The abelian Zilber–Pink conjecture

Proof. Fix integral homomorphisms f_1,\ldots,f_b forming a real basis of \Hom(G,G')_{\mathbb R}. For each j, ample domination of f_j^*L' by a power of L gives \widehat h_{L'}(f_j(x))\le C_j\widehat h_L(x) for all algebraic points x. The same inequality holds on \mathcal H(G), by its quadratic-form interpretation on every finite-dimensional span. Hence

The constants are independent of the fields of definition of the points. This proves the assertion about operator norms.

The underlying real Lie representation identifies with the real extension of the representation on rational homology. It is faithful, also after scalar extension from \mathbb Q to \mathbb R. The polarization adjoint belongs to rational homomorphisms, and therefore extends to real homomorphisms. For a real homomorphism f, the operator f^*f is nonnegative and self-adjoint on \Lie(G). Choose a real polynomial p that is zero at zero and equals 1/t at every positive eigenvalue t of f^*f. Then

is a real homomorphism and f^\dagger f=P_{(\ker f)^\perp}, ff^\dagger=P_{\operatorname{im}f}. Faithfulness makes these identities identities of real homomorphisms, so they also hold on the height spaces. The last assertion follows by applying the fixed bounded operator f^\dagger.□

Proof. If the point heights are bounded, use A itself as the bounded factor. Otherwise take a subsequence with H_i=\|x_i\|^2\to\infty. Among fixed quotient homomorphisms \pi:A\to Q for which \|\pi(x_i)\|^2=o(H_i) on some subsequence, choose one with maximal \dim Q, and restrict to that subsequence. The zero quotient is allowed. A quotient of dimension \dim A is an isogeny and cannot have this property, by height comparison. Poincaré reducibility provides an isogeny A\to Q\times W' whose first component is \pi; harmless fixed isogenies of the targets can be absorbed here.

Write p'_i for the W' coordinate. Suppose a fixed nonzero homomorphism g on W' satisfies

Take a further subsequence on which this ratio tends to zero. The homomorphism whose coordinates are \pi and g on the complementary factor has image of dimension \dim Q+\dim g(W')>\dim Q, and its point heights are o(H_i). Quotienting by the identity component of its kernel changes its image only by an isogeny. This contradicts the maximality of Q. Thus W' is a high factor with scale H_i and the required nondegeneracy for every fixed homomorphism.

If Q=0, stop. Otherwise continue on Q with the points \pi(x_i). If their heights are bounded, this is the low factor; if not, extract a diverging scale and repeat. At each step the remaining quotient has smaller dimension and its squared heights are little-oh of the preceding scale. The process therefore terminates. Reverse the order of the extracted factors, so that the scales are increasing. Composing the finitely many fixed isogenies gives \alpha. Fixed height comparisons preserve all the asserted bounds and ratios.□

Proof. Let X\subseteq A be irreducible and contained in no proper torsion coset. If \dim A=0, there are no torsion cosets of codimension \dim X+1, so there is nothing to prove. Suppose otherwise that X\cap A^{[\dim X+1]} is Zariski dense. Since \Qbar is countable, choose a sequence in this intersection that eventually avoids every fixed proper closed subset of X defined over \Qbar. Apply Lemma 2.4 and replace A,X and the sequence by their isogeny images. Dimensions of subvarieties and codimensions of torsion cosets are preserved by this finite map. Its image X is still contained in no proper torsion coset. We have now written A=\prod_{\ell\in I}W_\ell and x_i=(p_{i\ell})_\ell.

For each i, choose a torsion coset B_i+\tau_i containing x_i with \codim_A B_i>\dim X. After extraction its codimension is constant. In \Lie(A) let

The orthogonal projector P_i onto U_i is a real endomorphism by Lemma 2.1, applied to the scaled subgroup inclusion. All these projectors have the same rank. A subsequence converges to an orthogonal projector P of the same rank. Moreover P\in\End(A)_{\mathbb R}, because this finite-dimensional subspace is closed in the space of real linear maps on \Lie(A). Put U=\operatorname{im}P. Then

For the second assertion, torsion vanishes in the height space, so 1-P_i kills the indicated normalized vector exactly. Those vectors are bounded, and Lemma 2.1 converts convergence of the projectors into degree-independent operator-norm convergence.

To obtain the local fiber bounds in the sequence theorem, we retain copies of all slower coordinates. For each \ell\in I set

Regard W_\ell as the last coordinate in A_\ell. Let m_\ell=\dim Y_\ell-\dim Y_{\ell^-}, where Y_{\ell^-} is the preceding projection and is a point for the first index. These integers are nonnegative and sum to \dim X. The local dimension of the fiber of Y_\ell\to Y_{\ell^-} equals m_\ell on a nonempty open subset. The sequence eventually lies in that subset: the inverse image in X of its complement is a fixed proper closed set. Thus (2.1) holds after discarding finitely many terms.

For a high factor, every slower coordinate is o(\sqrt{H_{i\ell}}) in norm. A fixed homomorphism on A_\ell nonzero on its last W_\ell coordinate is the sum of a nonzero homomorphism on that coordinate and fixed homomorphisms on the slower ones. The triangle inequality and Lemma 2.4 prove (2.2). If a low factor occurs, its image of X is contained in no proper torsion coset: the inverse image of such a coset is a finite union of proper torsion cosets, one of which would contain irreducible X. The same pullback argument proves eventual avoidance for the low sequence.

Finally let j:A\to\cA insert each W_\ell into the distinguished last coordinate of A_\ell, and let \rho:\cA\to A be the coordinate retraction. Put V=j(U). Since \rho j=1, the real endomorphism jP\rho is idempotent with image V, and

Write q_i^0=(p_{i\ell}/\sqrt{H_{i\ell}})_\ell for the normalized vector before enlargement, and q_i for the enlarged vector in Theorem 2.3. Their difference has only copied slower coordinates. With product norms,

If \Phi is the complementary projector for V, then \Phi j=\Phi j(1-P). Equation (2.4) and the displayed error bound therefore give \|\Phi q_i\|\to0. We have constructed (2.3), a contradiction to Theorem 2.3.□

Proof. Suppose such subvarieties and integers exist and restrict to the given subsequence. Let \pi_\ell:A_\ell\to A'_\ell=A_\ell/C_\ell be the quotient, let W'_\ell=\pi_\ell(W_\ell), and let e_\ell be the dimension of the image of C_\ell in D_\ell=A_\ell/W_\ell. We first construct fixed irreducible parents Y'_\ell\subseteq A'_\ell for the projected points with local W'_\ell-fiber bounds

Fix one factor and temporarily omit its index. In Y let S be the intersection of the open locus where local fiber dimension over A/W is at most m and the closed locus where local fiber dimension over A/C is at least a. Local fiber dimension is upper semicontinuous, so S is locally closed and contains every marked point under consideration. Every nonempty fiber of S\to A/C has dimension at least a. Indeed, through each of its points there is an irreducible component of the original C-fiber of dimension at least a. That whole component belongs to the closed high-fiber locus, and its intersection with the open low-fiber locus is a nonempty open subset of the same dimension.

Put E=A/(W+C)=A'/W'. The map from a fiber of S\to E to A/W has image in an e-dimensional coset, and all of its nonempty fibers have dimension at most m. Consequently every fiber of S\to E has dimension at most m+e. Let Q be the constructible image of S in A'. The fibers of S\to Q have dimension at least a, so the dimension inequality for fibers, applied over every point t\in E, gives

This can also be seen by applying the fiber dimension theorem to each irreducible locally closed stratum of Q_t: one of the finitely many strata in its inverse image dominates it with generic fiber dimension at least a. In particular m+e-a\ge0 when a marked point exists.

Partition Q into finitely many irreducible locally closed subsets. After taking a subsequence, the projected points lie in one such subset Q^\circ. Set Y'=\overline{Q^\circ}. Because Q^\circ is open in its closure, its local germ at any of the selected points equals the germ of Y'. The local fiber dimension of Y'\to E at that point is therefore at most m+e-a. Performing this construction in each of the finitely many factors, with successive subsequences, proves (3.2). All these parents are defined over \Qbar.

The new parents provide the local fiber bounds. We next check the small-projection condition and the strict codimension inequality. Let \pi=\prod_\ell\pi_\ell and V'=\pi(V)\subseteq\Lie(W'). This is the image of a real homomorphism, so Lemma 2.1 supplies its orthogonal projector. If \Phi' is the complementary projector, then

since \pi(V)=V'. Thus the projected normalized points satisfy the required small-projection condition. The height upper bounds are preserved by the fixed quotient homomorphisms.

The gain in fiber bounds compensates for the rank lost under projection:

Here \dim(W\cap C)=\dim C-\sum e_\ell and \ker(\Phi|_{\Lie(C)})=V\cap\Lie(C).

It remains to obtain nonzero distinguished factors and check the scale conditions. Discard each factor with W'_\ell=0. In such a factor V' has zero coordinate, so deletion does not change the codimension inside \Lie(W') and only decreases the sum of the nonnegative bounds m'_\ell. The small-projection condition on the remaining product follows by composition with its coordinate projection. At least one factor survives, since otherwise (3.3) would say that zero is strictly greater than a sum of nonnegative integers. The scales on the remaining ordered list still have successive ratios tending to zero. For a surviving high factor, a homomorphism nonzero on W'_\ell composes with \pi_\ell to one nonzero on W_\ell, proving (2.2). If the low factor survives, then W'_0=A'_0 and its torsion-coset avoidance follows by pulling a proper torsion coset back under \pi_0.

We have obtained counterexample data of smaller total ambient dimension because C\ne0. If the low factor is present both before and after projection, or absent both before and after, this contradicts minimality. If it has disappeared, it contradicts the already-established all-high case. This proves the exclusion.□

Proof. We will replace the subgroup supplied by Ax’s theorem by a product subgroup. The degeneration used for this replacement cannot decrease the tangent-intersection dimension. Its g-rank must also be bounded by the rank appearing in Ax’s inequality. We arrange this comparison first by choosing factor scalings that maximize the rank on every abelian subgroup; only then do we apply Ax’s theorem.

Write D(\lambda)=\operatorname{diag}(\lambda_\ell) for \lambda\in(\mathbb C^\times)^I. For every abelian subvariety B\subseteq A, the rank of gD(\lambda)|_{\Lie(B)} attains its maximum on a nonempty Zariski open subset of (\mathbb C^\times)^I. There are only countably many abelian subvarieties of A: their rational homology groups determine distinct rational subspaces of H_1(A,\mathbb Q). The Baire category theorem therefore gives one \lambda at which all these ranks are maximal. Fix such a \lambda and set h=gD(\lambda).

At a smooth point of Z, its tangent space is the direct sum of the tangent spaces of its factors. It is consequently preserved by D(\lambda). The failure of tangent injectivity for g, which is equivalent to P_Z(g)=0, thus also holds for h at every smooth point. This is the same product property underlying (3.5).

Choose a very general smooth point z\in Z. We require that the rank of h on T_zZ be maximal, and that, for every abelian product subgroup C\subseteq A, the differential of Z\longrightarrow A/C have its generic rank at z. These requirements exclude only countably many proper closed subsets of Z, so they can be imposed simultaneously. On a sufficiently small logarithmic chart around z, the map obtained by applying h to the logarithm has constant rank less than \dim Z. Its fiber through z therefore has a positive-dimensional irreducible analytic germ. Let J be a sufficiently small irreducible representative in the logarithmic chart, let U_1 be the Zariski closure of \exp(J), and let z+C^* be the smallest coset containing U_1. Writing \widetilde z for the logarithm of z, we have

We use Ax’s theorem [1], in the precise analytic-subset formulation [13, Theorem 5.3]. If an irreducible analytic subset K of an open subset of a translate of a complex vector space U has exponential image with Zariski closure B, that theorem gives

where \langle B\rangle_{\mathrm{geo}} denotes the smallest abelian coset containing B. Applied to J and U=\Lie(C^*)\cap\ker h, it gives

Since U_1\subseteq Z\cap(z+C^*) contains z, the Zariski tangent space at z gives

No smoothness assumption on U_1 at z is needed for this inequality.

We now replace C^* by a product direction. Order the index set I and choose strictly increasing integers w_\ell. Put S=\Lie(C^*) and F_\ell=\bigoplus_{j\geq\ell}\Lie(A_j). A basis of S adapted to the filtration S\cap F_\ell shows that the Grassmannian limit

exists. Indeed, after dividing an adapted basis vector by the power of t corresponding to its first nonzero coordinate, its limit is the projection to that coordinate. Each summand is the Lie algebra of an abelian subvariety C_\ell\subseteq A_\ell: take the identity component of the intersection of C^* with the product subgroup corresponding to F_\ell, and project to A_\ell.

The product tangent space T_zZ is preserved by all the displayed diagonal scalings. Upper semicontinuity of intersection dimension on the Grassmannian therefore gives

Set R=\rank(h|_S). By the choice of \lambda, for every t\neq0,

The condition that the restriction of a fixed linear map have rank at most R is closed on the Grassmannian. Hence \rank(g|_{S_0})\leq R. Together with (3.7), this proves

By the choice of z, the dimension on the left is the generic fiber dimension of Z\longrightarrow A/\prod_\ell C_\ell. Local fiber dimension at every point is at least this generic value. Since both Z and the subgroup are products, local fiber dimensions add over the factors. This proves (3.6) for every point of Z.□

Proof. Choose an irreducible component D_\ell of Z_\ell\cap(y_\ell+C_\ell) through y_\ell of dimension d_\ell. Write its smallest containing coset as y_\ell+B_\ell; then B_\ell\subseteq C_\ell. Among the subvarieties of Y_\ell containing D_\ell and having geodesic defect at most \delta_{\mathrm{geo}}(D_\ell), choose one, denoted E_\ell, of maximum dimension. Such a choice is possible because D_\ell is an admissible choice and dimensions are bounded integers. It is geodesic-optimal: a proper enlargement of no greater defect would contradict its choice.

Write \langle E_\ell\rangle_{\mathrm{geo}}=y_\ell+H_\ell. The inclusion D_\ell\subseteq E_\ell implies B_\ell\subseteq H_\ell. The defect comparison gives

Rank increases by at most the dimension added to the domain. Therefore

Since E_\ell\subseteq Y_\ell\cap(y_\ell+H_\ell) contains y_\ell, this implies (3.8). Moreover, a positive-dimensional D_\ell forces H_\ell\neq0.

By [13, Proposition 6.1], the directions of the geodesic closures of geodesic-optimal subvarieties of a fixed Y_\ell belong to a finite set. Take this set as \mathcal H_\ell. The cited proposition applies to arbitrary subvarieties of complex abelian varieties and imposes no condition on the translating points. Thus the sets are independent of y_\ell, Z_\ell, C_\ell, and g.□

Proof. Suppose first that the first assertion fails. Choosing successively smaller neighborhoods and larger indices produces a subsequence, relabeled by i, maps g_i\longrightarrow\Phi, and subproducts Z_i=\prod_\ell Z_{i\ell} through the marked points such that P_{Z_i}(g_i)=0. Lemma 3.2 gives product subgroups C_i=\prod_\ell C_{i\ell} satisfying

The sum on the left is positive. Apply Lemma 3.3 with \varepsilon=1 and pass to a subsequence on which all the resulting directions H_\ell are fixed. Their product H is nonzero. Passing again to a subsequence, we may also fix the integers

They are bounded between 0 and \dim Y_\ell. We obtain \rank(g_i|_{\Lie(H)})\leq\sum_\ell a_\ell-1. Since a rank upper bound is closed under limits of linear maps,

This contradicts the assumed exclusion of (3.9).

For the second assertion, suppose that there are arbitrarily large indices i and subproducts Z_i through the marked points with P_{Z_i^r}(\Delta\Phi)=0. Apply Lemma 3.2 to the product whose factors are indexed by both \ell\in I and j\in\{1,\ldots,r\}. For a fixed i, suppress i from the notation, write C_j=\prod_\ell C_{j\ell} for the resulting subgroup in copy j, and set

The consecutive-difference map on \bigoplus_j E_j has kernel the diagonal copy of \bigcap_j E_j. Its rank is consequently \sum_j\dim E_j-\dim\bigcap_j E_j, and the lemma gives

There is an index j such that C_j\neq0 and a_j\geq\dim E_j. Indeed, if some C_j=0, then the intersection term vanishes and (3.10) forces a strict surplus in another copy. If all C_j are nonzero and every a_j<\dim E_j, the integral deficits sum to at least r=\dim\cA+1, whereas \dim\bigcap_jE_j\leq\dim\cA, again contradicting (3.10).

Select one such copy for each i. If its a_j is zero for any index, then its nonzero product subgroup is contained in \ker\Phi. That one fixed subgroup satisfies (3.9) for every index, since all local fiber dimensions are nonnegative. This is impossible. Thus the selected sum is positive. Apply Lemma 3.3 with g=\Phi and \varepsilon=0. It supplies a nonzero product direction in a fixed finite set and satisfying the parent inequality. Passing to a fixed direction along an infinite subsequence contradicts (3.9) once more.□

Proof. Put n=\dim\cA and q=\rank_{\mathbb Z}\End(\cA), and choose an integral basis e_1,\ldots,e_q. The polynomial

has degree 2d\le 2n: pullback of a Hermitian form is quadratic in the real coordinates of the map. At integral arguments it is an integer, because it is the top intersection number of the pullback of L by an actual endomorphism.

Set K=2n. In the multivariate Newton expansion,

all finite differences are integers. Since \prod_j\alpha_j! divides K!, every coefficient of p_Z has denominator dividing Q=K!. This denominator is independent of Z.

The coefficients also have absolute value at most C D_Z for a fixed C. Indeed, on the fixed interpolation grid \{0,\ldots,K\}^q, the semipositive forms g^*\omega_L are bounded above by a fixed multiple of \Omega=\sum_\ell\operatorname{pr}_\ell^*\omega_{L_\ell}. Consequently

Fixed-grid interpolation gives the coefficient bound. There are only finitely many possible dimension tuples, so the constants are uniform. These integrations can be taken over the regular locus, and the same argument therefore includes singular subvarieties.

Here is the real-algebraic step which turns these bounds into a lower bound for a positive minimum. Let p_c(x) be the universal polynomial of degree at most K, whose coefficients form the free parameter vector c. Apply quantifier elimination over the real numbers to

leaving c and t free. The resulting formula is a Boolean combination of sign conditions on finitely many polynomials R_j(c,t)\in\mathbb Z[c,t]; denominators from the rational box have been cleared. Quantifier elimination with free variables, including its coefficient-ring formulation, is recalled in [4, arXiv version, Section 2.5.2, Theorem 2.27].

Specialize to the coefficient vector c(Z), and put m_Z=\min_{\mathcal B}p_Z>0. At least one specialized polynomial R_j(c(Z),t) which is not identically zero must vanish at m_Z. Otherwise the signs of every nonzero specialized polynomial would be constant in a neighborhood of m_Z, and the identically zero ones would have constant sign zero there. The formula describing the range would then be locally constant. This is impossible: it holds at m_Z and fails immediately to its left.

Let b=\max_j\deg_c R_j and h=\max_j\deg_t R_j. Multiplication by Q^b makes each specialization an integer polynomial, of degree at most h and coefficient size at most C_1D_Z^b. Choose a nonzero one which vanishes at m_Z and remove its maximal power of t. Its constant coefficient is now a nonzero integer. If m_Z\le1, its root equation implies

For m_Z>1 a lower bound is immediate. Enlarging fixed integer constants gives (4.1). The case d=0, where p_Z=1, is included directly.□

Proof. We first fix a positivity neighborhood and the constant in Dill’s inequality. For an arbitrary approximation accuracy, we then balance the height scales by integer multipliers and construct the required bundles. The final comparison will determine how small the accuracy must be.

Suppose that all-high data violate Theorem 2.3, and choose a counterexample with \dim\cA minimal. The projection exclusion and Proposition 3.4 then apply. Every parent Y_\ell has positive dimension: if it were a point, the positive normalized lower bound (2.2), applied to the identity homomorphism, would contradict H_{i\ell}\to\infty. Choose a closed rational box \mathcal B which contains \Phi in its interior and lies in the neighborhood supplied by Proposition 3.4. After discarding finitely many indices, P_Z(g)>0 on this box for every subproduct through y_i in the fixed parents. Lemma 4.1 gives constants \theta,\omega_1 valid for all these products.

If there is only one factor, use two subsequences n_i,m_i\to\infty with H_{n_i}/H_{m_i}\to0. They exist by choosing H_{m_i}\ge i^2H_{n_i}. On the two copies use the maps g\oplus g and the external product target polarization. For subvarieties Z_1,Z_2 through the corresponding points,

Thus the required bound holds with \theta^2 and the same \omega_1. The block diagonal limiting projector still sends the normalized point pairs to zero. From now on there are at least two factors. In the duplicated case all approximations remain in this block diagonal family; no new minimality argument in the larger ambient is needed. We relabel the doubled data and replace \theta by \theta^2 in this case, retaining the notation already introduced.

Choose a positive integer t_1, fixed independently of the rational approximation, so large that

is positive definite for every g\in\mathcal B (or for the indicated block diagonal family). Compactness of the box makes this possible. With the parent embeddings and \theta,\omega_1,t_1 fixed, Theorem 4.2 has a fixed multiplicative constant c_1.

Let 0<\eta<1/2. The following construction is available for every such \eta, with c_1 already fixed. Choose a rational homomorphism f/e in the box with \|f/e-\Phi\|\le\eta, where e>0 is even and f is an integral homomorphism divisible by two.

We want the multiplied points in every factor to have the same height scale: v_{i\ell}\sqrt{H_{i\ell}} should be close to one number R_i. At the same time, the multiplication maps must have polynomial formulas whose coefficient heights grow at most quadratically in v_{i\ell}. Both requirements are met by a dyadic family. Choose an integer N>2/\eta and put U=\{N,N+1,\ldots,2N\}. The numbers u2^k, with u\in U and k\ge0, approximate every real number at least N to relative error at most \eta. Taking R_i\ge N\max_\ell\sqrt{H_{i\ell}}, choose

so that

We verify the coefficient-height requirement below.

On the ambient product set

The choice of t_1 makes its invariant curvature positive definite, hence makes this bundle ample [11, Theorem 1.1.4]. Writing e=2e_0 and f=2f_0, symmetry identifies it with the fourth power of the ample bundle

It is therefore globally generated [11, Theorem 2.2.3]. Fix a generating system z_1,\ldots,z_J for \mathcal Q_0.

Let [v_i] denote factorwise multiplication by the chosen integers, and define

The bundle \mathcal M_i is nef. Symmetry identifies \mathcal P_i\otimes\mathcal M_i^{-1} with [v_i]^*\mathcal Q_0|_{\prod Y_\ell}, which is generated by the pullbacks of the z_j.

Choose a coordinate section \sigma of the target L which is nonzero at f([v_i]y_i). Multiplication by (f\circ[v_i])^*\sigma gives the injection

It is an isomorphism on the nonvanishing open set of that section, which contains y_i. Thus we may take t_2=t_1 in Theorem 4.2. Before pullback, each product z_jf^*\sigma is a section of \boxtimes_\ell L_\ell^{t_1e^2}. Projective normality of the ambient factor embeddings, together with the external-product formula for global sections, represents these sections by multihomogeneous polynomials of multidegree (t_1e^2,\ldots,t_1e^2). There are only finitely many systems, one for each choice of target coordinate \sigma.

We next check that the dyadic choice (4.4) gives the uniform coefficient bound required by Dill. Symmetry and projective normality give polynomial coordinate systems of degree four for [2] and of degree u^2 for each [u], u\in U. Their compositions give the required degree v_{i\ell}^2 formulas.

Fix a number field containing the ambient data, f, and the chosen generating systems. At a finite place use the maximum coefficient norm, and at an infinite place use the sum of absolute coefficient values. If b_w(k) is the logarithm of the maximum of one and the coefficient norm after k iterations of the degree-four system, submultiplicativity gives

Thus b_w(k)=O_w(4^k). Composition with one of the finitely many [u] gives O_w((u2^k)^2). At finite places the constants vanish outside a fixed finite set. Substitution in the finitely many fixed systems representing z_jf^*\sigma now bounds their joint coefficient height by

Consequently we may fix \delta_\ell=\max(1,C_{f,e,U}) for every \ell, independently of i and of the chosen nonvanishing coordinate. The generator count J is also fixed. Dependence on the approximation and on U changes only the height thresholds.

For every subproduct Z through y_i, the scaling identity for P_Z and (4.1) give

This verifies the last geometric hypothesis of Dill’s theorem.

Height comparison and choice of accuracy.

All geometric and coefficient conditions have now been checked for each fixed \eta. The balancing of the multipliers will make the source height large and the image height small after division by e^2R_i^2.

Let \Xi_i be the full monomial coordinates for \mathcal P_i, and let \mathcal Z_i be the pulled-back generating system just constructed. Then

Indeed, the monomial height is exactly t_1\sum_\ell a_{i\ell}h(y_{i\ell}). Multiplying all generator values by the same nonzero injection section does not change their projective height. Their height is therefore the fixed generator height of \mathcal Q_0 at [v_i]y_i. As \mathcal Q_0 is symmetric, this equals

The bounded difference between projective and canonical heights on each fixed factor proves (4.5).

By the high-scale condition applied to the identity, there are constants \kappa_\ell>0, independent of the approximation, such that

for all sufficiently large i. The normalized vectors q_i are bounded. Operator-norm comparison for real homomorphisms therefore gives a constant C independent of \eta, f/e, and the multipliers: the maps f/e lie in the fixed box, and the balancing ratios lie in [1/2,3/2]. Let D_i be the diagonal real endomorphism with entries v_{i\ell}\sqrt{H_{i\ell}}/R_i. In the real height space,

Writing this as the sum of \Phi q_i, (f/e-\Phi)D_iq_i, and \Phi(D_i-1)q_i gives

For the weights, separation of successive scales gives

Moreover

Once \eta, the approximation, and U are fixed, all the height thresholds in Theorem 4.2 are fixed. The projective heights tend to infinity and the weight ratios cross the fixed ratio threshold, so (4.3) applies for all sufficiently large i. Dividing it by e^2R_i^2 and using (4.5)–(4.7) yields

The construction and this inequality hold for every 0<\eta<1/2, whereas c_1, C, and the \kappa_\ell are independent of \eta. Choose \eta so that c_1C^2\eta^2<\frac14\sum_\ell\kappa_\ell. Keeping its associated f,e,U fixed and letting i tend to infinity gives the contradiction. All constants depending on this approximation enter only fixed thresholds or errors which vanish after normalization.□

Proof. Let q=\binom{N+D}{D} and form the rows of evaluations of the degree-D monomials at points of \mathcal S. Each row is a projective vector of height at most DT. Select a maximal independent set of these rows; there are at most q of them. Their kernel is the required polynomial space. Choose pivot columns and give the free variables their standard basis values. The resulting kernel vectors have entries which are ratios of minors of the selected matrix. The determinant height inequalities bound their coefficient heights by O_{N,D}(T).

Although the rows can have entries in an extension of arbitrarily large degree, the resulting vectors lie in F. Indeed the kernel is \operatorname{Gal}(\Qbar/F)-invariant, and its normalized basis with the chosen free coordinates is unique. Each automorphism therefore fixes that basis. Both the number of rows and all height estimates are independent of their fields of definition.□

Proof. Linear equations defining the linear span of Z supply its normal directions in \mathbb P^N. We may therefore work in that span and assume Z nondegenerate. Put d=\dim Z and c=N-d. There is nothing to prove if c=0, and the hypersurface equation proves the case c=1.

Suppose c\geq2 and let z be smooth. Choose a hyperplane H containing the projective tangent space at z. Since Z is nondegenerate, H does not contain Z. The closed join of z and Z has dimension at most d+1, and its intersection with H has dimension at most d. A general (c-2)-plane \Lambda in H therefore avoids this join and the tangent space. Projection from \Lambda gives a morphism

It is finite because \pi^*\mathcal O(1)=\mathcal O_Z(1) is ample. It has just one point over \pi(z) and is immersive at z. The scheme-theoretic fiber has length one: its tangent space is zero and its residue field is the ground field. For the finite local algebra over the local ring of Z' at \pi(z), Nakayama’s lemma now says that 1 generates. Since Z' is the scheme-theoretic image, the local map is an isomorphism. In particular Z' is smooth at \pi(z).

The hypersurface Z' has degree at most D. Its equation pulls back to an equation of Z whose differential at z defines H: the inverse image of the tangent hyperplane is the span of \Lambda and the tangent space of Z, which is H. Varying H supplies all conormal directions.□

Proof. Suppose the first assertion fails for fixed D,B. Choose violating hypersurfaces along an infinite subsequence, pass to a fixed factor and a fixed degree, and denote their fields by F_i. Lemma 5.1, applied to the conjugates of y_{i\ell} over F_i, replaces each equation by one of height O(H_{i\ell}) which still vanishes at the marked point and does not vanish identically on Z_{i\ell}. Here we used the height bound in the original sequence data, rather than any bound on the degree of the marked point.

Take a geometric irreducible component through y_{i\ell} of the intersection with this hypersurface. Its dimension is strictly smaller than d_\ell, and its degree is bounded by ordinary Bézout. Its conjugates over F_i are also components of this intersection. The degree bound therefore bounds the number of conjugates, and Galois descent defines the chosen component over an extension of F_i of bounded degree.

We check the dense height condition on that component. Use the standard projective adelic metrics, with Fubini–Study metrics at infinity. For effective cycles use the additive absolute arithmetic intersection degree. The normalized height of an n-dimensional irreducible variety is this degree divided by the product of n+1 and its geometric degree. The dense height condition, the fundamental height inequality, and the bounded geometric degrees therefore bound the arithmetic degree of Z_{i\ell} by O(H_{i\ell}). The arithmetic divisor intersection formula bounds the arithmetic degree of its intersection with a degree-bounded polynomial by the polynomial degree times the original arithmetic degree, plus the geometric degree times the absolute weighted sum of logarithmic sup norms of that polynomial. The latter sum is at most its coefficient height plus a constant depending only on the degree and the projective space. Thus the intersection has arithmetic degree O(H_{i\ell}). Additivity and nonnegativity give the same bound for the selected component. Dividing by its positive geometric degree and dimension factor, and using the other direction of the successive minima inequality, gives a dense set of points on it of height O(H_{i\ell}); see [31, Theorem 1.10], or [28, Theorem 5.3.3] with absolute normalization. The arithmetic divisor estimate used here is the usual intersection formula for a section, equivalently the arithmetic divisor formula of [5, Proposition 3.2.1(iv), Equation (3.2.2)]; see also the divisor identity in the proof of [28, Corollary A.4.3]. We have produced an admissible product of smaller total dimension, a contradiction.

For the second assertion apply Lemma 5.1 to the dense set of bounded-height points on Z_{i\ell}. Its evaluation kernel is the space of equations of Z_{i\ell}, hence is defined over k_i. In degrees up to the bounded degree of Z_{i\ell}, these bases span the conormal at every smooth point by Lemma 5.2.

Suppose a marked point were singular for infinitely many indices. Choose an affine chart containing that point. Among the Jacobian minors of size equal to the codimension, formed from the bounded equations just obtained, some minor is nonzero generically on the variety. Every such minor vanishes at a singular point: otherwise the corresponding equations would cut out a smooth germ of the same dimension containing the germ of the reduced variety, which would itself then be smooth. Homogenizing a suitable minor gives a bounded-degree hypersurface over k_i containing the marked point and not the variety. This contradicts the first assertion. The codimension-zero case has no singular points.

Finally H_{i0}=1. The low-factor equations just constructed have bounded degrees and bounded coefficient heights over fields of bounded degree. Northcott’s theorem gives a finite set of possible coefficient vectors. The conormal assertion at a smooth point shows that each Z_{i0} is an irreducible component of a zero locus of these bounded-degree equations. There are only finitely many such zero loci and components. This proves the last assertion.□

Proof. If d_0=0, the fixed low variety is a point p, so y_{i0}=p for every i. Eventual avoidance implies that p lies in no proper torsion coset of A_0.

First V projects onto \Lie(A_0). Otherwise let P be the orthogonal projector onto the orthogonal complement of that projection. This is a nonzero element of \End(A_0)_{\mathbb R}. Writing \pi_0 for the low projection, the identity P\pi_0=P\pi_0\Phi and the small-projection hypothesis give P(p)=0 in the real height space. But evaluation at p is injective on \End(A_0)_{\mathbb Q}: a nonzero rational endomorphism taking p to torsion places p in a proper torsion coset. Tensoring this injection with \mathbb R proves the same assertion for real endomorphisms. This contradicts P\ne0. If there are no high factors, surjectivity already contradicts the strict codimension hypothesis.

Put L=\Lie(A_0), H=\bigoplus_{\ell\geq1}\Lie(W_\ell), and write a=\dim L, b=\dim H, and c=\codim_{L\oplus H}V. For a positive integer h define

Kernels and images realize this as the image of a real homomorphism, hence of a real idempotent. The space before forgetting v_0 has dimension a+h(b-c), since V\to L is surjective. The kernel of forgetting v_0 has dimension t=\dim(V\cap(L\oplus0)). Consequently

Since c>\sum_{\ell=0}^s m_\ell, a sufficiently large fixed h makes this strictly greater than h\sum_{\ell\geq1}m_\ell.

Choose h subsequences of indices so that, in the ordered list of their high factors, the largest scale in one copy divided by the smallest scale in the next tends to zero. This is possible recursively because all the high scales tend to infinity. Take the corresponding high marked points, retaining their original parents and fiber bounds. All the high nonvanishing hypotheses remain valid on these subsequences.

Their normalized tuples satisfy the small-projection condition for V_*. Write A_+=\prod_{\ell\ge1}A_\ell and consider the fixed real homomorphism whose Lie action is

Let K=\ker T and let \pi_+ forget the low coordinate. Since \ker\Phi=V\subseteq L\oplus H, we have \pi_+(K)=V_*. Let \Phi_* be the orthogonal projector with kernel V_* in the full high ambient space \Lie(A_+)^h. The generalized inverse from Lemma 2.1 gives the identity

because 1-T^\dagger T projects onto K, whose image under \pi_+ is killed by \Phi_*. This identity also holds in the real height spaces. At the tuple formed from p and the selected normalized high points, T tends to zero. Applying the fixed bounded operator \Phi_*\pi_+T^\dagger therefore proves the required small-projection condition. The high points need only lie in \mathcal H(A_+)^h; the codimension inequality is measured in H^h. We have obtained forbidden all-high sequence data, contrary to Theorem 4.3.□

Proof. We first compute two distinct limiting measures geometrically. We then show that a vanishing subsequence of the arithmetic intersection quotients would force those same measures to agree.

The two measures and their geometric limits.

Put X_i=Z_i^r, u=rd, and let p_0:\cA^r\to A_0^r denote projection to the low coordinates. For j=1,2, define a probability measure on A_0^r(\mathbb C) by

Curvatures in this formula are taken at the chosen complex embedding. Both measures are supported on the fixed low product Z_0^r. The geometric comparison follows the product-versus-pullback argument of the Bogomolov method [22, Section 4]; the consecutive-difference construction is explained in [30, Section 3], following Faltings.

Let \omega=c_1(\bar L) at the fixed complex embedding. For a real homomorphism g, let \beta_1(g) be the invariant form on \cA^r obtained by summing the pullbacks of \omega under g in the r separate copies. Let \beta_2(g) be the sum of the pullbacks under the consecutive differences of those outputs. These definitions use only linear maps on Lie algebras and therefore make sense also for g=\Phi.

On a product tangent space, block scaling multiplies the top form by B_i^r. For any smooth real function f on A_0^r(\mathbb C), multiplication by p_0^*f does not alter that pointwise identity. It follows that

No multiplication map on points or change of the test function is being used in this identity.

Write Q_i=\prod_{\ell=1}^s Z_{i\ell}, allowing a point when there are no high factors. Integration of an invariant form along Q_i^r gives an invariant form on A_0^r: in invariant coordinates its coefficients are integrals of fixed invariant forms over Q_i^r, hence depend only on its homology class. Choose one of the products Q_i as a fixed representative Q. Integrating \beta_j(g)^u over Q^r produces a smooth invariant form \gamma_j(g) of bidegree (rd_0,rd_0) on A_0^r. Its coefficients are polynomial in those of g. Thus \gamma_j(\varphi_i)\to\gamma_j(\Phi) smoothly, and (6.5) shows that \mu_{i,j} converges to the probability measure \mu_j given by

Both denominators are positive by the tangent positivity already used in (6.3). Singularities cause no problem in these integrations: one may integrate over resolutions, and the resulting densities on (Z_0^{\mathrm{sm}})^r are smooth.

The first tangent integral supplies smooth points z\in Z_0 and q\in Q such that \Phi is injective on T_zZ_0\oplus T_qQ, where the tangent spaces are translated into the ambient Lie algebras. At the diagonal low tuple (z,\ldots,z), the density \gamma_1(\Phi)|_{Z_0^r} is strictly positive. Indeed, at (q,\ldots,q) the separate-output form is positive on the full product tangent space, and it remains positive on a nonempty open set of high tuples, whose integration gives a positive density.

At this same low tuple the density \gamma_2(\Phi)|_{Z_0^r} is zero. Choose a nonzero tangent vector v\in T_zZ_0, possible because d_0>0. For every smooth high tuple, the tangent vector which has low part (v,\ldots,v) and zero high parts is killed by every consecutive difference. The top power of the semipositive form \beta_2(\Phi) therefore vanishes on that full product tangent space; its integral over the high factors vanishes as well.

After their positive normalizations, the two smooth densities still differ at (z,\ldots,z). Their difference has a fixed positive sign on a sufficiently small neighborhood in the smooth locus. Choose a nonnegative smooth function f_* on A_0^r(\mathbb C), supported in a coordinate neighborhood of this point and positive on a smaller neighborhood. We have proved

Figure 1 records this comparison on the low product.

Small points from a vanishing arithmetic intersection. We now suppose, contrary to the proposition, that the nonnegative quotient in (6.4) tends to zero along a subsequence. We recall the two arithmetic inequalities used here and in the small-section construction below. For a nef adelic bundle \bar D on a projective variety X of dimension n, with D geometrically big, the fundamental inequalities read

Here e_1 is the essential minimum of the point height. We use the nef, geometrically big form of these inequalities [28, Theorem 5.3.3]; the ample projective case is [31, Theorem 1.10]. We also use the arithmetic Siu inequality

for nef adelic bundles, with the same absolute normalization [28, Theorem A.5.1(2)]. These statements apply after a projective resolution; pullback preserves the intersection numbers and essential minima used here. Resolution in characteristic zero can be performed over the original field k_i, by smooth-center blowups of a smooth projective ambient variety, with centers disjoint from the original smooth locus of X [26, Theorems 1.0.2–1.0.3 and Section 5.7]. Thus using a resolution requires no extension of the bounded-degree field k_i. A bundle of positive arithmetic volume has a nonzero section of some positive power with supremum norm at most one at every place. Its point height is therefore nonnegative outside a proper closed subset.

By (6.7) and (6.3), each Z_i has a dense set of points of \bar E_i-height at most \epsilon_iM_i^2, where \epsilon_i\to0. Products of these sets give a dense subset \mathcal S_i\subset X_i(\Qbar) such that

For the difference bundle this follows from \|a-b\|^2\le2\|a\|^2+2\|b\|^2. We will use these common dense sets to show that, for every smooth real function f on A_0^r(\mathbb C),

Applied to f_*, this will contradict (6.6). The required metric variation must be uniform in both the varieties and their defining fields.

Uniform bounds for metric variation. Let v_i be the complex place of k_i determined by the chosen embedding, and put \lambda_i=2/[k_i:\mathbb Q]. Thus \lambda_i is bounded below by a positive constant. Write \overline{\mathcal O}_{v_i}(f) for the trivial bundle whose negative logarithmic norm is p_0^*f at this embedding, its conjugate at the conjugate embedding, and zero at all other places. Write \bar{\mathbf 1} for the trivial bundle with negative logarithmic norm one at every archimedean place and zero at the finite places. Its absolute point height is one.

Fix \eta>0. For either j, abbreviate \bar D_i=\bar E_{i,j}|_{X_i}, \mu_i=\mu_{i,j}, and set

In particular |a_i|\le\|f\|_\infty+\eta. There are nef bundles \bar U_{i,1},\bar U_{i,2}, pulled back from the low ambient product, such that

where \bar H is a fixed canonical symmetric ample bundle on A_0^r, and C,K are fixed positive constants. To obtain this, choose an integer C so that C c_1(\bar H)+dd^cf is semipositive, and then choose K>\|f\|_\infty+\sup_i|a_i|.

We spell out why this is nef in the model sense required by (6.8). Start with a relatively ample projective model of a power of H, give it the invariant archimedean curvature, and add a constant so that it is nef. Normalized pullbacks by multiplication, on graph models, converge to \bar H; their archimedean curvatures are all the same invariant form. Adding the metric in (6.11) leaves every vertical curve degree unchanged. At infinity the new curvature is semipositive by the choice of C. The added negative-log-norm function is nonnegative at each archimedean place by the choice of K, so the arithmetic degree of every horizontal integral curve can only increase. This verifies the nef criterion for hermitian models: semipositive curvature and nonnegative degrees on all integral arithmetic curves. The perturbed nef models converge to \bar U_{i,1}. The construction also applies to \bar U_{i,2}, and its constants are independent of i and of the field extension.

Every arithmetic mixed intersection of total order u+1 formed from

is bounded by

with one constant for all these finitely many types of intersections. For a direct proof, let p_{a\ell} be the coordinate projection from \cA^r to factor \ell in copy a, and put

on X_i. This is nef and geometrically ample. Remove the factors b_{i\ell} on each product tangent space. The remaining forms for D_i are bounded by (6.1); those for M_i^2U_{i,a} are bounded because their geometric parts use only the low coordinates, where b_{i0}=M_i. Consequently

On the products of the dense factorwise sets of height O(H_{i\ell}), all the point heights just displayed are O(M_i^2). For D_i, use boundedness of \varphi_i to get

The same estimate holds for \bar T_i. The heights of \bar U_{i,1},\bar U_{i,2} are bounded on these sets because the low point heights and the metric shifts are bounded. Thus e_1(X_i,\bar S_i)=O(M_i^2), and the upper inequality in (6.7) yields (\bar S_i^{u+1}\cdot X_i)=O(M_i^2B_i^r). Mixed intersections of nef adelic bundles are nonnegative. Each term in (6.12) occurs with a positive coefficient in this last self-intersection, which proves the assertion. Absolute normalization has introduced no field-dependent constant.

Comparison against a low-factor test function. The metric intersection formula gives

Apply (6.8) to the two nef bundles \bar D_i+tM_i^2\bar U_{i,1} and tM_i^2\bar U_{i,2}, where t>0 is rational. Expansion and (6.12) give

The first term is nonnegative. By (6.3), a sufficiently small fixed t>0, depending on f,\eta but not on i, makes the right side positive for all sufficiently large i. Hence the perturbed height is nonnegative outside a proper closed subset of X_i.

Repeat this argument for both j and for f and -f. Choose x_i\in\mathcal S_i outside the four resulting proper closed subsets. Let \operatorname{av}_{i,x_i}(f) be the average of f on the low projections of the conjugates of x_i over k_i at the chosen embedding. Then

Divide the perturbed height inequalities by tM_i^2, and use (6.9). For j=1,2 this yields

Since \inf_i\lambda_i>0 and \eta is arbitrary, this proves (6.10). This is the only point in this metric-variation argument that needs the bounded degrees of the fields k_i; the degrees of the points x_i are unrestricted.

Taking f=f_* in (6.10) contradicts (6.6). This proves (6.4).□

Proof. By Proposition 6.1 and (6.3), there are constants \delta,C>0 with

Choose 0<c<\delta/((d+1)C). The constant bundle cM_i^2\bar{\mathbf 1} is nef, and (6.8) shows that \bar E_i-cM_i^2\bar{\mathbf 1} has positive arithmetic volume. After pulling back to the resolution, take a nonzero section of a positive power with norms at most one for this shifted metric. Restoring the original metric gives exactly (6.13).□

Proof. For a first nonzero Taylor coefficient, Cauchy’s inequality and the product formula will compare the cost of differentiation, \sum_\ell|\alpha_\ell|H_{i\ell}, with the small-section gain n_iM_i^2. The relation b_{i\ell}^2H_{i\ell}\asymp M_i^2 will turn this comparison into the asserted weighted index. We first choose the algebraic parameters and coefficient to which this argument will apply, then bound the analytic radii and metric variation.

An algebraic coefficient and its local estimates. Enlarge the field of the section to a number field E containing y_i and the equations to be used below. At a place v of E write \nu_v=[E_v:\mathbb Q_{v|\mathbb Q}]/[E:\mathbb Q], with the usual absolute values. All heights and sums over places below use these weights. The constants do not depend on [E:\mathbb Q].

Choose an affine projective-coordinate chart at y_{i\ell}. From the equations in the first hypothesis choose a maximal invertible Jacobian minor. Center the coordinates at y_{i\ell}, write U_\ell for the coordinates in that minor and T_\ell for the remaining coordinates, and multiply the equations by the inverse minor matrix. They become

where Q_\ell has no terms of degree less than two. All degrees and the number of coefficients are bounded. Translation and matrix inversion are bounded-degree rational operations on the original coefficients and the affine point coordinates. Consequently every coefficient in (7.3) has absolute height O(H_{i\ell}). Normalizing a nonzero coefficient of each original equation to one justifies this assertion for coefficient vectors initially specified only projectively.

Fix an algebraic local frame e, and a multi-index \alpha attaining (7.1) in the factorwise parameters T_\ell just chosen. If a_\alpha is its coefficient, then

is a nonzero algebraic fiber element. In any analytic frame its coefficient times the value of that frame to the power n_i equals a_{\mathrm{fib}}. Indeed a change of frame multiplies the coefficient by its constant term, while all other contributions involve coefficients of strictly smaller weighted order and vanish. The multi-index is kept fixed here; only the index itself is asserted to be independent of a change of parameters.

We will realize the parameters T_\ell on analytic polydisks of radii \rho_{i\ell v} and choose analytic frames e_v there. If

Cauchy’s coefficient inequality, or its nonarchimedean Gauss-norm counterpart, gives

The estimate is first applied on smaller closed polydisks and then passed to the indicated radii. Because the left side uses the same algebraic fiber element at every place, the product formula says

It remains to bound the two losses on the right of (7.4): the inverse radii and the metric variation.

Analytic radii from the conormal equations. Let S_{i\ell v} be the maximum of one and the absolute values of the coefficients of (7.3). Then

Fix tolerances 0<\gamma_v\le1, determined by the place of a fixed field of definition of the ambient data, equal to one outside its infinite and finitely many finite places. We may use the radii

where c_v=1 at finite places and a fixed sufficiently large constant at infinite places.

Here is the analytic justification, including uniformity over extensions. At a finite place, on a smaller closed T_\ell-polydisk of radius s<\rho_{i\ell v}, solve U=-\beta T-Q(T,U) in the Gauss-norm ball \|U\|\le S_{i\ell v}s. The quadratic term has norm at most S_{i\ell v}^3s^2, and its Lipschitz constant in U is at most S_{i\ell v}^2s<1. The map therefore preserves the ball and is a contraction. At an infinite place use instead the ball \|U\|\le C_1S_{i\ell v}s, with a fixed C_1 larger than twice the maximum number of entries in a row of \beta_\ell. This absorbs the row sums in its linear term. Bounds for the quadratic term and its Lipschitz constant now acquire only fixed factors depending on C_1, the degrees and the number of monomials. Increasing c_v makes the quadratic contribution smaller than half the ball radius and makes the Lipschitz constant less than one. The solutions for smaller disks agree and give an analytic parametrization on the open polydisk of radius \rho_{i\ell v}. All affine coordinate displacements have norm less than \gamma_v, and the Jacobian in U remains invertible. At the center these equations define the germ of Z_{i\ell}: they define a smooth complete intersection of its dimension containing that germ. Every further equation of Z_{i\ell} thus vanishes as a power series after substitution, and hence on the whole analytic branch. The invertible minor shows that this branch remains in the smooth locus. Finally (7.5) gives the second assertion of (7.6). Notice that sums of weights over the places above a fixed place do not change under extension of E.

Metric variation at the infinite places. We next choose the tolerances so that the metric varies little on the product of these branches. At infinity, fix a small \xi>0. The bound on affine coordinate displacements gives a uniform bound on projective distance: for x=[1:a] and x'=[1:a+u], the chordal distance is at most a fixed constant times \|u\|, because both homogeneous coordinate norms are at least one. Compactness of the fixed complex abelian varieties and local exponential charts therefore allow us to make each group difference from the center have an analytic logarithm of norm at most \xi. Its image under F_i has a logarithmic lift of norm O(\xi\sum_\ell b_{i\ell})=O(\xi M_i). The same coefficient bound holds at every infinite embedding: all norms on the fixed finite-dimensional real homomorphism spaces are comparable.

On the target universal cover, centered at a lift of F_i(y_i), the canonical curvature has a fixed quadratic potential Q. There is a holomorphic frame e_v whose negative log norm differs from its value at the origin by Q: subtracting this potential leaves a flat metric, with a unitary holomorphic frame on the simply connected cover. Pull this frame back along the logarithmic lift of our polydisk. We obtain

The constant is independent of the center. The image of the polydisk need not lie in an injectivity neighborhood on the target: the construction takes place on its universal cover.

Metric variation at the finite places. At a finite place we need a different estimate, independent of all integer coefficients of F_i. Outside a fixed finite set of places, the varieties, group laws, polarization data and basis homomorphisms extend over good models. Differences of reduction zero remain of reduction zero under every integral linear combination of the basis homomorphisms. The choice \gamma_v=1 therefore suffices there.

At one of the remaining places, work over the completed algebraic closure. Arbitrarily small analytic balls at the identity are subgroups. Indeed, in fixed smooth group parameters addition is u+v plus terms of degree at least two, and inversion is -u plus terms of degree at least two; on a sufficiently small ball the nonarchimedean inequality makes both operations preserve that ball. In particular the ball is stable under multiplication by every integer. Choose a target subgroup ball small enough that translation by any of its points preserves the reduction class in the fixed projective embedding. Such a choice is uniform in the point being translated. Here is a uniform projective estimate for this assertion. For normalized homogeneous coordinates, write

In a common affine chart, coordinate displacements of norm less than \gamma_v\le1 imply d_v(x,x')<\gamma_v, even when the affine coordinates of the center are unbounded. A fixed projective morphism f has finitely many homogeneous polynomial presentations which together have no base point on its source. A homogeneous Nullstellensatz identity gives a fixed \varepsilon_v>0 such that at every normalized source point one presentation has value of norm at least \varepsilon_v. Polynomial difference estimates give a fixed C_v' such that, when d_v(x,x') is below a fixed threshold, the same presentation remains nonzero and

To apply the polynomial estimate, rescale nearby normalized coordinate vectors so that their coordinate differences are bounded by their projective distance; this is possible in a coordinate which is a unit at both points. All constants are in the fixed coefficient field and remain valid over its completed algebraic closure. Apply the estimate to the difference morphism and to translation A\times A\to A, comparing (a,b) with (a,0). Thus the source coordinate tolerances uniformly control identity neighborhoods, and translation by a sufficiently small identity neighborhood preserves projective reduction for every center.

The finitely many basis homomorphisms carry sufficiently small source identity neighborhoods into this target subgroup ball. Every integral linear combination does so too. We can therefore choose the input tolerances \gamma_v independently of i so that the target remains in the reduction class of F_i(y_i) throughout the product branch. A homogeneous coordinate having maximal absolute value at this center defines a nonvanishing frame with constant projective metric norm on the branch. The comparison between this metric and the canonical metric gives

where C_v=0 outside the fixed bad places. These constants remain unchanged upon extension of the point field. This is the step that prevents a loss depending on the sizes of the integers in F_i.

Summing over the places. The constructed frames satisfy \kappa_v\le C\xi^2M_i^2 at infinity and \kappa_v\le C_v at finite places. The section norm bound applies on all the branches because they lie in the resolution’s isomorphic smooth locus. Sum (7.4), using the product formula, the small-section bound (7.2), the radius estimate (7.6), and the two metric estimates (7.7) and (7.8). We obtain

Choose \xi so that C\xi^2<c/2, then let i tend to infinity. The comparison b_{i\ell}^2H_{i\ell}\asymp M_i^2 now yields a fixed positive lower bound for (7.1).□

Proof. Put c_\ell=\codim_{\mathbb P^{d_\ell}}J_\ell and c=\sum_\ell c_\ell>0. Both index loci are defined over k, so every conjugate of J is again a common component. All conjugates have the same factor codimensions. Choose general linear spaces \Lambda_\ell\simeq\mathbb P^{c_\ell} and put \Lambda=\prod_\ell\Lambda_\ell. Each conjugate of J meets \Lambda transversely in \prod_\ell\deg J_\ell reduced points. We may choose the slices so that all these sets are disjoint and none of their points belongs to any other component of \mathcal I_\sigma.

Here is the incidence justification for this simultaneous choice. The intersection of a component J' with another component of \mathcal I_\sigma is a proper closed subset of J', hence has dimension less than \dim J'. For a fixed point, the condition that it belong to \Lambda has codimension \sum_\ell(d_\ell-c_\ell)=\dim J' in the product of the Grassmannians parametrizing the slices. Thus a general product slice misses every such proper subset. There are only finitely many components and conjugates. The same argument, together with generic transversality in each factor in characteristic zero, avoids the singular loci. It follows that the indicated q\prod_\ell\deg J_\ell points are isolated in \Lambda\cap\mathcal I_\sigma. They all belong to the higher index locus.

Multiply each derivative defining \mathcal I_\sigma by all monomials of the missing multidegree, so that every resulting equation has multidegree D. This operation does not change their common zero set: at any point, some monomial of each specified nonnegative multidegree is nonzero. Near a point of the higher index locus, differentiating by weighted order less than \sigma leaves weighted order at least \epsilon. Multiplying by a monomial cannot lower this order. Restriction to \Lambda is factorwise and cannot lower it either.

Choose c general linear combinations of these restored equations so that their restrictions to \Lambda have isolated intersections at every indicated point. This is possible because the common base locus is isolated there: successive general combinations avoid the finitely many associated components of positive dimension through these finitely many points. In completed local coordinates t_{\ell,1},\ldots,t_{\ell,c_\ell} at one such point, their ideal is contained in the monomial ideal generated by monomials of weighted order at least \epsilon. The local intersection multiplicity is therefore at least the number of monomials of weight less than \epsilon.

In particular consider monomials for which every exponent in the \ellth block is strictly less than \epsilon D_\ell/(2c). Their total weight is less than \epsilon/2, and their number is at least

Indeed there are \lceil x\rceil\ge x nonnegative integers strictly less than a positive real number x.

On the other hand, the sum of the isolated intersection multiplicities of c divisors of multidegree D on \Lambda is at most

where H_\ell denotes the hyperplane class from the \ellth factor. The bound remains valid if other intersection components have positive dimension. Namely, perturb the divisors in their basepoint-free complete linear systems to make the intersection proper; at each original isolated complete intersection the local length is conserved, and its contribution is part of the total degree (8.8). There are q\prod_\ell\deg J_\ell of the isolated points counted above. Combining their individual multiplicity bound with the total bound gives

The powers of the multidegrees cancel, proving (8.6) independently of D. For example, after harmless enlargement the constant is bounded by N!(2N/\epsilon)^N.

Finally, the stabilizer of J in the absolute Galois group of k has index q. Galois descent defines J over its fixed field, a degree-q extension of k. Each J_\ell is the image of J under a factor projection, so it is defined over the same field. All factor degrees are positive integers; hence (8.6) bounds them separately as well.□

Proof. The finite projections and norms above give G_i and z_i with (8.3). Pass to constant dimensions of the projective factors and let N be their sum. Apply the preceding component construction with \eta=\eta_1. Its N+2 levels and their gap \epsilon are fixed independently of i. It gives a proper component J_i through z_i common to two adjacent index loci; the adjacent pair may depend on i.

For at least two factors, (8.4) eventually implies (8.5), so the product theorem gives J_i=\prod_\ell J_{i\ell}. With one factor this conclusion requires no theorem. Lemma 8.1 then gives a common field of definition k_i' with [k_i':k_i] bounded and all \deg J_{i\ell} bounded. The finitely many possible choices of adjacent levels do not affect these bounds.

Because J_i is proper, at least one factor J_{i\ell} is proper in \mathbb P^{d_\ell}. After extraction fix this index \ell. A proper projective variety of bounded degree admits a nonzero homogeneous equation of bounded degree over its field of definition. One can obtain it by projecting generally over that field to a hypersurface in a projective space of one larger dimension than the variety and pulling back its equation; the image degree does not exceed the original degree. Let R_i be such an equation for J_{i\ell} over k_i'.

Pulling back R_i through the linear projection \pi_{i\ell} gives a bounded-degree hypersurface over k_i' containing y_{i\ell}. It does not contain Z_{i\ell}, because that projection is surjective onto \mathbb P^{d_\ell} and R_i is nonzero. The degrees [k_i':\mathbb Q] are bounded, since the degrees [k_i:\mathbb Q] were bounded in choosing the minimal products. This is exactly the hypersurface excluded by Proposition 5.3, a contradiction.□

Completion of the proof of Theorem 2.3. Theorem 4.3 handles the case with only unbounded scales. For a counterexample with a low factor, choose one of minimal ambient dimension. The projection and tangent arguments apply in this minimal counterexample. Proposition 5.3 then provides its auxiliary products. If d_0=0, Lemma 5.4 produces an all-high counterexample, already impossible. If d_0>0, Proposition 6.1 and Lemma 6.2 provide the small sections; Proposition 7.1 gives their positive index; and Proposition 8.2 contradicts minimality of the auxiliary products. Thus there is no counterexample with a low factor either.□

Proof. Fix an abelian variety A, and nonnegative integers m,e. Theorem 1.9 of Barroero and Dill [2] has the following implication. Suppose that, for every abelian subvariety B\subseteq A and every irreducible Z\subseteq B of dimension at most m which is not contained in a proper special subvariety of B, the set

is not dense in Z. Then each irreducible subvariety of A of dimension at most m has only finitely many optimal subvarieties of defect at most e.

The displayed intersection is a subset of Z\cap B^{[\dim Z+1]}, so Theorem 1.2 verifies the hypothesis for every B and Z in this statement. All these varieties are defined over number fields after finite extension. The same non-density theorem is available for quotients and isogenous varieties as well, so no ambient restriction is introduced by the reduction. Take m=\dim X and e=\dim A. Every special defect in A lies between 0 and \dim A, which proves the proposition.□

Proof. Let Y be maximal atypical, witnessed by a special T\subseteq S. Since \langle Y\rangle_{\mathrm{sp}}\subseteq T, inequality (1.2) gives

Suppose Y\subsetneq U\subseteq X and \delta(U)\le\delta(Y). Put T'=\langle U\rangle_{\mathrm{sp}}; then T'\subseteq S by (1.1). Choose an irreducible component C of X\cap T' containing U. We have

Thus C is atypical and strictly contains Y, a contradiction.□

Proof of Theorem 1.1. Write S=B+\tau with \tau torsion. Translation by -\tau identifies S with the abelian variety B, preserves dimensions and special subvarieties, and sends X to a subvariety with special closure B. The torsion point, B and this translated subvariety are all defined over a number field after a finite extension. We may therefore work inside B.

Proposition 9.1 and Lemma 9.2 show that the set of maximal atypical subvarieties is finite. Each is proper: if X were a component witnessing its own atypicality, the witnessing special subvariety would contain X and hence equal its special closure, making the strict dimension inequality impossible. Starting from any atypical subvariety, a sequence of strict atypical enlargements increases dimension and must terminate. Thus every atypical subvariety lies in a maximal one. Their finite union is a proper closed subset of the irreducible variety X.

If \dim X=0, a special subvariety containing its only point must contain its special closure, so there are no atypical subvarieties. This also covers the zero-dimensional ambient case. The assertions and the possibility of an empty exceptional list follow in every case.□

Proof of Corollary 1.3. Put r=\dim_{\mathbb Q}(\Gamma\otimes_{\mathbb Z}\mathbb Q) and choose a_1,\ldots,a_r\in\Gamma whose images form a basis of this vector space. Let D\subseteq A^r be the Zariski closure of the subgroup generated by (a_1,\ldots,a_r). Choose a positive integer m that annihilates the finite component group D/D^0. Multiplication by m is surjective on the abelian variety D^0, so [m]D=D^0. It is also a finite, hence closed, morphism on A^r. Consequently

Replace each a_i by ma_i, write a=(a_1,\ldots,a_r), and put C=D^0. The images of the new a_i still form a basis of \Gamma\otimes_{\mathbb Z}\mathbb Q, and the cyclic subgroup generated by a is Zariski dense in the connected abelian subvariety C. When r=0, this construction uses the empty tuple and the zero abelian variety. For every \gamma\in\Gamma there are integers n>0 and n_1,\ldots,n_r such that

Indeed, a relation after tensoring with \mathbb Q has only a torsion error, which a further positive multiple kills.

Set B=A\times C and Y=X\times\{a\}. Both are defined over \Qbar; B is abelian and Y is irreducible of dimension d. We check that Y is contained in no proper algebraic subgroup of B. If an algebraic subgroup J\subseteq B contains Y, fix x_0\in X(\Qbar). Then J contains (x-x_0,0) for every x\in X(\Qbar). The Zariski closure of the subgroup of A generated by these differences is all of A: otherwise its translate by x_0 would be a proper translate containing X. It follows that A\times\{0\}\subseteq J. The projection of J to C is a closed algebraic subgroup containing a, and hence is C. These two facts imply J=B. Since every proper torsion coset lies in a proper algebraic subgroup of the same dimension, Y is contained in no proper torsion coset. Theorem 1.2 therefore shows that Y(\Qbar)\cap B^{[d+1]}(\Qbar) is not Zariski dense in Y.

It remains to verify the inclusion

Let x=g+\gamma belong to the intersection on the left, with g\in G(\Qbar) for an algebraic subgroup G\subseteq A of codimension at least d+1 and \gamma\in\Gamma. Choose n,n_i as in (10.1), and define homomorphisms

The algebraic subgroup

contains (x,a), since

Here [n]_B^{-1} denotes preimage under multiplication by n on B. The kernel of \psi is contained in C[n], and so is the preimage in C of \psi(C)\cap(G\times\{0\}). Thus both the kernel and this intersection are finite. Multiplication by n on B is finite as well. Therefore

The algebraic-subgroup description of B^{[d+1]} now gives (x,a)\in B^{[d+1]}(\Qbar), proving (10.2). The isomorphism X\longrightarrow Y, x\longmapsto(x,a), and the non-density already proved for the right side yield the asserted non-density in X. When d=1, the intersection is contained in a proper closed subset of the irreducible curve X, and every such subset is finite.□