Completing Song–Tian's analytic minimal model program: the Kähler–Ricci flow through birational surgeries and metric completions, canonical measures without finite generation, and analytic abundance

Complete the analytic minimal model program for the weak Kähler–Ricci flow: (i) run the weak Kähler–Ricci flow through its finite-time singularities via the birational surgeries predicted by the minimal model program, proving that the metric completion of the flow at each singular time is the corresponding divisorial contraction, flip, or Fano-fibration base, and that the surgery sequence terminates unconditionally at a minimal model or Mori fiber space; (ii) establish uniqueness and regularity of the canonical measure on the canonical model without assuming finite generation of the canonical ring; (iii) derive algebraic consequences (abundance, finite generation) by analytic means.

Background

Algebraic background. On a complex projective variety $X$ of dimension $n$ the canonical bundle $K_X$ is the top exterior power of the cotangent bundle; its global sections over all powers $mK_X$ are the pluricanonical forms. The growth rate of $H^0(X,mK_X)$ defines the Kodaira dimension $\kappa(X)$, a value in $\{-\infty,0,1,…,n\}$, with $\kappa(X)=n$ the case of general type. The graded ring $R(X)=⊕_{m≥0}H^0(X,mK_X)$ is the canonical ring; when it is finitely generated, its projective spectrum is the canonical model $X_{can}$ and the induced rational map $X⇢X_{can}$ is the Iitaka fibration. A line bundle is semi-ample if some power is globally generated and hence defines a morphism. A minimal model of $X$ is a birational model with mild singularities on which $K_X$ is nef (non-negative on every curve); a Mori fiber space is a fibration with $-K_X$ relatively ample, also called a Fano fibration. The minimal model program (MMP) modifies $X$ into a minimal model or Mori fiber space through a sequence of elementary surgeries: divisorial contractions, which contract a divisor, and flips, which replace a small contraction by another birational model on which the canonical class is again positive in the contracted curves. The singularities are taken Kawamata log terminal (klt), a standard numerical measure of mildness defined via discrepancies. Two algebraic inputs are central here: the theorem of Birkar–Cascini–Hacon–McKernan (BCHM) that canonical rings of varieties of log general type are finitely generated, together with the existence of the needed flips; and the abundance conjecture, asserting that a nef $K_X$ on a minimal model is semi-ample, which is open in dimension at least four, as is termination of the MMP there.

Analytic background. A Kähler metric on a complex manifold is a Hermitian metric whose associated real $(1,1)$-form $ω$ is closed; its Ricci form $\mathrm{Ric}(ω)$ represents the first Chern class $c_1(X)=-c_1(K_X)$. The (unnormalized) Kähler–Ricci flow evolves the metric by the parabolic equation $∂_tω(t)=-\mathrm{Ric}(ω(t))$; the normalized flow uses $∂_tω=-\mathrm{Ric}(ω)-ω$. The flows preserve Kähler classes, which move linearly along the ray $[ω(t)]=[ω_0]+t·c_1(K_X)$, equivalently $[ω_0]-t·c_1(X)$, in the unnormalized case. By work of Cao building on Yau's solution of the Calabi conjecture, the normalized flow converges to a Kähler–Einstein metric ($\mathrm{Ric}(ω)=-ω$) whenever one exists. On a projective variety with $K_X$ pseudo-effective, the moving ray eventually leaves the nef cone at a finite time; by the MMP the corresponding boundary point corresponds to a birational contraction or fibration, and the equation for the flow potential becomes a degenerate complex Monge–Ampère equation. On singular varieties and in degenerating classes the flow is interpreted weakly, as a family of currents whose potentials solve the parabolic Monge–Ampère equation in the pluripotential sense (following Eyssidieux–Guedj–Zeriahi's theory of singular Kähler–Einstein metrics) or the viscosity sense. Gromov–Hausdorff convergence is convergence of compact metric spaces up to uniformly small distortion; by the metric completion of $(X_{reg},ω(t))$ as $t→T_1$ one means the completion of the path metric induced on the smooth locus.

Song–Tian's canonical measures. When $K_X$ is semi-ample with Iitaka fibration $π:X→X_{can}$, Song and Tian proved that the normalized Kähler–Ricci flow collapses along the fibers and converges to a twisted singular Kähler–Einstein metric $ω_{can}$ on the canonical model satisfying $\mathrm{Ric}(ω_{can})=-ω_{can}+ω_{WP}$, where $ω_{WP}$ is the Weil–Petersson form measuring the complex variation of the fibration. They constructed the associated canonical measure $Ω_{can}$ as a monotone limit of volume forms built from pluricanonical forms, the parabolic analogue of Tsuji's analytic Zariski decomposition (a singular Hermitian metric on $K_X$ with semipositive curvature current encoding positivity analytically), and proved uniqueness and birational invariance of $Ω_{can}$ under the hypothesis that the canonical ring is finitely generated. The introduction of their paper notes that the flow potentials converge away from singular fibers and poses the local $C^{1,1}$ convergence (potential differentiable in time with spatially Lipschitz derivative) as an open problem; their final section relates the flow to the minimal model program.

The analytic minimal model program. Song–Tian conjectured that the Kähler–Ricci flow itself performs the MMP: at the singular time $T_1$, when the moving ray leaves the nef cone, the metric completion of $(X_{reg},ω(t))$ should be homeomorphic to, and carry a natural singular Kähler metric on, the birational model $X_1$ predicted by the MMP (the target of the divisorial contraction, the flip, or the base of the Fano fibration); the weak flow should continue uniquely on $X_1$, including through flips, where the continuation must be defined via a common resolution and shown independent of choices; and, iterating, finitely many surgeries should bring the flow to a minimal model or Mori fiber space, on which it exists for all time and the underlying metric spaces converge in the Gromov–Hausdorff sense. These statements are formalized as Conjectures 6.1–6.7 in their companion paper on the flow through singularities, which proves existence and uniqueness of the weak flow on projective Q-factorial klt varieties and its unique continuation through a divisorial contraction or a flip conditional on the existence of that surgery. The surface case of the weak passage through surgeries was carried out by Song–Tian using surface classification.

Regularity and abundance questions. The canonical measure $Ω_{can}$ is constructed without any finite-generation hypothesis, but its uniqueness was proved only under finite generation; for intermediate Kodaira dimension $0<κ(X)<n$ that hypothesis is not available, since finite generation there is essentially the abundance problem, which is open in dimension at least four. Finally, going back to Tsuji and pursued by Siu, there is the hope that analytic objects (analytic Zariski decompositions, singular Kähler–Einstein metrics, multiplier ideals) could prove the algebraic statements themselves: abundance and finite generation of canonical rings. This is the analytic abundance direction.

Problem Statement

Of the four directions of the program, the higher-regularity direction is fully resolved (Hein-Lee-Tosatti 2025) and the endgame convergence statements of the surgery direction (Gromov-Hausdorff convergence on semi-ample minimal models) are resolved (Jian-Song 2026; Lee-Tosatti-Zhang 2026); these are excluded from what is asked. The Fano endpoint of the first-singular-time split, $\dim Y=0$ (Conjecture 6.7), is excluded as well: it is stated for the normalized flow and a Kähler-Ricci soliton limit, not for the metric completion of the weak unnormalized flow, so that all three branches of the source's $\dim Y$ split are either asked for below or excluded here. The surviving problem, whose common objective is a Ricci-flow-analytic construction and completion of the minimal model program, has four components.

(1) Surgeries and metric completions (Song-Tian, Invent. Math. 2017, Conjectures 6.1, 6.5, 6.6, in the numbering of the preprint arXiv:0909.4898; and JAMS 2012, final section). Let $X$ be a projective $\mathbb{Q}$-factorial variety with klt singularities, let $H$ be a big semi-ample $\mathbb{Q}$-divisor on $X$ such that $H+\varepsilon K_X$ is ample for sufficiently small $\varepsilon>0$, and let the weak unnormalized Kähler-Ricci flow $\partial_t\omega=-\mathrm{Ric}(\omega)$ start from an initial current $\omega_0$ in Song-Tian's admissible class $K_{H,p}(X)$ for some $p>1$ (closed positive $(1,1)$-currents in the class $c_1(H)$ with $L^p$ Monge-Ampère mass; on a smooth $X$ this includes every Kähler form in the class). The flow class follows the ray $c_1(H)+t\,c_1(K_X)$; let $T_1=\sup\{t\geq 0:\ H+tK_X\ \text{is nef}\}$ be the first time it leaves the nef cone (the threshold called $T_0$ in the source). Prove: (a) for every $t\in(0,T_1)$, the metric completion of the smooth locus $X_{\mathrm{reg}}$ by the path metric of $\omega(t)$ is homeomorphic to $X$, and the Ricci curvature of $\omega(t)$ is bounded from below, as conjectured in the source (arXiv:0909.4898, Conjecture 6.1); (b) if $T_1<\infty$ and the semi-ample divisor $H+T_1K_X$ induces a birational contraction $\pi:X\to Y$ with $\dim Y=\dim X$, then as $t\to T_1$ the metric spaces $(X,\omega(t))$ converge in the Gromov-Hausdorff sense to a metric space $(X^+,\omega_{X^+})$ such that $\omega(t)$ converges smoothly outside a proper subvariety $S$ of $X$ and $(X^+,\omega_{X^+})$ is the metric completion of the smooth limit of $(X\setminus S,\omega(T_1,\cdot))$, where $X^+$ is a general flip of $X$ over $Y$ (a normal projective variety with a diagram $X\dashrightarrow X^+\to Y$ in which $X^+$ is ample over $Y$; a divisorial contraction is the special case where $X^+=Y$) - so the metric limit is the projective variety produced by the MMP surgery and carries the singular Kähler structure of the flow; (c) if $0<\dim Y<\dim X$, so that $X$ admits a Fano fibration over $Y$, then $(X,\omega(t))$ converges in the Gromov-Hausdorff sense to a metric space $(Y',\omega_{Y'})$ on which the program restarts, where $Y'$ is the base $Y$ of this Fano fibration, carrying the limiting current $\omega_{Y'}$. The unique continuation of the weak flow through divisorial contractions and flips, conditional on the existence of the surgery, is already proved by Song-Tian (2017) and may be used; the open content of this component is the metric identification of the limits.

(2) Unconditional termination. Prove that the sequence of surgeries performed by the flow is finite, ending at a minimal model (on which the flow exists for all time and converges as in the already-resolved convergence theorems) or at a Mori fiber space, without assuming termination of the MMP with scaling or any other unproved algebraic input. In general type the algebraic input is available (MMP with scaling terminates by BCHM) and Song-Tian accordingly prove that the weak flow replaces $X$ by its minimal model after finitely many surgeries there, for a good initial divisor (preprint arXiv:0909.4898, Theorem 5.9); the demand bites in the remaining cases, where algebraic termination is open in dimension at least four.

(3) Canonical measures without finite generation. For projective varieties of intermediate Kodaira dimension $0<\kappa(X)<n$, prove uniqueness, birational invariance, and the regularity asserted by the source's conditional theorem (Song-Tian, JAMS 2012, Theorem C.2 = Theorem 4.4) - continuity of $\Omega_{\mathrm{can}}$ on $X$, smoothness on a Zariski open set, and existence of the associated bounded-potential twisted singular Kähler-Einstein current $\omega_{\mathrm{can}}$ on the canonical model with $(\omega_{\mathrm{can}})^{\kappa(X)}=\Phi_*\Omega_{\mathrm{can}}$ ($\Phi$ the Iitaka map) and $\mathrm{Ric}(\omega_{\mathrm{can}})=-\omega_{\mathrm{can}}+\overline{\omega}_{WP}$ ($\overline{\omega}_{WP}$ the generalized Weil-Petersson form of the source's Section 4.5, Definition 4.2, which agrees with $\omega_{WP}$ on a Zariski open set of the canonical model) - for the canonical measure $\Omega_{\mathrm{can}}$ constructed by the Song-Tian monotone limit procedure, without assuming finite generation of the canonical ring or semi-ampleness of $K_X$. The construction of $\Omega_{\mathrm{can}}$ is unconditional for positive Kodaira dimension; where the Iitaka fibration to the canonical model is used to place the target metric, either bypass the semi-ampleness hypothesis or derive semi-ampleness.

(4) Analytic abundance. Give a rigorous analytic derivation of the algebraic statements the program targets: semi-ampleness of the canonical class on minimal models (the abundance conjecture, open in dimension at least four) or finite generation of canonical rings in the cases where it is open.

The conjectures above are posed for projective varieties of arbitrary dimension. On surfaces, the weak passage through surgeries is complete (Song-Tian 2007), and Song-Weinkove proved that the flow performs canonical surgical contractions - Gromov-Hausdorff convergence and smooth convergence away from the exceptional divisors through each contraction of exceptional curves, with unique continuation - for any projective surface with rational initial class, until the minimal model is reached or the volume collapses (Duke Math. J. 2013; the orbifold cases, where the Gromov-Hausdorff limit is moreover identified with the metric completion of the limiting metric, in Proc. Lond. Math. Soc. 2014); but the metric-completion identification of Conjecture 6.5 is open already there outside the orbifold cases (Song-Weinkove retract it in general), as is the Ricci lower bound of Conjecture 6.1, and no flip case is verified in any dimension. No dimension is excluded from what is asked.

References

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