Noncommutative Minimal Model Program

• Noncommutative MMP is a framework extending classical minimal model techniques to noncommutative orders, Brauer pairs, and derived categorical settings using b-log methods.
• It employs methodologies such as b-log discrepancies, Brauer class analysis, and semiorthogonal decompositions to study birational transformations and singularity resolutions.
• The program informs practical applications in arithmetic surfaces, Fano varieties, and noncommutative projective surfaces through derived equivalences and stability conditions.

The Noncommutative Minimal Model Program (noncommutative MMP) is a broad generalization of the classical minimal model program for algebraic varieties, extending its scope to noncommutative and categorical geometries. It encompasses b-log MMPs for noncommutative orders—such as Brauer pairs and arithmetic surfaces with Brauer classes—as well as categorical and derived enhancements via semiorthogonal decompositions and Bridgeland stability conditions. The noncommutative MMP provides foundational methods for understanding birational transformations, minimal models, and the structure of derived categories in a noncommutative context.

1. b-Log MMP and Noncommutative Orders

The b-log minimal model program extends the classical log MMP to b-log varieties, which are pairs $(X, \mathbf{D})$ of a proper normal variety $X$ and a fractional b-divisor $\mathbf{D}$. A b-divisor $\mathbf{D}$ assigns to each model $X$ (normal proper birational models of a function field $K$ over an algebraically closed field of characteristic $0$) a $\mathbb{Q}$-Weil divisor $\mathbf{D}_X$, compatible under birational pushforward. The fractional condition requires coefficients $d_\Gamma\in [0,1)$ for all geometric valuations $\Gamma$ of $K$ (Chan et al., 2017).

The canonical divisor in this setting is $K_X + \mathbf{D}_X$, with discrepancies and singularity types defined analogously to the classical theory. For a birational morphism $f: Y\to X$, one defines b'-discrepancies $b'(E;X,\mathbf{D})$ for $f$-exceptional divisors $E$, and reweights them via the ramification indices $r_E=1/(1-d_E)$ to obtain b-discrepancies $b(E;X,\mathbf{D})= r_E b'(E;X,\mathbf{D})$. This allows a uniform treatment of classical, equivariant, and noncommutative settings.

When specialized to noncommutative orders—such as maximal orders in central simple algebras over $K$—the program singles out Brauer pairs $(X, \alpha)$, consisting of a normal proper model $X$ and a Brauer class $\alpha\in \mathrm{Br}(K)$. The ramification of $\alpha$ is reflected in the fractional b-divisor $\mathbf{D}_\alpha = \sum_{D} (1-1/r_D) D$, where $r_D$ is the local ramification index at each prime $D \subset X$. The "noncommutative canonical divisor" is $K_X + \mathbf{D}_{\alpha,X}$, and the corresponding minimal model program for Brauer pairs is fully encoded within the b-log MMP structure (Chan et al., 2017).

Key Theorems

The main theorems for the b-log MMP hold for noncommutative settings:

• Existence: If $K_X+\mathbf{D}_X$ is not $\pi$-nef ($\pi:X\to U$ projective), extremal contractions and flips exist.
• Preservation: b-terminal, b-canonical, b-log-terminal, and b-log-canonical singularities are preserved throughout each step.
• Termination: Termination of the classical log MMP for $(X, \mathbf{D}_X)$ implies termination for $(X,\mathbf{D})$.
• Uniqueness: Minimal models for the same b-pair are isomorphic in codimension one and differ by b-flops only.
• Specialization: Ordinary and equivariant MMPs arise as special cases.

For Brauer pairs, any such $(X,\alpha)$ admits a projective birational $f:Y\to X$ with $(Y,\mathbf{D}_\alpha)$ b-terminal and $Y$ $\mathbb{Q}$-factorial; the noncommutative MMP is run on such a $Y$, and, upon termination, yields a minimal model with a maximal order $\Lambda_Z$ and nef/semiample noncommutative canonical divisor $K_Z+\mathbf{D}_{\alpha,Z}$ (Chan et al., 2017).

2. Arithmetic Surfaces with Brauer Classes

The noncommutative MMP is also formulated for arithmetic surfaces enriched by a Brauer class $\beta$. Here, $X$ is a normal, integral, separated, 2-dimensional, excellent scheme quasi-projective over a finite field base, and $\beta \in \mathrm{Br} K(X)'$ is a Brauer class of order $p>5$ prime to all residue characteristics. Maximal orders correspond to central simple algebras over the function field $K(X)$, and the program studies pairs $(X,\beta)$ (or associated orders $\mathcal{A}$), termed $\beta$-surfaces (Chan et al., 2021).

The log discrepancy $a_E$ for a divisor $E$ over $X$ is modified to the b-discrepancy $b_E = a_E + (1-1/n_E)$, adding the contribution of the ramification index $n_E$ of $\beta$ along $E$. $\beta$-terminality ($b_E > 0$ for all $E$), $\beta$-canonicity ($b_E \geq 0$), and their roles in the stepwise process follow the commutative paradigm, but drastically new phenomena occur, such as $\beta$-twisted blowups and contractible ramified $(-p)$-curves, which do not appear in the classical MMP.

The three-step process is:

1. $\beta$-terminal resolutions: Existence of birational models with $\beta$-terminality.
2. $\beta$-Castelnuovo contractions: Contraction of $K_{X,\beta}$-negative irreducible curves, with subtle behavior in the ramified case.
3. Zariski factorization: Any birational map between two $\beta$-terminal models factors uniquely as a sequence of $\beta$-Castelnuovo contractions.

Classification results fully describe local types of $\beta$-terminal singularities and contractions, exhibiting behaviors such as non-regular terminal surfaces and twisted blowup-sequences non-coincident with the commutative ones (Chan et al., 2021).

3. Categorical and Derived Noncommutative MMP

The categorical extension of the MMP, as formulated by Halpern-Leistner, seeks a parallel between birational geometry and semiorthogonal decompositions (SOD) of the bounded derived category $D^b(\mathrm{Coh} X)$. Each birational contraction $\pi:X\to Y$ predicts a canonical SOD:

$$D^b(\mathrm{Coh}X) = \langle D^b(\mathrm{Coh}Y), \mathcal{A} \rangle,$$

where $\mathcal{A}$ measures the non-triviality of the contraction (Halpern-Leistner, 2023).

The central mechanism relates categorical wall-crossing and SOD to the geometry of the Bridgeland stability manifold $\mathrm{Stab}(X)$. Birational modifications correspond to quasi-convergent paths in $\mathrm{Stab}(X)$, constructed so that as the central charges evolve along a solution to the quantum differential equation (QDE), the derived category undergoes associated canonical decompositions.

The quantum MMP conjecture asserts that for each contraction, such paths exist, producing canonical SODs compatible with further contraction sequences and matching the deformation behaviors predicted by quantum cohomology and stability conditions. In this way, the noncommutative/categorical MMP extends classical MMP principles to derived and noncommutative geometries (Halpern-Leistner, 2023).

4. Examples: Blowups, Fano Varieties, and Surface Case Studies

Blowups and Surfaces

For blowup surfaces, the noncommutative MMP is realized via quasi-convergent paths in $\mathrm{Stab}(\widetilde X)$. For a blowup $\pi:\widetilde X=\mathrm{Bl}_p X \to X$, the derived category admits Orlov’s SOD, and the categorical flow, governed by a truncated QDE, yields stepwise SOD mutations which mirror the contraction of the exceptional divisor in the commutative MMP. Explicit wall-crossings correspond to the loss of exceptional objects (e.g., $\mathcal{O}_{C}(-1)$) from the SOD, ultimately reducing $D^b(\widetilde X)$ to $D^b(X)$. These flows confirm that the noncommutative minimal model is achieved as a limit along a quasi-convergent path (Karube, 2024).

Fano Varieties

For Fano varieties (Grassmannians, quadrics, cubic threefolds/fourfolds), recent advances construct holomorphic lifts of Iritani’s quantum cohomology central charge to paths of stability conditions. The asymptotic behavior of these paths is proven to be quasi-convergent, and the resulting SODs match those expected from quantum geometry, such as the Kapranov and Kuznetsov decompositions. The noncommutative MMP thus produces the expected atoms of the derived category, consistently with quantum cohomology data, and verifies conjectures such as one direction of Dubrovin's semisimplicity conjecture (Karube et al., 28 Jan 2026).

5. Noncommutative Surfaces and Homological MMP

For noncommutative projective surfaces (e.g., Sklyanin algebras, Van den Bergh quadrics), the minimal model program is formulated in terms of maximal orders and birational overrings. Rogalski–Sierra–Stafford establish that the "minimal" models are precisely those with no proper connected graded noetherian overrings inside the graded quotient ring—this encapsulates the notion of minimality in the noncommutative regime (Rogalski et al., 2018).

A noncommutative Castelnuovo blowdown theorem is proven: every overring of an elliptic algebra $T$ (coordinate ring for a noncommutative surface) is obtained by blowing down finitely many line modules of self-intersection $-1$, paralleling the contraction of $(-1)$-curves in the commutative MMP. This theorem ties the algebraic structure of $T$ to its birational geometry and provides an explicit noncommutative MMP in this domain.

In threefold birational geometry, Wemyss’s Homological MMP formalizes the minimal model program using derived equivalences between minimal models via tilting bundles and mutation of endomorphism algebras. Contractibility of configurations of curves corresponds to finite-dimensionality of factor algebras, and mutations correspond precisely to Bridgeland–Chen flop functors, yielding a categorical algorithm for navigating the geography of minimal models in noncommutative threefold settings (Wemyss, 2014).

6. Significance, Open Problems, and Interrelations

The noncommutative Minimal Model Program encompasses a vast reach:

• It subsumes ordinary, equivariant, and twisted (Brauer) settings as special cases of the b-log MMP.
• It provides a unifying categorical framework connecting birational geometry, quantum cohomology, and derived categories.
• For Fano and rational varieties, it directly relates the geometry of quantum differential equations to canonical decompositions of $D^b(\mathrm{Coh}X)$.
• For noncommutative surfaces and threefolds, it provides precise algebraic and categorical criteria for minimality, contractions, and mutations, underpinned by homological invariants and stability conditions.

Current open directions include classification of all minimal models in birational classes of noncommutative surfaces, extension of MMP philosophy to higher-dimensional and singular noncommutative varieties, deepening the relationship with stability condition moduli, and further investigation of singularity theory in the noncommutative context (Chan et al., 2017, Chan et al., 2021, Halpern-Leistner, 2023, Karube et al., 28 Jan 2026, Rogalski et al., 2018, Wemyss, 2014).