths in the space of Bridgeland stability conditions on the equivariant derived category $D^b_G(X)$, whose limits cut $D^b_G(X)$ into $G$-linear semiorthogonal decompositions with functorial refinement properties (Conjecture 3.8), and that these paths can be built with central charges given by equivariant periods $\mathrm{ev}_{\mathbf{s}}\int_X^{eq}\Phi^G_t(v^G(-))$ of a fundamental solution of the $G$-truncated quantum differential equation (Proposal 3.11). Both statements are verified only in restricted regimes — finite-group lifts of known non-equivariant solutions and $\mathbb{T}$-equivariant projective spaces — and the case of trivial $G$ is Halpern-Leistner's original noncommutative minimal model program, itself open in general. The problem is to prove or refute these conjectures in the stated generality.

Background

The minimal model program (MMP) is the birational-geometric procedure that simplifies a projective variety by contracting curves spanning extremal rays (the edge rays of the cone of effective curves, along which a contraction morphism exists) until one reaches a minimal model or a Mori fiber space (a fiber-type end product, such as a projective bundle over a lower-dimensional base); when an algebraic group $G$ acts on the variety, the $G$-equivariant MMP carries out the same program while preserving the group action. Bondal and Orlov observed that birational transformations occurring in the MMP, such as blow-ups, induce semiorthogonal decompositions of the bounded derived category $D^b(X)$ of coherent sheaves. A semiorthogonal decomposition (SOD) $D^b(X)=\langle\mathcal{A}_1,\dots,\mathcal{A}_n\rangle$ expresses the category as generated by full subcategories with $\mathrm{Hom}(\mathcal{A}_j,\mathcal{A}_i)=0$ for $j>i$; an exceptional collection of objects — a sequence $E_1,\dots,E_n$ with $\mathrm{Hom}(E_j,E_i[k])=0$ for $j>i$ and all shifts $k$, and $\mathrm{End}(E_i)\cong\mathbb{C}$ — is the most familiar source. Halpern-Leistner's noncommutative minimal model program (NcMMP) reverses the perspective: instead of deriving decompositions from geometry, it proposes to run a categorical MMP that produces them canonically, with birational geometry recovered as a shadow.

The engine of the NcMMP is the theory of Bridgeland stability conditions. A Bridgeland stability condition on a triangulated category $\mathcal{D}$ is a pair $\sigma=(Z,\mathcal{P})$ consisting of a group homomorphism $Z:K(\mathcal{D})\to\mathbb{C}$ (the central charge, satisfying a support property bounding masses) and a slicing of $\mathcal{D}$ by abelian categories of semistable objects of constant phase; stability conditions form a complex manifold $\mathrm{Stab}(\mathcal{D})$. The NcMMP predicts that to a contraction one can attach a quasi-convergent path in $\mathrm{Stab}(X)/\mathbb{C}$: a family of stability conditions $\sigma_t$ for $t\to\infty$ such that, in the limit, every object of $D^b(X)$ admits a filtration whose factors are limit semistable of strictly decreasing phase, so the limiting behavior splits the category into a semiorthogonal decomposition — the categorical counterpart of the contraction's exceptional locus.

The second input is quantum cohomology. For a Fano variety $X$, the quantum product deforms the cup product using Gromov–Witten invariants (counts of curves subject to prescribed incidence conditions), and the associated quantum differential equation governs the flat sections of the quantum connection (the flat connection on cohomology whose horizontal sections are the quantum periods). Dubrovin's conjecture and its refinement Gamma Conjecture II predict that the asymptotic behavior of a fundamental solution of this equation encodes full exceptional collections in $D^b(X)$. Halpern-Leistner's central proposal accordingly predicts that the central charges along the conjectured quasi-convergent paths are periods $\int_X\Phi_t(v(-))$ of a fundamental solution $\Phi_t$ of the truncated quantum differential equation, where $v$ is a twisted Chern character on the numerical $K$-group.

Wu and Zhang's $G$-NMMP extends this framework equivariantly. For a smooth projective variety $X$ with an action of an algebraic group $G$, the equivariant derived category $D^b_G(X)$ has as objects $G$-equivariant coherent sheaves, and one studies its stability manifold $\mathrm{Stab}(X_G)$; a $G$-contraction is a projective surjective $G$-equivariant morphism $\pi:X\to Y$ with connected fibers. For a connected reductive group with maximal torus $\mathbb{T}=(\mathbb{C}^*)^m$, they introduce $\mathbb{T}$-stability conditions: stability conditions on the $\mathbb{T}$-category $\mathcal{D}_{\mathbb{T}}$ (the equivariant category with its $m$ commuting shift autoequivalences) that are compatible with the torus action up to the action of characters, generalizing the $q$-stability conditions studied for Calabi–Yau-$\mathbb{X}$ categories (categories whose Serre functor is modeled as a formal shift $\mathbb{X}$). The conjectured central charges take the equivariant periods $Z^G_t(\mathcal{E})=\mathrm{ev}_{\mathbf{s}}\int_X^{eq}\Phi^G_t(v^G(\mathcal{E}))$, where $\Phi^G_t$ is a fundamental solution of the $G$-truncated quantum differential equation, $v^G$ is the $G$-equivariant twisted Chern character, and $\mathrm{ev}_{\mathbf{s}}$ evaluates the equivariant cohomology parameters at a generic point $\mathbf{s}\in\mathbb{C}^m$; the genericity is needed because the equivariant periods are a priori meromorphic in these parameters. A stability condition is geometric if all $G$-point sheaves $\mathrm{Ind}^{G}_{G_x}(\mathcal{O}_x)$ (skyscraper sheaves induced from stabilizers) are stable of the same phase; the conjectured paths should ideally start at such points.

What is known: Halpern-Leistner verified the non-equivariant proposal for $\mathbb{P}^1$ via the stability manifold of $\mathbb{P}^1$; Zuliani handled projective spaces from small quantum cohomology; Karube treated certain blow-up surfaces; and Karube–Robotis–Zuliani recently extended the construction to Grassmannians, smooth quadrics, and smooth cubic threefolds and fourfolds. The general mechanism relating quasi-convergent paths to semiorthogonal decompositions is developed by Halpern-Leistner–Jiang–Robotis. Wu and Zhang prove that non-equivariant solutions lift to finite group actions by induction on equivariant exceptional collections, and they construct $\mathbb{T}$-equivariant quasi-convergent paths for projective spaces using Cotti–Varchenko's Stokes bases of the equivariant quantum differential equation (flat-section bases adapted to its exponential asymptotics); geometric basepoints are obtained only for $\mathbb{P}^2$ with $\mathbb{T}=(\mathbb{C}^*)^3$. Their birational applications — a $G$-equivariant $D$-equivalence statement for Calabi–Yau and pluricanonically base-point-free varieties, and a $G$-equivariant one-way implication of Dubrovin's conjecture — are proved only conditionally on Conjecture 3.8 and Proposal 3.11. Everything outside these regimes is open, and for $G$ trivial the conjectures reduce exactly to Halpern-Leistner's original open ones.

Problem Statement

Prove or refute the central conjectures of the $G$-equivariant noncommutative minimal model program, in the generality stated by Wu and Zhang (arXiv:2602.20335), which together form one coherent objective: constructing the conjectured canonical stability-theoretic mechanism for $G$-equivariant contractions and identifying its quantum-cohomological central charges.

(1) Conjecture 3.8. Let $\pi:X\to Y$ be a $G$-contraction (a projective surjective $G$-equivariant morphism with connected fibers) of a smooth projective variety $X$.

(A) One can associate to $\pi$ a canonical family of quasi-convergent paths $\{\sigma^{\pi,\psi}_t\}$ in $\mathrm{Stab}(X_G)/\mathbb{C}$, indexed by $\psi=\omega+\mathrm{i}B\in H^2(X;\mathbb{C})$ with $\omega$ relatively ample (the parameter of the $G$-truncated quantum endomorphism $E^G_\psi$), such that generic values of the parameter $\psi$ yield a $G$-linear semiorthogonal decomposition of $D^b_G(X)$, and different generic values give mutation-equivalent decompositions (related by mutations, the elementary operation of exchanging two adjacent semiorthogonal components).

(B) For generic $\psi$, the semiorthogonal factors are closed under tensor product with $\pi^*(E)$ for $E\in\mathrm{Perf}_G(Y)$, the category of perfect $G$-equivariant complexes on $Y$.

(C) Given a further $G$-contraction $Y\to Y'$, the decomposition associated with $X\to Y'$ refines the one associated with $X\to Y$.

(D) For a $G$-equivariant morphism $\pi:X\to X'$ of smooth varieties with $R\pi_*(\mathcal{O}_X)=\mathcal{O}_{X'}$, the decomposition of $D^b_G(X)$ associated to $X\to Y$ refines the decomposition obtained by combining the decomposition of $\pi^*(D^b_G(X'))\cong D^b_G(X')$ associated to $X'\to Y$ with the Bondal–Orlov decomposition $D^b_G(X)=\langle\ker(R\pi_*),\pi^*(D^b_G(X'))\rangle$.

(2) Proposal 3.11. Let $\pi:X\to Y$ be a $G$-contraction of a smooth projective variety of dimension $d$ and let $\Phi^G_t\in\mathrm{End}(H^*_G(X;\mathbb{C}))$ be a fundamental solution of the $G$-truncated quantum differential equation. Then for any generic tuple $\mathbf{s}=(s_1,\dots,s_m)\in\mathbb{C}^m$ (one parameter per torus character; here $m=\mathrm{rank}\,\mathbb{T}$) there exists a quasi-convergent path $\sigma^G_t=(Z^G_t,\mathcal{P}^G_t)$, $t\in[t_0,\infty)$, in $\mathbb{T}\mathrm{Stab}_{\mathbf{s}}(X)\subseteq\mathrm{Stab}(X_G)/\mathbb{C}$ whose central charges are

$Z^G_t(\mathcal{E})=\mathrm{ev}_{\mathbf{s}}\int_X^{eq}\Phi^G_t(v^G(\mathcal{E})),\qquad\mathcal{E}\in K^G(X),$

including the well-definedness of the evaluation $\mathrm{ev}_{\mathbf{s}}$ for generic $\mathbf{s}$ and the asymptotic separation of the leading exponential terms required for quasi-convergence.

Both statements must be established beyond the regimes already verified — finite-group lifts of the known non-equivariant solutions ($\mathbb{P}^1$, projective spaces, certain blow-up surfaces, Grassmannians, smooth quadrics, smooth cubic threefolds and fourfolds) and $\mathbb{T}$-equivariant projective spaces. Since the case $G=1$ reduces exactly to Halpern-Leistner's original NcMMP conjectures, any general proof must in particular settle those. The required result is to prove or refute both statements (1) and (2) in the stated generality: since Conjecture 3.8 is the conjunction of its parts (A)–(D), a proof of (1) requires all four parts, while a refutation of any one part refutes that conjecture, and a complete answer must in addition resolve Proposal 3.11; results addressing only some parts of Conjecture 3.8, or only one of the two statements, are partial progress and do not by themselves complete the task. Three further open directions recorded in the source belong to the same objective but are optional and are not required for a complete answer: extending the $\mathbb{T}$-stability framework to settings where $K(\mathcal{D}_{\mathbb{T}})$ is not free of finite rank over the representation ring $R(\mathbb{T})$ (the ring of isomorphism classes of finite-dimensional algebraic representations of $\mathbb{T}$), and to reductive groups beyond reduction to a maximal torus with Weyl-group invariance; producing geometric basepoints (stability conditions under which all $G$-point sheaves $\mathrm{Ind}^{G}_{G_x}(\mathcal{O}_x)$ are stable of the same phase) for general classes of equivariant Fano varieties, beyond the $\mathbb{P}^2$ case of Theorem 5.22; and verifying in concrete equivariant geometries the auxiliary hypotheses of Proposition 6.5, namely that $\mathrm{ch}_G:K^G(X)\otimes\mathbb{Q}\to H^*_G(X;\mathbb{Q})$ is an isomorphism and that $E_\psi(1)$ is semisimple with distinct eigenvalues for some $\psi$.

References

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