Limit mixed Hodge structure and nilpotent orbit

(The flat canonical extension, without assuming extension of F.) Use the marked quotient convention (z,v)∼(z+1,Tv), which is the convention $\mathrm{\Phi }(z+1)=T\mathrm{\Phi }(z)$. The vectors $e^{zN}{e}_a$ form an equivariant holomorphic frame, where $(e_a)$ is a constant marked basis. Coefficients in this frame are obtained by $e^{-zN}$, so the Hodge filtration in this frame is exactly Ψ. Extend the frame freely across $t=0$. Differentiating $e^{zN}$ gives connection matrix N $dz=(N\mathrm{/}(2\pi i))dt\mathrm{/}t$; its residue is nilpotent. This fixes both the bundle convention and the scalar/sign without changing the stated period-map convention.

Here is uniqueness of the nilpotent-residue logarithmic lattice, not an assumption of regular singularity. The transition G(t) between two such logarithmic frames satisfies $t{G}^{\mathrm{\prime }}=A(t)G-GB(t)$, with A,B holomorphic and with nilpotent residues at zero. Integrating the norm inequality along radii gives $\mathrm{\mid }\mathrm{\mid }G(t)\mathrm{\mid }\mathrm{\mid }\le C\mathrm{\mid }t{\mathrm{\mid }}^{-b}$; the same equation for $G^{-1}$ gives its polynomial bound, uniformly in the angle on one fixed circle. Hence G and its inverse are meromorphic at zero, by multiplying by a sufficiently large integral power and applying the removable-singularity theorem entry by entry.

Let a be the least integer with $t^a{E}_1\subset E_2$. The image of $t^a{E}_1$ in E₂/tE₂ is nonzero by minimality and is stable under the logarithmic operator. Its residue has the single eigenvalue a, because the original residue on E₁ is nilpotent, but also has eigenvalue zero, because it is a subquotient of the nilpotent residue on E₂. Therefore $a=0$. Interchanging the lattices gives both inclusions. This proves the uniqueness of the canonical logarithmic extension with nilpotent residue. It does not yet assert that its Hodge filtration extends.

(Coarse Hodge norm bounds from the existing Schwarz root.) Use the decomposition $\mathrm{\nabla }=D+\theta +\theta †$ from the actual curvature supplier, and write $\theta =A(z)dz$. That supplier proves ||A(z)||≤c/y. For a flat vector v, $\mathrm{\nabla }v=0$ gives $D_{z}v=-Av$ and $D_{\bar{z}}v=-A†v$. Metric compatibility of D, with h linear in its first argument, gives ${\mathrm{\partial }}_z$ $h_z(v,v)=h_z(D_{z}v,v)+h_z(v,D_{\bar{z}}v)=-2h_z(Av,v)$ and ${\mathrm{\partial }}_{\bar{z}}{h}_z(v,v)=-2h_z(A†v,v)$.

Since ${\mathrm{\partial }}_y=i({\mathrm{\partial }}_z-{\mathrm{\partial }}_{\bar{z}})$, Cauchy–Schwarz applied to the two identities in step 4.1 gives $\mathrm{\mid }{\mathrm{\partial }}_y$ log $h_z(v,v)\mathrm{\mid }\le 4\mathrm{\mid }\mathrm{\mid }A(z)\mathrm{\mid }\mathrm{\mid }\le c_1\mathrm{/}y$.

Integrating on each vertical ray, and using compactness of 0≤x≤1 at $y=Y$, gives

c $y^{-b}\mathrm{\mid }\mathrm{\mid }v\mathrm{\mid }{\mathrm{\mid }}_0^2\le h_z(v,v)\le C$ $y^b\mathrm{\mid }\mathrm{\mid }v\mathrm{\mid }{\mathrm{\mid }}_0^2$.

The nilpotent matrices $e^{\pm zN}$ and their inverses have polynomial growth in y for bounded x. Consequently in the untwisted canonical frame the Hodge metric and its inverse are bounded by powers of y. Exterior-power metrics have the same property, with enlarged exponents. No sharp monodromy-weight asymptotic is being inferred from these coarse bounds.

(Determinant Hodge lines have controlled scalar curvature.) Fix p and $r=\mathrm{rank}$ $F^p$. The Plücker line ${\mathrm{\ell }}_p=\det$ $F^p$ is a holomorphic line subbundle of ${\wedge }^r$ of the canonical flat bundle on Δ*. Give it the determinant of the Hodge metric. Its scalar curvature coefficient in the z coordinate has absolute value at most $C_p\mathrm{/}{y}^2$.

To check this, the Chern connection of the holomorphic flat bundle with its Hodge metric is D−θ+θ†: its (0,1) part is the given holomorphic structure and its off-diagonal parts are opposite adjoints. Flatness and $D_{\bar{z}}A=0$ give curvature −2[A,A†]. The second fundamental form of $F^p$ is the boundary lowering block of −θ. The subbundle curvature formula is obtained by multiplying the two block connection matrices; taking its trace bounds the determinant curvature by a fixed dimension multiple of ||A||². Thus for any nonzero holomorphic local frame s of ${\mathrm{\ell }}_p$,

$\mathrm{\mid }{\mathrm{\Delta }}_z$ log $\mathrm{\mid }\mathrm{\mid }s\mathrm{\mid }{\mathrm{\mid }}_h^2\mathrm{\mid }\le C_p\mathrm{/}{y}^2$.

This calculation only uses the finite-dimensional connection identities already established for the variation. It does not posit ambient period-domain curvature or positivity of this individual determinant line.

(A scalar cylinder lemma.) Let u(x,y) be smooth and 1-periodic for $y>Y$, and suppose |Δu|≤C/y². Then

$u(x,y)=a$ y+Re $H(e^{2\pi iz})+O(\log y)$,

where H is holomorphic on the punctured disk. More precisely there exists a smooth periodic $v=O(\log y)$ with $\mathrm{\Delta }v=\mathrm{\Delta }u$; u−v is harmonic and has the displayed linear-plus-holomorphic form.

Here is the construction. Let $f=\mathrm{\Delta }u$ and $f_n(y)$ its Fourier coefficients. Its mean has |f₀(y)|≤C/y². Choose v₀ with $v_0^{\mathrm{\prime }\mathrm{\prime }}=f_0$ and $v_0^{\mathrm{\prime }}(y)=-{\int }_y^{\mathrm{\infty }}{f}_0(s)ds$, so $v_0=O(\log y)$. For n≠0 solve $v_n^{\mathrm{\prime }\mathrm{\prime }}-(2\pi n{)}^2{v}_n=f_n$ on the half-line with the decaying Green kernel

$v_n(y)=-(4\pi \mathrm{\mid }n\mathrm{\mid }{)}^{-1}{\int }_Y^{\mathrm{\infty }}{e}^{-2\pi \mathrm{\mid }n\mathrm{\mid }\mathrm{\mid }y-s\mathrm{\mid }}$ $f_n(s)ds$.

The series of nonzero modes converges uniformly, bounded by a constant times $y^{-2}+e^{-c(y-Y)}$, because the integral contributes another factor $\mathrm{\mid }n{\mathrm{\mid }}^{-1}$; split the s integral at y/2 to see the stated bound uniformly. Local elliptic regularity, or differentiating the locally rapidly convergent Fourier series of the smooth right side, makes v smooth and gives $\mathrm{\Delta }v=f$. The harmonic periodic function u−v has a harmonic conjugate after subtracting its one real period, represented by a y term. The remaining conjugate is periodic, so it is the real part of a single-valued holomorphic H(t). This proves the lemma, including the possible essential harmonic terms; those are not being assumed absent.

(Extension of each Hodge filtration.) A holomorphic line bundle on Δ* has a nowhere-zero holomorphic section. An elementary construction suffices here: choose a smooth nonzero frame on the cylinder, solve its scalar ∂bar equation, and multiply by the resulting exponential. The scalar ∂bar equation on Δ* can be solved by a locally finite annular decomposition and Cauchy transforms; subtract finite Laurent polynomials on earlier compact annuli so that the series converges there with all derivatives. This uses the ordinary Laurent expansion of a holomorphic function on an annulus, not a Hodge extension theorem.

Choose such a frame s(t) of ${\mathrm{\ell }}_p$. Apply the scalar cylinder lemma in steps 13.1–18.1 to $u=\log \mathrm{\mid }\mathrm{\mid }s\mathrm{\mid }{\mathrm{\mid }}_h^2$. Multiplying s by the nowhere-zero holomorphic gauge exp(−H(t)/2) removes the harmonic term. The resulting holomorphic frame satisfies

$\mathrm{\mid }\mathrm{\mid }s(t)\mathrm{\mid }{\mathrm{\mid }}_h^2=e^{ay}$ $y^{O(1)}$.

By the coarse norm bounds in steps 4.1–8.1 its coefficient vector in the trivial exterior-power canonical frame has Euclidean norm at most C $e^{ay\mathrm{/}2}$ $y^b$. This is bounded by an integral power of $\mathrm{\mid }t{\mathrm{\mid }}^{-1}$. Every coefficient is therefore meromorphic at zero. Multiply by a common power of t, then remove their common zero of finite order, obtaining a holomorphic vector with a nonzero value at zero. Its line defines the holomorphic extension of the Plücker map. Decomposability is preserved because the Grassmannian is closed in its Plücker embedding. Thus $F^p$ extends as a holomorphic subbundle of the canonical extension.

All incidences $F^{p+1}\subset F^p$ and Q-isotropy identities hold on the punctured disk and hence at zero by holomorphic continuation. The result is a holomorphic map Ψ:Δ→check D, with $F_{\mathrm{\infty }}=\mathrm{\Psi }(0)$. Logarithmic Griffiths transversality gives N $F_{\mathrm{\infty }}^p\subset F_{\mathrm{\infty }}^{p-1}$: multiply the connection equation by t, extend the holomorphic subbundles, and set $t=0$. The nonzero scalar 1/(2πi) in the residue does not affect this inclusion.

(A polynomial lower bound for distance to the period-domain boundary.) The coarse metric bounds also control how near Φ(z) can lie to the boundary of D in the compact dual. Fix F₀∈D with Hodge metric h₀ and Weil operator C₀. For any F∈D, let S be defined by $h_F(u,v)=h_0(Su,v)$. It is positive self-adjoint and belongs to the real Q-isometry group: $S=C_0^{-1}{C}_F$, with the signs dictated by Q's parity. Also $C_{0}S{C}_0^{-1}=S^{-1}$. The positive square root gives $g=S^{-1\mathrm{/}2}$ in that group and $g{C}_0{g}^{-1}=C_F$. Indeed in an h₀-orthogonal eigenbasis, Q pairs only reciprocal eigenvalues, so every real power of S preserves Q.

Therefore $g^{-1}F$ has Weil operator C₀ and lies in the compact orbit K·F₀, where K is the h₀-unitary subgroup of the Q-isometry group. Its stabilizer is the compact product preserving the fixed Hodge blocks. The compact orbit lies entirely inside D. The coarse norm bounds in steps 4.1–8.1 bound ||g|| and $\mathrm{\mid }\mathrm{\mid }{g}^{-1}\mathrm{\mid }\mathrm{\mid }$ by powers of y for $F=\mathrm{\Phi }(z)$.

For the fixed projective Plücker embedding, the Lipschitz constant of the action of a matrix g and of its inverse is bounded by a fixed power of $\mathrm{\mid }\mathrm{\mid }g\mathrm{\mid }\mathrm{\mid }+\mathrm{\mid }\mathrm{\mid }{g}^{-1}\mathrm{\mid }\mathrm{\mid }$. This follows by differentiating its exterior-power coordinate matrices and using the lower bound $\mathrm{\mid }\mathrm{\mid }gv\mathrm{\mid }\mathrm{\mid }\ge \mathrm{\mid }\mathrm{\mid }{g}^{-1}\mathrm{\mid }{\mathrm{\mid }}^{-1}\mathrm{\mid }\mathrm{\mid }v\mathrm{\mid }\mathrm{\mid }$ in normalized projective coordinates. As g preserves D and its boundary, and K·F₀ has a positive distance from that boundary, there are constants $c,b>0$ with

dist(Φ(z),∂D)≥c $y^{-b}$, 0≤x≤1,

for the fixed compact-dual metric. This is a finite-dimensional metric calculation, not an SL₂-orbit estimate.

(The nilpotent orbit and the exponential correction.) The holomorphic extension lies in an opposite-cell Lie-algebra chart around F∞. In that chart uniquely write

$\mathrm{\Psi }(t)=e^{\mathrm{\Gamma }(t)}{F}_{\mathrm{\infty }}$, $\mathrm{\Gamma }(0)=0$,

with Γ holomorphic. Its Taylor expansion gives ||Γ(t)||≤C|t| on a smaller disk. Since $e^{\pm zN}$ have polynomial matrix growth,

$dist(\mathrm{\Phi }(z),e^{zN}{F}_{\mathrm{\infty }})\le C$ $y^b$ $e^{-2\pi y}$, 0≤x≤1.

For large y this is smaller than the boundary-distance lower bound in steps 24.1–28.1. The nilpotent-orbit point is therefore inside D. Real translations $e^{xN}$ preserve D, so membership holds for all x once it holds on one fundamental strip. This proves the analytic extension, opposite-cell correction and nilpotent-orbit-membership parts of the target, with the asserted chart estimate O(|t|).

(The canonical monodromy weight filtration.) First, $\exp (aN)=T^a$ preserves Q for every integer a; since exp(aN) is polynomial in a, this identity holds for every real a. Differentiating gives $Q(Nu,v)+Q(u,Nv)=0$. Choose finite nilpotent Jordan strings e,N $e,\dots ,N^m$ e. Give $N^j$ e weight m−2j and let $W_l$ be the span of all vectors of weight at most l. Then $N{W}_l\subset W_{l-2}$, and $N^l:G{r}_l^W\to G{r}_{-l}^W$ is an isomorphism for l≥0.

These conditions uniquely determine W. If $N^r=0$ and $N^{r-1}\mathrm{\ne }0$, all graded pieces outside [−r+1,r−1] vanish, because the required isomorphisms there are zero maps. Moreover $W_{-r+1}=im$ $N^{r-1}$ and $W_{r-2}=\ker$ $N^{r-1}$. Pass to ker $N^{r-1}\mathrm{/}im$ $N^{r-1}$, whose nilpotence index is smaller, and induct. The Jordan construction restricts to exactly that smaller centred filtration, proving uniqueness and independence of the chosen strings. This argument also proves functoriality under maps commuting with N.

Q makes the filtration self-dual:

$(W_l{)}^{\perp }=W_{-l-1}$.

Indeed the dual filtration defined by the right side has the same lowering and $N^l$-isomorphism properties, since N is Q-skew. Uniqueness identifies it with W.

(A real Q-skew grading and its SL₂ representation.) A Jordan grading H₀ acts by m−2j on each $N^j$ e. It splits W and satisfies $[H_0,N]=-2N$. Any two such gradings H₀,H₁ are conjugate by a unique exp B where $[B,N]=0$ and B lowers W strictly. To prove this, decompose H₁−H₀ by the negative weights of ad H₀. At its highest remaining negative weight a, conjugation by $B_a=-(H_1-H_0{)}_a\mathrm{/}a$ removes that term. This commutes with N because its weight components do separately. Continue through the finitely many negative weights, using the finite nilpotent exponential/BCH formulas. Uniqueness follows from the first nonzero weight of a putative stabilizing B.

Write † for the Q-adjoint. Self-duality of W shows that −H₀† is another such grading. Let $-H_0†=e^B$ H₀ $e^{-B}$. Adjointing and using the uniqueness just proved gives $B†=B$. Thus

$H=e^{B\mathrm{/}2}{H}_0{e}^{-B\mathrm{/}2}$

satisfies $H†=-H$, is real, splits W and has $[H,N]=-2N$. This is an explicit finite construction, not an invocation of Jacobson–Morozov.

The grading and N determine a unique raising operator $X_R$ with $[H,X_R]=2X_R$ and $[X_R,N]=H$. On a primitive highest string of length m+1 it is

$X_R$ $N^j$ $e=j(m-j+1)N^{j-1}e$.

The $N^l$-isomorphisms prove the primitive-string decomposition by induction, so these formulas define the operator on the whole space. The Q-adjoint relations and uniqueness give $X_R†=-X_R$.

Set $B_Q(u,v)=i^{k}Q(u,barv)$, the flat Hermitian polarization form. Then H is $B_Q$-skew, while

$Y=iN$, $X=-i{X}_R$

are $B_Q$-self-adjoint and satisfy $[H,X]=2X$, $[H,Y]=-2Y$, $[X,Y]=H$. This fixes the factors of i used in the later primitive polarization calculation.

(A boundedness lemma for a monodromy-invariant flat vector.) For a graded Hermitian flat bundle with $\mathrm{\nabla }=D+\theta +\theta †$, write $\theta =A$ dz and let v be flat and nonzero. The same finite connection identities as in the existing horizontal-curvature root give

${\mathrm{\partial }}_z$ $h(v,v)=-2h(Av,v)$,

${\mathrm{\partial }}_z\mathrm{\partial }ba{r}_z$ $h(v,v)=2(\mathrm{\mid }\mathrm{\mid }Av\mathrm{\mid }{\mathrm{\mid }}^2+\mathrm{\mid }\mathrm{\mid }A†v\mathrm{\mid }{\mathrm{\mid }}^2)$.

Therefore log h(v,v) is subharmonic: Cauchy–Schwarz bounds |h(Av,v)|² by both ||Av||²h(v,v) and ||A†v||²h(v,v). Also |d log h(v,v)|≤C||A||. These identities do not require a real lattice; they hold for an auxiliary complex graded Hermitian flat bundle as well.

Suppose v is monodromy invariant and ||A||≤C/y. Then $u=\log$ h(v,v) is 1-periodic in x, has |du|≤C'/y, and is subharmonic. Its mean u₀(y) satisfies u₀''≥0 and |u₀'|≤C'/y. Thus u₀' increases to zero and is nonpositive; u₀ is bounded above. Moreover |u−u₀|≤C'/y on a period by the x-derivative bound. Hence the Hodge norm of this invariant flat vector is uniformly bounded near the puncture.

For two such metrics on isomorphic flat bundles, apply this to the invariant identity section in their Hom bundle and then to its inverse. The Hom Higgs operator is the difference of the two Higgs actions, so its norm is bounded by their sum. Its flat connection and induced Hermitian metric have the same D+θ+θ† identities. The Hilbert–Schmidt norm bounds the operator norm. Consequently the two metrics are uniformly mutually bounded. This comparison does not assume that the flat isomorphism preserves a Hodge filtration or a polarization.

(Explicit auxiliary models and the sharp weight bound.) On the two-dimensional complex space with basis e₊,e₋ take $H=\mathrm{diag}(1,-1)$, Y $e_{+}=e_{-}$, Y $e_{-}=0$ and the Hermitian form with matrix $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$. Its split filtration is $F^1=C{e}_{+}$, $F^0=V$. At $z=x+iy$, $y>0$, the filtration

$F(z)=e^{-izY}F$

is a polarized complex Hodge structure of weight one: at $x=0$ its positive line is C(e₊+y e₋), its negative line is C(e₊−y e₋), and the Hodge metric is $\mathrm{diag}(y,y^{-1})$. Real x acts by an isometry of the flat Hermitian form. The period map is horizontal, its connection has the required graded harmonic decomposition, and its Higgs norm is a constant times 1/y, directly from this matrix. Its monodromy logarithm is −iY.

Take its m-th symmetric power. In the standard string basis its metric at iy is diagonal, with positive constant multiples of $y^{m-2j}$. Direct differentiation gives the same 1/y Higgs bound. Relabel the second Hodge index by k−m, multiplying the flat Hermitian form by the corresponding sign; this changes its weight to k without changing its positive metric or connection. These auxiliary models are complex graded Hermitian variations; no artificial integral lattice or real polarization is asserted for them.

Use the N,H primitive-string decomposition to identify the original complex flat bundle with a direct sum of these models, adjusting the constant string scalings to make their monodromy logarithm exactly N. At $x=0$ the resulting model metric is $y^H$ in a fixed positive weight-orthogonal reference metric. Apply the invariant Hom comparison in steps 49.1–54.1 to the invariant flat identity in both directions. The original Higgs norm is bounded by C/y by the existing Schwarz root, and the model bound is explicit. We obtain constants $c,C>0$ with

c $h_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}(y)\le h_{\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}(y)\le C$ $h_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}(y)$.

In particular, for every nonzero constant vector,

$v\in W_l\setminus W_{l-1}$ iff $h_y(v,v)$ is bounded above and below by positive multiples of $y^l$.

More strongly the rescaled metrics $h_y(y^{-H\mathrm{/}2}-,y^{-H\mathrm{/}2}-)$ are uniformly positive and bounded on the whole fixed vector space. For bounded x the same statement follows by $e^{xN}$ and $y^{H\mathrm{/}2}{e}^{xN}{y}^{-H\mathrm{/}2}=e^{xN\mathrm{/}y}$. This is the actual sharp growth root, proved from the explicit model and invariant-vector comparison, not from an assumed asymptotic SL₂ theorem.

(The rescaled limit is inside the period domain.) Let Ψ(t) be the holomorphic extended untwisted filtration from the analytic root. The real Q-isometry $y^{H\mathrm{/}2}$ takes Φ(iy) to

$y^{H\mathrm{/}2}\mathrm{\Phi }(iy)=e^{iN}{y}^{H\mathrm{/}2}\mathrm{\Psi }(e^{-2\pi y})$.

The model comparison in steps 59.1–63.1 says that these filtrations have uniformly bounded positive Hodge metrics. The compact-fibre/polar-decomposition argument of the analytic root therefore puts them in a compact subset of D, not just in its compact dual.

The algebraic one-parameter action $y^{H\mathrm{/}2}$ on a flag has a limit as y→∞. In an H-eigenbasis this is obtained by taking the leading weights of each Plücker coordinate, or equivalently by Gaussian elimination against the increasing W filtration. Denote

$F_H={\lim }_{y\to \mathrm{\infty }}{y}^{H\mathrm{/}2}{F}_{\mathrm{\infty }}$.

It is H-invariant, and on $V_l=E_l(H)$ its filtration identifies with the one induced by F∞ on $G{r}_l^W$. The exponential error $\mathrm{\Psi }(e^{-2\pi y})-F_{\mathrm{\infty }}$ is still negligible after this polynomial rescaling. Thus the displayed rescaled period maps converge to $e^{iN}{F}_H$. Compact containment proves $e^{iN}{F}_H\in D$. Passing the closed incidence $N{F}_{\mathrm{\infty }}^p\subset F_{\mathrm{\infty }}^{p-1}$ through the rescaling also gives $N{F}_H^p\subset F_H^{p-1}$.

We now prove the finite recognition assertion needed at $F_H$, rather than import it. Let $H,X,Y$ be an ${\mathfrak{s}\mathfrak{l}}_2$ representation with $[H,Y]=-2Y$, $[X,Y]=H$, and Hermitian adjoints $H^{†}=-H$, $X^{†}=X$, $Y^{†}=Y$ for a nondegenerate Hermitian form $B$. Let $HF=F$, $Y{F}^p\subseteq F^{p-1}$ and suppose $e^{Y}F$ is a pure complex Hodge filtration of weight $k$ polarized by $B$; on type $(p,q)$ our sign convention is $(-1{)}^{q}B>0$. We shall identify its total split filtration and its primitive forms by finite matrices.

Decompose $Y$ and $H$ by Hodge degree for $e^{Y}F$. Transversality and $Y^{†}=Y$ imply $Y=Y_{-1}+Y_0+Y_1$, with $Y_{-1}^{†}=Y_1$ and $Y_0^{†}=Y_0$. Since $e^{Y}H{e}^{-Y}=H+2Y$ preserves $e^{Y}F$, its negative-degree part vanishes. Skew-adjointness gives $H=-2Y_{-1}+H_0+2Y_1$ and $H_0^{†}=-H_0$. Taking the degree-zero part of $[Y,H]=2Y$ gives $2Y_0=4[Y_{-1},Y_1]+[Y_0,H_0]$. These identities come directly from the two displayed commutators, including the factor four.

The invariant subspace $V^{{\mathfrak{s}\mathfrak{l}}_2}=\ker H\cap \ker Y$ is a pure Hodge subspace for $e^{Y}F$. Indeed for an invariant $v={\sum }_p{v}_p$ choose its least nonzero first Hodge index $p$. The lowest components of $Yv=Hv=0$ give $Y_{-1}{v}_p=0$ and $H_0{v}_p=-2Y_0{v}_p$. Hermitian self-adjointness of $Y_0$ and skew-adjointness of $H_0$ imply $B(Y_0{v}_p,v_p)=0$. Pairing the last identity in step 71.1 with $v_p$ gives $0=4B(Y_1{v}_p,Y_1{v}_p)-4B(Y_0{v}_p,Y_0{v}_p)$. The two terms have opposite polarization signs because their Hodge indices differ by one. Both positive norms therefore vanish. Thus $Y{v}_p=H{v}_p=0$; subtract $v_p$ and repeat. This proves the assertion. On invariants $F$ and $e^{Y}F$ agree and $B$ restricts to their pure polarization.

For the standard representation $S_1$ take $Ya=b$, $Xb=a$, $Ha=a$, $Hb=-b$, with Hermitian pairing $B_1(a,b)=1$ and $B_1(a,a)=B_1(b,b)=0$. Give the split lines $\mathbb{C}a,\mathbb{C}b$ total types $(1,1),(0,0)$ respectively, and use their first indices for $F$. Its $e^{Y}F$ has first line $\mathbb{C}(a+b)$, with positive $B_1$, and complementary line $\mathbb{C}(a-b)$, with negative $B_1$; it is pure polarized of weight one. On $S_m={\mathrm{Sym}}^m{S}_1$, the monomial $v_j=a^{m-j}{b}^j$ has $H$-weight $m-2j$ and total type $(m-j,m-j)$, and the induced pairing has $B_m(v_j,v_{m-j})=j!(m-j)!\mathrm{/}m!$, all other pairs zero. Symmetrized tensor products prove these formulas and the pure polarization of $e^{Y}F$ without a representation-theoretic Hodge theorem.

On $\mathrm{Hom}(S_m,V)$ put $F^a\mathrm{Hom}=\{f:f(F^p{S}_m)\subseteq F^{p+a}V\}$ and let $H,X,Y$ act by commutators with the source and target actions. These formulas show that $H$ preserves this filtration and $Y$ lowers it by one. The Hermitian form $B_{\mathrm{H}\mathrm{o}\mathrm{m}}(f,g)=(m+1{)}^{-1}\mathrm{tr}(g^{†}f)$ has the same adjoint identities. Its $e^Y$-filtration is exactly the Hom filtration of the two pure structures, because $(e^{Y}f)(s)=e^{Y}f(e^{-Y}s)$. In Hodge-orthogonal frames its alternating signs are the differences of the source and target second indices and its positive metric is the normalized Hilbert–Schmidt metric. It is consequently a pure polarized structure of weight $k-m$.

Apply step 72.1 to this Hom space. Its invariant space $P_m={\mathrm{Hom}}_{{\mathfrak{s}\mathfrak{l}}_2}(S_m,V)$ is a polarized pure Hodge structure of weight $k-m$. The finite string decomposition gives the bijective evaluation map ${⨁}_m{S}_m\otimes P_m\to V$. For two intertwining maps $f,g$, their composite $g^{†}f$ is a scalar on $S_m$: its highest-weight line is one-dimensional and the lowering operator generates the string. The scalar equals the normalized trace. Different isotypical terms are orthogonal by the same intertwiner argument. Thus evaluation preserves the tensor Hermitian forms.

Evaluation preserves the pure $e^Y$ Hodge structures by the definition of the Hom filtration, and a bijective pure-Hodge map is strict: its kernel and image are direct sums of its Hodge components. Undo $e^Y$ on both sides to obtain equality of their total filtrations $F$. Hence every $H$-weight-$l$ space has a pure complex Hodge structure of weight $k+l$, and $Y$ and $X$ have total types $(-1,-1)$ and $(1,1)$. This proves the recognition assertion from steps 70.1–75.1.

Its primitive polarization is equally explicit. On the highest string vector $a^m$ one has $Y^m{a}^m=m!b^m$, so $B_m(a^m,(-Y{)}^m{a}^m)=(-1{)}^{m}m!$. The highest vector's second Hodge index has increased by $m$ from that of its multiplicity space. Therefore $B(u,(-Y{)}^{m}v)$ has the required sign $(-1{)}^{q+m}$ and is a polarization on the primitive $H$-weight-$m$ space; its positive form is $m!$ times the multiplicity metric. Orthogonality and nondegeneracy follow from the evaluation decomposition. This also proves that the primitive kernel is a pure Hodge subspace, since the raising/lowering maps have the stated types.

Apply this construction with $B(u,v)=i^{k}Q(u,\bar{v})$, $Y=iN$, $X=-i{X}_R$, and $F=F_H$. The adjoint identities were checked in steps 42.1–48.1. Step 69.1 supplies precisely $H{F}_H=F_H$, $Y{F}_H^p\subseteq F_H^{p-1}$ and $e^Y{F}_H\in D$. Thus every conclusion of steps 76.1–77.1 is available for this actual normalized limit.

We must identify its second filtration with actual real conjugation; a complex pure recognition alone would not suffice. Since $F_H$ is a limit of real $Q$-isometry translates of $F_{\mathrm{\infty }}$, it lies in $\check{D}$. Its rank and isotropy identities give $(F_H^{k-q+1}{)}^{{\perp }_Q}=F_H^q$. Hence the Hermitian-annihilator filtration ${\tilde{F}}_H^q:=\{v:B(v,F_H^{k-q+1})=0\}$ equals $\overline{F_H^q}$. The second pure filtration for $e^Y{F}_H$ is $e^{-Y}{\tilde{F}}_H$: Hermitian self-adjointness of $Y$ transfers its exponential across $B$, and the pure annihilator identity then gives this formula. It also equals $\overline{e^Y{F}_H}$, since $\bar{Y}=-Y$.

The evaluation construction respects both filtrations: on the elementary factor its Hermitian-annihilator total second filtration is the one with second index $m-j$ on $v_j$ from step 73.1; on each pure multiplicity space it is its usual second Hodge filtration. Tensor products and the pairing-preserving evaluation map therefore identify the recognition construction's total second filtration with ${\tilde{F}}_H={\bar{F}}_H$. Consequently each real $H$-weight-$l$ space, stable under conjugation, has opposed filtrations $F_H$ and ${\bar{F}}_H$ of weight $k+l$. This proves the required real, rather than merely complex, pure structures on the graded spaces.

By step 69.1, $E_l(H)\cong {\mathrm{Gr}}_l^{W(N)}$ identifies the filtration of $F_H$ with that induced by $F_{\mathrm{\infty }}$. The same holds for conjugates because $H$ is real. The opposed filtrations in step 80.1 therefore prove that $(W(N)[-k],F_{\mathrm{\infty }})$ is a real mixed Hodge structure. Its weight filtration is rational when the input is integral, since the canonical kernel/image formula for $W(N)$ is rational and $N=\log T$ is rational. Thus this is the claimed rational limiting structure as well.

For the primitive part in degree $m$, the Hermitian primitive form in step 77.1 is $B(u,(-Y{)}^{m}v)=i^{k+m}Q(u,N^m\bar{v})$: the scalar in the second argument is conjugated. It is therefore exactly the Hermitian form associated with the real bilinear form $Q_m(u,v)=Q(u,N^{m}v)$ in weight $k+m$. Its sign calculation in step 77.1 proves the positive polarization condition in the repository convention $h=Q_m(C\cdot ,\bar{\cdot })$. The parity is $(-1{)}^{k+m}$ by $Q$-skewness of $N$.

The primitive forms descend to ${\mathrm{Gr}}_m^{W(N)}$ by self-duality of $W(N)$ and its lowering identity, as in the canonical filtration supplier; their primitive kernels and filtrations identify under $E_m(H)\cong {\mathrm{Gr}}_m^{W(N)}$. Thus step 82.1 proves their nondegeneracy and positivity for the actual limiting filtration, not only for an auxiliary model.

Logarithmic transversality from step 23.1 gives $N{F}_{\mathrm{\infty }}^p\subseteq F_{\mathrm{\infty }}^{p-1}$; the real conjugate inclusion and $N{W}_l\subseteq W_{l-2}$ make $N$ a morphism of mixed Hodge structures of type $(-1,-1)$. Together with the primitive polarizations just proved, this is the polarized limiting mixed Hodge structure with the stated centre $k$.

The analytic extension was constructed in steps 19.1–23.1 without assuming the limiting mixed Hodge conclusion. Its normalized interior limit was proved from explicit invariant-vector comparison and finite models in steps 49.1–69.1. The recognition argument in steps 70.1–84.1 therefore has no circular dependence on a nilpotent-orbit or limiting-MHS theorem.

The opposite-cell map $\mathrm{\Gamma }$ from steps 29.1–31.1 is holomorphic with zero constant term, so $\mathrm{\parallel }\mathrm{\Gamma }(t)\mathrm{\parallel }=O(\mathrm{\mid }t\mathrm{\mid })$ in its fixed finite-dimensional chart. This is the exact chart asymptotic in the Statement.

Step 33.1 proves that $e^{zN}{F}_{\mathrm{\infty }}$ is in the period domain for all sufficiently large $\mathrm{Im}z$. Together with the canonical-frame identity in step 1.1 and the chart formula it gives $\mathrm{\Phi }(z)=e^{zN}{e}^{\mathrm{\Gamma }(e^{2\pi iz})}{F}_{\mathrm{\infty }}$.

All constructions apply to the real flat bundle; integrality is used only for the rational structure on the weight filtration in step 81.1. In particular real positive combinations of commuting generators may use this analytic and real limiting-structure argument without assuming that combination is integral.

The holomorphic extension, polarized limiting mixed Hodge structure, nilpotent-orbit membership and chart asymptotic prove every assertion of the Statement.