The minimal model program reduces an algebraic variety to a birational model whose canonical divisor is as positive as the geometry permits, by contracting along canonical-negative rays and flipping small contractions. For curves the answer is classical, for surfaces it is the Enriques-Kodaira classification, and for threefolds it rests on Mori's flip theorem. The result is that every smooth connected complex projective variety of dimension at least three whose canonical divisor is pseudo-effective of numerical dimension one admits a projective Q-factorial terminal minimal model with nef canonical divisor, with no hypothesis on the Euler characteristic.
https://doi.org/10.1090/S0894-0347-1988-0924704-X
https://doi.org/10.1017/CBO9780511662560
Throughout, a variety is a separated scheme of finite type over the complex numbers; all varieties and morphisms are over C, and all birational models are projective. A divisor D on a normal variety is effective when D is a nonnegative integral combination of prime divisors, Cartier when it is locally cut out by a single equation, and Q-Cartier when some positive integer multiple is Cartier; Q-factorial means every Weil divisor is Q-Cartier. Intersection numbers of rational Cartier divisors are defined after clearing denominators, and on an integral surface that need not be normal, intersections are taken with the fundamental cycle. The canonical divisor K_X of a smooth variety is the divisor of a global top-degree differential form; for a smooth variety over C its first Chern class is -c_1(T^*X). A divisor is nef when it has nonnegative degree on every integral curve, and pseudo-effective when its numerical class lies in the closure of the cone generated by effective divisors. A divisor is big when sections of its large multiples grow like m^n on an n-fold quotient, the maximal possible rate.
https://doi.org/10.1007/978-1-4757-3849-0
https://doi.org/10.1007/978-1-4612-1700-8
https://doi.org/10.1007/978-3-642-18808-4
For a normal variety U and a boundary Delta (a Q-divisor with K_U+Delta Q-Cartier), and for a prime divisor F over U, presented on a resolution r: U~ -> U carrying F, the log discrepancy is
a(F; U, Delta) = 1 + coeff_F ( K_{U~} - r^*(K_U+Delta) ).
An exceptional prime has log discrepancy 1. The pair (U,Delta) is terminal when a(F;U,Delta) > 1 for every exceptional prime F, and Kawamata log terminal (klt) when a(F;U,Delta) > 0 for every exceptional prime and the coefficients of Delta are strictly below 1. A terminal variety is smooth in codimension two. These classes and the discrepancy calculus are developed systematically by Kollar and Mori.
https://doi.org/10.1017/CBO9780511662560
For a Cartier divisor D on a smooth projective n-fold X, measure its positivity by the section-growth invariant
kappa_sigma(X,D) = max { k in {0,1,...,n} : there is an ample Cartier divisor A
with limsup_{m->oo} h^0(X, O_X(mD+A)) / m^k > 0 };
the maximum of the empty set is taken to be -infinity. The ample twist A is fixed before the limit is taken, so the definition allows the quadratic growth to appear only along the multiples of one fixed positive integer. The invariant kappa_sigma(X,D) = 1 forbids, for every fixed ample Cartier twist, a positive quadratic limsup; it does not by itself impose the upper bound h^0(X,mD+A) = O(m). The invariants and their interaction with divisorial Zariski decompositions are Nakayama's.
https://doi.org/10.2969/msjmemoirs/014010000
THEOREM. Let X be a smooth connected complex projective variety of dimension n >= 3. Suppose K_X is pseudo-effective and kappa_sigma(X,K_X) = 1. Then there are a normal projective Q-factorial terminal variety Y with K_Y nef, and a finite sequence phi: X --> Y of K-negative divisorial contractions and K-flips whose inverse extracts no divisors, such that on a common smooth projective resolution r: W -> Y and u: W -> X the compatible canonical divisors satisfy
u^*K_X = r^*K_Y + E,
where E is an effective r-exceptional Q-divisor whose support contains the strict transform of every prime divisor of X contracted by phi. Consequently every smooth connected complex projective variety of dimension at least three with pseudo-effective canonical divisor of numerical dimension one has a minimal model, including every smooth fivefold, without any condition on chi(X, O_X).
https://doi.org/10.1017/CBO9780511662560
When log abundance holds -- the assertion that a nef canonical divisor of a log canonical pair is semiample, conjectured in general and proved for varieties of general type and for klt pairs with big boundary -- the endpoint Y is a good minimal model, meaning K_Y is semiample and some positive multiple of K_Y is globally generated. The reduction of the good-minimal-model statement to log abundance is routine; the substance is the existence of the minimal model and the explicit comparison of canonical divisors. Abundance and its known cases are surveyed by Kollar and Mori and by Birkar, Cascini, Hacon, and McKernan.
https://doi.org/10.1017/CBO9780511662560
birkar-cascini-hacon-mckernan-2010.pdf
The threefold program was assembled by Reid, Benveniste, Kawamata, Shokurov, Kollar, Miyaoka, and others, with Mori's flip theorem supplying the central existence step. For log pairs, Birkar formulated the existence of log minimal models alongside the Mori fiber space alternative, and Birkar, Cascini, Hacon, and McKernan proved that varieties of general type have minimal models and that klt pairs with big boundary admit finite minimal model programs with scaling. The perturbations lie inside that established range: their boundary transforms remain big, though they need not remain ample.
https://doi.org/10.1090/S0894-0347-1988-0924704-X
birkar-2010.pdf
birkar-cascini-hacon-mckernan-2010.pdf
At numerical dimension zero, Druel proved termination of directed programs for divisors with vanishing numerical dimension, and Gongyo extended the method to minimal-model existence for Q-factorial divisorial log terminal pairs of numerical log Kodaira dimension zero. On the section side, Lazic and Peternell established nonvanishing for already minimal projective terminal varieties of numerical dimension one with nonzero Euler characteristic, and Liu and Xu proved abundance in numerical dimension at most one for smooth projective varieties of dimension at most five under nonnegative Kodaira dimension, and in dimension at most four with 0 <= kappa_sigma <= 1. Those statements differ from the existence result for numerical dimension one, and none of their nonvanishing or abundance conclusions is used as a proof input.
As Liu and Xu put it, "The abundance conjecture is usually considered as (one of) the most important and difficult conjectures in birational geometry."
druel-2011.pdf
gongyo-2011.pdf
lazic-peternell-2018.pdf
liu-xu-2025.pdf
The two positivity mechanisms on which the proof rests are classical. Nadel's multiplier-ideal vanishing theorem, in the standard multiplier-ideal formulation, produces the vanishing that extends sections from a very general surface while the perturbation shrinks with the section degree; multiplier ideals and their vanishing are presented in Lazarsfeld's books and short course. Surface Riemann-Roch, in Hartshorne's treatment, then converts a positive square into quadratic section growth. A second surface, drawn through a point of a hypothetical negative curve, turns a vanishing-order estimate into a fixed exceptional curve, and the Hodge index theorem makes its square uniformly negative. Keeping one surface resolution and one exceptional curve fixed while the rational perturbations vary is what yields the contradiction.
https://doi.org/10.2307/1971429
https://doi.org/10.1007/978-3-642-18810-7
lazarsfeld-multiplier-ideals.pdf
Characteristic-zero projective resolutions and common resolutions are used throughout, following Hironaka's resolution of singularities.
https://doi.org/10.2307/1970486
PROPOSITION (positive perturbations on a fixed model). Let X be a smooth projective variety of dimension n >= 3 with K_X pseudo-effective. There exist an effective ample Q-divisor B on X, an integer s >= 0, and a finite sequence of K-negative divisorial contractions and K-flips
X = V_0 --> V_1 --> ... --> V_s = Y
with the following properties. (i) The composition extracts no divisors, Y is projective Q-factorial and terminal, and for every rational t > 0 the divisor K_Y + t B_Y is big, where B_Y is the transform of B. (ii) With t_j = 2^{-j} and M_j = K_Y + t_j B_Y, some positive multiple d_j M_j is Cartier and base free on the complement of a closed subset Z_j of Y of codimension at least three. (iii) On a common smooth projective resolution r: W -> Y and u: W -> X, the compatible canonical divisors satisfy u^*K_X = r^*K_Y + E, with E effective and r-exceptional, whose support contains the strict transform of every prime divisor of X contracted by the composite. The construction does not use the numerical-dimension hypothesis; it only needs K_X pseudo-effective.
https://doi.org/10.1017/CBO9780511662560
The proof has three parts: a discrepancy estimate for a single step, a stabilization lemma showing that after finitely many steps the maps are isomorphisms outside codimension three, and the construction of the perturbations together with the canonical comparison. Start with a single step.
LEMMA (strict discrepancy increase). Suppose U --> U+ is a K-negative divisorial contraction or a K-flip between normal projective Q-Gorenstein varieties. Write h: U -> R for the contraction and h+: U+ -> R for the other morphism, with h+ the identity in the divisorial case. For every prime divisor F over these varieties,
a(F,U) <= a(F,U+),
and the inequality is strict whenever the center of F on R lies in the locus where either h or h+ has a positive-dimensional fiber.
Proof. Take a common smooth projective resolution W carrying F, with projections p: W -> U and q: W -> U+, and set g = h o p = h+ o q for the common map to R. Consider the divisor D = p^*K_U - q^*K_{U+}. It is q-exceptional, because the step extracts no divisors. On every curve contracted by g the divisors -p^*K_U and q^*K_{U+} have nonnegative degree, and in the divisorial case q^*K_{U+} has degree zero; hence -D is g-nef and therefore q-nef. The negativity lemma gives D >= 0.
For the support assertion, let r be a closed point of R with a positive-dimensional fiber on either side. There is a curve in g^{-1}(r) mapping onto a curve in that fiber: take a component dominating that curve and cut it by sufficiently general ample divisors. The relative ampleness signs give D. C < 0 for this lifted curve, so g^{-1}(r) meets Supp D.
Use the fiber-support form of the negativity lemma: the support of an effective divisor that is antinef over a proper birational morphism to a normal variety either contains or is disjoint from each fiber. The same conclusion can be seen directly on the reduced irreducible components of the fiber. If such a component T is not contained in Supp D but meets it, a Cartier multiple of D restricts to a nonzero effective Cartier divisor on the integral projective variety T; its degree against sufficiently many ample hyperplanes is positive, producing a curve contracted by g with positive D-degree, contradicting antinefness. The argument needs T neither normal nor the fiber reduced, only the support being at issue; zero-dimensional components already lie in the support when they meet it. The fibers of g are connected because R is normal and g is proper and birational, and containment therefore propagates through their reduced irreducible components.
It follows that every such fiber lies in Supp D. If the center of F on R lies in this locus, then F is contained in Supp D on the chosen resolution. Finally a(F,U+) - a(F,U) = coeff_F D, which proves both claims. QED
https://doi.org/10.1017/CBO9780511662560
https://doi.org/10.1007/978-1-4757-3849-0
The first step is under control. The next lemma shows that after finitely many steps the sequence stops disturbing codimension two, so a very general surface later passes through every stage unchanged. Two mechanisms do the work: a discrepancy count removes the codimension-two pieces of the target of each flip, and a homological count removes the codimension-two pieces of the source.
LEMMA (stabilization in codimension two). Let
X = U_0 --> U_1 --> U_2 --> ...
be a finite or infinite sequence of K-negative divisorial contractions and K-flips of projective Q-factorial varieties, starting from a smooth n-fold with n >= 3. After finitely many steps every remaining map is an isomorphism outside closed subsets of codimension at least three on both sides.
Proof. The standard minimal model properties give preservation of Q-factoriality, invariance of the Picard number under a flip, and a drop of one under an elementary divisorial contraction. There are therefore only finitely many divisorial contractions. Since the steps extract no divisors, only finitely many prime divisors of X can disappear; denote this finite set by D.
Consider a flip whose target U_{i+1} carries a codimension-two component T of its flipped locus. A terminal variety is smooth in codimension two, so blowing up T at its generic smooth point defines a prime valuation F with a(F, U_{i+1}) = 2. The center of F on the base R lies in the locus of positive-dimensional fibers of the flipped contraction, so the strict-discrepancy-increase lemma gives a(F,X) <= a(F,U_i) < 2 = a(F,U_{i+1}). Thus F is not exceptional over X: it is an original prime divisor, hence a member of D. After this occurrence its discrepancy is at least two forever, so it can never again satisfy the strict inequality. Each such flip therefore consumes a distinct member of the finite set D, and after a finite prefix all steps are flips and every flipped locus has codimension at least three.
For the source side put d = n - 2 and let A_d(V) be the real span of the classes [T] of closed integral d-folds T in V, inside the homology H_{2d}(V,R). For a remaining flip U --> U+ over R, remove from R the images of both exceptional loci and take inverse images; this gives a common open subset U0. Each exceptional locus consists of positive-dimensional fibers, so its image has dimension at most one less than its own; the source contraction is small and the target exceptional locus has dimension at most n - 3, so the closed complements Z in U and Z+ in U+ satisfy dim Z <= d and dim Z+ <= d - 1, the extra removed points lying in an isomorphism locus of dimension at most n - 3.
Restriction gives maps
A_d(U) --> H^{BM}_{2d}(U0,R) <-- A_d(U+)
with the same image: intersect an integral d-fold with U0 and take its closure on the other model, while a d-fold contained in the complement restricts to zero. The right-hand map is injective by the localization sequence and H_{2d}(Z+,R) = 0, so dim A_d(U+) <= dim A_d(U). If the flipping locus has a d-dimensional component T, then [T] lies in the kernel of the left restriction and is nonzero, since for an ample Cartier divisor A one has <c_1(A)^d,[T]> = A^d . T > 0. The inequality is then strict. A nonnegative integer can drop only finitely many times, so eventually the flipping loci also have codimension at least three, and the same construction of the common open gives the codimension bound on both complements. QED
https://doi.org/10.1017/CBO9780511662560
alexeev-hacon-kawamata-2007.pdf
https://doi.org/10.1007/978-1-4612-1700-8
The discrepancy count is the four-dimensional termination argument of Alexeev, Hacon, and Kawamata adapted to (n-2)-cycle classes in degree 2n-4; it removes only finitely many codimension-two pieces per mechanism. Cycle classes, Borel-Moore localization, and their compatibility with restriction are Fulton's, and the description of a proper birational morphism as an isomorphism over its quasi-finite locus is standard.
LEMMA (terminal varieties are smooth in codimension two). A terminal variety is smooth in codimension two.
Proof. This is Kollar and Mori, Corollary 5.18: at a singular point of a terminal variety, the log discrepancy of the exceptional divisor of the blow-up is greater than one, contradicting terminality once the singular locus has codimension two. QED
https://doi.org/10.1017/CBO9780511662560
It remains to build the perturbations and the fixed model. Choose an effective ample rational divisor B on X with (X,B) klt and K_X + B ample: take a sufficiently positive general smooth very ample divisor and multiply it by a rational number strictly between zero and one. Since K_X is pseudo-effective, K_X + t B is big for every rational t > 0.
Global sections persist on every model. If a birational map X --> U extracts no divisors, every prime divisor of U corresponds to a prime divisor of X, and the divisorial criterion for regularity of a rational section gives an injection H^0(X,kD) -> H^0(U,kD_U) whenever the multiples are Cartier. Hence effectiveness and bigness of B_U, and bigness of K_U + t B_U, persist on every model constructed subsequently; in particular B_U need not be ample.
Finite stages. Put t_j = 2^{-j} for every integer j >= 0 and start with V_0 = X. Obtain V_{j+1} from V_j by running the minimal model program for K_{V_j} + Delta_j with Delta_j = t_{j+1} B_{V_j}, with scaling of C_j = (t_j - t_{j+1}) B_{V_j}. The input is the termination theorem of Birkar, Cascini, Hacon, and McKernan: for a projective Q-factorial klt pair (V,Delta) with big boundary Delta and C >= 0 such that (V,Delta+C) is klt and K_V + Delta + C is nef, the minimal model program with scaling of C terminates. The inductive hypothesis is that (V_j, t_j B_{V_j}) is klt with nef adjoint. The smaller pair is klt because B_{V_j} is effective; its boundary is big by section persistence; the larger pair is exactly the inductive hypothesis.
birkar-cascini-hacon-mckernan-2010.pdf
The adjoint remains big by section persistence, so a Mori fiber space cannot occur: an effective representative of a positive multiple cannot have negative degree on curves covering general fibers. Each stage is therefore a finite, possibly empty, sequence ending at a nef adjoint, and standard preservation gives the next klt pair and a projective Q-factorial model. The rational divisor K_{V_j} + t_j B_{V_j} is nef and big, hence semiample by the klt base point free theorem.
https://doi.org/10.1017/CBO9780511662560
Canonical steps. To invoke stabilization in codimension two, every adjoint step must be identified as a canonical step: canonical negativity on the contracted ray, and canonical ampleness on the flipped side. On the source U let D = K_U + t_{j+1} B_U. The contracted ray R_0 satisfies D . R_0 < 0 and (D + lambda C) . R_0 = 0 for some lambda > 0, where C = (t_j - t_{j+1}) B_U is the current scaling divisor. Therefore B_U . R_0 > 0 and K_U . R_0 < 0. This ray calculation, standard for a directed program, already identifies each divisorial step as a K-negative divisorial contraction.
For a small contraction h: U -> R the adjoint flip makes D+ relatively ample; it must be shown that K_{U+} is relatively ample as well. Take the positive rational b = (K_U . R_0)/(D . R_0). The divisor K_U - b D is numerically trivial over R and descends up to rational linear equivalence: a Cartier multiple is relatively semiample by the relative klt base point free theorem for the pair (U, t_{j+1} B_U), subtracting its adjoint D gives relative ampleness because -D is relatively ample, its semiample morphism is constant on each connected projective fiber since its degree on every fiber curve is zero, and the image is proper and quasi-finite over R, hence finite and birational over the normal R, hence equal to R. Thus K_U - b D ~_Q h^*A for a rational Cartier divisor A on R. Both morphisms in the adjoint flip are small, so this transforms to K_{U+} - b D+ ~_Q (h+)^*A. Since D+ is relatively ample and b > 0, so is K_{U+}: the adjoint flip is a K-flip, and all steps of the finite stages satisfy the hypotheses of stabilization in codimension two.
https://doi.org/10.1017/CBO9780511662560
The fixed model. Concatenate the finite stages and apply stabilization in codimension two. Choose an endpoint Y = V_s after the resulting finite prefix, or after the last nonempty stage if only finitely many occur. For every j > s the finite composition Y --> V_j is an isomorphism outside closed subsets of codimension at least three on both models; this is preserved under finite composition, since on a common open one removes the next bad set and takes its closure on the preceding model, whose dimension does not increase. Let Z_j be the resulting closed subset of Y.
The divisor M_j = K_Y + t_j B_Y agrees on this common open with K_{V_j} + t_j B_{V_j}. Choose a multiple that is Cartier on both varieties and base free on V_j. A line bundle on a normal variety has the same global sections after deleting a subset of codimension at least two, because it is reflexive; the two spaces of sections agree, so the complete system on Y is generated outside Z_j. This proves (ii). Bigness is preserved under divisibility, giving (i).
The comparison. Finally Y is projective, Q-factorial, and terminal by construction, and the finite composition extracts no divisors. On a common smooth projective resolution of the finite chain the effective differences telescope to u^*K_X = r^*K_Y + E. A divisor mapping onto a prime divisor of Y also corresponds to a prime divisor of X, so its coefficient in E is zero; thus E is r-exceptional. For an original prime divisor contracted on Y, terminality gives log discrepancy greater than one on Y against one on X, so its strict transform has positive coefficient in E. This proves (iii) and completes the construction. QED
The fixed model Y now carries a whole family of perturbations K_Y + t_j B_Y whose multiples are generated away from arbitrarily small bad sets. The numerical part of the argument decides whether L = K_Y is nef, using only this data; the minimal model program no longer appears. Two independent tests are carried out on surfaces cut out of Y. The first takes a very general surface S: if L^2 . H^{n-2} is positive, sections on S grow quadratically, and transferring this to X contradicts kappa_sigma(X,K_X) = 1, forcing L^2 . H^{n-2} = 0. The second draws a surface T through a point of a hypothetical curve C_0 with L . C_0 < 0; vanishing forced along C_0 produces a fixed exceptional curve on a resolution of T, and the Hodge index theorem then makes the square of L on T positive, contradicting the zero square just found. Both tests need one precise statement about the perturbations.
APPROXIMATION HYPOTHESIS. Let Y be a normal projective n-fold over C with n >= 3 and codim_Y (Sing Y) >= 3. Let L and B_Y be Q-Cartier, let t_j be positive rationals with t_j -> 0, and set M_j = L + t_j B_Y. For each j there are a closed subset Z_j of Y with codim_Y Z_j >= 3 and a positive integer d_j such that d_j M_j is Cartier and the complete linear system |d_j M_j| is generated by global sections on Y minus Z_j.
Every positive multiple of d_j keeps the same generation property, and the integers d_j may depend on j. Restrictions of rational Cartier divisors mean restrictions as rational line bundles. Nothing in the hypothesis assumes B_Y is effective or positive, and no uniform Cartier index along the sequence is required.
LEMMA (surface growth with a fixed twist). Assume the approximation hypothesis, let H be a very ample Cartier divisor on Y, and fix a projective resolution pi: Yhat -> Y that is an isomorphism over the smooth locus. There are a smooth complete-intersection surface S in Y, cut out by n - 2 members of |H|, and a Cartier divisor P on Yhat with the following properties, identifying S with its inverse image in Yhat. First, L|_S is nef, so L^2 . H^{n-2} >= 0. Second, if d L is Cartier, then for every positive integer m divisible by d the restriction map
H^0(Yhat, m pi^*L + P) --> H^0(S, m L|_S + P|_S)
is surjective. Third, as m -> oo through the multiples of this one fixed integer d,
h^0(Yhat, m pi^*L + P) >= (m^2/2) L^2 . H^{n-2} + O(m),
where the error term depends only on the fixed surface and its divisors.
Proof. Put r = n - 2. Very general members of |H| have a smooth complete-intersection surface S avoiding Sing Y and every Z_j: each of these countably many closed sets has dimension at most n - 3, so the condition that r hyperplanes meet any one of them is a proper closed incidence condition, and over C these can be avoided simultaneously together with the closed conditions excluded by Bertini's theorem. The inverse image of S is then exactly the smooth complete intersection cut out by the r pulled-back equations, with no extra components over the singular locus. Each M_j|_S is semiample, hence nef, and passing to the limit on every curve of S shows that L|_S is nef, proving L^2 . H^{n-2} >= 0.
Write A = pi^*H, choose an ample Cartier divisor A_0 on Yhat, and make the fixed choice P = K_Yhat + r A + A_0. Since A is nef and ampleness is open, there is eta > 0 such that A_0 + (r - i)A - epsilon pi^*B_Y is ample whenever 0 <= i <= r and 0 <= epsilon < eta.
For a given positive m with d | m, first choose j = j(m) so that m t_j < eta, then choose a multiple k of d_j with k > m. The pullback of |k M_j| is generated near S, and a general member D_m is smooth there by Bertini, with R_m = (m/k) D_m >= 0, R_m ~_Q m pi^*M_j, and multiplier ideal J(R_m) = O near S; the last local triviality follows from m/k < 1 and the smoothness of D_m there. The divisors m pi^*L + P - iA are integral Cartier, and
m pi^*L + P - iA - K_Yhat - R_m ~_Q A_0 + (r - i)A - m t_j pi^*B_Y
is ample for 0 <= i <= r. Nadel vanishing then gives
H^q(Yhat, O(m pi^*L + P - iA) (x) J(R_m)) = 0 for q > 0, 0 <= i <= r.
Set F = O(m pi^*L + P) (x) J(R_m). Tensoring the Koszul complex of the r equations with F yields the exact complex
0 -> F(-rA) -> ... -> F(-A)^{oplus r} -> F -> O_S(m L|_S + P|_S) -> 0.
At points of S the equations form a regular sequence and F is locally free; outside S one equation is a unit, so the Koszul complex is contractible even after tensoring with an arbitrary sheaf. Splitting the complex into short exact sequences and using Nadel vanishing gives surjectivity on global sections onto its last term, and the inclusion F in O(m pi^*L + P) then gives the restriction surjectivity.
Adjunction and the chosen twist give P|_S - K_S = A_0|_S, so m L|_S + P|_S - K_S is ample. Kodaira vanishing (the zero-boundary ample case of Nadel vanishing) and surface Riemann-Roch give
h^0(S, m L|_S + P|_S) = (m^2/2)(L|_S)^2 + (m/2) L|_S . (2P|_S - K_S)
+ (1/2) P|_S . (P|_S - K_S) + chi(S, O_S).
If S is disconnected, read the formula componentwise and sum. All coefficients are fixed before m, j, k, and D_m vary, and together with the restriction surjectivity this proves the quadratic growth. QED
lazarsfeld-multiplier-ideals.pdf
https://doi.org/10.1007/978-3-642-18810-7
https://doi.org/10.1007/978-1-4757-3849-0
https://doi.org/10.1007/978-3-642-18808-4
The first surface excludes positive total square. The second reads a bad curve through the order to which sections vanish at a point. Order of vanishing is measured in the local ring of the ambient variety, so it survives even when the curve itself is singular; the point of the next lemma is a uniform bound in which the constant depends on the curve and the variety, not on the line bundle.
LEMMA (uniform ambient multiplicity). Let V be a smooth projective complex variety, let Gamma be an integral curve in V, and let iota: N -> V be the map from its normalization. For a section s write ord_z(s) for its order in the regular local ring of V at z. There is a positive constant g, depending only on this map, such that for every line bundle D on V, every nonzero section s in H^0(V,D), and every closed point z in Gamma,
ord_z(s) >= - (deg_N iota^*D) / g.
Proof. Choose a line bundle G of positive degree on the fixed smooth curve N such that iota^*(Omega^1_V) embeds as a subbundle of a direct sum G^{oplus a} for some a. Such a choice follows by global generation of (iota^*(Omega^1_V))^* (x) G for a sufficiently ample G, followed by dualization; set g = deg G > 0.
For a nonzero section s the integer b = min over closed p in Gamma of ord_p(s) is finite and attained. The leading ambient jet gives a nonzero section of Sym^b(iota^*(Omega^1_V)) (x) iota^*D. This holds even where Gamma is singular: for b >= 1 the bundles of principal parts on the smooth variety V carry the locally split exact sequence
0 -> Sym^b(Omega^1_V) (x) D -> P^b(D) -> P^{b-1}(D) -> 0,
and at a point where the order equals b the b-th jet is nonzero, so the graded piece contributes a nonzero section along Gamma. Taking symmetric powers of the subbundle inclusion embeds this leading jet into a direct sum of copies of G^{otimes b} (x) iota^*D; a nonzero component is a nonzero section of this line bundle on N, so 0 <= b g + deg_N iota^*D. Since ord_z(s) >= b for every z in Gamma, the bound follows. QED
https://doi.org/10.1007/978-1-4757-3849-0
The second lemma is the surface-side converse. If a fixed curve e on a resolution of a surface T always receives a definite fraction of the fixed part of a section system, then the surface square of the divisor cannot shrink to zero: the fixed part is negative by the Hodge index theorem, the moving part is nonnegative, and their sum is the surface square.
LEMMA (surface fixed-part estimate). Let T be an integral projective surface, let f: T_1 -> T be a projective birational morphism from a smooth surface, and fix a curve e in T_1 contracted by f. For each j let D_j be a rational Cartier divisor on T and k_j > 0 an integer with k_j D_j Cartier. Suppose a nonzero linear system of k_j f^*D_j is generated outside the inverse image of a finite subset of T, with fixed divisor F_j. If for one fixed c > 0,
coeff_e F_j >= c k_j for all j,
then there is epsilon > 0, independent of j, such that
D_j^2 . [T] >= epsilon for all j.
No normality assumption on T is needed.
Proof. Every curve in Supp F_j maps to a point of T, by the generation assumption. Fix a very ample Cartier divisor H_T on T and an ample Cartier divisor J on T_1, and put Q = f^*H_T. The projection formula gives Q^2 = H_T^2 . [T] > 0 and Q . F_j = f^*D_j . F_j = 0. The Hodge index theorem on the fixed smooth surface T_1 makes the negative intersection form positive definite on Q^perp inside N^1(T_1)_R, so there is one constant K > 0 with (J . F)^2 <= K (-F^2) for every F in Q^perp. Effectiveness of F_j gives J . F_j >= c k_j (J . e), and therefore, with the fixed positive number epsilon = c^2 (J . e)^2 / K, F_j^2 <= -epsilon k_j^2.
After removal of F_j the residual linear system has no fixed curve, and its square is nonnegative: choose two members with no common curve and take their effective intersection, or use a nowhere vanishing member if the system is trivial. By the orthogonality and the projection formula,
0 <= (k_j f^*D_j - F_j)^2 = k_j^2 (D_j^2 . [T]) + F_j^2.
Now F_j^2 <= -epsilon k_j^2 proves D_j^2 . [T] >= epsilon. QED
https://doi.org/10.1007/978-1-4757-3849-0
Now the two lemmas combine. Suppose the total square vanishes. Then no curve can have negative degree for L, and L is nef. That is the criterion on which the whole proof rests.
PROPOSITION (nefness criterion). Assume the approximation hypothesis and suppose L^2 . H^{n-2} = 0. Then L is nef, including on curves lying in the singular locus of Y.
Proof. Suppose some curve C_0 satisfies L . C_0 < 0; it may lie inside Sing Y, which has codimension at least three. Take a very general surface through one of its points. Choose n - 2 very general members H_1,...,H_{n-2} of |H|, all passing through a point x of C_0. They meet properly, and their intersection meets each Z_j and Sing Y in at most finitely many points, since the hyperplanes through x have no other base point and the countably many conditions can be imposed simultaneously over C. Write the intersection cycle as
H_1 ... H_{n-2} . [Y] = sum_i a_i [T^{(i)}], a_i > 0,
where each T^{(i)} is an integral surface; the dimension theorem guarantees a component through x, and one such component is fixed and called T.
On every T^{(i)} the system induced by |d_j M_j| has at most a finite base locus, so its square is nonnegative, including when T^{(i)} is nonnormal: choose a section nonzero at the generic point, then a second section not vanishing identically on any curve of the first zero divisor, and the resulting Cartier intersections with [T^{(i)}] form an effective zero-cycle. Hence M_j^2 . [T^{(i)}] >= 0 for every i, and
sum_i a_i (M_j^2 . [T^{(i)}]) = M_j^2 . H^{n-2} -> 0,
using L^2 . H^{n-2} = 0 and M_j -> L. In particular M_j^2 . [T] -> 0.
Let Gamma be a component of the inverse image of C_0 in a resolution of Y, normalized by iota: N -> Yhat, with a closed point z above x. Resolve the surface T by a smooth projective surface T_0 -> T, choose a point w in T_0 above z, and blow it up; let T_1 be the resulting surface, e in T_1 the exceptional curve of the blow-up, and f: T_1 -> T the composite. The surface T_1, the curve e, and f are fixed for the rest of the argument.
The projection formula and M_j -> L give, for one fixed delta > 0,
deg_N iota^* pi^* M_j <= -delta
for all sufficiently large j; this degree equals the positive degree of N -> C_0 times M_j . C_0. For each such j choose any positive multiple k_j of d_j. The uniform ambient multiplicity lemma, applied on the fixed Yhat and Gamma, shows that every nonzero pullback section s' in H^0(Yhat, k_j pi^* M_j) of a section on Y satisfies
ord_z(s') >= c k_j, c = delta / g > 0,
with c depending only on the fixed normalized curve and the margin delta.
The sections on Y induce a nonzero linear system of k_j f^*(M_j|_T), generated outside f^{-1}(T intersect Z_j); nonzeroness follows from generation at points of T minus Z_j. Every nonzero section in this induced system has order at least c k_j at w before the last blow-up, because the local ring map O_{Yhat,z} -> O_{T_0,w} sends m_z^b into m_w^b, and the uniform bound applies to a section inducing it; sections whose restriction is identically zero do not contribute. Thus the fixed divisor F_j of the induced system satisfies coeff_e F_j >= c k_j. All the hypotheses of the surface fixed-part estimate hold with D_j = M_j|_T, so that estimate gives a fixed epsilon > 0 with
M_j^2 . [T] >= epsilon
for all sufficiently large j, contradicting M_j^2 . [T] -> 0. Therefore no curve of negative L-degree exists, and L is nef. QED
The main result now follows by assembling the pieces. Apply the positive-perturbations proposition to obtain a finite canonical sequence from X to a terminal model Y with L = K_Y, B_Y, and M_j = L + t_j B_Y, whose multiples are generated outside closed subsets Z_j of codimension at least three. The variety Y is terminal, hence smooth in codimension two, which is exactly the standing hypothesis of the numerical results.
https://doi.org/10.1017/CBO9780511662560
Fix a very ample Cartier divisor H on Y and a projective resolution pi: Yhat -> Y that is an isomorphism over the smooth locus. The surface-growth lemma gives a fixed Cartier divisor P on Yhat. Choose a positive integer r_0 with r_0 L Cartier. Then
h^0(Yhat, m pi^*L + P) >= (m^2/2) L^2 . H^{n-2} + O(m) for r_0 | m,
and also L^2 . H^{n-2} >= 0.
Suppose this intersection number is positive. Choose a smooth projective common resolution W dominating Yhat and every model in the finite prefix, with maps rho: W -> Yhat and u: W -> X. The positive-perturbations proposition gives u^*K_X - rho^* pi^* L >= 0. For every positive multiple m of r_0 this effective difference gives an injection
H^0(Yhat, m pi^*L + P) --> H^0(W, m u^*K_X + rho^*P).
Set P_X = u_* rho^* P. This is a fixed integral Weil divisor, hence Cartier on the smooth X. A rational section f in the right-hand space satisfies div_W(f) + m u^*K_X + rho^*P >= 0, and pushing this inequality forward gives div_X(f) + m K_X + P_X >= 0. So there is an injective linear map
H^0(W, m u^*K_X + rho^*P) --> H^0(X, m K_X + P_X).
Choose a single ample Cartier divisor A_X for which A_X - P_X has a nonzero section. Multiplication by that fixed section gives
h^0(X, m K_X + A_X) >= h^0(Yhat, m pi^*L + P).
By the quadratic growth, the left side is bounded below by c m^2 for a fixed c > 0 along all sufficiently large multiples of r_0. Hence
limsup_{m->oo} h^0(X, m K_X + A_X) / m^2 > 0,
contradicting kappa_sigma(X,K_X) = 1 with the fixed ample twist A_X. It follows that L^2 . H^{n-2} = 0. The nefness criterion then implies that L = K_Y is nef, including on curves in the singular locus. The remaining conclusions, including the effective exceptional comparison u^*K_X = rho^*K_Y + E and its support condition, are supplied by the positive-perturbations proposition. This completes the proof of the theorem.
https://doi.org/10.1017/CBO9780511662560
When log abundance is granted, the endpoint Y is a good minimal model: K_Y is nef and, by log abundance, semiample, so some positive multiple of K_Y is globally generated. The reduction of semiampleness to log abundance is routine, and the substance is the existence of the minimal model together with the explicit comparison of canonical divisors.
https://doi.org/10.1017/CBO9780511662560
birkar-cascini-hacon-mckernan-2010.pdf
Two features keep the argument within reach. First, the perturbations stay on one model: the discrepancy count and the cycle-class decrease show that after finitely many steps the minimal model program stops moving codimension two, so one very general surface passes through every later stage, and every section estimate is made on a single fixed resolution with a single fixed twist. Second, the contradiction is numerical: positive surface square forces quadratic growth of sections, which numerical dimension one forbids, while a negative curve forces a positive surface square, which the vanishing square forbids. The same pair of ideas reaches smooth threefolds through Mori's classification, fourfolds through the work of Alexeev, Hacon, and Kawamata, and the numerical-dimension-one stratum in every dimension. The result covers every smooth fivefold with pseudo-effective canonical divisor of numerical dimension one, with no condition on the Euler characteristic, and the comparison of canonical divisors is effective.
https://doi.org/10.1090/S0894-0347-1988-0924704-X
alexeev-hacon-kawamata-2007.pdf