Cattani–Deligne–Kaplan algebraicity of Hodge loci

(Actual-VHS analytic bridge and parameterized meromorphic removal.) Work locally with an actual polarized variation on (Δ*)^r×Δ^m and commuting unipotent marking actions exp $N_i$. Write $\mathrm{\Phi }(z)=\exp (\sum z_i{N}_i)\mathrm{\Psi }(s)$, $s_i=\exp (2\pi i{z}_i)$, in the joint canonical flat frame. (Parameterized meromorphic removal, proved by Cauchy moments.) We first record the needed elementary analytic lemma. Suppose a projective-valued map is holomorphic near {|t|=ρ}×B, and for each fixed b its restriction continues meromorphically throughout |t|<ρ. Then its projective coordinates extend jointly meromorphically inside that circle times B. In particular this applies to a map on Δ*×B whose fixed-parameter curves extend at the puncture.

Near a chosen parameter b₀, take homogeneous linear forms $l_0,\dots ,l_M$ and a small circle |t|=ρ so that l₀ is nonzero on the circle image at b₀. Such a circle exists: choose l₀ not identically zero on the fixed holomorphic curve and avoid its discrete zeros. Compactness of the circle gives one parameter neighborhood on which all ratios $f_j=l_j\mathrm{/}{l}_0$ are holomorphic near that circle. Their negative Laurent moments $a_n(b)$, n≥1, are holomorphic functions of b by the Cauchy integral. For each fixed b the meromorphic extension in t has finitely many poles in |t|<ρ. Its negative Laurent sequence therefore has finite Hankel rank: the principal part is a finite sum of derivatives of 1/(t−a), whose coefficients satisfy the recurrence with characteristic polynomial the product of (X−a) to the pole orders.

The loci where all (d+1)-row Hankel minors vanish are closed, and their countable union covers B. Baire gives one such locus with nonempty interior. The minors are holomorphic, so the identity theorem makes them vanish on B. Choose a generically nonzero maximal minor. Cramer's rule gives a finite recurrence for all moments, with meromorphic parameter coefficients; the shifted recurrence follows from the vanishing of the next minors. Clear their holomorphic denominators. We obtain a nonzero polynomial q(t,b), with holomorphic coefficients, such that q $f_j$ has no negative Laurent terms. Its nonnegative part is the Cauchy integral of q $f_j$ on |t|=ρ, hence a holomorphic $p_j(t,b)$ for |t|<ρ and b near b₀. Thus $f_j=p_j\mathrm{/}q$ is meromorphic across $t=0$. Repeating for the finitely many ratios gives the meromorphic projective map. The argument also works if the fixed-parameter meromorphic property was initially known only on a dense open parameter set: Baire there and then the identity theorem give the same minors on B.

For completeness, finite Hankel rank gives the asserted recurrence without an extra theorem: choose the first maximal independent collection of shifted columns; the next column is its unique linear combination, with coefficients given by the chosen invertible minor. Every further row obeys the same relation because adjoining that row still has rank at most d. Shifting gives the relation for every subsequent column. Multiplication of the Laurent series by its characteristic polynomial cancels all negative coefficients. Rank zero means all negative coefficients vanish and requires no denominator.

Choose an affine line direction a all of whose boundary-coordinate components are nonzero, and write nearby points as b+ta with b in a transverse polydisk. At $b=0$ its punctured line lies in the VHS domain, and a small circle has all boundary coordinates nonzero; the same holds uniformly near that circle for b small. For every fixed b, each boundary coordinate on the line has at most one zero and is never identically zero. Restrict the actual VHS to this line, punctured at those finitely many zeros. At any such point its monodromy is a product of the commuting unipotent local monodromies, hence unipotent. The one-variable theorem extends its untwisted filtration. The line-coordinate untwisting differs from the original joint untwisting only by the holomorphic invertible exponential of the logs of the nonvanishing coordinate units. Thus the original Ψ restricted to that line also extends holomorphically at each of its finitely many holes. Apply the Cauchy-moment lemma to its Plücker maps, using this family of lines. It gives a meromorphic Ψ on a full neighborhood of the corner. The flag-incidence equations persist by meromorphic continuation. This argument uses line restrictions, so it does not assume joint holomorphic dependence of one-variable limits. It remains to rule out indeterminacy; Hartogs for holomorphic projective maps alone would not do that.

(The only analytic surface modification needed.) A meromorphic map from a smooth germ has a finite homogeneous representative $f_0,\dots ,f_M$ with no common divisorial factor. If it has a base point, a general smooth two-dimensional linear section through a general point of an indeterminacy component has isolated base ideal at that point. The restriction still has a base point: its ideal has height two and hence no common curve factor, whereas a holomorphically extendable projective map would factor as a common holomorphic scalar times a vector having a unit component; that would make the ideal principal. A principal ideal of height two is impossible. A general section avoids all other local indeterminacy components. This reduces the issue to an isolated base ideal in C{x,y}.

This analytic surface ideal admits a resolution by finitely many ordinary point blowups, with exceptional fibre a connected tree of projective lines. Here is a construction using the actual pinned supplier, without applying a finite-type algebraic statement directly to a convergent analytic ring. Complete at the point, so $A=\mathbb{C}[[x,y]]$. Blow up the completed ideal, take its finite normalization, and apply lem-normalized-point-blowups-dominate-local-normal-surface-modifications with $S=\mathrm{Spec}$ A. Its hypotheses are exactly complete equicharacteristic normal dimension two; since S is regular the resulting dominating sequence is ordinary point blowups. It makes the pulled-back completed ideal invertible.

Each centre above the closed point has residue C. Reproduce that finite sequence in the convergent analytic charts: ordinary point blowups have the explicit charts $y=xv$ or $x=yu$, and each chosen infinitely near point is specified by complex constants in these charts. The completed local chart is the same in both constructions. At every point of the compact exceptional fibre the analytic ideal has principal completion; faithful flatness of Noetherian local completion and Nakayama imply that the analytic ideal itself is principal there. Shrinking the original neighborhood makes it invertible throughout the required inverse image. Thus the projective map lifts holomorphically on the analytic point-blowup sequence. C{x,y} is Noetherian by Weierstrass preparation and division: choose a nonzero ideal element regular in y, divide by its monic Weierstrass polynomial of degree d, and identify the remainders with a submodule of the Noetherian C{x}-module C{x}^d; finitely many lifted remainder generators together with the chosen polynomial generate the ideal. The same induction works in any finite dimension. These rings are regular local, with the coordinate functions as a regular system of parameters. The actual regular-local UFD theorem thm-nonaffine-regular-local-ring-is-ufd therefore justifies removal of a common divisorial factor in the preceding paragraph. Completion is faithfully flat by thm-faithful-flatness-of-jacobson-adic-completion. No formal model was declared automatically algebraized: all blowups and all centres were explicitly reproduced, and principal generation was descended stalkwise.

The exceptional fibre begins with P¹; a point blowup adds a P¹ meeting the earlier fibre at one point, or subdivides one crossing. It stays connected with projective-line components. For a general two-plane section of the original normal-crossing boundary its boundary branches are distinct lines through the point. The first point blowup separates these lines; later point blowups preserve the SNC total support. Each exceptional component has strictly positive integral vanishing orders for every original coordinate vanishing at the base point.

(A common weight filtration without assuming multivariable extension.) Freeze all nonvanishing coordinates in one flat branch. For a positive integer vector a, restrict the actual VHS to $s_i=t^{a_i}$. The one-variable theorem and sharp metric lemma give W(N(a)), $N(a)=\sum a_i{N}_i$, and identify its weights by the powers of y along the ray Im $z_i=a_i$ y. The uniform horizontal Schwarz bound compares the Hodge metrics on any two such rays by a constant independent of y, by integrating along coordinate paths and using $\int d{y}_i\mathrm{/}{y}_i$. Their growth exponents, and hence their weight filtrations, agree. Call the common filtration W.

Every $N_i$ lowers this W by two. Indeed every positive integer combination N(a) does, and $N_i$ is the difference of the combinations with coefficients all one and with the i-th coefficient increased by one. This avoids assuming that $N_i$ itself is horizontal for an actual, non-nilpotent period map. The same comparison works for curves $s_i=t^{a_i}{u}_i(t)$, $u_i(0)\mathrm{\ne }0$: their logarithmic coordinates differ from the ray by bounded constants plus a term tending to zero, which changes the metric by a bounded factor. Thus every arc approaching this stratum with positive coordinate orders has the same W. All the original $N_i$ act trivially on each $G{r}^W$.

Let q be the analytic surface modification in the preceding construction and let Ξ be the holomorphic lifted untwisted filtration. At a point p of its exceptional fibre, take any small curve through p, otherwise avoiding the total boundary. Its coordinate orders $a_i$ are positive integers. Its one-variable limiting filtration is

$\exp ({\sum }_i(\log u_i(0)\mathrm{/}(2\pi i))N_i)\mathrm{\Xi }(p)$.

The prefactor is W-unipotent and acts trivially on $G{r}^W$. Therefore Ξ(p) itself defines a mixed Hodge structure with weight W (shifted by the original weight k). This assertion holds at all points, including crossings, not just at a generic point of the exceptional component.

The induced Hodge filtrations on $G{r}^W$ are holomorphic subbundles along the exceptional fibre. To verify the rank assertion explicitly, put $A_{p,l}=\dim (F^p\cap W_l)$ for the finite relevant indices. These are upper semicontinuous. At every point the graded Hodge numbers $h_{a,b}$ are symmetric, with fixed marginal dimensions for F and for W. The finite sum ${\sum }_{p,l}{A}_{p,l}$ equals ${\sum }_{a,b}(a-p_{\min }+1)(l_{\max }-a-b+1)h_{a,b}$. It is constant: the only mixed quadratic term is Σ $a(a+b)h_{a,b}=\frac{1}{2}\sum (a+b{)}^2{h}_{a,b}$, by symmetry; all other terms are determined by the fixed marginals. Since each $A_{p,l}$ is upper semicontinuous and their finite sum is constant, they are all locally constant. The analytic constant-rank theorem now gives the required holomorphic intersections and quotients.

(Every exceptional component has constant filtration.) Fix an exceptional component $E=P^1$ and let $N_E=\sum a_i{N}_i$ be its residue combination. At a generic point, a transverse curve has precisely these orders, so its one-variable LMHS polarizes (W,Ξ) by $N_E$. The residue of logarithmic transversality along E gives $N_E{\mathrm{\Xi }}^p\subset {\mathrm{\Xi }}^{p-1}$ on all of E. $N_E$ has the fixed Lefschetz filtration W, and the graded Hodge filtrations have constant ranks by the preceding graded/analytic construction. Consequently the primitive polarization forms remain nondegenerate at every point of E. Their generic positivity continues as nonnegativity and, with nondegeneracy, as positivity. Thus the graded primitive pieces are polarized pure Hodge bundles over the whole projective line, including its boundary intersections.

On the open stratum E° obtained by removing its boundary-crossing points, the local system and its commuting residues are locally constant, so their common monodromy filtration W is flat. Since each $N_i$ lowers W by two (step 11.1), every local monodromy induces the identity on each fixed quotient $G{r}_{\mathrm{\ell }}^W$. Thus each $G{r}_{\mathrm{\ell }}^W$ local system extends over the removed points with trivial local monodromy; because E≅P¹ is simply connected, that extended local system is trivial. The primitive sub-local systems, defined as kernels of powers of the flat residue $N_E$ on these graded local systems, are trivial as well. Their period maps are horizontal by Griffiths transversality. Restrict one such map to C⊂P¹ and apply the horizontal Schwarz bound on disks of radius R; letting R→∞ forces its derivative to vanish. Combining primitive strings gives constancy of every $G{r}^W$ Hodge filtration. This proves constancy of the primitive period maps without asserting that the regular tangential connection matrix is zero.

With its W-graded filtration fixed, the remaining holomorphic filtration Ξ|E varies in an affine space. Explicitly split W once, and successively choose the graph maps from each graded F summand into lower W degrees. Gaussian elimination with diagonal entries one identifies this fibre of the flag space with a quotient of the W-lowering unipotent group by its filtration stabilizer, and the ordered lower-degree graph coordinates identify that quotient with affine space. Thus a holomorphic map from the compact P¹ to this fibre is constant. This proves that Ξ is constant on E. The components meet in a connected tree, so their constants agree on the whole exceptional fibre.

The map therefore descends holomorphically to the original surface neighborhood. To check descent directly, choose an affine target neighborhood containing the image of the exceptional fibre. Properness shrinks the source neighborhood so its whole inverse image maps into that affine chart. Holomorphic functions constant under point blowups descend: away from the centre they are functions on a punctured bidisk and extend by the elementary Cauchy Hartogs theorem; equality on the dense complement identifies their pullbacks. Iterate through the finite point sequence. The descended target coordinates satisfy its defining equations by the identity theorem. This contradicts the supposed isolated base point.

The generic two-plane reduction of the preceding construction therefore rules out every indeterminacy point. The meromorphic Ψ is holomorphic on the full polydisk with values in the compact dual. Its subspaces are holomorphic subbundles because it is a flag-valued map. This proves the claimed several-variable Hodge-filtration extension.

The preceding steps prove Ψ holomorphic at the whole corner. Set $F=\mathrm{\Psi }(0)$; we next construct the actual nilpotent orbit and partial estimates. (Polarization on every ray and the genuine nilpotent orbit.) Fix any positive real vector c. Along $z_i=cz$, $y=Im$ z→∞, the horizontal Schwarz estimate and the connection identity give polynomial upper and lower Hodge-metric bounds in y. The nilpotent matrix exp(zN(c)), $N(c)=\sum c_i{N}_i$, has polynomial growth. Since Ψ is holomorphic, its difference from F on this ray is exponentially small, O(exp(−2π min $c_i$ y)). Consequently the compact-dual distance between the actual period and exp(zN(c))F is a polynomial in y times this exponential.

The elementary polar-decomposition argument of the one-variable analytic proof gives a polynomial lower bound for the distance of the actual period to the boundary of D: its Hodge metric and inverse are polynomially bounded. Conjugating by its positive square-root Q-isometry puts it in a fixed compact Hodge-metric fibre, and that conjugation and inverse have polynomial matrix growth. The exponential error remains negligible there. Thus exp(zN(c))F belongs to D for y large, and its Hodge metric is uniformly equivalent to the actual metric on this ray, with ratio tending to one. Real translations preserve D, so this is a polarized real one-variable nilpotent orbit.

Apply the completed one-variable argument to that real orbit. It proves that (W(N(c))[−k],F) is polarized by N(c). Its sharp growth statement and the metric comparison in the preceding paragraph identify W(N(c)) with the growth weights of the actual period on this ray. Integrating the uniform Schwarz bound along coordinate paths compares any two positive rays by constants independent of y, so W(N(c)) is independent of c. Logarithmic transversality of the now extended Hodge subbundles gives $N_i{F}^p\subset F^{p-1}$; writing $N_i$ as differences of positive combinations gives its weight lowering by two. The same reasoning on every face, with complementary parameters fixed in their interior, gives all face weight filtrations.

The algebraic part of the completed SL₂ proof needs precisely this common polarized MHS on each positive cone, the transverse commuting generators, and the canonical face weights. Its relative-face proof constructs its relative filtrations from this data, not from any pre-assumed several-variable estimate. Its remaining finite-grading and primitive-polarization construction therefore gives the normalized SL₂ comparison and an interior limit F₀ for every fixed ordered face chain of the candidate $\exp (\sum z_i{N}_i)F$. In each sufficiently ordered sector the exact normalized expression R(u), $u_j=({\tau }_{j+1}\mathrm{/}{\tau }_j{)}^{1\mathrm{/}2}$, $u_d={\tau }_d^{-1\mathrm{/}2}$, lies near F₀∈D. This proves membership of the candidate orbit there without assuming its global membership as an input.

These sectors imply membership whenever min $y_i$ is sufficiently large. Otherwise take a sequence of non-members with min $y_i\to \mathrm{\infty }$. After a subsequence every such sequence has an ordered decomposition $y=\sum {\tau }_j{\theta }^j+b$ with fixed directions, successive ratios tending to infinity and b bounded. Here is a finite construction: normalize y and pass to a limiting unit direction θ¹; subtract its exact orthogonal projection. If the residual is unbounded, normalize it and pass to a limit θ² orthogonal to θ¹, subtract its projection, and continue. At most r orthogonal directions occur; the final residual is bounded and each earlier coefficient dominates the next. A coordinate first appearing in a direction has positive coefficient, because the original $y_i$ are positive. Fixed triangular additions of earlier directions to later ones make all directions nonnegative and preserve the divergent ratios. Every coordinate appears, since otherwise its unbounded $y_i$ would survive in the bounded residual. The completed sector construction now puts the supposed non-members inside D eventually, a contradiction. Thus the candidate is a genuine multivariable nilpotent orbit.

(The boundary centralizer identity from horizontality.) Use the explicit Deligne bigrading of the limiting MHS (W[−k],F). Write $g=\mathrm{End}(V_C)=⨁g^{p,q}$ and $q={⨁}_{p<0}{g}^{p,q}$. It is a nilpotent algebra complementary to the filtration stabilizer. The opposite-cell chart gives uniquely

Ψ(s)=exp Γ(s)F, Γ(s)∈q, Γ(0)=0.

Each $N_i$ has bidegree (−1,−1), hence lies in $q_{-1}$. The left logarithmic derivatives of $\exp (\sum z_i{N}_i)\exp$ Γ(s) lie in q; Griffiths transversality says that they have only first Hodge degree −1. In the i-th direction the derivative is

$\exp (-\mathrm{ad}\mathrm{\Gamma })N_i+\exp (-\mathrm{\Gamma })(2\pi i{s}_i{\mathrm{\partial }}_{s_i})\exp \mathrm{\Gamma }$.

Set $s_i=0$. The second summand vanishes, so $\exp (-\mathrm{ad}{\mathrm{\Gamma }}_i)N_i$ has only degree −1, where ${\mathrm{\Gamma }}_i=\mathrm{\Gamma }{\mathrm{\mid }}_{s_i=0}$. Its degree −2 part is $-[{\mathrm{\Gamma }}_{i,-1},N_i]$, hence zero. Its degree −3 part then reduces to $-[{\mathrm{\Gamma }}_{i,-2},N_i]$, also zero. Continue through the finitely many negative Hodge degrees. This proves

$[\mathrm{\Gamma }{\mathrm{\mid }}_{s_i=0},N_i]=0$.

For a face I, ${\mathrm{\Gamma }}_I(s)=\mathrm{\Gamma }(0_I,s_{I^c})$ therefore commutes with every $N_i$ for i∈I. Restricting the same logarithmic derivative equations also proves horizontality of the partial map

${\mathrm{\Phi }}_I(z)=\exp (\sum z_i{N}_i)\exp$ Γ_I(s)F.

No nilpotent-orbit centralizer statement was assumed here.

(Uniform exponential control in a fixed ordered sector.) Fix an ordered sector and use its commuting real gradings $H_j$ and normalization $\beta =\prod t_j^{H_j\mathrm{/}2}$ from the completed SL₂ construction. Let $I_j$ be the positive face at the j-th step. Decompose Γ into the joint ad $H_j$ eigenspaces, with integral weights $a=(a_1,\dots ,a_d)$.

If $a_j>0$, then the a-component of Γ vanishes on the face $s_{I_j}=0$. Indeed Γ restricted there commutes with the face generators by the preceding construction, hence commutes with their positive sum and preserves its canonical weight filtration $W_j$. All positive ad $H_j$ components are therefore zero. Since Γ is holomorphic on a fixed polydisk, Cauchy estimates give

||Γ_a(s)||≤C exp(−2π ${\min }_{i\in I_j}{y}_i$)

when $a_j>0$, while Γ(0)=0 gives the ordinary bound C exp(−2π min_i $y_i$) for every component.

Conjugation by β multiplies that component by $\prod t_j^{a_j\mathrm{/}2}$. If all $a_j\le 0$ there is no amplification. Otherwise let j be the first positive index. Earlier factors are ≤1. Later factors, including the j-th, are bounded by a fixed power of τ_j, while ${\min }_{i\in I_j}{y}_i\ge c{\tau }_j$ in this fixed sector. An exponential in that latter minimum absorbs this polynomial. Therefore, for some $c,C>0$ independent of the divergent ratios in the sector,

||Ad(β)Γ(s)||≤C exp(−c min_i $y_i$).

The completed nilpotent-orbit normalization has image in a fixed compact neighborhood of F₀ for all sufficiently large $t_j$. Inserting exp(Ad(β)Γ) perturbs that normalized filtration by the displayed exponentially small amount. The local period-domain and Hodge-metric maps are real analytic on the compact neighborhood. Consequently actual Φ and its nilpotent orbit have intrinsic distance and relative Hodge-metric difference bounded by C exp(−c min_i $y_i$) in this sector. Differentiating Γ uses Cauchy bounds; derivatives of β introduce only fixed powers of t and τ, absorbed by slightly reducing c. This proves the corresponding fixed-order differentiated estimates too.

For the partial map, replace Γ by Γ_I. It has the same positive-weight vanishing on each face $I_j$ and tends to zero in the remaining variables; if none remain it is zero. Its normalized image is therefore in the same compact neighborhood and Φ_I is a genuine period map in the sufficiently ordered region. Applying the same bad-sequence extraction used in the preceding meromorphic/ray construction shows that Φ_I is a genuine period map whenever all imaginary coordinates are sufficiently large: any sequence of counterexamples has a fixed ordered subsequence, on which this normalized argument proves membership. Thus later translations to a different ordered subsector do not use a partial map outside its proven period-map domain.

The difference $D=\mathrm{\Gamma }-{\mathrm{\Gamma }}_I$ vanishes on $s_I=0$. Its positive-$a_j$ component also vanishes on $s_{I_j}=0$, by the centralizer argument applied to both terms. Holomorphic estimates thus bound it by both exponentials, hence by their geometric mean:

$\mathrm{\mid }\mathrm{\mid }{D}_a\mathrm{\mid }\mathrm{\mid }\le C$ exp(−π ${\min }_{i\in I}{y}_i-\pi {\min }_{i\in I_j}{y}_i$).

As above, the second exponential absorbs every positive normalization factor. Components with no positive weights require no absorption. Thus

||Ad(β)(Γ−Γ_I)||≤C exp(−c ${\min }_{i\in I}{y}_i$).

The finite BCH/exponential maps have bounded derivatives in the normalized compact neighborhood, so this gives the actual partial-orbit estimate

$d_D(\mathrm{\Phi }(z),{\mathrm{\Phi }}_I(z))\le C$ exp(−c ${\min }_{i\in I}{y}_i$),

with the same type of relative Hodge-metric bound and fixed-order derivative bounds. The constants are uniform in the entire sufficiently ordered part of this fixed sector, including arbitrarily separated growth rates in the other coordinates. In particular, for its fastest block, ${\min }_{i\in I}{y}_i$ is comparable to max_i $y_i$; this is exactly the exponential error required by the CDK induction.

Bounded real parts and bounded remainder terms are handled by the same finite normalized exponentials as in the completed SL₂ proof. Auxiliary unpunctured parameters are handled by working on a compact parameter polydisk, or by puncturing their nonzero strata with trivial monodromy. No estimate in min y has been incorrectly substituted for an estimate in the fastest-block minimum.

We now prove bounded-norm finiteness in weight zero. Put $Y=\max y_i$ and $0\le x_i\le 1$. A nonzero real v is exponentially close to $F^0$ when its Hodge distance from $F^0$ is at most exp(−αY)||v|| for a fixed α>0. We prove that, for fixed K,α, only finitely many integral v with ||v||²≤K can satisfy that condition sufficiently near the corner; they lie in $W_0$, and a fixed such v occurring along a corner-approaching sequence belongs to some limiting filtration $\exp (\sum z_i{N}_i)F\mathrm{\infty }$. The zero vector is immediate. The induction will also prove that the eventual constant vector is killed by every direction $T_j$ of its sector. (Rational simultaneous gradings and two finite-dimensional facts.) Extract an ordered decomposition $y={\sum }_{j=1}^d{\tau }_j{\theta }^j+b$ as in the analytic component, with θ¹≤⋯≤θᵈ after fixed triangular changes, linearly independent directions and divergent ratios $t_j={\tau }_j\mathrm{/}{\tau }_{j+1}$, $t_d={\tau }_d$. Let $I_j$ be the support of θ^j, $W^j$ its rational face weight filtration and $T_j={\sum }_i{\theta }_i^j{N}_i$. The supports are nested and cover every coordinate. The installed SL₂ construction gives commuting factors with weight operators $Y_j$, lowering operators $L_j$ and normalizer $\beta =\prod {\tau }_j^{Y_j\mathrm{/}2}$; $L_1=T_1$. The normalized actual period tends to F♯∈D, including for partial nilpotent replacements.

The $W^j$ admit a rational simultaneous grading $A^{\mathrm{\ell }}$, with $W_w^j$ the sum over ℓ_1+⋯+ℓ_j≤w. Indeed the real commuting grading proves that the finite lattice of these subspaces is distributive; choose rational complements to the successive intersections in that finite distributive lattice, by finite Gaussian elimination over Q. Their dimensions are the real multi-graded dimensions, so the chosen rational summands form the desired direct sum. Any two such splittings differ by an automorphism which is identity on the iterated graded and strictly decreases at least one, and increases none, of the cumulative weights. Conjugating that automorphism by β makes every off-diagonal entry a product of nonpositive powers of $t_j$ with at least one strictly negative power. It tends to the identity. Thus, also in the rational grading,

c Σ_ℓ ${\tau }^{\mathrm{\ell }}\mathrm{\mid }{v}_{\mathrm{\ell }}{\mathrm{\mid }}^2\le \mathrm{\mid }\mathrm{\mid }v\mathrm{\mid }{\mathrm{\mid }}^2\le C$ Σ_ℓ ${\tau }^{\mathrm{\ell }}\mathrm{\mid }{v}_{\mathrm{\ell }}{\mathrm{\mid }}^2$.

The constants are uniform along the sector. Every nonzero projection of Λ to $A^{\mathrm{\ell }}$ has a fixed positive lower bound in the chosen Euclidean norm, since the splitting is rational. Conjugations, projections, β and their inverses grow at most polynomially in Y. Exponential errors in Y therefore remain exponential after these operations.

First finite-dimensional fact: in a pure weight-zero Hodge structure carrying a Hodge-compatible real sl₂ representation, a real vector in $W_0(L)\cap F^0$ is sl₂-invariant. Decompose into ${\mathrm{Sym}}^m(R^2)\otimes P_m$. A real $F^0$ vector has type (0,0). In a component of $P_m$ type (−p,−q), $p+q=m$, its symmetric-power factor is a multiple of (e+if)^p(e−if)^q. For $m>0$ this polynomial has a nonzero $e^m$ coefficient, whereas $W_0$ requires divisibility by $f^{\mathrm{ceil}(m\mathrm{/}2)}$. It must therefore vanish. Only the $m=0$ factor remains. This proves the assertion, also in direct sums with arbitrary Hodge numbers.

Second fact: if $(W[-w],F)$ is an MHS, a nonzero real vector in $W_l\cap F^0$ requires $w+l\ge 0$ (here $W$ is zero-centred). Its highest nonzero $W$-graded image is real and in both induced filtrations $F^0$ and ${\bar{F}}^0$; opposition in a pure structure of negative total weight makes that intersection zero. In the applications below $(W^1[-w],F^{\mathrm{♯}})$ really is an MHS. To see this from the constructed comparison, write $F^{\mathrm{♯}}=\exp (i{\sum }_j{L}_j)F_s$ and regard $z\mapsto \exp (z{L}_1)\exp (i{\sum }_{j>1}{L}_j)F_s$ as a polarized one-variable orbit; the tensor-string construction proves membership for $\mathrm{Im}z>0$. Its limiting MHS has weight $W(L_1)[-w]=W^1[-w]$. Multiplication by $\exp (i{L}_1)$ acts trivially on its $W^1$-graded pieces, so the same induced opposed filtrations occur at $F^{\mathrm{♯}}$. The corresponding assertion for the moving family concerns the fastest-block partial-limit map ${\mathrm{\Phi }}_{I_1}$ of steps 32.1–42.1. For each fixed choice of the slower coordinates its untwisted fastest-block limit $G$ gives an MHS $(W^1[-w],G)$ by the one-variable theorem along a positive fastest direction. Its actual partial value is $\exp ({\sum }_{i\in I_1}{z}_i{N}_i)G$; these $N_i$ lower $W^1$ by two, so this exponential acts trivially on ${\mathrm{Gr}}^{W^1}$ and the partial value induces exactly the same opposed graded filtrations. It is therefore still an MHS for $W^1[-w]$. The real map $\beta$, which preserves $W^1$, transports it to another such MHS. No MHS assertion about an arbitrary full actual-VHS flag is needed. Its normalized flags converge to $F^{\mathrm{♯}}$ by the established partial estimate. To control ranks, choose an ambient neighborhood of $F^{\mathrm{♯}}$ in which all $\dim (W_l^1\cap F^p)$ are bounded above by their values at $F^{\mathrm{♯}}$. Induct on $l$: equality at $l-1$ gives an upper bound on $d_p=\dim F^p{\mathrm{Gr}}_l^{W^1}$, while the actual MHS property gives $d_p+d_{w+l-p+1}=\dim {\mathrm{Gr}}_l^{W^1}$. Comparing with $F^{\mathrm{♯}}$ forces equality of both bounds, proving every intersection rank constant. Gaussian elimination using fixed nonzero minors at $F^{\mathrm{♯}}$ now makes these intersections and the induced graded flags continuous on this rank stratum. Shrink the neighborhood until the determinants expressing their graded opposition stay bounded away from zero. Its smaller closed neighborhood, intersected with the rank stratum, is compact: an intersection-rank jump in a limit would contradict the ambient upper bound just imposed. Its flags are still MHSs by the uniform opposition determinants. Finally $W_{-3}^1\cap (F^{\mathrm{♯}}{)}^{-1}\cap \overline{(F^{\mathrm{♯}}{)}^{-1}}=0$ in weight zero; the minimum distance from the real unit sphere in $W_{-3}^1$ to $(F^{\mathrm{♯}}{)}^{-1}$ is positive. Continuity gives the uniform real angle bound throughout that compact neighborhood. Exponentially close full actual flags inherit the same angle bound. This supplies the precise mixed-family, constant-rank and angle hypotheses used below.

(Projection of almost-Hodge vectors and the position inequalities.) For a fixed nonzero v, its position ℓ is the lexicographically largest index of a nonzero component in a simultaneous grading. It is independent of the splitting: the change just described decreases cumulative weights, hence decreases the lexicographic index. Passing to a subsequence makes the position of v(n) fixed. Then $v(n)\in W_{{\mathrm{\ell }}_1}^1$ and its image $v_1(n)$ in $G{r}_{{\mathrm{\ell }}_1}^{W^1}$ is nonzero and has remaining position (ℓ_2,…,ℓ_d).

We need a generalized assertion. Suppose the variation has weight w≥0, the same exponential closeness to $F^0$, and ||v(n)||≤Cτ_1^{-w/2}. Then

w+ℓ_1+⋯+ℓ_j≥0 for every j.

Replace the actual VHS by its partial nilpotent orbit in the fastest variables $I_1$. The preceding analytic estimate is exp(−c ${\min }_{I_1}{y}_i$), with ${\min }_{I_1}{y}_i$ comparable to Y, so the hypotheses persist after shrinking α if needed. The new period map is exactly nilpotent in those variables. It defines a mixed Hodge variation with weight $W^1$, and its graded piece of degree ℓ_1 is a polarized pure variation Φ_1 of weight w+ℓ_1 independent of the fastest variables.

Normalize by β and take a limit of the real ray spanned by βv(n). Its norm-one limit is nonzero, lies in $W_{{\mathrm{\ell }}_1}^1$, and lies in F♯^0 and its conjugate. The second finite-dimensional fact gives w+ℓ_1≥0.

The simultaneous norm formulas on V and on this graded variation give

$\mathrm{\mid }\mathrm{\mid }{v}_1(n)\mathrm{\mid }\mathrm{\mid }\le C{\tau }_1^{-{\mathrm{\ell }}_1\mathrm{/}2}\mathrm{\mid }\mathrm{\mid }v(n)\mathrm{\mid }\mathrm{\mid }\le C{\tau }_1^{-(w+{\mathrm{\ell }}_1)\mathrm{/}2}$.

The projected vector is still exponentially close to Φ_1^0. Here is the needed relative-error verification. Write the partial map as $\exp (\sum z_i{N}_i)\exp {\mathrm{\Gamma }}_{I_1}(s)F\mathrm{\infty }$. The boundary centralizer identity makes ${\mathrm{\Gamma }}_{I_1}$ preserve $W^1$. Untwist by $\exp (-\sum (z_i-z_i^0)N_i)$, with $z^0$ fixed and sufficiently high, and then by the conjugate of exp ${\mathrm{\Gamma }}_{I_1}$, which tends to the identity and preserves $W^1$. The resulting fixed reference filtration is $\exp (\sum z_i^0{N}_i)F\mathrm{\infty }$. A vector in $W_{{\mathrm{\ell }}_1}^1$ exponentially close to its $F^0$ is exponentially close to their intersection, by a fixed finite-dimensional angle bound. Project this intersection to $G{r}_{{\mathrm{\ell }}_1}^{W^1}$. The projected nonzero lattice vector has norm at least an inverse polynomial in Y after untwisting, because all inverse matrices are polynomially bounded. Dividing the exponentially small absolute error by that lower bound still gives exp(−α′Y). Undo the polynomially bounded transformations. This proves relative exponential closeness in the quotient; no arbitrary intersection projection was assumed to preserve a relative angle.

The quotient sector has d−1 levels τ_2,…,τ_d. Since w+ℓ_1≥0 and τ_1≥τ_2, its norm bound is at most Cτ_2^{-(w+ℓ_1)/2}. Induct on d, including the case of dependent projected directions by deleting them. This proves all the asserted partial-sum inequalities. For $d=1$ only the first inequality is needed, so this induction has a base.

Apply it with $w=0$ and bounded norm. In the rational grading, the nonzero position component gives

K≥c ${\tau }^{\mathrm{\ell }}=c{\prod }_{j=1}^d{t}_j^{{\mathrm{\ell }}_1+\cdots +{\mathrm{\ell }}_j}$.

Every exponent is nonnegative and every $t_j$ diverges. They must all be zero. Thus ℓ=0 and, in particular, $v(n)\in W_0^1$ eventually.

(The constant first graded value is killed by the fastest generator.) The main strengthened theorem is now proved by induction on d. Project v(n) to $G{r}_0^{W^1}$. The preceding norm and closeness estimates put it under the induction hypothesis for d−1 levels. If there are no remaining variables, this is simply a bounded lattice in a fixed positive metric. After a subsequence its value is a constant $v_1$. The induction also says that $v_1$ is killed by $T_j$, j≥2, lies in the remaining zero weight steps, and belongs to some limiting Hodge filtration of the quotient variation.

We prove $T_1{v}_1=0$. Since βv(n) is bounded, pass to a limit $v_0\in W_0^1\cap F{\mathrm{♯}}^0$, real. The first finite-dimensional fact makes v₀ invariant under the first sl₂ factor. Its projection to $G{r}_0^{W^1}$ is the joint zero remaining-weight component of $v_1$: all other components have nonpositive cumulative remaining weights and at least one strictly negative one, hence their β factors tend to zero. Therefore the all-zero remaining-weight component of $T_1{v}_1$ is zero. $T_1=L_1$ has first-factor degree −2 and degree zero for every other factor. Thus every remaining component of $T_1{v}_1$ has nonpositive cumulative remaining weights with at least one strict negative inequality. It follows that ${\beta }_{\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{g}}(T_1{v}_1)\to 0$.

On the other hand $T_1{v}_1$ is killed by the remaining $T_j$ and lies in $F^{-1}$ for the corresponding limiting filtration, because $T_1$ is a (−1,−1) map from $G{r}_0^{W^1}$ to $G{r}_{-2}^{W^1}$. After a Tate twist by −1, it is a real type-(0,0) vector for the remaining nilpotent orbit at one fixed limiting point z*. As the $T_j$ kill it, it remains type (0,0) at z*+iΣ_{j≥2}τ_jθ^j for every sufficiently large ordered τ. Its squared Hodge norm there is therefore the constant value of its fixed polarization form. But the target graded/Tate sector norm comparison makes that norm tend to zero, since ${\beta }_{\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{g}}(T_1{v}_1)\to 0$. Positivity of a type-(0,0) vector forces $T_1{v}_1=0$. For this target use the split filtration and top real grading induced from the original split object, with the primitive-string polarization on the graded piece constructed in the relative-face proof of the SL₂ supplier. They satisfy the same polarized split hypotheses; uniqueness of the subsequent relative gradings gives exactly the remaining original factors. The bounded z* shift disappears after normalization. Thus the same target comparison applies, without assuming that a polarization-dependent initial splitting was functorial for an arbitrary map. This proves the required kernel statement without assuming a Hodge limit for an unnormalized vector.

Consequently $T_{1}v(n)$ lies in $W_{-3}^1$: its possible degree −2 image is $T_1{v}_1=0$. It is real and exponentially close to $F^{-1}$, by transversality and the polynomial bound for $T_1$. Normalize. The filtrations tend to F♯ and

$W_{-3}^1\cap F{\mathrm{♯}}^{-1}\cap \overline{F{\mathrm{♯}}^{-1}}=0$,

because total weights at most −3 cannot have both Hodge indices at least −1. Compactness gives a uniform positive angle for real vectors in $W_{-3}^1$ to the moving $F^{-1}$. Thus $T_{1}v(n)$ itself is exponentially small, in either Hodge or fixed norm. This is a full exponential conclusion, not just convergence to zero.

(Remove the fastest scale and finish.) The partial map is exactly nilpotent in $T_1$. Apply exp(−iτ_1T_1), whose matrix norm is polynomial in Y. Exponential smallness of $T_{1}v(n)$ implies exp(−iτ_1T_1)v(n)−v(n) is exponentially small. Hence v(n) is exponentially close to the Hodge filtration at

z(n)−iτ_1(n)θ^1.

For a real almost-$F^0$ vector in weight zero, its nonzero Hodge-index components are exponentially small in relative norm: negative-index components are small by $F^0$ proximity and positive-index components are their conjugates. Therefore Q(v,v)/||v||² tends to one. The polarization value Q(v(n),v(n)) was bounded at the original point, so the Hodge norms at the new points are bounded too.

If $d>1$, the new imaginary parts are Σ_{j≥2}τ_jθ^j+b. Monotonicity of the directions ensures that all coordinates still tend to infinity. Apply the main induction hypothesis to this d−1-level sequence for the partial variation. It makes v(n) constant on a subsequence, places it in $W_0$ and in a limiting Hodge filtration, and kills it by every remaining $T_j$. Since its $T_1$ image is exponentially small and it is now constant, $T_1$ also kills it.

If $d=1$, subtract a fixed constant from τ_1 so that the residual imaginary parts remain in a compact subset of the admissible interior. The translated period map is the full nilpotent orbit, since the fastest block is all variables. The new bounded Hodge norm in a compact positive family bounds the fixed Euclidean norm. Only finitely many lattice vectors occur; take a constant subsequence and a limiting translated point. It gives membership in a limiting Hodge filtration. Membership in $W_0$ was already proved. The kernel conclusion follows from the preceding exponentially small $T_1$ image.

This closes the induction. If any of the stated finiteness, $W_0$, or fixed-vector limiting-Hodge assertions failed uniformly near the corner, choose an offending sequence with min $y_i\to \mathrm{\infty }$ (distinct vectors for finiteness). Extract its ordered sector and apply the completed induction. The resulting constant subsequence with all the required properties contradicts that failure. Thus all three strengthened assertions, and in particular exact bounded-norm Hodge-class finiteness and limiting-MHS membership, hold.

(The local logarithmic equations.) On a sufficiently small punctured polydisk, the bounded-norm theorem gives finitely many integral classes in the fundamental strip which can be Hodge, and each is in $W_0$ and in $F_v^0$ for some limiting filtration $F_v=\exp (\sum z_i^0{N}_i)F\mathrm{\infty }$. Fix one such v and translate the logarithmic coordinates by $z^0$. In these coordinates the limiting filtration is $F_v$, and the canonical untwisted map is still holomorphic at the origin. Choose its opposite-cell logarithm Γ(s) using the Deligne bigrading of the MHS $(W,F_v)$. Since v is real and in $W_0\cap F_v^0$, the map Z(0)→V sending 1 to v is an MHS morphism; functoriality puts v in $I^{0,0}$.

Write $V^p={⨁}_q{I}^{p,q}$ and $q={⨁}_{p<0}\mathrm{End}(V{)}^p$. The generators $N_i$ have degree −1 and Γ has negative degrees. Thus $g(z)=\exp (\sum z_i{N}_i)\exp$ Γ(s) is in the corresponding unipotent group. Since $v\in V^0$, the condition $v\in g(z)F_v^0$ is equivalent to $g(z{)}^{-1}v=v$: every other term has strictly negative p degree. Taking its degree −1 part gives

$(\sum z_i{N}_i+{\mathrm{\Gamma }}_{-1}(s))v=0$. (L)

The Hodge locus is a union of connected components of this logarithmic-equation locus. Here is the converse on a component. The equation says that v is in $F^0+U$, where $U=g(z)({⨁}_{p\le -2}{V}^p)$ is a holomorphic complement to $F^{-1}$. Project the flat section v onto U along $F^{-1}$. Griffiths transversality gives $\mathrm{\nabla }{F}^0\subset {\mathrm{\Omega }}^1{F}^{-1}$; on the equation locus the projected section therefore solves the homogeneous induced connection equation. If it is zero at one point, uniqueness for the linear ODE along each piecewise smooth path makes it zero throughout that connected component. Analytic spaces are locally path connected; this also follows by taking finite local irreducible branches and their connected regular loci, so singular points introduce no extra continuation assumption. Thus if one point of a connected component satisfies the full Hodge condition, every point does.

In a rational basis, (L) is a finite system

$\sum {\nu }_{\alpha i}{z}_i+{\gamma }_{\alpha }(s)=0$,

where ν_{αi} are rational ($N_i$ and v are rational) and γ_α are holomorphic at the origin. Clear denominators and exponentiate using $s_i=\exp (2\pi i{z}_i)$. This gives equations

$a_{\alpha }(s){\prod }_{n_{\alpha i}\ge 0}{s}_i^{n_{\alpha i}}-{\prod }_{n_{\alpha i}<0}{s}_i^{-n_{\alpha i}}=0$,

with integers $n_{\alpha i}$ and holomorphic nowhere-zero $a_{\alpha }$. Constants introduced by $z^0$ are included in $a_{\alpha }$; their value at zero is nonzero. These equations define a closed analytic subspace Y of the full polydisk.

On the punctured polydisk, lifting Y to logarithmic coordinates gives discrete logarithm levels. Each connected lift component has one fixed level, so a connected component satisfying (L) projects onto a connected component of Y in the punctured polydisk. A deck translation m preserving that lift component has $\sum m_i{N}_{i}v=0$, by subtracting the logarithmic equations. Its monodromy $\exp (\sum m_i{N}_i)$ therefore fixes v. Hence v gives a single-valued class section on the image component, with no unproved finite-monodromy assertion.

After shrinking, Y has finitely many local irreducible branches; deleting the coordinate boundary from each non-boundary branch leaves a connected space. One elementary justification for the latter standard analytic fact is to take a local finite Weierstrass projection. A disconnected regular cover would partition its roots; the elementary symmetric functions of one partition are holomorphic off the discriminant and locally bounded, hence extend across it by one-variable Cauchy removal in Weierstrass slices. They would factor the irreducible local Weierstrass equation, a contradiction. The regular locus is therefore connected. Removing a further complex hypersurface does not disconnect that complex manifold, since a path can be perturbed off its real-codimension-at-least-two subsets. Closure gives the same assertion for the analytic branch. This also proves the path-continuation assertion used above.

The full Hodge conditions select a union of these finitely many traces, by the connection argument. Their closures are closed analytic branches. Doing this for the finite list of v proves: locally near the corner, the space of integral type-(0,0) classes with Q(v,v)≤K is a finite disjoint union of traces of closed analytic subspaces of the polydisk. The disjointness is in the class-labelled space, even when two image subspaces in the base intersect. This is the full local bounded-norm assertion, not only analyticity away from the boundary.

(Quasi-unipotent monodromy and local finite descent.) Use the completed quasi-unipotent monodromy theorem and the actual SNC projective compactification supplier. Near a boundary point choose the finite local power cover $s_i=t_i^m$ that makes all commuting local monodromies unipotent. The preceding local theorem applies to its pullback. No global algebraic covering theorem is needed for this local construction.

The images of the resulting finitely many analytic branches downstairs are closed analytic. This finite-map fact is explicit for the power cover: its local holomorphic algebra is free over the base with basis $t_1^{a_1}\cdots t_r^{a_r}$, $0\le a_i<m$. For an invariant analytic ideal its contraction is coherent by finite-module kernels. Equivalently, to detect the image set take an equation f vanishing on the chosen branch and form ∏_{g∈G}g(f) for the finite deck group G; it is invariant and therefore holomorphic in s. If a finite fibre misses the branch, a generic linear combination of its finitely many local generators is nonzero at every point of that fibre, so its product is nonzero. These invariant products cut out exactly the finite image. No general proper-mapping theorem or GAGA is hidden in the descent.

Compactness of the projective compactification now gives a finite cover by such neighborhoods. In each, the class-labelled space has only finitely many local irreducible traces, also after the finite power descent. Consequently it has only finitely many global irreducible components for this fixed K: each global component contains one of the finitely many local traces, and each such trace belongs to one global component. Its closure image in the compactified base is locally a finite union of exactly those closed analytic branches whose interior trace belongs to that component. Thus that closure image is a closed analytic subset of the projective compactification. The same holds for every individual irreducible component, not merely for the union over all bounded classes (which would include the zero class and lose the class-locus information).

(Projective analytic sets are algebraic: the finite-projection proof used here.) We prove the precise algebraicity fact needed in the preceding construction, in arbitrary finite projective dimension. Let $Z\subset P^N$ be a closed reduced analytic set, and work with one pure-dimensional part of dimension d. If $d=N$, that part is a closed analytic subset of full dimension in connected $P^N$, hence equals $P^N$ and is algebraic. Assume $d<N$. A general projective linear centre L of dimension N−d−1 misses Z. To see existence, form the incidence pairs (z,L) with z∈Z∩L. On each regular analytic stratum its dimension is one less than that of the Grassmannian of centres; countably many such coordinate strata have measure-zero images. Hence some centre lies outside the incidence image. The elementary analytic stratification here follows by successive Jacobian-rank loci and induction on dimension, using Weierstrass preparation; no algebraicity of Z is used.

Projection from L gives $\pi :Z\to P^d$. It is proper because Z is compact, and has finite fibres. Indeed each fibre sits in an affine space (the projective projection fibre with its hyperplane L removed). A compact positive-dimensional analytic set in an affine space is impossible: each coordinate holomorphic function is constant by the maximum principle on its regular branches and continuation, forcing dimension zero. Here is the analytic finite-algebra step explicitly. In a source embedding near a fibre point, write $A=C\{x,w\}\mathrm{/}I$, where x are the base coordinates and the zero fibre is isolated. The fibre quotient A/(x) has finite length. This elementary isolated-zero consequence follows by Weierstrass induction: a positive-dimensional prime quotient admits a finite injection from $C\{t_1,\dots ,t_e\}$, e≥1, obtained by preparing a nonzero equation monic in the last variable and repeating on its contraction in the remaining variables. Roots of the resulting finitely many monic equations stay near zero for small base values and give complex points over a punctured open base; it could not have an isolated zero. Thus the zero-dimensional fibre quotient is Artinian and its maximal ideal is nilpotent. There is N with $w^{\nu }=\sum x_j{b}_{\nu j}$ modulo I for every |ν|=N. These relations prove A finite over C{x} by a convergent division argument: equip power series on a smaller polydisk with the norm $\sum \mathrm{\mid }{a}_{\alpha \beta }\mathrm{\mid }{\epsilon }^{\alpha }{\rho }^{\beta }$. Split each high w monomial into one chosen degree-N factor times a remainder. This gives bounded division operators $D_{\nu }$ of norm at most ρ^{−N}, and a degree-less-than-N remainder operator R of norm at most one. Substitution of the displayed relations writes $g=Rg+Tg$ modulo I, with $T=\sum x_j{b}_{\nu j}{D}_{\nu }$. Shrink ε so ||T||<1/2. Then ${\sum }_{k\ge 0}R{T}^{kg}$ converges and is a polynomial in w of degree less than N with holomorphic x coefficients; the corresponding sums of the relation multipliers converge too. Hence it equals g modulo I. Those finitely many monomials generate A as a C{x}-module. Properness supplies finitely many such neighborhoods over a small base neighborhood, proving the finite holomorphic algebra assertion for π.

A reduced finite algebra is generically a product of finite separable field extensions, because characteristic is zero. Clearing a finite set of denominators gives a monic primitive-element equation with nonzero discriminant on a dense open set. Thus away from an analytic discriminant π is a finite unramified cover of degree m. Its image is all $P^d$ for the pure d-dimensional part: a proper finite image is locally cut out by its finite-module annihilator, and a same-dimensional analytic subset of the connected smooth $P^d$ is the whole space. Lower-dimensional parts will be treated separately below.

The complement $P^N\setminus L$ is the vector bundle O(1)^{N−d} over $P^d$. Write w for its fibre coordinate. For a linear functional λ in that fibre, form over the unramified locus

$P_{\lambda }(x,T)={\prod }_{z\in {\pi }^{-1}(x)}(T-\lambda (w(z)))$,

with the generic multiplicities. Its coefficients are holomorphic sections of O(j), 1≤j≤m. They extend across the discriminant: properness bounds the finitely many fibre coordinates in every local trivialization, hence bounds all elementary symmetric functions; bounded holomorphic functions extend across an analytic hypersurface by the one-variable Cauchy formula on Weierstrass slices and then separate holomorphy. At higher-codimension strata the same Cauchy Hartogs formula supplies the unique extension.

Every holomorphic section of O(j) on $P^d$ is a homogeneous polynomial of degree j. Pull it back to $C^{d+1}\setminus \{0\}$; it is a homogeneous holomorphic function of degree j. For d≥1 Hartogs extends it at the origin; its Taylor series and homogeneity leave only the degree-j polynomial. For $d=0$ the assertion is just a constant. Thus all coefficients of $P_{\lambda }$ are algebraic. They also depend polynomially on λ, because they are elementary symmetric functions of the linear expressions λ(w(z)); this identity continues after their extension.

Require $P_{\lambda }(x,\lambda (w))=0$ for every λ. Equating its finitely many polynomial coefficients in λ gives finitely many algebraic homogeneous equations, hence an algebraic subset A of O(1)^{N−d} and of $P^N$. Over the unramified locus its fibre is exactly Z's finite fibre: if w is not one of its m points, choose λ separating it from all of them, and the required equation fails. Z is contained in A everywhere by holomorphic continuation. Remove from A the irreducible components supported over the discriminant and take the closure of its remaining part. This algebraic set is exactly Z: over the unramified locus they agree; every pure-dimensional component of finite Z meets that locus and is its analytic closure. Conversely a Zariski open dense subset of an algebraic component is analytically dense, by the identity theorem on its dense regular locus. Hence the two closures coincide.

Lower-dimensional parts are treated by decreasing dimension; compact Noetherian analyticity supplies only finitely many such parts. This proves the required projective analytic-set algebraicity without a dimension-limited GAGA invocation.

(The claimed Hodge loci.) Apply the preceding graded/analytic construction to the closed analytic closure image of each irreducible bounded-class component from the preceding construction, in a fixed projective embedding of the compactification. Its restriction to S is a closed algebraic subvariety.

For a fixed local class, each irreducible local Hodge-locus branch is the image of one irreducible class-labelled branch. Its global component has the same dimension, since the projection is a closed immersion on each local class sheet. Its algebraic image therefore has that branch as an irreducible component of its germ: a same-dimensional irreducible analytic branch contained in an algebraic germ cannot be a proper subbranch. This proves the algebraicity of the fixed-class local germ in precisely the sense of the CDK statement. If one asks for all determinations of a fixed flat class, take the components whose labels lie in its monodromy orbit; there are only finitely many for $K=Q(v,v)$, so their algebraic images form a closed algebraic subset.

Finally let K range over the positive integers after scaling the rational polarization to an integral one. Every integral Hodge class has nonnegative polarization norm and is in some such bounded set. For each K only finitely many algebraic images occur, so the global Hodge locus is a countable union of closed algebraic subvarieties. Tate twisting weight 2p by p returns type (p,p) to type (0,0); it changes neither the class locus nor these algebraicity conclusions. Torsion classes map to zero in the rational Hodge filtration and cause no additional locus condition. No step asserts algebraicity of an individual Hodge class as an algebraic cycle.

Together with the actual-VHS extension/partial estimate and the bounded-norm induction, this proves the full public CDK target. It does not close any independent K3 period/Torelli or SYZ assertion.

These constructions prove the fixed-class algebraicity and countable closed algebraic union asserted in the Statement, without asserting the Hodge conjecture.