Arithmetic Stein-degree bounds for log Calabi–Yau pairs

Proof. The extension k_V/k is finite and separable, and k(V)/k_V is regular. Its embeddings in \kb give precisely the geometric components. Properness makes global functions finite over k. On a normal variety every element of k(V) algebraic over k is integral over every local ring and hence regular, proving (i).

The regular extension k(Z)/k is linearly disjoint from k_V/k. Thus k_Vk(Z) is a degree-[k_V:k] extension of k(Z) inside k(V) and is contained in the relative algebraic closure of k(Z) there. This proves (ii). The first assertion of (iii) follows from (i). For a contraction, global functions on the base are k and localizing f_*\OO_X=\OO_Z gives the asserted generic-fibre function field. Normality is preserved by localization, and a normal variety over a perfect field is geometrically normal; the assertion follows over k(Z) as well.□

Proof. Write the specified multiple as the divisor of a rational function. Since the map extracts no divisors, pushing forward gives the same principal-divisor identity on the new model. Its two pullbacks to a common resolution are that same principal divisor. Thus the pullbacks agree, which proves crepancy and the discrepancy assertions. Without a specified multiple, choose one first.□

Proof. Apply bounded complements to (V_{\kb},bS_{\kb}) and multiply the index by the denominator of b. The geometric complement has the form bS_{\kb}+D/n, where

This linear system is defined over k. For clarity, its underlying space is the space of rational sections of the integral Weil divisor -nK_V-nbS; it can be computed on the smooth locus and extended across codimension two. Sections commute with field extension. The divisor is \Q-Cartier, although it need not be Cartier.

A general geometric member of (3.1) gives an lc pair. To see this, resolve both (V,bS) and the rational map of this linear system. On a further resolution the pullbacks of its members have the form F+M, with F fixed and rational and M varying in a base point free Cartier system. Indeed, differences of pullbacks are divisors of ratios of sections, so eliminating the base ideal gives precisely this decomposition. Arrange that the support of F and the crepant boundary B_W of (V,bS) has simple normal crossings. The existence of one lc member implies that every coefficient of B_W+F/n is at most one: adding the effective moving part could not remove an excessive coefficient. By Bertini, a general M is smooth and transverse to this support, and its coefficient 1/n is at most one. The resulting log smooth pair is lc.

The argument describes a nonempty geometric open subset of the projective space of sections. Since k is infinite, its k-rational points are Zariski dense even after extension to \kb. Choose a member D over k in this open subset and set C=bS+D/n. Then n(K_V+C)=nK_V+nbS+D is principal over k, as required.□

Proof. Let T be the centre. If fewer than \operatorname{codim}T components of H pass through its generic point, complete their local equations by a smooth hypersurface containing T, keeping simple normal crossings. The enlarged reduced divisor is lc, and the additional hypersurface has positive value under v. The discrepancy for H would therefore be positive. Thus the components through the generic point of T have number \operatorname{codim}T, and T is a stratum.

We prove uniqueness by a sequence of blowups that is determined by the weights. If T is already a divisor, the normalized valuation centred at its generic point is its order of vanishing. Otherwise choose two components through T, with values p_1,p_2>0, and blow up their codimension-two intersection near the generic point of T. The new reduced total divisor is simple normal crossings and the blowup is log crepant. The exceptional divisor has value \min(p_1,p_2); the values of the two strict transforms are

with a zero value indicating that the centre is not on that strict transform. The other values are unchanged.

The new centre is again a stratum. These data specify it uniquely over the generic point of T: in the exceptional \Pj^1-bundle, the two strict transforms give the two distinguished sections. Unequal weights choose one section; equal weights choose the full fibre, since a different proper subvariety would not be a stratum. Intersecting with the remaining components through T imposes the remaining prescribed conditions. This is also immediate in the two blowup charts. The sum of the positive weights decreases by \min(p_1,p_2). Repeating therefore reaches a divisorial centre, where the valuation is unique. Tracing back proves the assertion.□

Proof. First work over an algebraic closure and take a log resolution with crepant boundary B_W. Let H=\sum H_i be a reduced simple normal crossings divisor containing its support, and write B_W=\sum b_iH_i, allowing zero and negative coefficients. Since the pair is klt, b_i<1. For a valuation v with discrepancy less than one,

The first term on the right is a nonnegative integer, so it is zero. The centre is therefore a stratum by Lemma 3.2. At each stratum, the positive integral weights satisfy \sum_i(1-b_i)v(H_i)<1, which allows only finitely many choices. There are finitely many strata, and Lemma 3.2 gives uniqueness. Over the original field, each divisorial valuation extends to a geometric one with the same discrepancy, so finiteness follows there as well.□

Proof. We explain both the finite-type parametrization and the field assertion. A reduced pure-dimensional projective locus of dimension j and degree at most D_0 in a fixed \Pj^N is cut out set-theoretically by equations of bounded degree. Indeed, given a geometric point outside the locus, choose a linear projection to \Pj^{j+1} whose centre misses the locus and whose fibre through that point misses it. Such a choice exists by the usual dimension count. The projection is finite on the locus; its image is a hypersurface of degree at most D_0. Pulling back its equation separates the point. Multiplying by forms nonvanishing at that point puts these equations in one fixed degree. The codimension-one case uses the identity projection; the empty and full loci require no argument. A basis of the space of equations in this fixed degree has bounded size. These spaces of equations descend to the field of the locus, because extension of the ground field commutes with their defining linear conditions.

Apply this to V and D. Tuples of equations of bounded size and degree give parameter spaces of finite type over \Q containing all the embedded pairs over their original fields. Stratify to take fibrewise reductions and to impose geometric normality, integrality, dimension, and the divisor condition. This can be done by spreading the reductions at generic points and shrinking, using characteristic zero and generic flatness, and then proceeding by Noetherian induction.

At the generic point of each stratum, resolve the pair over that point’s own function field. Resolution and principalization in characteristic zero may be chosen to give geometric simple normal crossings, with no extension of that function field [BMT11, Theorems 1.3 and 6.1, including the Addendum]. Spread the projective morphism, the reduced divisor, and a dense open on which the morphism is an isomorphism. After shrinking, the resolving variety is smooth and projective over the stratum, the morphism is fibrewise birational, and the divisor has geometric simple normal crossings in every fibre and contains the required exceptional and marked loci. To verify the last condition, spread the inverse image of the closed bad locus and the smoothness and expected codimensions of intersections of the divisor components; the components can be considered after an auxiliary finite étale cover of the parameter stratum for this verification. The resolution itself remains defined on the original stratum. Noetherian induction on its complement yields finitely many families.

Every embedded pair over a field is a point of one of these strata, so pulling back the corresponding family gives a resolution over that same field. Smooth proper base change bounds the second Betti numbers. Applying it also to intersections of components, after the finite étale covers just used, bounds the numbers of their geometric connected components and hence the numbers of strata. Taking maxima over the finitely many families gives R and M.□

Proof. A bounded polarization after a bounded extension. By BAB, Q_{\kb} has a very ample line bundle L of bounded degree. Its space of sections also has bounded dimension: for a nondegenerate q-dimensional projective variety, degree is at least codimension plus one, so h^0(L)\leq L^q+q.

We claim that \Pic(Q_{\kb}) is free abelian of bounded rank. First a numerically trivial line bundle N on Q_{\kb} is trivial. Indeed, klt singularities are rational, so Euler characteristics can be computed on a resolution; Riemann–Roch there gives \chi(N)=\chi(\OO_{Q_{\kb}}). Klt Kawamata–Viehweg vanishing applies to both bundles, since N-K_Q and -K_Q are ample. Hence h^0(N)=h^0(\OO_{Q_{\kb}})=1. The divisor of a nonzero section of N is effective and numerically trivial. Its intersection with a power of an ample divisor forces it to be zero, so N is trivial.

Pullback to a resolution therefore embeds \Pic(Q_{\kb}) into the Néron–Severi group modulo numerical equivalence of that resolution: numerical triviality of a pullback implies numerical triviality downstairs by lifting curves and the projection formula. This is a free abelian group of rank at most the second Betti number. Lemma 3.4, applied to the bounded embedding with empty marking, bounds that number. This proves the claim.

The Galois action on this lattice has finite image, because its finitely many generators are defined over finite extensions. Finite subgroups of \operatorname{GL}_r(\Z) have bounded order for bounded r: reduction modulo 3 is injective on each such subgroup. To recall why, the kernel has no nontrivial finite-order element. If I+3^aM with a\geq1 and M\not\equiv0\pmod3 had prime order, the binomial expansion at that prime would have a nonzero lowest 3-adic term; every finite-order element has a power of prime order. Thus an extension F/k of bounded degree fixes the class of L.

Invariance of a class need not descend a line bundle, so one more step is necessary. Put r=h^0(L). Choose a finite Galois field of definition of L over F, and isomorphisms between its conjugates. Their failure to satisfy the cocycle condition consists of scalar factors. On

these factors cancel: a scalar acts to the rth power on both factors. This gives effective descent data. The required isomorphisms exist over the chosen field of definition because two line bundles that become isomorphic over an algebraic closure are already isomorphic over that field on a proper geometrically integral variety: apply base change to the sections of their quotient and its inverse. Consequently (3.3) descends to a line bundle A on Q_F whose geometric class is L^{\otimes r}. It is very ample and has bounded degree and bounded h^0(A).

A bounded marked support. The principal-divisor identity implies that n\Lambda is integral, since K_Q is an integral Weil divisor. Thus every positive coefficient of \Lambda is at least 1/n. On the geometric variety,

For the last inequality, intersect q-1 general members of |A|. When q>1 they give a smooth curve avoiding the singular locus of the normal variety, and adjunction says

For q=1 the same formula is the usual degree formula on a smooth curve. Hence the embedded pair (Q_F,(\Supp\Lambda)_\red) satisfies the hypotheses of Lemma 3.4 with uniform bounds.

The orbit of a valuation. Choose the resolution supplied there over F, and write K_W+B_W=p^*(K_Q+\Lambda). Its geometric reduced divisor H contains the support of B_W. Since (Q,\Lambda) is lc, B_W\leq H. For every geometric extension \bar v of v,

The left-hand discrepancy is integral, hence zero. By Lemma 3.2, \bar v is determined by its centre stratum and its weights on the components of H. The Galois subgroup over F that fixes all the geometric components and strata fixes each such valuation: its centre and weights cannot change. There are at most M objects in this finite set, so each orbit over F has size at most M!. Its full orbit over k has size at most [F:k]M!. These are uniformly bounded, proving the proposition.□

Proof. Take a projective resolution and run a relative K-MMP with ample scaling over X. In this birational setting every divisor class, including the zero boundary, is big over X: the generic fibre is a point. The big-boundary MMP applies. A Mori fibre space over X cannot be an output, since its total space is birational to X, whereas its base would have smaller dimension and still dominate X. We therefore obtain a \Q-factorial klt minimal model h:Y\to X. Write

Minimality says that -B_Y is nef over X, and h_*B_Y=B\geq0. The negativity lemma gives B_Y\geq0. Crepancy proves the remaining assertions.□

Proof. Set D=K_X+B-bS\simq-bS. It is not pseudo-effective, since its intersection with a fixed ample divisor to the power d-1 is negative. Choose an ample rational divisor H and \delta>0 rational and small enough that D+\delta H remains outside the pseudo-effective cone.

There is an effective klt boundary \Delta with K_X+\Delta\simq D+\delta H. Indeed, (X,B-bS) is lc and X is klt, so (X,(1-\eta)(B-bS)) is klt for 0<\eta\ll1. The class \delta H+\eta(B-bS) is ample for sufficiently small \eta. Represent it by a general effective rational divisor with small coefficients, preserving klt, and add this divisor to (1-\eta)(B-bS).

Run the MMP for K_X+\Delta with scaling by a sufficiently large multiple mH. At a ray R used by the program, including the final Mori fibre ray, write L_i=D_i+\delta H_i for the transformed class. The scaling condition gives

It follows that \lambda_i>0, H_i\cdot R>0, and hence D_i\cdot R<0. Thus every step is positive on S_i. The divisor S_i cannot be contracted by a divisorial step: if it were, it would be an effective exceptional divisor nef over that contraction, in contradiction with the negativity lemma. Flips do not contract divisors.

The resulting Mori fibre contraction has relative Picard number one, so positivity on its extremal ray makes S_1 relatively ample. Lemma 2.2 gives the assertions about (X_1,B_1). If the base is a point, the function-field condition makes it \Spec k. Now \rho(X_1/k)=1. The nonzero effective divisor B_1\simq-K_{X_1} is ample, and the effective divisor B_1-bS_1\simq-K_{X_1}-bS_1 is nef, as asserted.□

Proof. First -P is pseudo-effective over Z. Choose an effective Cartier divisor on Z containing the proper closed image of P. A sufficiently large multiple of its pullback has coefficient at least one along P, so subtracting P leaves an effective divisor.

Choose an effective klt boundary \Delta for which L=-(K_U+\Delta) is ample over Z. For a sufficiently small rational a>0, the class L-aP is also relatively ample. After twisting a sufficiently divisible multiple by a divisor from Z, choose a general member and divide by that multiple to obtain an effective rational divisor D_a\simq L-aP over Z with (U,\Delta+D_a) klt. Then \Psi=\Delta+D_a is big over Z, and

The big-boundary MMP with ample scaling terminates in a minimal model U', since this class is relatively pseudo-effective. Its steps are (-P)-negative. As in Lemma 4.2, the negativity lemma prevents the contraction of P. The steps are isomorphisms over a nonempty open subset of Z: there P is absent and its negative has zero intersection with every curve in a fibre. In particular, every horizontal prime survives.

Put L'=K_{U'}+\Psi', which is nef over Z. The boundary \Psi' is still big over Z, so write \Psi'\simq A'+D' over Z with A' relatively ample and D'\geq0. For sufficiently small rational \eta>0, the effective boundary

is klt and satisfies

Klt base point freeness applies to a sufficiently divisible nef Cartier multiple of L' [LM26, Theorem 11.1]. Thus L', and consequently -P', is semiample over Z. The relative theorem can also be applied geometrically and descended: relative generation of sections is detected by faithfully flat base change.

Let f:U'\to Z' be the associated contraction. On the generic fibre over Z, the line bundle of a multiple of P' is trivial and the field of global functions is k(Z). Hence Z'\to Z is birational. Choose m>0 such that mP' is Cartier and \OO_{U'}(-mP')=f^*\mathcal A for a line bundle \mathcal A on Z'. The canonical section of \OO_{U'}(mP')=f^*\mathcal A^{-1} descends, by f_*\OO_{U'}=\OO_{Z'} and the projection formula, to a section of \mathcal A^{-1}. Its divisor is a nonzero effective Cartier divisor T, and its pullback is exactly mP', proving (4.1).

A horizontal prime E dominates Z' and, being proper, maps onto Z'. Thus it meets f^{-1}(T). It is not contained there, so pulling back T to its normalization gives a nonzero effective Cartier divisor. Dividing by m proves the final assertion.□

Proof. Divisorial adjunction gives an effective lc different on the normalization, geometrically and over k; its residue construction is compatible with field extension. We verify that it retains the specified index n, rather than an unspecified multiple. For the construction see [Kol10, Definition 122 and equations (122.7)–(122.10)] and [Kol13].

Choose a nonzero rational top differential \omega on U, and write K_U=\operatorname{div}(\omega). The relation in the hypothesis provides a rational n-canonical form \tau with

At the generic point of E, the variety is regular and E has coefficient one. The logarithmic n-fold residue of \tau along E therefore defines a nonzero rational n-canonical form \rho on E^\nu, over k. Residue commutes with powers. After passing to \kb and taking any sufficiently divisible power, the usual pluri-residue definition of the different gives

Indeed, the identity for that power is the positive integral multiple of this identity. It consequently holds already for \rho over k. This proves the principal-divisor assertion at index n, and also that nC_{E^\nu} is integral. There is no gluing across different normalizations here; the construction is on the single normalized prime E^\nu.

For (4.2), remove \beta P from C. The remaining pair is effective and lc and still has coefficient one along E, so its different is effective. After a common Cartier multiple, the two log pluricanonical sheaves differ by the tensor factor \OO_U(m\beta P), and their residues differ by its restriction to E^\nu. Dividing by m shows that adding \beta P adds exactly \beta P|_{E^\nu} to the different. This comparison does not require K_U+E to be \Q-Cartier. Distinct primes over k have disjoint sets of geometric prime components, so this restriction is defined on every geometric component of E^\nu. This proves the inequality.□

Proof. Take general very ample hyperplane sections down to a surface, and choose a point where the surface meets a suitable dense open subset of the given codimension-two locus. The surface is normal and klt, and its restricted boundary is lc. These assertions can be checked on a log resolution: general hyperplanes meet its simple normal crossings divisors transversely, adjunction preserves the discrepancy coefficients, and the images of exceptional loci and boundary intersections are met in the expected dimensions. Each prime being counted restricts to a curve germ through the chosen point, with its original coefficient, and the germs are distinct.

It remains to bound the number of these germs at a klt surface singularity. Descend the finite data and a log resolution to an embeddable characteristic-zero field and work over \mathbb C. A klt complex surface singularity is a quotient singularity [KM98, Proposition 4.18]; it has a finite smooth analytic uniformizing cover étale in codimension one. The crepant pullback of the pair is lc by the finite-cover discrepancy formula. Each curve germ pulls back to at least one curve through the point upstairs, with at least its original coefficient, and different germs have no common curve component. Blowing up this smooth surface point gives the log discrepancy

If there are r original germs, their contributions give rb\leq\operatorname{mult}_{0}(C_{\mathrm{up}})\leq2. Thus r\leq2/b, as required.□