Minimal model program

The Minimal Model Program (MMP) is a foundational research program in algebraic geometry that seeks to classify smooth projective varieties up to birational equivalence by constructing simplified models through a series of controlled birational transformations, ultimately producing either a minimal model with a nef canonical divisor or a fibration whose general fibers are of Fano type.^[1] This program extends classical classification results from lower dimensions—such as Riemann's trichotomy for curves based on the sign of the Euler characteristic and the Italian school's work on surfaces—to higher-dimensional varieties, addressing the structure of their canonical divisors to reveal intrinsic geometric properties.^[2] Historically, the MMP traces its origins to the early 20th-century classification of algebraic surfaces by Castelnuovo and Enriques, who introduced minimal models by contracting exceptional curves of negative self-intersection, a process later formalized by Kodaira in the 1960s for complex surfaces.^[3] The program's modern formulation emerged in the 1980s through Shigefumi Mori's development of the cone theorem, which decomposes the Mori cone of effective curves into extremal rays amenable to contraction, enabling the classification of threefolds and earning Mori the 1990 Fields Medal.^[1] Key challenges in higher dimensions, such as the existence of flips—birational maps that replace small contractions to preserve volume and progress toward minimality—were resolved in the mid-2000s by Birkar, Cascini, Hacon, and McKernan, whose work established the existence of minimal models for varieties of general type in arbitrary dimensions.^[4][5] Central to the MMP are concepts like the nef canonical class, where the canonical divisor $K_X$KX​ pairs non-negatively with every curve, ensuring no further "negative" contractions are possible, and the Mori cone, a polyhedral cone in the Néron-Severi space parameterizing curve classes that guides the contraction process.^[3] The program proceeds algorithmically by running contractions of extremal rays until reaching a minimal model or a Mori fiber space, a fibration over a lower-dimensional base with Fano fibers, thereby providing a birational invariant decomposition of the variety.^[1] While fully resolved for surfaces and threefolds, the MMP in higher dimensions remains an active area, with recent extensions to log canonical pairs, positive characteristic, and arithmetic settings highlighting its versatility in modern geometry.^[6]

Overview

Goals and Principles

The Minimal Model Program (MMP) is a systematic approach in algebraic geometry that constructs a sequence of birational morphisms to simplify the structure of algebraic varieties defined over fields of characteristic zero. This process aims to classify projective varieties up to birational equivalence by producing models with controlled geometric properties, focusing on those with Kawamata log terminal (klt) singularities. The primary goal of the MMP is to obtain a minimal model for a given variety, where the canonical divisor $K_X$KX​ (or more generally $K_X + \Delta$KX​+Δ for a boundary $\Delta$Δ) becomes nef, meaning it pairs non-negatively with every curve on the variety. If no such minimal model exists—typically when $K_X$KX​ is not pseudo-effective—the program identifies a Mori fiber space, which generalizes the minimal model by contracting extremal rays to yield a fibration whose fibers have simpler structure, often leading to the Iitaka fibration that encodes the Kodaira dimension. This classification refines birational equivalence classes by highlighting invariants like the sign of the canonical class. Central principles underlying the MMP include the abundance conjecture and termination. The abundance conjecture posits that for a minimal model, the canonical divisor is not only nef but semi-ample, ensuring the canonical ring is finitely generated and the Iitaka fibration is well-defined, thus resolving the equality between the Kodaira dimension $\kappa(K_X)$κ(KX​) and the numerical dimension $\nu(K_X)$ν(KX​). Termination asserts that the sequence of birational modifications—such as contractions and flips—halts after finitely many steps, yielding a good minimal model or fiber space without infinite descent. These principles enable the MMP to study Kodaira dimension by stratifying varieties into classes of general type (where minimal models exist with nef $K_X$KX​), non-general type (leading to fiber spaces), and the role of mild singularities, as the program preserves birational properties while resolving or controlling them through log terminal models.

Key Components

The canonical divisor $K_X$KX​ of a normal variety $X$X is a Weil divisor such that its associated sheaf $\mathcal{O}_X(K_X)$OX​(KX​) corresponds to the dualizing sheaf on the smooth locus of $X$X, up to linear equivalence.^[7] In the minimal model program (MMP), $K_X$KX​ plays a central role in measuring the "positivity" or "negativity" of curves and divisors, guiding birational modifications to achieve a minimal model where $K_X$KX​ is nef.^[8] A divisor $D$D on $X$X is nef (numerically effective) if its intersection number with every irreducible curve $C$C on $X$X satisfies $D \cdot C \geq 0$D⋅C≥0.^[9] This condition ensures that $D$D generates a morphism or semiample line bundle under certain assumptions, forming a boundary for the MMP's goal of producing models with nef canonical divisors.^[7] The Kleiman-Mori cone of curves, denoted $\overline{NE}(X)$NE(X) or $NE(X/S)$NE(X/S) for a morphism $\pi: X \to S$π:X→S, is the closed convex cone in the vector space of 1-cycles modulo numerical equivalence, generated by classes of effective curves mapped to points in $S$S by $\pi$π.^[9] It captures the numerical behavior of curves on $X$X and supports the cone theorem, which decomposes the cone into extremal rays where $K_X$KX​ has negative intersection.^[8] An extremal ray $R$R is a one-dimensional face of this cone such that the locus of contraction—the set of points in $X$X whose fiber under the associated contraction morphism has positive dimension—is of positive dimension, allowing the ray to be contracted birationally or via a fibration.^[7] These rays identify "directions" in the cone where the variety can be simplified, essential for progressing through the MMP.^[9] Elementary contractions in the MMP are morphisms associated to extremal rays, classified by their type. A fiber-type contraction $\phi: X \to Y$ϕ:X→Y has positive-dimensional fibers over a lower-dimensional base $Y$Y, such as in Mori fiber spaces where the general fiber is of Fano type.^[8] A divisorial contraction contracts an irreducible divisor to a subvariety of codimension at least 2 in $Y$Y, preserving birational equivalence while resolving singularities or reducing complexity.^[7] Small contractions, in contrast, contract a locus of codimension at least 2 without an exceptional divisor, necessitating a flip—a birational map resolving discrepancies in the canonical class—to continue the program.^[9] The termination of flips conjecture posits that, in characteristic zero, any sequence of flipping and flopping contractions in the MMP terminates after finitely many steps, yielding a minimal model with nef canonical divisor or a Mori fiber space.^[7] This ensures the program's finite length for varieties of general type, though it remains open in dimensions greater than 4.^[8] Flops, which preserve the canonical class up to numerical equivalence, complement flips in maintaining the descent toward minimality.^[9]

Historical Development

Foundations in Surface Theory

The foundations of the minimal model program for algebraic surfaces were laid in the early 20th century through efforts to classify compact complex surfaces, emphasizing the role of minimal models as those free of exceptional curves. In the 1910s, Federigo Enriques initiated a classification scheme for algebraic surfaces based on birational invariants such as the geometric genus and irregularity, distinguishing classes like rational, ruled, and irregular surfaces, where minimal models were identified as smooth surfaces without (-1)-curves that could be contracted.^[10] This approach was extended and culminated in the Enriques-Kodaira classification in the 1960s through Kunihiko Kodaira's work on compact complex surfaces, including non-algebraic ones, using the growth rate of plurigenera $\dim H^0(X, mK_X)$dimH0(X,mKX​) for the canonical divisor $K_X$KX​, later known as the Kodaira dimension. Kodaira's work divided surfaces into categories based on Kodaira dimension (-\infty, 0, 1, or 2). A cornerstone result enabling this classification is Castelnuovo's theorem from 1908, which asserts that every birational equivalence class of smooth projective surfaces over the complex numbers contains a unique minimal model obtained by successively contracting all (-1)-curves, rational curves with self-intersection -1 that arise from blow-ups. This theorem guarantees the existence and uniqueness of a minimal surface in each class, providing a canonical object for studying invariants like the canonical class and plurigenera.^[11] Central to obtaining these minimal models is the resolution of singularities, a process that replaces singular points on a surface with smooth exceptional divisors via blow-ups at points or curves, yielding a smooth birational model. For algebraic surfaces, this was achieved in the mid-20th century; notably, Joseph Lipman's 1966 work provided an algorithm for desingularizing two-dimensional schemes over fields of characteristic zero by successive normalizations and blow-ups, ensuring the exceptional locus has normal crossings and simple normal crossings support. Earlier contributions, such as Hans Grauert's 1961 resolution for analytic surface singularities, complemented this by handling embedded resolutions in complex analytic spaces. These resolutions are minimal in the sense that no further contractions of exceptional (-1)-curves are possible without introducing singularities, aligning with the minimal model paradigm. In the 1970s, Shigeru Iitaka advanced the theory by generalizing the Kodaira dimension to logarithmic settings and establishing fibration structures for surfaces with ample canonical divisors. Iitaka introduced the concept for pairs (X, D) where D is a boundary divisor, defining the logarithmic Kodaira dimension as the growth rate of $\dim H^0(X, m(K_X + D))$dimH0(X,m(KX​+D)), and proved that for surfaces with positive Kodaira dimension, the Iitaka fibration $\phi_{|mK_X|} : X \dashrightarrow Y$ϕ∣mKX​∣​:X⇢Y contracts subvarieties of relative dimension zero to yield a fibration onto a base Y of dimension equal to the Kodaira dimension, facilitating the study of minimal models via fiber structures. This fibration theorem underpins the classification by revealing the structure of surfaces of general type as fibrations over curves or points with general fibers of lower Kodaira dimension.

Mori Theory and Higher Dimensions

In the late 1970s and early 1980s, Shigefumi Mori introduced the bend-and-break technique, a powerful method for deforming families of rational curves on projective varieties to analyze their homology classes. This technique demonstrated that every extremal ray in the cone of effective curves on a smooth projective variety is spanned by the class of a rational curve, thereby extending foundational results from surface theory to higher dimensions. Using bend-and-break, Mori established the contraction theorem, which asserts that for any extremal ray on a smooth projective threefold, there exists a unique contraction morphism to a normal variety, contracting precisely the curves in that ray.^[12] Building on Mori's foundational contributions in the early 1980s, including the cone and contraction theorems, the minimal model program for smooth projective threefolds was established in the 1980s by Mori, Yujiro Kawamata, Vladimir Shokurov, and others, showing that any such threefold is birationally equivalent to either a minimal model with nef canonical divisor (if of general type) or a Mori fiber space (a fibration with rationally connected general fibers otherwise).^[1] In the 1980s and 1990s, Kawamata and Shokurov extended the framework by establishing the existence of flips for threefolds, allowing the minimal model program to proceed through small birational modifications when simple contractions are impossible. Their work also addressed the abundance conjecture for threefolds, proving that for a minimal threefold with big canonical divisor, some multiple of the canonical class is effective, thereby completing key aspects of the program in dimension three.^[13] These contributions culminated in Mori receiving the Fields Medal in 1990 for his groundbreaking work in birational geometry.^[14]

Core Machinery

Extremal Ray Contractions

In the minimal model program, extremal ray contractions constitute a fundamental step where an extremal ray of the Kleiman-Mori cone of effective curves on a projective variety is mapped via a birational morphism to a lower-dimensional model or a variety with resolved singularities, facilitating the simplification of the canonical divisor. These contractions are defined for extremal rays, which are the extremal faces of the cone NE(X)_{\mathbb{R}} such that the locus of curves generating the ray is a proper closed subset of X.^[7] Mori's contraction theorem establishes that for a smooth projective variety X, every extremal ray R admits a contraction morphism \phi: X \to Y to a normal variety Y, where curves generating R are mapped to points in Y, and the exceptional locus or fibers are determined by the geometry of R. This morphism is proper, surjective, and satisfies \mathcal{O}Y \cong \phi* \mathcal{O}_X, ensuring Y inherits key properties from X, such as Q-factoriality when applicable. The theorem, originally proved for smooth threefolds and extended to higher dimensions, relies on vanishing theorems like Kawamata-Viehweg to guarantee the existence of the contraction.^[15][7] Contractions are classified based on their type, with fiber-type contractions occurring when \dim Y < \dim X, resulting in a fibration where the general fibers have dimension at least 1 and -K_X is relatively ample over Y, often yielding Fano fibrations in the case of K_X-negative rays. In such contractions, the morphism \phi: X \to Y is equidimensional on the complement of the locus of R, and the fibers are connected varieties with ample anticanonical bundles, providing a structure theorem for varieties admitting such rays.^[7] Divisorial contractions, on the other hand, are birational morphisms where the exceptional locus is an irreducible prime divisor E \subset X, which is contracted to either a point or a lower-dimensional subvariety in Y, thereby resolving certain types of singularities or simplifying the model. These contractions preserve the dimension of the variety and are characterized by the property that E \cdot C < 0 for curves C in R, with Y normal and often exhibiting quotient singularities. For K_X-negative rays, divisorial contractions play a key role in reducing the complexity of non-nef canonical divisors by blowing down exceptional divisors with negative intersections.^[15][7] The length of an extremal ray R quantifies its negativity and determines the contraction type, defined as $$l(R) = \min \left\{ -K_X \cdot C \;\middle|\; [C] \in R, \, C \text{ an irreducible curve} \right\}.$$l(R)=min{−KX​⋅C∣[C]∈R,C an irreducible curve}. This length satisfies 0 < l(R) \leq \dim X + 1 for K_X-negative rays, with l(R) > 0 precisely when R is K_X-negative, indicating that the contraction reduces the numerical effectiveness of K_X along that ray. In the case where K_X is nef, extremal rays have l(R) = 0, and contractions map them without altering the nefness but may simplify the model further if needed. The bound on l(R) arises from bend-and-break techniques applied to rational curves spanning the ray.^[15][7]

Flips and Divisorial Contractions

In the minimal model program, small contractions arise when the exceptional locus of an extremal ray contraction has codimension at least 2 in the domain variety $X$X. In such cases, the contraction map $f: X \to Y$f:X→Y is birational but does not contract any prime divisor, preventing a standard divisorial contraction that would reduce the Picard number by 1.^[16] Instead, the locus of curves contracted by $f$f must be replaced through a more subtle birational modification to continue the program while preserving the relative ampleness conditions on the canonical divisor.^[11] A flip addresses this by providing a birational map $\phi: X \dashrightarrow X'$ϕ:X⇢X′, realized as a pair of proper birational morphisms $f: X \to Y$f:X→Y and $f^+: X^+ \to Y$f+:X+→Y (with $X' = X^+$X′=X+), where both exceptional loci have codimension at least 2. The map $f$f contracts a $K_X$KX​-negative extremal ray, meaning $-K_X$−KX​ is $f$f-ample, while the inverse map $\phi^{-1}$ϕ−1 contracts a $K_{X^+}$KX+​-positive extremal ray, meaning $K_{X^+}$KX+​ is $f^+$f+-ample. This "flips" the sign of the intersection with the canonical class for the relevant curves, allowing the minimal model program to proceed without altering the Picard number of the models.^[16] Flips exist in dimension 3, as established through detailed classification of contraction types, and more generally under assumptions like the existence of minimal models in lower dimensions.^[17] Small flips are the generic case in higher dimensions, often requiring resolution of non-Q-Cartier singularities on $Y$Y.^[11] In the 1990s, V. V. Shokurov formulated the polyhedral conjecture as part of the broader framework for termination in the minimal model program, asserting that the negative part of the Kleiman-Mori cone $\overline{NE}(X)$NE(X) is locally rational polyhedral and that the number of flips in a chain is bounded in terms of log discrepancies of the singularities involved. This conjecture ensures finite termination of flip sequences by controlling the descent of discrepancies along the process, preventing infinite chains. It was proven in dimension 3 using induction on the index of the canonical divisor and detailed analysis of log terminal pairs, confirming that flip chains terminate after finitely many steps for Q-factorial threefolds with reduced boundary supports.^[17]

Application to Surfaces

Minimal Models of Surfaces

The minimal model of a surface is constructed by beginning with an arbitrary smooth projective surface and iteratively contracting all exceptional curves of the first kind until none remain. An exceptional curve of the first kind is an irreducible rational curve $C$C satisfying $C^2 = -1$C2=−1. Castelnuovo's contraction theorem guarantees that such a curve can always be contracted via a birational morphism to a smooth point on another smooth projective surface, preserving smoothness. Each contraction reduces the Picard number by 1, ensuring the process terminates after finitely many steps. The resulting surface, called the minimal model, contains no exceptional curves of self-intersection $-1$−1. The structure of the minimal model is determined by the Kodaira dimension $\kappa$κ of the original surface, which remains invariant under birational morphisms. For $\kappa = -\infty$κ=−∞, encompassing rational and ruled surfaces, the minimal models are $\mathbb{P}^2$P2 and Hirzebruch surfaces $\mathbb{F}_n$Fn​ for $n \neq 1$n=1 (for rational ruled surfaces) or $\mathbb{P}(E) \to C$P(E)→C where $C$C is a curve of genus $g \geq 1$g≥1 and $E$E is a rank-2 vector bundle of degree 0 with no $-1$−1 sections (for non-rational ruled surfaces). For $\kappa = 0$κ=0, representative minimal models include K3 surfaces and Enriques surfaces, provided they admit no $-1$−1 curves. For $\kappa = 1$κ=1, the minimal models are elliptic surfaces featuring an elliptic fibration to a curve with generic fiber an elliptic curve and no contractible $-1$−1 components in the fibers. For $\kappa = 2$κ=2, corresponding to surfaces of general type, the minimal models have nef canonical bundle $K_S$KS​ and no $-1$−1 curves. In all cases, the minimal model has no exceptional curves of self-intersection $-1$−1. For instance, the Hirzebruch surface $\mathbb{F}_n$Fn​ for $n \geq 2$n≥2 has no $-1$−1 curves and is minimal, while $\mathbb{F}_1$F1​, the blow-up of $\mathbb{P}^2$P2 at one point, has a $-1$−1 curve and contracts to $\mathbb{P}^2$P2. Similarly, blow-ups of $\mathbb{P}^2$P2 at $r \leq 8$r≤8 general points have only the exceptional $-1$−1 curves and contract successively to $\mathbb{P}^2$P2.^[18] Every birational class of smooth projective surfaces admits a unique minimal model, as established by Castelnuovo.

Classification via MMP

The Minimal Model Program (MMP) for algebraic surfaces over the complex numbers achieves the Enriques–Kodaira classification by iteratively contracting exceptional curves of negative self-intersection until reaching a minimal model where the canonical divisor $K_X$KX​ is nef (numerically effective).^[19] This process resolves the birational equivalence class of any surface into one of a finite number of types, determined primarily by the Kodaira dimension $\kappa(X)$κ(X), which measures the asymptotic growth of the plurigenera $P_n(X) = h^0(X, \mathcal{O}_X(nK_X))$Pn​(X)=h0(X,OX​(nKX​)) as $n \to \infty$n→∞.^[20] Specifically, $\kappa(X) = -\infty$κ(X)=−∞ if $P_n(X) = 0$Pn​(X)=0 for all $n \geq 1$n≥1, $\kappa(X) = 0$κ(X)=0 if $P_n(X)$Pn​(X) is bounded but positive, $\kappa(X) = 1$κ(X)=1 if $P_n(X)$Pn​(X) grows linearly, and $\kappa(X) = 2$κ(X)=2 if it grows quadratically.^[19] Surfaces fall into nine classes grouped by Kodaira dimension, with minimal models distinguishing the non-general-type cases. For $\kappa = -\infty$κ=−∞, the minimal models are rational surfaces such as $\mathbb{P}^2$P2 or Hirzebruch surfaces $\mathbb{F}_n$Fn​ for $n \neq 1$n=1, or non-rational ruled surfaces over a base curve of genus $g \geq 1$g≥1, including non-minimal ruled surfaces over elliptic curves, whose minimal models are $\mathbb{P}^1$P1-bundles without $-1$−1 curves.^[8][20] For $\kappa = 0$κ=0, the minimal models include K3 surfaces (with trivial canonical bundle and $p_g = 1$pg​=1) and Enriques surfaces (with $2K_X \sim 0$2KX​∼0 and $p_g = 0$pg​=0), alongside abelian, bielliptic, and hyperelliptic types.^[19] For $\kappa = 1$κ=1, the minimal models are elliptic surfaces, fibered over a base curve with generic elliptic fibers.^[20] The remaining classes encompass additional ruled variants and general type surfaces with $\kappa = 2$κ=2.^[8] For surfaces of general type ($\kappa = 2$κ=2), the MMP yields a unique minimal model where $K_X$KX​ is nef and big, and by Noether's theorem, this coincides with the canonical model obtained by contracting any $K_X$KX​-trivial rational curves of self-intersection $-2$−2.^[19] Noether's theorem further relates the invariants via the formula $K_X^2 + c_2(X) = 12 \chi(\mathcal{O}_X)$KX2​+c2​(X)=12χ(OX​), ensuring the minimal model captures the birational invariants essential for classification.^[8] The geometric genus $p_g = \dim H^0(X, K_X)$pg​=dimH0(X,KX​) (the first plurigenus $P_1(X)$P1​(X)) distinguishes subclasses, particularly for $\kappa = 0$κ=0: $p_g = 0$pg​=0 for Enriques and bielliptic surfaces, and $p_g = 1$pg​=1 for K3 and abelian surfaces.^[20] This invariant, stable under birational equivalence, aids in identifying the type after reaching the minimal model.^[19] The completeness of the classification stems from the termination of the MMP: every birational class of surfaces is resolved into one of these minimal types, providing a comprehensive geometric and arithmetic description without further birational modifications.^[8]

Higher-Dimensional Cases

Threefolds

The minimal model program for threefolds, initiated by Shigefumi Mori in 1982, proposes a systematic process of birational modifications—specifically, extremal ray contractions followed by flips—to transform a given projective threefold into a model where the canonical divisor $K_X$KX​ is numerically effective (nef) or yields a small model suitable for further analysis. This approach builds on the contraction of extremal rays in the Mori cone, ensuring that each step reduces the negativity of $K_X$KX​ along curves, with the program terminating due to the finite-dimensional nature of the cone in dimension three. Key advancements in the 1980s by Yujiro Kawamata and Miles Reid established foundational results on contractions and the existence of minimal models for threefolds. Kawamata's contraction theorem guarantees that any extremal ray on a smooth threefold contracts to either a fiber-type, divisorial, or small contraction, while Reid introduced the concept of minimal models for canonical threefolds, showing under certain hypotheses that a nonsingular model exists with $K$K nef. The full minimal model theorem for threefolds, asserting the existence of a minimal model with $K$K nef for non-uniruled threefolds, was completed by Vyacheslav Shokurov and Mori, with Shokurov proving termination of the program for terminal singularities in 1985. Central to the program's success in dimension three is the flip theorem, proved by Mori in 1988, which resolves small contractions via a birational map called a flip, preserving the canonical class up to numerical equivalence. The flipping book refers to the finite sequence of such flips needed to navigate from a small contraction to a model with $K$K nef, with termination guaranteed in dimension three due to the bounded length of these sequences. For Fano threefolds, where $-K_X$−KX​ is ample, the Sarkisov program extends the MMP by decomposing birational maps between Fano models into links of types such as blow-ups and contractions, achieving a complete birational classification. A representative example is the smooth quintic threefold, a Calabi-Yau hypersurface of degree 5 in $\mathbb{P}^4$P4, whose canonical bundle is trivial and thus nef, making the threefold itself a minimal model without requiring further modifications.

Arbitrary Dimensions

The minimal model program (MMP) in arbitrary dimensions extends the foundational techniques developed for surfaces and threefolds to projective varieties of dimension $n \geq 4$n≥4, aiming to produce either a birational model with the canonical divisor $K_X$KX​ nef or a Mori fiber space when $K_X$KX​ is not pseudo-effective. Central to this generalization is the MMP with scaling, which addresses challenges arising from log discrepancies in pairs $(X, \Delta)$(X,Δ) with Kawamata log terminal (klt) singularities by introducing a scaling factor $\lambda > 0$λ>0. Specifically, for a nef divisor $C$C on a $\mathbb{Q}$Q-factorial klt pair $(X, B)$(X,B), the process runs an MMP with respect to $K_X + B + \lambda C$KX​+B+λC, adjusting $\lambda$λ downward until the coefficients of $B + \lambda C$B+λC stabilize, ensuring the program terminates with a model where $K$K is nef relative to the scaling. This framework, formalized in higher dimensions, allows for the controlled contraction of extremal rays while preserving birational equivalence. Recent progress includes effective termination results for MMPs of general type in dimensions at most five.^[21] A landmark result establishing the viability of the MMP in arbitrary dimensions is the theorem of Birkar, Cascini, Hacon, and McKernan, which proves the existence of minimal models for varieties of log general type. For a projective klt pair $(X, \Delta)$(X,Δ) of dimension $n$n with $K_X + \Delta$KX​+Δ nef and big, there exists a birational map $f: X \dashrightarrow X'$f:X⇢X′ to a $\mathbb{Q}$Q-factorial klt pair $(X', \Delta')$(X′,Δ′) such that $K_{X'} + \Delta'$KX′​+Δ′ is nef, and the pair admits a good minimal model where the canonical ring is finitely generated.^[22] This theorem relies on the MMP with scaling to produce a sequence of flips and contractions that terminate, yielding the desired model without assuming dimension-specific bounds beyond the klt condition. The result holds uniformly for all $n$n, contrasting with the more resolved cases in lower dimensions like threefolds, where additional termination tools are available. Termination of the MMP in higher dimensions is ensured for klt pairs through the boundedness of flips, a key component of the scaling technique. In particular, for a sequence of $(K_X + B + \lambda C)$(KX​+B+λC)-flips with $\lambda$λ decreasing, the discrepancies and lengths of the flip graphs are bounded independently of the initial pair, preventing infinite cascades and guaranteeing that the program ends in a finite number of steps. This boundedness follows from the existence of effective bounds on the coefficients of exceptional divisors in the minimal model construction. Despite these advances, the abundance conjecture remains open in dimensions greater than three: for a minimal model $(X, \Delta)$(X,Δ) where $K_X + \Delta$KX​+Δ is nef and big, it conjectures that $K_X + \Delta$KX​+Δ is semi-ample, implying the existence of an Iitaka fibration. While verified in dimensions up to three, counterexamples or obstructions are not known in higher dimensions, but progress has been limited to special cases, such as when the numerical dimension equals the Kodaira dimension. The resolution of abundance would complete the MMP by providing a full birational classification via canonical models in arbitrary dimensions.

Extensions and Variations

Relative Minimal Model Program

The relative minimal model program (MMP) extends the absolute MMP to the setting of a projective morphism $f: X \to Y$f:X→Y, where $X$X is a normal variety with $\mathbb{Q}$Q-factorial terminal singularities and $Y$Y is the base variety. In this framework, birational transformations—such as divisorial contractions and flips—are performed relative to $Y$Y, meaning they are isomorphisms over $Y$Y and preserve the fibers of $f$f. The goal is to obtain a relative minimal model $Z \to Y$Z→Y, birational to $X$X over $Y$Y, such that the canonical divisor $K_Z$KZ​ is nef relative to $Y$Y (i.e., $K_Z \cdot C \geq 0$KZ​⋅C≥0 for every curve $C$C in a fiber of $Z \to Y$Z→Y). If $K_X$KX​ is relatively big over $Y$Y, a relative canonical model exists where $K_Z$KZ​ is ample over $Y$Y.^[23] A central component of the relative MMP is the existence and termination of relative flips. For a small birational morphism $\phi: X \dashrightarrow X^+$ϕ:X⇢X+ that contracts an extremal face of the relative Mori cone over $Y$Y, a relative flip exists under the assumption that $(X, B)$(X,B) is a Kawamata log terminal (klt) pair with $B$B an effective $\mathbb{Q}$Q-divisor and $K_X + B$KX​+B relatively nef over $Y$Y. These flips terminate after finitely many steps when running the MMP with scaling, where a scaling divisor $H$H (relatively ample over $Y$Y) is introduced to ensure the process halts, yielding either a relative minimal model or a relative Mori fiber space (a contraction whose fibers are of lower dimension). This termination relies on the relative version of the Kawamata-Viehweg vanishing theorem and holds in characteristic zero for klt pairs of arbitrary relative dimension.^[23][24] The relative MMP has significant applications in constructing moduli spaces of families of varieties. For instance, it enables the production of relative minimal models for families of curves over a base, such as in the study of the moduli space of stable curves, where fibers are simplified birationally while preserving the base structure. A concrete example arises in the case of a fibration $X \to \mathbb{P}^1$X→P1, where the relative MMP contracts exceptional curves in the fibers (e.g., rational curves with negative self-intersection) to obtain a fibered minimal model, ensuring each fiber is a smooth curve of genus at least two or a stable curve, without altering the base $\mathbb{P}^1$P1. This approach underpins the birational classification of fibrations and extends to higher-dimensional families in moduli problems. In positive characteristic, progress has been made for fourfolds with p > 5, though counterexamples exist for certain cases like 1-foliations.^{[23][11][25][26]}

Logarithmic Minimal Model Program

The logarithmic minimal model program (log MMP) extends the classical minimal model program to log pairs $(X, \Delta)$(X,Δ), where $X$X is a normal variety and $\Delta$Δ is an effective $\mathbb{R}$R-divisor such that $K_X + \Delta$KX​+Δ is $\mathbb{R}$R-Cartier.^[27] In this setting, the program proceeds by iteratively performing contractions of extremal rays of the effective cone of $K_X + \Delta$KX​+Δ, aiming to produce a model where $K_X + \Delta$KX​+Δ is nef or ample relative to a fixed morphism.^[27] Log pairs are classified by their singularities via log discrepancies: a pair is log canonical if all discrepancies $a(E, X, \Delta) \geq -1$a(E,X,Δ)≥−1 for valuations $E$E, and log terminal if $a(E, X, \Delta) > -1$a(E,X,Δ)>−1.^[27] These discrepancies measure how singularities worsen under birational modifications while preserving the log structure.^[27] Central to the log MMP are log flips, which are birational modifications $\phi: (X, \Delta) \dashrightarrow (X^+, \Delta^+)$ϕ:(X,Δ)⇢(X+,Δ+) that are isomorphisms over the smooth locus, preserve the coefficients of the boundary in the sense that the pushforwards match appropriately, and ensure $K_{X^+} + \Delta^+$KX+​+Δ+ is the pushforward of $K_X + \Delta$KX​+Δ.^[27] Such flips arise from small projective contractions of $K_X + \Delta$KX​+Δ-negative extremal rays and are essential for navigating non-divisorial contractions while maintaining log canonical or log terminal properties.^[27] The existence of log flips for log terminal pairs was conjectured by Shokurov, positing that they admit log flips in any dimension, enabling the full log MMP.^[28] This conjecture was proven in dimension three by Shokurov himself.^[29] In higher dimensions, the existence for klt pairs (a subclass of log terminal) was established by Birkar, Cascini, Hacon, and McKernan, confirming the log MMP terminates with a log minimal model. The full conjecture for log canonical pairs remains open in dimensions greater than three. Applications of the log MMP extend to the study of stable pairs in birational geometry, where log minimal models provide compactifications of moduli spaces of pairs with controlled singularities.^[27] In particular, the program links to K-stability for log Fano pairs, as running the log MMP yields models whose stability can be tested via log discrepancies and Futaki invariants, facilitating criteria for the existence of Kähler-Einstein metrics on log canonical pairs.^[30]