[proofplan] This entry records the one-parameter form of Schmid's nilpotent orbit theorem as an external foundational result. We first unpack the normalization $q(z)=e^{2\pi i z}$ and the monodromy convention $T=e^N$, because these choices determine the sign in the nilpotent orbit $e^{zN}F_\infty$. The renormalized filtration $e^{-zN}\widetilde\Phi(z)$ is invariant under $z\mapsto z+1$ and therefore descends to the punctured disc; Schmid's theorem, in precisely this normalization, gives the limiting filtration, the eventual period-domain containment, and the Hodge-metric decay estimate. [/proofplan] [step:Lift the period map and record the monodromy normalization] Let $q:\mathfrak H\to\Delta^*$ denote the universal covering map defined by $q(z)=e^{2\pi i z}$. Let $\tau:\mathfrak H\to\mathfrak H$ be the deck transformation $\tau(z)=z+1$. Since \begin{align*} q(\tau(z))=e^{2\pi i(z+1)}=e^{2\pi i z}=q(z), \end{align*} the transformation $\tau$ represents one positive loop around the puncture. By hypothesis, the chosen lift $\widetilde\Phi:\mathfrak H\to D$ satisfies \begin{align*} \widetilde\Phi(\tau(z))=T\widetilde\Phi(z) \end{align*} for every $z\in\mathfrak H$. Since $T$ is unipotent, its logarithm \begin{align*} N=\log T=\sum_{k=1}^{r} \frac{(-1)^{k+1}}{k}(T-I)^k \end{align*} is a finite sum for any integer $r$ with $(T-I)^{r+1}=0$, and therefore $N$ is nilpotent and $T=e^N$. [/step] [step:Renormalize the lifted period map to obtain a single-valued map on the punctured disc] Define $\psi:\mathfrak H\to\check D$ by \begin{align*} \psi(z)=e^{-zN}\widetilde\Phi(z). \end{align*} Here $e^{-zN}$ denotes the finite exponential of the nilpotent endomorphism $-zN$, acting on the compact dual $\check D$ through the complexified automorphism group of the underlying polarized [vector space](/page/Vector%20Space). For every $z\in\mathfrak H$, using $\widetilde\Phi(z+1)=e^N\widetilde\Phi(z)$ and the commutativity of the powers of $N$, we compute \begin{align*} \psi(z+1)=e^{-(z+1)N}\widetilde\Phi(z+1). \end{align*} Substituting the monodromy relation gives \begin{align*} \psi(z+1)=e^{-zN}e^{-N}e^N\widetilde\Phi(z)=e^{-zN}\widetilde\Phi(z)=\psi(z). \end{align*} Thus $\psi$ is invariant under the deck transformation $\tau$. Consequently there is a unique holomorphic map $\varphi:\Delta^*\to\check D$ such that \begin{align*} \varphi(q(z))=\psi(z) \end{align*} for every $z\in\mathfrak H$. The uniqueness follows from the surjectivity of $q$, and the existence follows because $\psi$ is holomorphic and constant on the fibres of the covering map $q$. [guided] The purpose of the renormalization is to remove the monodromy. The lifted period map $\widetilde\Phi$ is not periodic in $z$; instead, translating $z$ by $1$ applies the monodromy operator $T$. Since $T=e^N$, multiplying by $e^{-zN}$ is exactly the correction that cancels this translation behaviour. We define \begin{align*} \psi(z)=e^{-zN}\widetilde\Phi(z) \end{align*} as a map from $\mathfrak H$ to the compact dual $\check D$. This expression is meaningful because $N$ is nilpotent, so $e^{-zN}$ is a finite polynomial in $N$, and the corresponding complexified [group action](/page/Group%20Action) acts holomorphically on $\check D$. Now we check invariance under the deck transformation $\tau(z)=z+1$. The hypotheses give \begin{align*} \widetilde\Phi(z+1)=T\widetilde\Phi(z)=e^N\widetilde\Phi(z). \end{align*} Therefore \begin{align*} \psi(z+1)=e^{-(z+1)N}\widetilde\Phi(z+1). \end{align*} Substituting the equivariance relation yields \begin{align*} \psi(z+1)=e^{-zN}e^{-N}e^N\widetilde\Phi(z). \end{align*} Since $e^{-N}e^N$ is the identity transformation, this becomes \begin{align*} \psi(z+1)=e^{-zN}\widetilde\Phi(z)=\psi(z). \end{align*} The covering map $q:\mathfrak H\to\Delta^*$ has fibres given by translating by integers. Because $\psi$ is invariant under $z\mapsto z+1$, it is constant on these fibres. Hence it descends to a unique holomorphic map $\varphi:\Delta^*\to\check D$ satisfying $\varphi(q(z))=\psi(z)$. This is the single-valued object to which Schmid's [extension theorem](/theorems/59) applies. [/guided] [/step] [step:Apply Schmid's nilpotent orbit theorem to the descended map] We now invoke Schmid's nilpotent orbit theorem in its one-parameter form. This is the external foundational theorem being recorded here: for an admissible polarized pure variation of Hodge structure over $\Delta^*$ with unipotent monodromy $T=e^N$, and for the covering convention $q(z)=e^{2\pi i z}$, the renormalized period map $e^{-zN}\widetilde\Phi(z)$ descends to a holomorphic map $\varphi:\Delta^*\to\check D$ that extends holomorphically across $0$, and its value at $0$ is the limiting filtration. The hypotheses apply here because the theorem statement assumes an admissible polarized pure variation over $\Delta^*$, unipotent local monodromy $T$, and the lift normalized by $\widetilde\Phi(z+1)=T\widetilde\Phi(z)$. Thus Schmid's theorem gives a holomorphic extension of the descended map $\varphi:\Delta^*\to\check D$ to a map $\overline\varphi:\Delta\to\check D$. Define \begin{align*} F_\infty=\overline\varphi(0). \end{align*} Schmid's theorem also asserts that there exists a real number $Y>0$ such that \begin{align*} e^{zN}F_\infty\in D \end{align*} for every $z\in\mathfrak H$ with $\operatorname{Im}(z)>Y$. [guided] At this point no elementary argument is being substituted for Schmid's theorem; this step is the citation point for the foundational nilpotent orbit theorem. The input data match Schmid's one-parameter hypotheses: the variation is polarized and admissible, the base is $\Delta^*$, the local monodromy operator is unipotent, and the chosen lift satisfies $\widetilde\Phi(z+1)=T\widetilde\Phi(z)$. With the convention $q(z)=e^{2\pi i z}$ and $T=e^N$, the monodromy-removing expression is $e^{-zN}\widetilde\Phi(z)$, so the limiting filtration is defined as the value at the puncture of the extension of this descended map. Schmid's theorem therefore supplies a holomorphic map $\overline\varphi:\Delta\to\check D$ whose restriction to $\Delta^*$ is the descended map $\varphi$. We define \begin{align*} F_\infty=\overline\varphi(0). \end{align*} The same theorem asserts the period-domain part of the nilpotent orbit statement: there is a real number $Y>0$ such that \begin{align*} e^{zN}F_\infty\in D \end{align*} for every $z\in\mathfrak H$ satisfying $\operatorname{Im}(z)>Y$. [/guided] [/step] [step:Use Schmid's metric estimate to obtain the nilpotent orbit asymptotic] The metric estimate included in Schmid's nilpotent orbit theorem states that, after possibly increasing $Y$, there exist constants $C>0$ and $\alpha>0$ such that for all $z\in\mathfrak H$ satisfying $0\le \operatorname{Re}(z)\le 1$ and $\operatorname{Im}(z)>Y$, \begin{align*} d_D\bigl(\widetilde\Phi(z),e^{zN}F_\infty\bigr)\le C\bigl(\operatorname{Im}(z)\bigr)^\alpha e^{-2\pi\operatorname{Im}(z)}. \end{align*} The right-hand side tends to $0$ as $\operatorname{Im}(z)\to\infty$, since exponential decay dominates polynomial growth. Hence \begin{align*} \lim_{\operatorname{Im}(z)\to\infty} d_D\bigl(\widetilde\Phi(z),e^{zN}F_\infty\bigr)=0 \end{align*} uniformly for $0\le \operatorname{Re}(z)\le 1$. Thus the map $z\mapsto e^{zN}F_\infty$ is a holomorphic map into $D$ for $\operatorname{Im}(z)>Y$, and it has precisely the asymptotic relation to the lifted period map asserted in the theorem. This map is therefore the associated nilpotent orbit in the normalization $q(z)=e^{2\pi i z}$. [/step]