Termination of generalized-canonical flips on compact Kähler fourfolds

Proof. Enlarge the finite portion to an initial segment starting at X_0. For one step, take the reduced component of X_i\times_{Z_i}X_{i+1} containing the graph over the common isomorphic open. It is projective over both models, and its normalization remains so. Combine these graphs and X'\to X_0 by successive dominating components of fibre products, then normalize and resolve. To include a prescribed projective modification of a member, take one further fibre product over that member before resolving. Base change, restriction, finite normalization, and composition show that all resulting morphisms to the members of the segment are projective.

Fix T. Choose a closed analytic subset of codimension at least two outside which the map is an isomorphism, T is smooth, and the boundary components are smooth and disjoint; include their singular loci and pairwise intersections in this subset. Principalize the pullback of its ideal and resolve the total divisor without changing this open set. The inverse image of the chosen subset is now a divisor whose components are exceptional over T; it contains the entire exceptional locus. Resolving together with the strict boundary gives the claimed normal-crossing condition. Pulling back M' preserves the fixed data.□

Proof. Suppose the singular locus has a codimension-two component. Near its general smooth point, choose a holomorphic projection to \C^2 submersive on that component. Take a log resolution which is an isomorphism over the regular locus. Choose a general fibre S of the projection: its inverse image \widetilde S is smooth, misses exceptional strata whose images have dimension less than two, and meets all remaining strata transversely. These choices follow by generic smoothness on the resolution and its finitely many strata. Write \rho:\widetilde S\to S for the induced resolution. Terminal singularities are klt, so the ambient space has rational singularities by [8, Theorem 3.12], and is Cohen–Macaulay [11, Remark 2.1]. The fibre has the expected codimension two. Its two defining parameters therefore form a regular sequence, so S is Cohen–Macaulay. It is regular away from isolated points by the general choice of fibre. Serre’s criterion makes S normal.

For the regular sequence, the Koszul calculation of the dualizing sheaf gives \omega_S\simeq\omega_T\otimes\mathcal O_S, where T denotes the ambient neighbourhood; the two slice equations trivialize the normal determinant. This uses the local analytic Ext description of the dualizing sheaf [17, pp. 3–4]. Away from the isolated intersections with the ambient singular locus, a local Cartier pluricanonical power restricts accordingly, and the identification extends reflexively over the normal surface. Applying adjunction also on the smooth resolution therefore restricts its relative canonical equation to

Here each exceptional curve is a transverse restriction of an exceptional divisor on the resolution; terminality gives its positive coefficient. A normal surface germ with this property is smooth. Indeed, if exceptional curves remain, negative definiteness [10, Section 4.8(e)] gives A^2<0, so A\cdot E_j<0 for some j. The pulled-back canonical divisor has degree zero on E_j, and therefore K_{\widetilde S}\cdot E_j<0. Since E_j^2<0, adjunction gives

Both negative intersection numbers are integers, so each equals -1. Thus E_j is a smooth rational (-1)-curve. Contract it to a smooth point by [10, Section 4.8(a)]. The map to the normal surface germ factors through this contraction, and pushforward of the relative canonical equation leaves positive coefficients on all remaining exceptional curves. Repetition ends with a finite bimeromorphic map to the normal surface germ, which is an isomorphism.

But the transverse slice at the chosen singular fourfold point has embedding dimension at least that of the fourfold germ minus two, hence greater than two. It is singular, a contradiction.□

Proof. The common exceptional image. For each of the two morphisms, the exceptional locus is precisely the inverse image of the locus of positive-dimensional fibres, by the discussion in Section 2.1. Off the union of these two images, the morphisms initially give isomorphic opens whose complements upstairs have codimension at least two. Thus prime divisors already correspond, before equality of the images is known.

Let U\subset Z be the open set where f is an isomorphism. Transport D_T|_{f^{-1}(U)} to an actual \Q-Cartier divisor D_U on U. The two adjoints agree under the identification of prime divisors induced by the small map: this holds for the canonical and boundary terms by their prescribed transforms, and for the nef traces by the pushforward rule for the fixed b-divisor. Hence on g^{-1}(U) the divisors D_{T^+} and g^*D_U agree away from the exceptional locus of g. That locus contains no divisor, so the two Weil divisors agree everywhere on g^{-1}(U). Clearing Cartier multiples and using normality gives equality as \Q-Cartier divisors as well.

If g had a positive-dimensional fibre over a point of U, its projectivity would provide a curve C in that fibre. The equality just proved would give D_{T^+}\cdot C=0, contrary to g-ampleness. Thus g is also an isomorphism over U. Interchanging f and g and using f-ampleness of -D_T proves the reverse inclusion of their isomorphism loci. This establishes the asserted common image A and the identification of the complements of L and L^+. Smallness gives \dim L,\dim L^+\leq2. Every fibre over A is positive dimensional, so the fibre-dimension theorem gives \dim A\leq1.

An effective divisor on the graph. Let G be the normalization of the component of T\times_ZT^+ dominating Z. Its projections fit into the commutative diagram

The morphisms p,q,h are projective and bimeromorphic, as in Proposition 2.1. Define the actual \Q-Cartier divisor

It vanishes off h^{-1}(A). Since both L and L^+ have codimension at least two, F is exceptional over both models; in particular, p_*F=0.

If C is a curve contracted by p, then q_*C is either zero or a curve contracted by g. The projection formula therefore gives

Thus -F is p-nef in the relative curve-degree sense. Applying Lemma 2.2 to a positive Cartier multiple of F proves that F\geq0.

There is a strict inequality on every curve contracted by h. Indeed, for such an irreducible curve C, the relative ample signs give

with strict inequality whenever the corresponding image is a curve. The two images cannot both be points: the map G\to T\times_ZT^+ is finite onto the graph component, so it cannot contract C. Hence

We next show that this strict inequality determines the whole support:

Fix z\in A. The fibre h^{-1}(z) is projective and connected; connectedness also follows directly from the proper bimeromorphic morphism h and normality of Z. It is positive dimensional because it surjects onto f^{-1}(z). Its reduction has finitely many irreducible components. A zero-dimensional irreducible component would be a point disjoint from every other component, contradicting connectedness. Therefore every point belongs to a positive-dimensional projective component. Intersecting such a component with hyperplanes through the point produces an irreducible curve through that point.

An effective \Q-Cartier divisor has nonnegative degree on every compact irreducible curve not contained in its support: a positive multiple is an effective Cartier divisor, whose canonical section restricts to a nonzero holomorphic section on the normalization of that curve. By (8), every curve in h^{-1}(z) is therefore contained in \Supp F. The preceding paragraph puts every point of the fibre in \Supp F. This proves h^{-1}(A)\subset\Supp F; the reverse inclusion was already noted.

Discrepancies of all places. Choose a sufficiently high smooth common model carrying the fixed nef data and dominating G, as supplied by Proposition 2.1. To compare a specified place E, further dominate a model representing it, so that E is a prime divisor on a smooth model W with morphism r:W\to G. The same canonical divisor K_W and the same nef trace M_W occur in both discrepancy equations:

Their difference is the equality of actual divisors

Taking the coefficient at E yields

Pullback of the effective \Q-Cartier divisor F is effective, proving (3).

If the centre of E on T lies in L, its centre on G lies in p^{-1}(L)=h^{-1}(A); the same conclusion holds if its centre on T^+ lies in L^+. By (9), this centre is contained in \Supp F. A local equation of a positive effective Cartier multiple of F vanishes on that centre. Its pullback has positive order along E: it is a nonzero holomorphic function on the inverse image of the coordinate neighbourhood and vanishes on the part of E lying there. Consequently the right-hand side of (10) is strictly positive. This proves strictness for an arbitrary place with either specified centre condition.

A place whose centre is a prime divisor on a normal model is represented by that divisor, since a proper bimeromorphic morphism to a normal space is an isomorphism at general points of each prime divisor on the target. Every prime divisor on T meets T\setminus L, and its corresponding prime divisor on T^+ meets T^+\setminus L^+. The common isomorphism identifies their orders. Thus a place is nonexceptional on one side exactly when it is nonexceptional on the other. Together with (3), this preserves positivity for all places and the lower bound 1 for exceptional places. For a place centred in L^+, strictness improves that lower bound to (4).

The dimension inequality. The step is nontrivial, so A\neq\varnothing. Equation (9) implies F\neq0. Every irreducible component P of \Supp F is a divisor on the four-dimensional normal space G, hence has dimension three. Its finite image under G\to T\times_ZT^+ is contained in L\times_ZL^+. Therefore

which proves (5). If L^+ contains no surface, then \dim L^+\leq1, while smallness gives \dim L\leq2. The inequality forces \dim L=2, so a two-dimensional irreducible component of L is the required surface.□

Proof. The pushforward of both r^*M_T and M_W is M_T, so J_W is r-exceptional. For every curve C contracted by r,

To see the last inequality, let s\colon W\to X' be the morphism to the fixed carrier. Since M_W=s^*M', its degree on C is zero if s(C) is a point, and otherwise is the degree of M' on s(C) multiplied by the degree of the induced map of normalized curves. Analytic nefness of M' implies that these curve degrees are nonnegative: restrict a sequence of Kähler classes converging to c_1(M') to the normalized curve and pass to the limit. We use this consequence of analytic nefness only for the relative degree calculation above. Lemma 2.2, applied to J_W and r_*J_W=0, now gives J_W\geq0.□

Proof. Compute at a general smooth point of E above the chosen coordinate neighbourhood. If t is a local equation of E, write x_j=t^{v(x_j)}u_j for 1\leq j\leq c, where u_j is a unit at the general point of E. Each v(x_j) is a positive integer. In expanding the pullback of dx_1\wedge\cdots\wedge dx_n, at most one factor in a nonzero term can contribute a differential dt. Thus its order along E is at least \sum_{j=1}^c v(x_j)-1. The remaining coordinate differentials are holomorphic and do not lower this order. Adding one to the coefficient of the relative canonical divisor proves (12).

For the boundary statement, each component with nonpositive coefficient contributes a nonnegative amount to the log discrepancy. Components not containing C have order zero along E. Subtracting just d\,v(x_1) from (12) therefore gives (13). If there is no positive component through C, nothing has to be subtracted.□

Proof. Choose a smooth modification r\colon W\to T, projective over T and carrying the nef data, such that its exceptional locus is divisorial and, together with the strict transform of \Supp B_T, has simple normal crossings. Starting with this simple normal crossing divisor, blow up the intersections of pairs of positive strict transforms until they are mutually disjoint. Each intersection is a finite disjoint union of smooth codimension-two subspaces. Blowing it up separates the chosen pair, preserves simple normal crossings, and cannot make a previously separated pair meet. Thus finitely many such blowups suffice.

Keep the notation W for the resulting model and \Delta_W for its boundary in (11). Its positive components are exactly the strict transforms of the positive components of B_T, with the same coefficients. They are smooth and pairwise disjoint. Every component exceptional over T has nonpositive coefficient by generalized canonicity; zero coefficients are allowed. On a higher smooth model u\colon V\to W, the nef terms cancel because M_V=u^*M_W, giving

Hence generalized discrepancies over T can be computed as ordinary discrepancies for (W,\Delta_W), including its negative coefficients.

Put all r-exceptional prime divisors on W into \mathcal F_T. There are finitely many, since W is compact. A further exceptional place E has centre C on W of codimension at least two: a place with divisorial centre on a normal model is represented by that divisor. Lemma 4.2 shows that a(E;W,\Delta_W)<2 forces C to have codimension two and to lie in one of the positive components, say H with coefficient d. The same lemma gives

If C is not contained in \Exc(r), then r is an isomorphism at its general point. Its image is the surface S required in the statement; smoothness and uniqueness of the positive component descend from W.

It remains to treat the centres contained in \Exc(r). Such a centre C, being codimension two and contained in H, must be an irreducible component of H\cap F for some exceptional divisor F on W. There are only finitely many such components, and each is smooth by simple normal crossings. We show that a fixed one contributes only finitely many places of log discrepancy below two.

Near C, the only positive component of \Delta_W is H. Deleting the nonpositive components can only lower discrepancies, so

for every place under consideration with centre C. We can therefore work with the smooth pair (W,dH) and the fixed smooth codimension-two centre C\subset H.

Set W_0=W and H_0=H. Blow up C to obtain W_1, with exceptional divisor F_1 and strict transform H_1. If E is not F_1, its centre on W_1 is contained in F_1 and has codimension at least two. The crepant boundary on W_1 has sole positive component dH_1; the coefficient of F_1 is -(1-d). Lemma 4.2 therefore forces the centre of E to have codimension two and to lie in H_1. It is consequently the entire intersection

This intersection is a smooth irreducible section of the exceptional \mathbb P^1-bundle over C, hence isomorphic to C.

Repeat this construction: after W_j has been obtained, blow up C_j=H_j\cap F_j to obtain W_{j+1}, with new exceptional divisor F_{j+1}. Writing C_0=C, the maps and their centres are

The new strict transform H_{j+1} meets F_{j+1} in a smooth section and is disjoint from the strict transform of F_j. Inductively, it misses every older exceptional divisor. The codimension-two blowup formula gives the coefficient

where K_{W_j}+\Delta_j is the pullback of K_W+dH. Indeed the new coefficient is the sum of the coefficients of H_j and F_j, minus one. This is -(j+1)(1-d).

Thus the only way a place of discrepancy below two can remain unextracted is for its centre to follow these successive intersections. At every stage, if E has not yet appeared, the argument just given forces its centre to be C_j. Choose local transverse coordinates x,y with H_j=(x=0) and F_j=(y=0), and put v=\operatorname{ord}_E. Retaining the negative coefficient of F_j in the differential estimate gives

For

the right side is at least two. Every place with discrepancy below two and centre C must therefore be one of F_1,\ldots,F_J. Equivalently, the possible indices in (17) satisfy j<1/(1-d); the strict endpoint is included in the cutoff calculation.

All these blowups have global smooth compact analytic centres. Thus the divisors just enumerated are global places. Conversely, an arbitrary place can be followed on common proper resolutions even when its original carrying model was not projective; its centres map onto the preceding centres. The argument forcing C_j therefore covers every place over C, not only those chosen in advance from this blowup chain.

Add the finitely many divisors obtained for all the finitely many intersections H\cap F to \mathcal F_T. Every place outside this set has the centre described earlier, and (15) gives (14). If the boundary has no positive component, Lemma 4.2 already excludes every further place of discrepancy below two, proving the last assertion. The proof used only nonpositivity of the exceptional coefficients on W, so exceptional places of log discrepancy exactly one cause no additional case.□

Proof. Choose a smooth model r\colon W\to T^+, projective over T^+ and carrying the fixed nef data, with divisorial exceptional locus as in Proposition 2.1. Using the divisors of (11), write the ordinary relative canonical equation as

The pullback r^*B_{T^+} is effective: after clearing a Cartier multiple, a local equation for the effective boundary is holomorphic by normality, and remains holomorphic after pullback. Lemma 4.1 gives J_W\geq0. Generalized canonicity therefore makes every exceptional coefficient of \Theta_W nonpositive. Its nonexceptional coefficients are zero, since it is an ordinary relative canonical divisor.

An exceptional prime P on W whose image contains S has image exactly S: its irreducible image has dimension at most two. Proposition 3.1 gives

because its target centre lies in L^+. Thus its coefficient in \Delta_W, and hence in \Theta_W, is strictly negative.

There are only finitely many exceptional prime divisors on W. Delete from T^+ the images of those with coefficient zero in \Theta_W, and call the resulting open set U. Proper mapping makes the deleted set closed analytic. None of these images contains S, so their finite union cannot contain the irreducible surface S; in particular, U\cap S is a dense open subset of S.

We check ordinary terminality on U using local analytic places. Restrict the resolution and (19) over U. Every local component of its exceptional divisor now has strictly negative coefficient in \Theta_W. If a local exceptional place E over U has divisorial centre on this resolution, it is the place of that divisor. The divisor must be exceptional over U, since otherwise E would not be exceptional. Its negative coefficient therefore gives ordinary log discrepancy greater than one.

For any remaining local exceptional place, take a common local resolution. Its centre on the smooth space r^{-1}(U) has codimension at least two. Since \Theta_W\leq0, Lemma 4.2 gives

This checks all local exceptional places and proves ordinary terminality. The canonical rational line bundle restricts to U, so Lemma 2.3 applies there. Its singular locus has codimension at least three and cannot contain the surface S\cap U. Thus T^+ is smooth at a general point of S.□

Proof. By Proposition 5.1, a general point of the surface S is a smooth point of both T^+ and S. Blow up the global coherent ideal of the reduced analytic subspace S. Over the open subset where T^+ and S are smooth, this is the blowup of a smooth codimension-two centre, with one exceptional prime dominating that open subset of S. After normalization the preimage of S is a compact analytic subset with finitely many irreducible components. One of these contains this exceptional divisor and dominates S, by proper mapping; the smooth calculation gives uniqueness. It is a global prime divisor, and we denote its place by E_S. Resolve further and use Proposition 2.1 to obtain a smooth model W, projective over T^+, carrying it and the nef data. The centre on T^+ is S, so it is exceptional there. Smallness of every step preserves exceptionalness, as in Proposition 3.1, and therefore it is exceptional over every X_j.

The ordinary log discrepancy of the blowup at a smooth codimension-two centre is two. Pulling back a boundary component H_{\ell} subtracts b_{\ell}\operatorname{mult}_S(H_{\ell}), where the generic multiplicity is a nonnegative integer and is zero unless S\subset H_{\ell}. The identity

then proves (21). This computation only uses the smooth general point of the centre. The boundary components may be singular or intersect one another along S.

Lemma 4.1 gives \operatorname{coeff}_{E_S}(J_W)\geq0. It also lies in \tfrac1m\Z. To verify this denominator claim, restrict to a smooth open subset of T^+ meeting a general point of S. By (20), mM_{i+1} is an integral Cartier divisor there, so its pullback has integral order along E_S. The divisor mM_W is integral Cartier as the pullback of mM'. Consequently the coefficient of

along E_S is an integer. This is a local calculation near the general point of S, and does not require mM_{i+1} to be Cartier elsewhere.

Every term subtracted from two in (21) is nonnegative and belongs to \tfrac1m\Z. Since the target centre is contained in L_i^+, strict increase in Proposition 3.1 and generalized canonicity on the source give

These facts prove (22). If S is contained in a component of coefficient b, its generic multiplicity is at least one, so (21) bounds the target discrepancy by 2-b. This proves (23); without that containment the same argument gives the strict source bound below two.□

Proof. Suppose i<j are two indices for which E_i=E_j=E. Monotonicity and the strict increase when E is used at step j give

Thus the target discrepancies at all uses of this fixed place are strictly increasing elements of \Lambda_m. This set has m elements, so there can be at most m uses. Summing this bound over \mathcal F proves the last assertion. No discreteness of the discrepancy at intervening steps is required.□

Proof. The restriction from Q will identify its fourth homology with the fourth Borel–Moore homology of the common open. We then compare the surface classes by their restrictions there.

Since L_Q has real dimension at most two, H_4(L_Q,\R)=H_3(L_Q,\R)=0. The degree-four part of (25) is therefore

Define the linear map

To determine \Phi on surface classes, we first check that strict transforms are analytic, without a normality hypothesis. Let S\subset P be an irreducible surface not contained in L_P. Removing a proper analytic subset from an irreducible analytic space leaves an irreducible space: its regular locus remains connected after removing an analytic subset of positive complex codimension [3, Section 3.3]. Thus S\cap U is irreducible. The analytic inverse image p^{-1}(S) has a unique irreducible component \widetilde S meeting the open where p and q are isomorphisms; there it agrees with S\cap U. This component is a compact irreducible surface. Remmert’s proper mapping theorem shows that

is an irreducible analytic surface, agreeing with S\cap U on U. It is the strict transform of S. The same construction with p and q interchanged gives strict transforms in the other direction.

The restrictions of [S] and [S^+] to U are the same oriented fundamental class, so (28) gives \Phi([S])=[S^+]. If instead S\subset L_P, its restriction vanishes and \Phi([S])=0. It follows that \Phi maps \mathcal C_2(P) into \mathcal C_2(Q). Every irreducible surface in Q meets U, since \dim L_Q\leq1; its strict transform in P therefore maps back to its class. The induced map of cycle spans is surjective. Finally, a nonzero surface class supported in L_P lies in its kernel. Finite-dimensionality gives the strict rank inequality, and (26) gives the stated Kähler case.□

Proof. Represent the place on a common higher model. Its image onto S is surjective by the definition of the centre and properness. The inverse image of the dense open portion of S on which the step is an isomorphism is nonempty and dense in E. There the two centre maps agree. The closure of this portion’s target image is both the strict transform of S and the whole target centre. The assertions follow.□

Proof. At every step i under consideration, choose one such flipped surface and its witness E_i from Proposition 5.2. The place is exceptional over X_N, and discrepancy monotonicity gives

For b=0 this is the witness bound for an arbitrary flipped surface.

Apply Lemma 4.3 on the fixed model X_N. If E_i\notin\mathcal F_{X_N}, its centre there is a surface contained in a positive boundary component of coefficient d, and

Thus d>b. By the hypothesis on flipping surfaces, this surface centre is not contained in L_N. Lemma 7.1 carries it to a surface in the corresponding coefficient-d component on X_{N+1}. At each successive step up to and including step i, the same hypothesis and lemma apply. Consequently the centre on X_{i+1} meets X_{i+1}\setminus L_i^+. This contradicts its definition as the chosen flipped surface, which is contained in L_i^+.

Every chosen witness therefore belongs to the fixed finite set \mathcal F_{X_N}. Lemma 5.3 permits each place to be used at most m times, since its discrepancies on witness targets are strictly increasing members of (1,2]\cap m^{-1}\Z. There are at most m\,\#\mathcal F_{X_N} steps under consideration.□

Proof of Theorem 1.1. Suppose the sequence is infinite. Proposition 3.1 preserves generalized canonicality: every exceptional place has log discrepancy at least one on every model. The strict bound greater than one used in the witness count holds on the target of its witness step by strict discrepancy increase. No terminality of the whole model is used.

Order the distinct positive boundary coefficients as b_1>\cdots>b_s>0. Inductively pass to tails on which no flipping or flipped surface lies in a component of any coefficient already treated. Start with the full sequence. For the next coefficient b=b_j, the induction hypothesis satisfies the assumption of Lemma 7.2. Passing beyond the finitely many steps it counts leaves a tail with no flipped surface in any coefficient-b component.

Index those components on the initial model by a finite set I_b, and write H_{i,\alpha} for their strict transforms on X_i, \alpha\in I_b. Smallness preserves this indexing. Each target component now satisfies

Otherwise this intersection would contain a flipped surface. The boundary-component application of Lemma 6.1 therefore gives

with strict inequality whenever H_{i,\alpha} contains a flipping surface. The hypotheses were checked in Section 6 for the reduced components, without normalizing them. In particular, Kähler positivity in X_i makes every removed surface class nonzero in H_4(H_{i,\alpha},\R).

It follows that the finite nonnegative integer

does not increase and decreases strictly at each step with a flipping surface in a coefficient-b component. Only finitely many such steps occur. Pass beyond them to finish this stage.

After all positive coefficients have been treated, the hypothesis of Lemma 7.2 holds with b=0. If B_0=0, the positive-coefficient induction is empty and this hypothesis already holds on the original sequence. Pass beyond the finitely many steps having a flipped surface. On the resulting tail, \dim L_i^+\leq1, whereas (29) forces L_i to contain a surface. Lemma 6.1, applied to the fourfolds themselves, gives

at every remaining step. These are finite nonnegative integers, so they cannot decrease strictly along an infinite tail. This proves termination.□