— or the reduction map $s_{f^*\tau} : Z \to S_{f^*\tau}$ coincides with $s_\nu : Z \to S_\nu$, where $\nu : \pi_1(Z) \to (H/\mathcal{D}H)(K)$ is the composition of $\tau$ with the group homomorphism $H \to H/\mathcal{D}H$. Here $H$ is the Zariski closure of $f^*\tau$.

If the first case happens, by Lemma 1.24 again we have $\int_{\Sigma} \{T_{\tau}\}^{\dim \Sigma} > 0$. we finish the proof of the induction for $\Sigma \subset S_{\tau}$ with $\dim \Sigma = r$. Assume that the second situation occurs. Since $H$ is assumed to be semisimple, it follows that $H/\mathcal{D}H$ finite. Therefore, $\nu$ is bounded and thus $S_{f^*\tau}$ is a point. This contradicts with the fact that $\dim S_{f^*\tau} = \dim \Sigma = r > 0$. Therefore, the second situation cannot occur. We finish the proof of the induction. The claim is proved. $\square$

This claim in particular implies that the *generic rank* $r$ of the multivalued holomorphic 1-forms on $Y$ induced by the differential of harmonic mappings of $\tau$ is equal to $\dim S_\tau$.

Since $G$ is almost simple, by [CDY22] we know that the Katzarkov-Eyssidieux reduction map $s_\tau : Y \to S_\tau$ is birational. Therefore $r = \dim Y$. On the other hand, we note that $r$ is less or equal to the dimension of the Bruhat-Tits building $\Delta(G)_K$, which is equal to $\mathrm{rank}_K G$. The theorem is proved. $\square$

**4.5. Constructing Kähler classes via representations into non-archimedean fields.** — Let $X$ be a smooth projective variety. In this subsection we will prove a more general theorem than Theorem 4.5.

**Theorem 4.21.** — Let $\mathcal{C}$ be absolutely constructible subset of $M_B(X, N)(\mathcal{C})$. Then there is a family of representations $\tau := \{\tau_i : \pi_1(X) \to \mathrm{GL}_N(K_i)\}_{i=1,\dots,M}$ where $K_i$ are non-archimedean local fields such that

— For each $i = 1, \dots, M$, $[\tau_i] \in \mathcal{C}(K_i)$;

— The reduction map $s_\tau : X \to S_\tau$ of $\tau$ coincides with $s_{\mathcal{C}} : X \to S_{\mathcal{C}}$ defined in Definition 3.1.

— For the canonical current $T_\tau$ defined over $S_{\mathcal{C}}$, $\{T_\tau\}$ is a Kähler class.

*Proof.* — Step 1. By Definition 3.1 and Lemma 1.28 there are non-archimedean local fields $L_1, \dots, L_\ell$ of characteristic zero and reductive representations $\tau_i : \pi_1(X) \to \mathrm{GL}_N(L_i)$ such that $[\tau_i] \in \mathcal{C}(L_i)$ and $s_{\mathcal{C}} : X \to S_{\mathcal{C}}$ is the Stein factorization of $(s_{\tau_1}, \dots, s_{\tau_\ell}) : X \to S_{\tau_1} \times \dots \times S_{\tau_\ell}$. Write $\tau := \{\tau_i\}_{i=1,\dots,\ell}$. We shall prove that we can add more reductive representations $\tau_{\ell+1}, \dots, \tau_M$ into non-archimedean local fields $L_i$ with $[\tau_i] \in \mathcal{C}(L_i)$ for each $i = \ell + 1, \dots, M$ such that $\{T_{\tau'}\}$ over $S_{\mathcal{C}}$ is Kähler for the new family $\tau' := \{\tau_i : \pi_1(X) \to T(L_i)\}_{i=1,\dots,M}$.

*Step 2.* By Theorem 1.12, it suffices to find extra $\tau_{\ell+1}, \dots, \tau_M$ such that $\{T_\tau\}^{\dim \Sigma} \cdot \Sigma > 0$ for every closed subvariety $\Sigma$ of $S_{\mathcal{C}}$.

Let $\dim \Sigma = 1$. Let $Z$ be the desingularization of an irreducible component in $s_{\mathcal{C}}^{-1}(\Sigma)$ which is surjective over $\Sigma$. Hence after we reorder $\tau_1, \dots, \tau_\ell$, one has $s_{\tau_1}(Z) = \Sigma_1$ is a curve. This implies that $\{T_{\tau_1}\} \cdot \Sigma_1 > 0$. Note that $e_{\tau_1} : \Sigma \to \Sigma_1$ is finite. Hence $\{e_{\tau_1}^* T_{\tau_1}\} \cdot \Sigma > 0$. Note that $T_\tau \ge e_{\tau_1}^* T_{\tau_1}$. Therefore, $\{T_\tau\} \cdot \Sigma > 0$. The case of curves is proved. We now make two inductions of dimension of closed subvarieties in $S_{\mathcal{C}}$ to prove the theorem.

**Induction One.** Assume that for every closed subvariety $\Sigma \subset S_{\mathcal{C}}$ of dimension $\le r-1$, one can add reductive representations $\{\tau_i : \pi_1(X) \to \mathrm{GL}_N(L_i)\}_{i=\ell+1,\dots,k}$ (depending on $\Sigma$) with $[\tau_i] \in \mathcal{C}(L_i)$ for each $i = \ell + 1, \dots, k$ such that $\{T_{\tau'}\}^{\dim \Sigma} \cdot \Sigma > 0$ for the new family $\tau' := \{\tau_i : \pi_1(X) \to T(L_i)\}_{i=1,\dots,k}$.

Let $\Sigma \subset S_{\mathcal{C}}$ be a closed subvariety of dimension $r$ such that $\{T_\tau\}^{\dim \Sigma} \cdot \Sigma = 0$. Let $\pi : X^{\mathrm{sp}} \to X$ be a ramified Galois cover which dominates the spectral covers associated to each $\tau_1, \dots, \tau_\ell$. Pulling back all the spectral forms associated with $\tau_i$ to $X^{\mathrm{sp}}$, we obtain a linear space $V \subset H^0(X^{\mathrm{sp}}, \Omega_{X^{\mathrm{sp}}}^1)$. We denote by $\mathcal{A}_{X^{\mathrm{sp}}}$ the Albanese variety of $X^{\mathrm{sp}}$. Then $s_{\pi^*\tau} : X^{\mathrm{sp}} \to S_{\pi^*\tau}$ is the Stein factorization of the partial Albanese morphism
$X^{\text{sp}} \to \mathcal{A}_{X^{\text{sp}},V}$ associated to $V$. Then we have a commutative diagram

<figure><img src="image_10.png" /></figure>

with $\sigma_\pi$ a finite surjective morphism. Take a closed subvariety $\Sigma' \subset S_{\pi^*\tau}$ which dominates $\Sigma$ via $\sigma_\pi$. Let $Y$ be the desingularization of the normalization of an irreducible component of $X^{\text{sp}} \times_W \Sigma'$ which dominates $\Sigma'$. Consider the pullback representation $\varphi^*\tau_i : \pi_1(Y) \to \text{GL}_N(L_i)$ and the reduction maps $s_{\varphi^*\tau_i} : Y \to S_{\varphi^*\tau_i}$. Then $s_{\varphi^*\tau} : Y \to S_{\varphi^*\tau}$ is the Stein factorization of the partial Albanese morphism associated to $\iota^*V \subset H^0(Y, \Omega_Y^1)$.

<figure><img src="image_11.png" /></figure>

Note that $\sigma_\iota(S_{\varphi^*\tau}) = \Sigma'$. By taking successive hyperplane sections in $Y$, we can find a morphism $Z' \to Y$ from a smooth projective variety $Z'$ which is birational into the image such that the composition $Z' \to S_{\varphi^*\tau}$ is generically finite surjective morphism.

<figure><img src="image_12.png" /></figure>

Then $S_{\phi^*\tau} \to S_{\phi^*\tau}$ is a finite surjective morphism by Lemma 1.24. It follows that $s_{\phi^*\tau} : Z' \to S_{\phi^*\tau}$ is a birational morphism. Note that for any reductive representation $\varrho : \pi_1(X) \to \text{GL}_N(K)$, its reduction map $s_\varrho : X \to S_\varrho$ factors through $s_\mathcal{E} : X \to S_\mathcal{E}$. Hence the reduction map $s_{\phi^*\varrho} : Z' \to S_{\phi^*\varrho}$ factors through $s_{\phi^*\tau} : Z' \to S_{\phi^*\tau}$ by Theorem 1.21. We assume that after we add reductive representations into non-archimedean local fields $\tau_{\ell+1}, \dots, \tau_k$ with $[\tau_i] \in \mathfrak{C}(L_i)$ for each $i = \ell + 1, \dots, k$, the generic rank of the multivalued holomorphic 1-forms on $Z'$ induced by the differential of harmonic mappings of $\{\phi^*\tau_i : \pi_1(Z') \to \text{GL}_N(L_i)\}_{i=1,\dots,k}$ achieves its maximum, which we denoted by $m$. We take a Galois cover $Z \to Z'$ which dominates all spectral covers of $\phi^*\tau_i$ for $i = 1, \dots, k$. We replace $Z$ by a desingularization and we pullback all the spectral forms of $\phi^*\tau_i$ for $i = 1, \dots, k$ to $Z$ to obtain $\mathbb{V} \subset H^0(Z, \Omega_Z^1)$. We still use the same notation $\tau$ to denote the increased family of representation $\{\tau_i\}_{i=1,\dots,k}$. Note that $s_\tau$ always coincides with $s_\mathcal{E}$ if we add $\tau_{\ell+1}, \dots, \tau_\ell$ with $[\tau_i] \in \mathfrak{C}(L_i)$ for each $i = \ell + 1, \dots, \ell'$. Therefore, the diagram below will stabilize
whenever we add such new reductive representations to {$\tau_1, \dots, \tau_k$}.

<figure><img src="image_13.png" /></figure>

Note that $s_{\psi^*\tau}: Z \to S_{\psi^*\tau}$ is the Stein factorization of the partial Albanese morphism $g_V: Z \to \mathcal{A}_Z, V$ associated with $V$. Note that $s_{\psi^*\tau}$ is birational as $s_{\phi^*\tau}$ is birational. Note that the generic rank of $V$ is $m$. Therefore, if $m = \dim Z$, according to Remark 4.16 the current $T_{\psi^*\tau}$ on $S_{\psi^*\tau}$ is big and by the functoriality of the canonical currents in Lemma 1.24, {$T_\tau$}^r \cdot \Sigma > 0$. The induction for subvarieties in $S_\mathcal{C}$ of dimension $r$ is thus proved.

Assume now $m < \dim Z$, which means that the generic rank of $V$ is less than $\dim Z$. We shall prove that this cannot happen.

**Case (1):** $m < \dim_{\mathbb{C}} V$. The proof is closed to Lemma 4.15. By Lemma 4.14 there is $W \subset V$ with $\dim_{\mathbb{C}} W \le m + 1$ such that $\mathrm{Im}[\Lambda^{\dim W}W \to H^0(Z, \Omega_Z^{\dim W})] = 0$, and for any hyperplane $W' \subset W$, we always have $\mathrm{Im}[\Lambda^{\dim W'}W' \to H^0(Z, \Omega_Z^{\dim W'})] \ne 0$.

Since we assume that $m < \dim Z$, it follows that $\dim W \le \dim Z$. By Theorem 4.13, there is a fibration $p: Z \to B$ with $B$ a projective normal variety with $\dim B = \dim W - 1 \le \dim Z - 1$ such that $W \subset p^*H^0(B, \Omega_B^1)$. Let $F$ be a general fiber of $p$ which is a proper closed subvariety of $Z$ such that $F$ is birational to $F := s_{\psi^*\tau}(F)$ via $s_{\psi^*\tau}$. Since $W \subset p^*H^0(B, \Omega_B^1)$, the generic rank of $W$ is equal to $\dim B$, and $\mathrm{Im}[\Lambda^{\dim W} \to H^0(Z, \Omega_Z^{\dim W})] = 0$, it implies that

$$ \mathrm{Im}[\Lambda^{\dim F}V \to H^0(F, \Omega_F^{\dim F})] = 0. $$

Therefore, {$T_{\psi^*\tau}$}^{\dim F} \cdot F = 0$ by Remark 4.16.

By the induction, we can add some new reductive representation $\tau_{k+1}, \dots, \tau_{k'}$ into non-archimedean local fields $L_i$ with $[\tau_i] \in \mathfrak{C}(L_i)$ for each $i = k + 1, \dots, k'$ such that for the new family $\tau' := \{\tau_i\}_{i=1,\dots,k'}$, one has {$T_{\tau'}$}^{\dim \sigma_\psi(F)} \cdot \sigma_\psi(F) > 0$. By Lemma 1.24, {$T_{\psi^*\tau'}$}^{\dim F} \cdot F > 0.

Note that $s_{\tau'}: X \to S_{\tau'}$ coincides with $s_\tau: X \to S_\tau$ by our definition of $s_\mathcal{C}: X \to S_\mathcal{C}$. Hence by Lemma 1.24 $s_{\psi^*\tau'}: Z \to S_{\psi^*\tau'}$ coincides with $s_{\psi^*\tau}: Z \to S_{\psi^*\tau}$. Since {$T_{\psi^*\tau'}$}^{\dim F'} \cdot F' > 0$, we conclude that the rank of multivalued one forms on $Z$ induced by $\psi^*\tau'$ has rank $\dim Z$. It implies that the rank of multivalued one forms on $Z'$ induced by $\phi^*\tau'$ has rank $\dim Z' = \dim Z$. This contradicts with our assumption that $m < \dim Z$. Hence Case (1) cannot happen. In the next Step we will deal with Case (2) using Theorems 4.5 and 4.17 and show that it can neither happen.

**Step 3.** Case (2): $m = \dim_{\mathbb{C}} V$.

**Claim 4.22.** — For any reductive representations {$\tau_i: \pi_1(X) \to \mathrm{GL}_N(L'_i)$}$_{i=k+1,\dots,k'}$ with $L_i$ non-archimedean local fields and $[\rho_i] \in \mathfrak{C}(L'_i)$, the new family $\tau' := \{\tau\} \cup \{\tau_i\}_{i=k+1,\dots,k'}$ satisfies that

$$ (4.3) \qquad CT_{\psi^*\tau'} \ge T_{\psi^*\tau} \ge C^{-1}T_{\psi^*\tau'} $$

for some constant $C > 0$.
*Proof.* — We may replace $Z$ by a Galois cover which dominates the spectral covers of $\psi^*\tau'$. Note that $T_{\psi^*\tau'} \ge T_{\psi^*\tau}$. Note that the rank of multivalued one forms on $Z$ induced by $\psi^*\tau'$ always has rank $m$ by our choice of $m$. Assume by contradiction that (4.3) does not happen. Then the dimension of global spectral forms $\mathbb{V}'$ induced $\psi^*\tau'$ will be greater than $\dim \mathbb{V}$. We are now in the situation of Case (1), which gives us the contradiction. The claim is proved. $\square$

**Claim 4.23.** — For any proper closed subvariety $V \subsetneq \Sigma$ (resp. $V \subsetneq S_{\psi^*\tau}$), one has $\{T_\tau\}^{\dim V} \cdot V > 0$ (resp. $\{T_{\psi^*\tau}\}^{\dim V} \cdot V > 0$).

*Proof.* — Indeed, by the induction, any proper closed subvariety $V \subsetneq \Sigma$ we can add some new reductive representation $\tau_{k+1}, \dots, \tau_{k'}$ into non-archimedean local fields $L_i$ with $[\tau_i] \in \mathfrak{C}(L_i)$ for each $i = k+1, \dots, k'$ such that we have $\{T_{\tau'}\}^{\dim V} \cdot V > 0$. By Lemma 1.24 this implies that $\{T_{\psi^*\tau'}\}^{\dim V'} \cdot V' > 0$ for any closed subvariety $V' \subset S_{\psi^*\tau}$ which dominates $V$. By Claim 4.22, it follows that $\{T_{\psi^*\tau}\}^{\dim V'} \cdot V' > 0$. The claim follows. $\square$

Let us denote by $H_i$ the Zariski closure of $\psi^*\tau_i(\pi_1(Z))$ for each $i = 1, \dots, k$, which is a reductive algebraic group over $L_i$. By [uh], there is some number field $k_i$ and some non-archimedean place $v_i$ of $k_i$ such that $L_i = (k_i)_{v_i}$ and $H_i$ is defined over $k_i$. Denote $T_i := H_i/\mathcal{D}H_i$. Consider the morphisms of affine $k_i$-schemes of finite type

$$ (4.4) \qquad \begin{array}{ccc} M_B(X, N) & \to & M_B(Z, N) \\ & \searrow & \\ M_B(Z, T_i) & \leftarrow & M_B(Z, H_i) \end{array} $$

Then by Theorem 1.19, $\mathfrak{C} \subset M_B(X, N)(\mathbb{C})$ is transferred via the diagram (4.4) to some absolutely constructible subset $\mathfrak{C}_i$ of $M_B(Z, T_i)$. Consider the reduction map $s_{\mathfrak{C}_i} : Z \to S_{\mathfrak{C}_i}$ defined in Definition 3.1. Denote by $f : Z \to S$ be the Stein factorisation of $s_{\mathfrak{C}_1} \times \dots \times s_{\mathfrak{C}_k} : Z \to S_{\mathfrak{C}_1} \times \dots \times S_{\mathfrak{C}_k}$.

**Claim 4.24.** — The reduction map $s_{\psi^*\tau} : Z \to S_{\psi^*\tau}$ factors through $Z \xrightarrow{f} S \xrightarrow{q} S_{\psi^*\tau}$.

*Proof.* — By Claim 4.23 the conditions in Theorem 4.17 are fulfilled for $\psi^*\tau$. Since we assume that $m < \dim Z$, which means that the generic rank of $\mathbb{V}$ is less than $\dim Z$. It implies that $\{T_{\psi^*\tau}\}^{\dim S_{\psi^*\tau}} \cdot S_{\psi^*\tau} = 0$. Hence the second possibility in Theorem 4.17 happens and thus we conclude that the reduction map $s_{\sigma_i} : Z \to S_{\sigma_i}$ coincides with $s_{\psi^*\tau_i} : Z \to S_{\psi^*\tau_i}$ where $\sigma_i : \pi_1(Z) \to T_i(L_i)$ is the composition of $\psi^*\tau_i : \pi_1(Z) \to H_i(L_i)$ with the group homomorphism $H_i \to T_i$. By (4.4) and the definition of $\mathfrak{C}_i$, $[\sigma_i] \in \mathfrak{C}_i(L_i)$. Therefore, $s_{\sigma_i}$ factors through $s_{\mathfrak{C}_i}$. Since $s_{\sigma_i} : Z \to S_{\sigma_i}$ coincides with $s_{\psi^*\tau_i} : Z \to S_{\psi^*\tau_i}$, it follows that $s_{\psi^*\tau}$ factors through $Z \xrightarrow{f} S \xrightarrow{q} S_{\psi^*\tau}$. $\square$

Since $T_i$ are all algebraic tori defined over number fields $k_i$, we apply Theorem 4.5 to conclude that there exists a family of reductive representations $\mathfrak{q}_i := \{\rho_{ij} : \pi_1(Z) \to T_i(K_{ij})\}_{j=1,\dots,n_i}$ with $K_{ij}$ non-archimedean local field such that

(1) For each $i = 1, \dots, k$; $j = 1, \dots, n_i$, $[\varrho_{ij}] \in \mathfrak{C}_i(K_{ij})$;

(2) The reduction map $s_{\mathfrak{q}_i} : Z \to S_{\mathfrak{q}_i}$ of $\tau_i$ coincides with $s_{\mathfrak{C}_i} : Z \to S_{\mathfrak{C}_i}$;

(3) for the canonical current $T_{\mathfrak{q}_i}$ over $S_{\mathfrak{C}_i}$ associated with $\mathfrak{q}_i$, $\{T_{\mathfrak{q}_i}\}$ is a Kähler class.

By the definition of $\mathfrak{C}_i$, there exist a finite extension $F_{ij}$ of $K_{ij}$ and reductive representations $\{\delta_{ij} : \pi_1(X) \to \mathrm{GL}_N(F_{ij})\}_{j=1,\dots,n_i}$ such that

(a) For each $i = 1, \dots, k$; $j = 1, \dots, n_i$, $[\delta_{ij}] \in \mathfrak{C}(F_{ij})$;

(b) the Zariski closure of $\psi^*\delta_{ij} : \pi_1(Z) \to \mathrm{GL}_N(F_{ij})$ is contained in $H_i$;

(c) $[\eta_{ij}] = [\varrho_{ij}] \in M_B(Z, T_i)(F_{ij})$, where $\eta_{ij} : \pi_1(Z) \to T_i(F_{ij})$ is the composition of $\psi^*\delta_{ij} : \pi_1(Z) \to H_i(F_{ij})$ with the group homomorphism $H_i \to T_i$.
Therefore, $\eta_{ij}$ is conjugate to $\varrho_{ij}$ and thus their reduction map coincides. It follows that the canonical currents $T_{\eta_{ij}}$ coincides with $T_{\varrho_{ij}}$. Let $R_i$ be the radical of $H_i$. Write $\eta'_{ij} : \pi_1(Z) \to (H_i/R_i)(F_{ij})$ to be the composition of $\psi^*\delta_{ij} : \pi_1(Z) \to H_i(F_{ij})$ with the homomorphism $H_i \to H_i/R_i$. Note that $H_i \to T_i \times H_i/R_i$ is an isogeny. It follows that the reduction map $s_{\psi^*\delta_{ij}}$ is the Stein factorization of $s_{\eta_{ij}} \times s_{\eta'_{ij}} : Z \to S_{\eta_{ij}} \times S_{\eta'_{ij}}$. Therefore, the reduction map $s_{\eta_{ij}} : Z \to S_{\eta_{ij}}$ factors through the reduction map $s_{\psi^*\delta_{ij}} : Z \to S_{\psi^*\delta_{ij}}$ with the finite morphism $q_{ij} : S_{\psi^*\delta_{ij}} \to S_{\eta_{ij}}$. Moreover, By Definition 1.23, one can see that

$$ (4.5) \qquad q_{ij}^* T_{\varrho_{ij}} = q_{ij}^* T_{\eta_{ij}} \leq T_{\psi^* \delta_{ij}}. $$

Consider the family of representations $\delta := \{\delta_{ij} : \pi_1(X) \to \text{GL}_N(F_{ij})\}_{i=1,...,k; j=1,...,n_i}$. By Items (2) and (c) the Stein factorization $f : Z \to S$ of $Z \to S_{\mathfrak{C}_1} \times \cdots \times S_{\mathfrak{C}_k}$ factors through the reduction map $s_{\psi^*\delta} : Z \to S_{\psi^*\delta}$. By Claim 4.24, $f : Z \to S$ coincides with $s_{\psi^*\delta} : Z \to S_{\psi^*\delta}$. Let $e_i : S \to S_{\mathfrak{C}_i} = S_{\varrho_i}$ be the natural map. Note that $e_1 \times \cdots \times e_k : S \to S_{\mathfrak{C}_1} \times \cdots \times S_{\mathfrak{C}_k}$ is finite. By Items (2) and (3), $\{\sum_{i=1}^k e_i^* T_{\varrho_i}\}$ is Kähler on $S = S_{\mathfrak{C}}$. By (4.5), we conclude that $\{T_{\psi^*\delta}\}$ is Kähler on $S_{\mathfrak{C}}$. According to Remark 4.16, it implies that the generic rank $m$ of the multivalued holomorphic 1-forms on $Z'$ induced by the differential of harmonic mappings associated with $\{\phi^*\delta_{ij} : \pi_1(Z') \to \text{GL}_N(F_{ij})\}_{i=1,...,k; j=1,...,n_i}$ is equal to $\dim Z$. This contradicts with our assumption that $m < \dim Z$. Hence Case (2) can neither happen. We prove Induction one.

**Step 4.** We now prove the theorem by another induction.

**Induction Two.** Assume that for every closed subvariety $\Sigma \subset S_{\mathfrak{C}}$ of dimension $\leq r-1$, one can add $\tau_{\ell+1}, \dots, \tau_p$ (depending on $\Sigma$) with $[\tau_i] \in \mathfrak{C}(L_i)$ for each $i = \ell+1, \dots, p$ such that for every closed subvariety $\Xi \subset \Sigma$, one has $\{T_{\tau}\}^{\dim \Xi} \cdot \Xi > 0$, where $\tau := \{\tau_i\}_{i=1, \dots, p}$.

Obviously, this induction is the same as Induction one for $\dim \Sigma = 1$ and thus it holds in this case. Let $\Sigma \subset S_{\mathfrak{C}}$ be a closed subvariety of dimension $r$. We shall prove that the induction holds for such $\Sigma$.

By Induction One, one can add reductive representations $\{\tau_i : \pi_1(X) \to \text{GL}_N(L_i)\}_{i=\ell+1, \dots, k}$ (depending on $\Sigma$) with $[\tau_i] \in \mathfrak{C}(L_i)$ for each $i = \ell+1, \dots, k$ such that $\{T_{\tau'}\}^{\dim \Sigma} \cdot \Sigma > 0$ for the new family $\tau' := \{\tau_i : \pi_1(X) \to T(L_i)\}_{i=1, \dots, k}$. We construct $\psi : Z \to X$ a diagram as (4.2). Then $\{T_{\psi^*\tau'}\}$ is a big class on $S_{\psi^*\tau'}$ by Lemma 1.24. We may replace $Z$ by a Galois cover which dominates the spectral covers of $\psi^*\tau'$. Let $V \subset H^0(Z, \Omega_Z^1)$ be the subspace generated by all spectral one forms induced by $\psi^*\tau'$. Note that there is a subspace $\mathbb{V} \subset H^0(S_{\psi^*\tau'}, \Omega_{S_{\psi^*\tau'}}^1)$ such that $s_{\psi^*\tau'}^*\mathbb{V} = V$. By Remark 4.16,

$$ \mathrm{Im} [\Lambda^{\dim S_{\psi^*\tau'}\mathbb{V}} \to H^0(S_{\psi^*\tau'}, \Omega_{S_{\psi^*\tau'}}^{\dim S_{\psi^*\tau'}})] \neq 0. $$

Pick some non-zero $\eta \in \mathrm{Im} [\Lambda^{\dim S_{\psi^*\tau'}\mathbb{V}} \to H^0(S_{\psi^*\tau'}, \Omega_{S_{\psi^*\tau'}}^{\dim S_{\psi^*\tau'}})]$. Let $Z_1, \dots, Z_c$ be all irreducible components of $(\eta = 0)$. Denote by $W_i := \sigma_{\psi}(Z_i)$. Since the image of $\sigma_{\psi} : S_{\psi^*\tau'} \to S_{\mathfrak{C}}$ is $\Sigma$, $W_i$'s are proper closed subvarieties of $\Sigma$. Let $\Xi$ be a proper closed subvariety in $\Sigma$ such that $\{T_{\tau}\}^{\dim \Xi} \cdot \Xi = 0$. We can take a closed proper subvariety $\Xi'$ such that $\sigma_{\psi}(\Sigma') = \Sigma$. Then by Lemma 1.24, $\{T_{\psi^*\tau'}\}^{\dim \Xi'} \cdot \Xi' = 0$. Therefore, $\Xi'$ must be contained in some $Z_i$. It follows that $\Xi$ is contained in some $W_i$.

Since $\dim W_i \leq r-1$, by Induction Two we can add more reductive representations $\tau_{k+1}, \dots, \tau_{k'}$ with $[\tau_i] \in \mathfrak{C}(L_i)$ for each $i = k+1, \dots, k'$ such that for the new family $\tau'' := \{\tau_i\}_{i=1, \dots, k'}$, one has $\{T_{\tau''}\}^{\dim \Xi} \cdot \Xi > 0$ for all closed subvarieties $\Xi$ contained in $\cup_{i=1}^c W_i$. Since $T_{\tau''} \geq T_{\tau'}$, it follows that $\{T_{\tau''}\}^{\dim \Xi} \cdot \Xi > 0$ for all closed subvarieties $\Xi$ contained in $\Sigma$. Induction Two is proved.

**Step 5.** We now apply Induction Two to $S_{\mathfrak{C}}$. Then we can add reductive representations $\tau_{\ell+1}, \dots, \tau_M$ with $[\tau_i] \in \mathfrak{C}(L_i)$ for each $i = \ell+1, \dots, M$ such that $\{T_{\tau'}\}^{\dim \Sigma} \cdot \Sigma > 0$ for every closed subvariety $Z$ of $S_{\mathfrak{C}}$, where $\tau' = \{\tau_i : \pi_1(X) \to \text{GL}_N(L_i)\}_{i=1, \dots, M}$. Hence $\{T_{\tau'}\}$ is Kähler by Theorem 1.12. We complete the proof of the theorem. $\square$
**4.6. Holomorphic convexity associated with absolutely constructible subsets.**

— In this subsection we will prove Theorem B. We shall use the notations and results proven in § 3.3 and Theorem 3.20 without recalling the details.

**Theorem 4.25.** — Let $X$ be a smooth projective variety. Let $\mathcal{C}$ be an absolutely constructible subset of $M_B(X, N)(\mathbb{C})$ defined in Definition 1.16. Assume that $\mathcal{C}$ is defined on $\mathbb{Q}$. Let $\pi : \tilde{X}_{\mathcal{C}} \to X$ be the covering corresponding to the group $\cap_{\varrho} \ker \varrho \subset \pi_1(X)$ where $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ ranges over all reductive representations such that $[\varrho] \in \mathcal{C}(\mathbb{C})$. Then $\tilde{X}_{\mathcal{C}}$ is holomorphically convex. In particular, if $\pi_1(X)$ is a subgroup of $\text{GL}_N(\mathbb{C})$ whose Zariski closure is reductive, then $\tilde{X}_{\mathcal{C}}$ is holomorphically convex.

*Proof.* — Let $H := \cap_{\varrho} \ker \varrho \cap \sigma$, where $\sigma$ is the $\mathbb{C}$-VHS defined in Proposition 3.12 and $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ ranges over all reductive representation such that $[\varrho] \in \mathcal{C}$. Denote by $\tilde{X}_H := \tilde{X}/H$. Let $\mathcal{D}$ be the period domain associated to the $\mathbb{C}$-VHS $\sigma$ defined in Proposition 3.12 and let $p : \tilde{X}_H \to \mathcal{D}$ be the period mapping. By (3.5), $H = \cap_{\varrho} \ker \varrho$, where $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ ranges over all reductive representation such that $[\varrho] \in \mathcal{C}$. Therefore, $\tilde{X}_{\mathcal{C}} = \tilde{X}_H$.

Consider the product

$$ \Psi = s_{\mathcal{C}} \circ \pi_H \times p : \tilde{X}_H \to S_{\mathcal{C}} \times \mathcal{D} $$

where $p : \tilde{X}_H \to \mathcal{D}$ is the period mapping of $\sigma$. Recall that $\Psi$ factors through a proper surjective fibration $\text{sh}_H : \tilde{X}_H \to \tilde{S}_H$. Moreover, there is a properly discontinuous action of $\pi_1(X)/H$ on $\tilde{S}_H$ such that $\text{sh}_H$ is equivariant with respect to this action. Write $g : \tilde{S}_H \to S_{\mathcal{C}} \times \mathcal{D}$ to be the induced holomorphic map. Denote by $\phi : \tilde{S}_H \to \mathcal{D}$ the composition of $g$ and the projection map $S_{\mathcal{C}} \times \mathcal{D} \to \mathcal{D}$. Since the period mapping $p$ is horizontal, and $\text{sh}_H$ is surjective, it follows that $\phi$ is also horizontal.

Recall that in Claim 3.27 we prove that there is a finite index normal subgroup $N$ of $\pi_1(X)/H$ and a homomorphism $\nu : N \to \text{Aut}(\tilde{S}_H)$ such that $\text{sh}_H : \tilde{X}_H \to \tilde{S}_H$ is $\nu$-equivariant and $\nu(N)$ acts on $\tilde{S}_H$ properly discontinuous and without fixed point. Let $Y := \tilde{X}_H/N$. Moreover, $c : Y \to X$ is a finite Galois étale cover and $N$ gives rise to a proper surjective fibration $Y \to \tilde{S}_H/\nu(N)$ between compact normal complex spaces. Write $W := \tilde{S}_H/\nu(N)$. Then $\tilde{S}_H \to W$ is a topological Galois unramified covering. Recall that the canonical bundle $K_{\mathcal{D}}$ on the period domain $\mathcal{D}$ can be endowed with a $G_0$-invariant smooth metric $h_{\mathcal{D}}$ whose curvature is strictly positive-definite in the horizontal direction. As $\phi : \tilde{S}_H \to \mathcal{D}$ is $\nu(N)$-equivariant, it follows that $\phi^*K_{\mathcal{D}}$ descends to a line bundle on the quotient $W := \tilde{S}_H/\nu(N)$, denoted by $L_G$. The smooth metric $h_{\mathcal{D}}$ induces a smooth metric on $L_G$ whose curvature form is denoted by $T$. Let $x \in W$ be a smooth point of $W$ and let $v \in T_{\tilde{S}_H, x}$. Then $|v|^2_\omega > 0$ if $d\phi(v) \neq 0$.

We fix a reference point $x_0$ on $\tilde{S}_H$. Define $\phi_0 := 2d_{\mathcal{D}}^2(\phi(x), \phi(x_0))$ where $d_{\mathcal{D}} : \mathcal{D} \times \mathcal{D} \to \mathbb{R}_{\ge 0}$ is the distance function on the period domain $\mathcal{D}$. By [Eys04, Theorem 3.3.2], we have

$$ (4.6) \qquad dd^c\phi_0 \ge \omega = q^*T. $$

where we $q : \tilde{S}_H \to \tilde{S}_H/\nu(N)$ denotes the quotient map.

We now apply Theorem 4.21 to find a family of representations $\tau := \{\tau_i : \pi_1(X) \to \text{GL}_N(K_i)\}_{i=1,...,m}$ where $K_i$ are non-archimedean local fields such that

— For each $i = 1, ..., m$, $[\tau_i] \in \mathcal{C}(K_i)$;

— The reduction map $s_{\tau} : X \to S_{\tau}$ of $\tau$ coincides with $s_{\mathcal{C}}$.

— For the canonical current $T_{\tau}$ defined over $S_{\mathcal{C}}$, $\{T_{\tau}\}$ is a Kähler class.
Consider

<figure><img src="image_14.png" /></figure>

Note that $p$ is a finite surjective morphism.

We fix a reference point $x_0$ on $\tilde{S}_H$. For each $i = 1, \dots, m$, let $u_i : \tilde{X}_H \to \Delta(GL_N)_{K_i}$ be the $\tau_i$-equivariant harmonic mapping from $\tilde{X}_H$ to the Bruhat-Tits building of $GL_N(K_i)$ whose existence was ensured by a theorem of Gromov-Schoen [GS92]. Then the function $\tilde{\phi}_i(x) := 2d_i^2(u_i(x), u_i(x_0))$ defined over $\tilde{X}_H$ is locally Lipschitz, where $d_i : \Delta(GL_N)_{K_i} \times \Delta(GL_N)_{K_i} \to \mathbb{R}_{\ge 0}$ is the distance function on the Bruhat-Tits building. By Proposition 1.25, it induces a continuous psh functions $\{\phi_i : \tilde{S}_H \to \mathbb{R}_{\ge 0}\}_{i=1,\dots,m}$ such that $dd^c\phi_i \ge r_i^*T_{\tau_i}$ for each $i$. By the definition of $T_\tau$, we have

$$ (4.7) \qquad dd^c \sum_{i=1}^{m} \phi_i \ge r_i^* T_\tau. $$

Therefore, putting (4.6) and (4.7) together we obtain

$$ (4.8) \qquad dd^c \sum_{i=0}^{m} \phi_i \ge q^*(f^*T_\tau + T). $$

As $f$ is a finite surjective morphism, $\{f^*T_\tau\}$ is also Kähler by Theorem 1.12.

By Claim 3.28, we know that $g : \tilde{S}_H \to S_\mathcal{C} \times \mathcal{D}$ has discrete fibers. Since $T$ is induced by the curvature form of $(K_\mathcal{D}, h_\mathcal{D})$, and $\phi : \tilde{S}_H \to \mathcal{D}$ is horizontal, we can prove that for every irreducible positive dimensional closed subvariety $Z$ of $W$, $f^*T_\tau + T$ is strictly positive at general smooth points of $Z$. Therefore,

$$ \{f^*T_\tau + T\}^{\dim Z} \cdot Z = \int_Z (f^*T_\tau + T)^{\dim Z} > 0. $$

Recall that $W$ is projective by the proof of Claim 3.29. We utilize Theorem 1.12 to conclude that $\{f^*T_\tau + T\}$ is Kähler.

Given that $\tilde{S}_H \to W$ represents a topological Galois unramified cover, we can apply Proposition 1.13 in conjunction with (4.8) to deduce that $\tilde{S}_H$ is a Stein manifold. Furthermore, since $\tilde{X}_H \to \tilde{S}_H$ is a proper surjective holomorphic fibration, the holomorphic convexity of $\tilde{X}_H$ follows from the Cartan-Remmert theorem. Ultimately, the theorem is established by noting that $\tilde{X}_H = \tilde{X}_\mathcal{C}$. $\square$

**4.7. Universal covering is Stein.** — We shall use the notations in the proof of Theorem 4.25 without recalling their definitions.

**Theorem 4.26.** — Let $X$ be a smooth projective variety. Consider an absolutely constructible subset $\mathcal{C}$ of $M_B(X, GL_N(\mathbb{C}))$ as defined in Definition 1.16. We further assume that $\mathcal{C}$ is defined over $\mathbb{Q}$. If $\mathcal{C}$ is considered to be large, meaning that for any closed subvariety $Z$ of $X$, there exists a reductive representation $\varrho : \pi_1(X) \to GL_N(\mathbb{C})$ such that $[\varrho] \in \mathcal{C}$ and $\varrho(\mathrm{Im}[\pi_1(Z^{\mathrm{norm}}) \to \pi_1(X)])$ is infinite, then all intermediate coverings between $\tilde{X}$ and $\tilde{X}_\mathcal{C}$ of $X$ are Stein manifolds.

*Proof.* — Note that $\mathrm{sh}_H : \tilde{X}_H \to \tilde{S}_H$ is a proper holomorphic surjective fibration.

*Claim 4.27.* — $\mathrm{sh}_H$ is biholomorphic.
Proof. — Assume that there exists a positive-dimensional compact subvariety Z of $\tilde{X}_H$ which is contained in some fiber of $\text{sh}_H$. Consider $W := \pi_H(Z)$ which is a compact positive-dimensional irreducible subvariety of X. Therefore, $\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(W^{\text{norm}})]$ is a finite index subgroup of $\pi_1(W^{\text{norm}})$. By the definition of $\tilde{X}_H$, for any reductive $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ with $[\varrho] \in \mathfrak{C}(\mathbb{C})$, we have $\varrho(\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(X)]) = \{1\}$. Therefore, $\varrho(\text{Im}[\pi_1(W^{\text{norm}}) \to \pi_1(X)]) = \{1\}$ is finite. This contradicts with out assumption that $\mathfrak{C}$ is large. Hence, $\text{sh}_H$ is a one-to-one proper holomorphic map of complex normal spaces. Consequently, it is biholomorphic. □

By the proof of Theorem 4.25, there exist

— a topological Galois unramified covering $q: \tilde{X}_H = \tilde{S}_H \to W$, where W is a projective normal variety;

— a positive (1, 1)-current with continuous potential $f^*T_\tau + T$ over W such that $\{f^*T_\tau + T\}$ is Kähler;

— a continuous semi-positive plurisubharmonic function $\sum_{i=0}^m \phi_i$ on $\tilde{S}_H$ such that we have

$$ (4.9) \qquad \mathrm{dd}^c \sum_{i=0}^{m} \phi_i \geq q^*(f^*T_\tau + T). $$

Let $p: \tilde{X}' \to \tilde{X}_H$ be the intermediate Galois covering of X between $\tilde{X} \to \tilde{X}_H$. By (4.6) we have

$$ (4.10) \qquad \mathrm{dd}^c \sum_{i=0}^{m} p^* \phi_i \geq (q \circ p)^* (f^* T_\tau + T). $$

We apply Proposition 1.13 to conclude that $\tilde{X}'$ is Stein. □

# Appendix A. Shafarevich conjecture for projective normal varieties

Ya Deng, Ludmil Katzarkov (a) & Katsutoshi Yamanoi

In this appendix, we aim to extend Theorems 4.25 and 4.26 to include singular normal varieties, and thus completing the proofs of Theorems B and C.

**A.1. Absolutely constructible subset (II).** — Let X be a projective normal variety. Following the recent work of Lerer [Ler22], we can also define absolutely constructible subsets in the character variety $M_B(X, N) := M_B(\pi_1(X), \text{GL}_N)$.

**Definition A.1.** — Let X be a normal projective variety, $\mu: Y \to X$ be a resolution of singularities, and $\iota: M_B(X, N) \hookrightarrow M_B(Y, N)$ be the embedding. A subset $\mathfrak{C} \subset M_B(X, N)(\mathbb{C})$ is called absolutely constructible if $\iota(\mathfrak{C})$ is an absolutely constructible subset of $M_B(Y, N)$ in the sense of Definition 1.16.

Note that the above definition does not depend on the choice of the resolution of singularities (cf. [Ler22, Lemma 2.7]). Moreover, we have the following result.

**Proposition A.2 ([Ler22, Proposition 2.8]).** — Let X be a normal projective variety. Then $M_B(X, N)$ is absolutely constructible in the sense of Definition A.1.

This result holds significant importance, as it provides a fundamental example of absolutely constructible subsets for projective normal varieties. It is worth noting that in [Ler22, Proposition 2.8], it is explicitly stated that $\iota(M_B(X, N))$ is $U(1)$-invariant, with $\iota$ defined in Definition A.1. However, it should be emphasized that the proof can be easily adapted to show $\mathbb{C}^*$-invariance, similar to the approach used in the proof of Proposition 3.32.

<footnote>a. E-mail: l.katzarkov@miami.edu, University of Miami, Coral Gables, FL, USA; Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria; NRU HSE, Moscow, Russia</footnote>
**A.2. Reductive Shafarevich conjecture for normal projective varieties.** —

**Theorem A.3.** — Let $Y$ be a projective normal variety. Let $\mathcal{C}$ be an absolutely con-
structible subset of $M_B(Y,N)(\mathbb{C})$, defined on $\mathbb{Q}$ (e.g. $\mathcal{C} = M_B(Y,N)$). Consider the cov-
ering $\pi : \tilde{Y}_{\mathcal{C}} \to Y$ corresponding to the subgroup $\cap_{\varrho} \ker \varrho$ of $\pi_1(Y)$, where $\varrho : \pi_1(Y) \to$
$\text{GL}_N(\mathbb{C})$ ranges over all reductive representations such that $[\varrho] \in \mathcal{C}$. Then the complex
space $\tilde{Y}_{\mathcal{C}}$ is holomorphically convex. In particular,

- The covering corresponding to the intersection of the kernels of all reductive representations of $\pi_1(Y)$ in $\text{GL}_N(\mathbb{C})$ is holomorphically convex;

- if $\pi_1(Y)$ is a subgroup of $\text{GL}_N(\mathbb{C})$ whose Zariski closure is reductive, then the universal covering of $Y$ is holomorphically convex.

*Proof*. — Let $\mu : X \to Y$ be any desingularization. Let $j : M_B(Y,N) \hookrightarrow M_B(X,N)$ the closed immersion induced by $\mu$, which is a morphism of affine $\mathbb{Q}$-schemes of finite type. Then by Definition A.1, $j(\mathcal{C})$ is an absolutely constructible in the sense of Definition 1.16. Since $\mathcal{C}$ is defined on $\mathbb{Q}$, so is $j(\mathcal{C})$. We shall use the notations in Theorem 3.20. Let $\tilde{X}_H$ be the covering associated with the subgroup $H := \cap_{\varrho} \ker \varrho$ of $\pi_1(X)$ where $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ ranges over all reductive representations such that $[\varrho] \in j(\mathcal{C})(\mathbb{C})$. In other words, $H := \cap_{\tau} \ker \mu^*\tau$ where $\tau : \pi_1(Y) \to \text{GL}_N(\mathbb{C})$ ranges over all reductive representations such that $[\tau] \in \mathcal{C}(\mathbb{C})$. Denote by $H_0 := \cap_{\tau} \ker \tau$ where $\tau : \pi_1(Y) \to \text{GL}_N(\mathbb{C})$ ranges over all reductive representations such that $[\varrho] \in \mathcal{C}(\mathbb{C})$. Therefore, $H = (\mu_*)^{-1}(H_0)$, where $\mu_* : \pi_1(X) \to \pi_1(Y)$ is a surjective homeomorphism as $Y$ is normal. Therefore, the natural homeomorphism $\pi_1(X)/H \to \pi_1(Y)/H_0$ is an isomorphism. Then $\tilde{X}_{\mathcal{C}} = \tilde{X}/H$ and $\tilde{Y}_H := \tilde{Y}/H_0$ where $\tilde{X}$ (resp. $\tilde{Y}$) is the universal covering of $X$ (resp. $X$). It induces a lift $p : \tilde{X}_H \to \tilde{Y}_{\mathcal{C}}$ such that

<figure><img src="image_15.png" /></figure>

**Claim A.4.** — $p : \tilde{X}_H \to \tilde{Y}_{\mathcal{C}}$ is a proper surjective holomorphic fibration with connected fibers.

*Proof.* — Note that Aut($\tilde{X}_H/X$) = $\pi_1(X)/H \simeq \pi_1(Y)/H_0$ = Aut($\tilde{Y}_{\mathcal{C}}/Y$). Therefore, $\tilde{X}_H$ is the base change $\tilde{Y}_{\mathcal{C}} \times_Y X$. Note that each fiber of $\mu$ is connected as $Y$ is normal. It follows that each fiber of $p$ is connected. The claim is proved. □

By Theorem 3.20, we know that there exist a proper surjective holomorphic fibration
sh<sub>H</sub> : X̃<sub>H</sub> → S̃<sub>H</sub> such that S̃<sub>H</sub> is a Stein space. Therefore, for each connected compact
subvariety Z ⊂ X̃<sub>H</sub>, sh<sub>H</sub>(Z) is a point. By Claim A.4, it follows that each fiber of p is
compact and connected, and thus is contracted by sh<sub>H</sub>. Therefore, sh<sub>H</sub> factors through a
proper surjective fibration f : X̃<sub>ε</sub> → S̃<sub>H</sub>:

<figure><img src="image_16.png" /></figure>

Therefore, $f$ is a proper surjective holomorphic fibration over a Stein space. By the Cartan-Remmert theorem, $\tilde{Y}_\mathcal{C}$ is holomorphically convex.

If we define $\mathcal{C}$ as $M_B(Y,N)$, then according to Proposition A.2, $\mathcal{C}$ is also absolutely
constructible. As a result, the last two claims can be deduced. Thus, the theorem is
proven. $\square$
**Theorem A.5.** — Let $Y$ be a projective normal variety. Let $\mathcal{C}$ be an absolutely constructible subset of $M_B(Y, N)(\mathbb{C})$, defined on $\mathbb{Q}$ (e.g. $\mathcal{C} = M_B(Y, N)$). Let $\mathcal{C}(\mathbb{C})$ be large in the sense that for any closed positive dimensional subvariety $Z$ of $Y$, there exists a reductive representation $\varrho: \pi_1(Y) \to \text{GL}_N(\mathbb{C})$ such that $[\varrho] \in \mathcal{C}(\mathbb{C})$ and $\varrho(\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(Y)])$ is infinite. Then all intermediate Galois coverings of $Y$ between $\tilde{Y}$ and $\tilde{Y}_{\mathcal{C}}$ are Stein spaces. Here $\tilde{Y}$ denotes the universal covering of $Y$.

*Proof.* — Let $\mu: X \to Y$ be any desingularization. In the following, we will use the same notations as in the proof of Theorem A.3 without explicitly recalling their definitions. Recall that we have constructed three proper surjective holomorphic fibrations $p, f$, and $\text{sh}_H$ satisfying the following commutative diagram:

<figure><img src="image_17.png" /></figure>

**Claim A.6.** — $f: \tilde{Y}_{\mathcal{C}} \to \tilde{S}_H$ is a biholomorphism.

The proof follows a similar argument to that of Claim 4.27. For the sake of completeness, we will provide it here.

*Proof of Claim A.6.* — As each fibers of $f$ is compact and connected, it suffices to prove that there are no compact positive dimensional subvarieties $Z$ of $\tilde{Y}_{\mathcal{C}}$ such that $f(Z)$ is a point. Let us assume, for the sake of contradiction, that such a $Z$ exists. Consider $W := \pi(Z)$ which is a compact positive-dimensional irreducible subvariety of $Y$. Therefore, $\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(W^{\text{norm}})]$ is a finite index subgroup of $\pi_1(W^{\text{norm}})$. By the definition of $\tilde{Y}_{\mathcal{C}}$, for any reductive $\varrho: \pi_1(Y) \to \text{GL}_N(\mathbb{C})$ with $[\varrho] \in \mathcal{C}(\mathbb{C})$, we have $\varrho(\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(X)]) = \{1\}$. Therefore, $\varrho(\text{Im}[\pi_1(W^{\text{norm}}) \to \pi_1(Y)])$ is finite. This contradicts with out assumption that $\mathcal{C}$ is large. Hence, $f$ is a one-to-one proper holomorphic map of complex normal spaces. Consequently, it is biholomorphic. $\square$

The rest of the proof is same as in Theorem 4.26. By the proof of Theorem 4.25, there exist

- a topological Galois unramified covering $q: \tilde{S}_H \to W$, where $W$ is a projective normal variety;

- a positive closed (1,1)-current with continuous potential $T_0$ over $W$ such that $\{T_0\}$ is Kähler;

- a continuous semi-positive plurisubharmonic function $\phi$ on $\tilde{S}_H$ such that we have

$$ (A.1) \qquad \mathrm{dd}^c \phi \geq q^*T_0. $$

By Claim A.6, $\tilde{Y}_{\mathcal{C}}$ can be identified with $\tilde{S}_H$. Let $p: \tilde{Y}' \to \tilde{Y}_{\mathcal{C}}$ be the intermediate Galois covering of $Y$ between $\tilde{Y}' \to \tilde{Y}_{\mathcal{C}}$. By (A.1) we have

$$ (A.2) \qquad \mathrm{dd}^c p^* \phi \geq (q \circ p)^* T_0. $$

We apply Proposition 1.13 to conclude that $\tilde{Y}'$ is Stein. $\square$

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Friday 2nd June, 2023

YA DENG • E-mail: ya.deng@math.cnrs.fr, CNRS, Institut Élie Cartan de Lorraine, Université de Lorraine, F-54000 Nancy, France. • Url: https://ydeng.perso.math.cnrs.fr

KATSUTOSHI YAMANOI • E-mail: yamanoi@math.sci.osaka-u.ac.jp, Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Url: https://sites.google.com/site/yamanoimath/