Proof. Let $ \phi \colon X \dashrightarrow Y $ be an almost holomorphic dominant rational map, and let $ \bar {\phi } \colon \Gamma \to Y $ be a resolution of its indeterminacy with a bimeromorphic morphism $ \tau \colon \Gamma \to X $ , as shown in the following diagram:

Set $ m := \dim Y> 0 $ . In [Reference MatsumuraMat22a, Theorem 3.1], we showed that if the canonical bundle $ K_Y $ is pseudo-effective, then the invertible sheaf

$$\begin{align*}L := \big( \tau_* (\mathcal{O}_\Gamma(\bar{\phi}^* K_Y)) \big)^{**} \end{align*}$$

defined by the reflexive hull (the double dual) is a Hermitian flat line bundle (equivalently, $c_{1}(L)=0$ ; see [Reference MatsumuraMat22a, page 767]). This is a crucial step and follows from [Reference MatsumuraMat22a, Claim 3.5], which is proved under the assumption that the holomorphic sectional curvature is semi-positive. This implies that the Kodaira dimension of $ \mathcal {O}_\Gamma (\bar {\phi }^* K_Y) $ cannot be greater than or equal to one.

However, the definition of special type requires us to consider the saturation

$$\begin{align*}\mathcal{O}_\Gamma(\bar{\phi}^* K_Y)_{{\mathrm{sat}}} \subset \Omega_\Gamma^m \end{align*}$$

of the pull-back $\bar {\phi }^* K_Y$ in the m-th exterior power $\Omega _\Gamma ^m$ . By essentially the same argument, if the above saturated sheaf is pseudo-effective, then the invertible sheaf

$$\begin{align*}M := \big( \tau_*\big( (\mathcal{O}_\Gamma(\bar{\phi}^* K_Y)_{{\mathrm{sat}}})\big) \big)^{**} \end{align*}$$

is also a Hermitian flat line bundle. Consequently, the Kodaira dimension of $ \mathcal {O}_\Gamma (\bar {\phi }^* K_Y)_{{\mathrm {sat}}} $ cannot be greater than or equal to one. Therefore, the manifold X does not admit any almost holomorphic dominant rational map $\phi \colon X \dashrightarrow Y$ of general type.

We briefly explain how to deduce the Hermitian flatness using the argument of [Reference MatsumuraMat22a]. In proving the above conclusion for L, we used only the properties that L is a pseudo-effective invertible subsheaf of $ \Omega ^m_X $ , and that L coincides with the pull-back $ \phi ^* K_Y $ on $X_1$ , where $ X_1 $ is the preimage of a Zariski open set $ Y_1 \subset Y $ such that

$$ \begin{align*}\phi \colon X_1:=\phi^{-1}(Y_{1}) \to Y_1\end{align*} $$

is a smooth fibration. These properties were used in [Reference MatsumuraMat22a, Claim 3.5] via [Reference MatsumuraMat22a, (3.1)]. Although [Reference MatsumuraMat22a, (3.1)] is formulated on $ X_0 $ (the locus where $ \phi $ is holomorphic), it suffices for it to hold on $ X_1 $ .

The invertible sheaf M satisfies the first property if the saturation $ \mathcal {O}_\Gamma (\bar {\phi }^* K_Y)_{{\mathrm {sat}}} $ is pseudo-effective. Furthermore, we have $M = \phi ^* K_Y$ on $X_1$ , since

$$ \begin{align*}\mathcal{O}_\Gamma(\bar{\phi}^* K_Y) \subset \Omega_\Gamma^m \end{align*} $$

is a subline bundle over the smooth locus $Y_1$ of $\phi \colon X \to Y$ without taking saturation and $\tau \colon \Gamma \to X$ is an isomorphism over $Y_{1}$ . Therefore, the same argument applied to M shows that $ M = (\tau _* ((\mathcal {O}_\Gamma (\bar {\phi }^* K_Y)_{{\mathrm {sat}}})))^{**} $ is a Hermitian flat line bundle, just like L.

We will apply the conclusion mentioned above to show that $ X $ is of special type. By the definition of varieties of special type, we can take an almost holomorphic dominant rational map $ \phi \colon X \dashrightarrow Y $ and a bimeromorphic morphism $ \tau \colon \Gamma \to X $ admitting a fibration $ \bar {\phi } \colon \Gamma \to Y $ of general type. According to the definition of fibrations of general type, the Kodaira dimension $ \kappa (\mathcal {O}_{\Gamma }(\bar {\phi }^* K_{Y})_{{\mathrm {sat}}}) $ is equal to $ \dim Y> 0 $ . In particular, the line bundle $ M $ is pseudo-effective. The conclusion mentioned above shows that $ M $ is a Hermitian flat line bundle. This contradicts the fact that $ \kappa (\mathcal {O}_{\Gamma }(\bar {\phi }^* K_{Y})_{{\mathrm {sat}}}) $ is greater than or equal to one.

Proof. Conclusion (1) is a direct consequence of [Reference CampanaCam04, Theorem 7.8] and Theorem 2.2.

Conclusion (2) has been implicitly proved in [Reference MatsumuraMat22a, Theorem 1.6]. For the reader’s convenience, we will specify the relevant parts of the proof. The Albanese map $ X \to Y := \mathrm {Alb}(X) $ is a fibration by [Reference CampanaCam04, Theorem 7.4]. Thus, by [Reference MatsumuraMat22a, Theorem 1.6], there exists a holomorphic orthogonal splitting of the tangent bundle $T_X$ :

(2.1) $$ \begin{align} T_X \cong T_{X/Y} \oplus \phi^{*}T_Y. \end{align} $$

The subbundle $ \phi ^{*}T_Y \subset T_X $ is an integrable foliation (see [Reference MatsumuraMat22a, page 772]), indicating that $ X \to \mathrm {Alb}(X) $ is a locally constant fibration (see, for example, [Reference MatsumuraMat22a, Lemma 2.5]). Moreover, any tangent vectors in $ \phi ^{*}T_Y \subset T_X $ are truly flat (see [Reference MatsumuraMat22a, page 772]). Thus, by the holomorphic splitting and the Gauss-Codazzi-type formula (see [Reference MatsumuraMat22a, (2.2)]), the induced metric $ g_{T_{X/Y}} $ on $ T_{X/Y} $ satisfies

$$ \begin{align*} \big \langle \sqrt{-1}\Theta_g(e, \bar{e})(e), e \big \rangle_g &= \big \langle \sqrt{-1}\Theta_{g_{T_{X/Y}}}(e, \bar{e})(e), e \big \rangle_{g_{T_{X/Y}}} \end{align*} $$

for any tangent vector $ e \in T_{X/Y} \subset T_X $ . Here, $\sqrt {-1}\Theta _h$ denotes the Chern curvature of a Hermitian vector bundle $(E, h)$ :

$$ \begin{align*}\sqrt{-1}\Theta_h := \sqrt{-1}\Theta_h(E) \in C^{\infty}(X, \Lambda^{1,1} \otimes \mathrm{End}(E)), \end{align*} $$

and $\langle \bullet , \bullet \rangle _h$ is the Hermitian inner product with respect to h. See [Reference MatsumuraMat22a, Section 2] for the precise notation and definition of truly flat tangent vectors.

The left-hand side equals the holomorphic sectional curvature $H_g([e])$ evaluated at $ e \in T_{X/Y} \subset T_X $ , which is semi-positive by assumption. This shows that the holomorphic sectional curvature of the induced metric $ g_F := g_{T_{X/Y}}|_F $ on $ T_F \cong T_{X/Y}|_F $ is semi-positive.

Proof. By [Reference Zhang and ZhangZZ, Theorem 1.4], there exists a holomorphic orthogonal decomposition $T_{X} \cong V \oplus W$ such that V is a nonzero vector bundle and any tangent vectors in $V \subset T_{X}$ are truly flat. In the same way as in the proof of Corollary 2.4, due to the holomorphic splitting, the induced metric $ g_{V} $ on $ V $ satisfies

$$ \begin{align*} \Big{\langle} \sqrt{-1}\Theta_{g}(v, \bar{v})(f), f \Big{\rangle}_g &= \Big{\langle}\sqrt{-1}\Theta_{g_{V}}(v, \bar{v})(f), f \Big{\rangle}_{g_{V}} \end{align*} $$

for any tangent vector $ f \in V \subset T_{X} $ and any tangent vector $v \in T_{X}$ . We can see that the left-hand side equals zero since $f \in V \subset T_{X}$ is a truly flat tangent vector. Hence, the induced metric on V is Hermitian flat, and thus V is a flat vector bundle.

Proof. For any finite étale cover $\nu \colon X' \to X$ , the Kähler metric $\nu ^{*} g$ defined by the pull-back has semi-positive holomorphic sectional curvature. Hence, the Albanese map $\phi \colon X' \to Y' := \mathrm {Alb}(X')$ is a locally constant fibration (in particular, surjective) by Corollary 2.4. This shows that

$$ \begin{align*}q(X') = \dim Y' \leq \dim X' = \dim X. \end{align*} $$

This argument proves Conclusion (1).

We now consider Conclusion (2). Once we know that X is a projective manifold, its rational connectedness is easily verified. Indeed, since Theorem 1.1 has been proved for projective manifolds, there exists an MRC fibration $X \to Y$ with the properties in Theorem 1.1. Then, we can easily see that $\dim Y \leq \hat {q}(X)$ . Hence, by the assumption $\hat {q}(X) = 0$ , we conclude that Y is a single point. This means that X itself is the fiber of the MRC fibration (i.e., it is rationally connected).

Suppose now that X is not projective. Then, there exists a nonzero holomorphic $2$ -form $\eta $ on X. Otherwise, the ample cone would coincide with the Kähler cone, which is nonempty, implying that X is projective (see, for example, [Reference HuybrechtsHuy05, Proposition 3.3.2 and Corollary 5.3.3]).

The strategy of the proof is to exploit the nonprojectivity of X to construct a foliation related to flatness. Similar ideas have appeared previously in the literature (see, for example, [Reference BrunellaBru06, Reference Höring and PeternellHP15, Reference Matsumura and QingMQ, Reference WuWu]). In our case, by applying Theorem 2.5, we obtain a splitting of the tangent sheaf, which yields the desired conclusion.

We consider the reflexive hull

$$ \begin{align*}\mathcal{F} := (\langle \eta \rangle_{\mathcal{O}_{X}})^{**} \end{align*} $$

of the sheaf $\langle \eta \rangle _{\mathcal {O}_{X}}$ generated by $\eta $ . This sheaf $\mathcal {F} \subset \Omega _{X}^{2}$ is an effective invertible subsheaf. Thus, by Theorem 2.5, the tangent bundle $T_{X}$ admits a holomorphic orthogonal decomposition $T_{X} \cong V \oplus W$ such that V is a nonzero flat vector bundle. Let

$$ \begin{align*}\rho \colon \pi_{1}(X) \to \mathrm{GL}(r\mathord{:}\,\mathbb{C})\end{align*} $$

be a $\mathrm {GL}$ -representation that determines the flat vector bundle V, where $r := \mathrm {rank} V> 0$ .

We will show that we may assume that $\mathrm {Im}(\rho) $ is an abelian group. By Corollary 2.4, there exists an abelian normal subgroup $A \subset \mathrm {Im}(\rho) $ of finite index. We consider the étale cover $X' \to X$ corresponding to the kernel of the composition

$$ \begin{align*}\pi_{1}(X) \to \mathrm{Im}(\rho) \to \mathrm{Im}(\rho) / A.\end{align*} $$

The cover $\nu \colon X' \to X$ is a finite étale cover since $\mathrm {Im}(\rho) / A$ is a finite group. Note that

$$ \begin{align*}0 \leq \hat{q}(X') \leq \hat{q}(X) = 0\end{align*} $$

holds by assumption. We now consider the induced $\mathrm {GL}$ -representation

$$ \begin{align*}\rho' \colon \pi_{1}(X') \to \pi_{1}(X) \xrightarrow{\rho} \mathrm{GL}(r\mathord{:}\,\mathbb{C}). \end{align*} $$

By construction, the image $\mathrm {Im}(\rho') $ is contained in A (in particular, it is an abelian group). Furthermore, we still have the holomorphic orthogonal splitting

(2.2) $$ \begin{align} T_{X'} = \nu^{*}T_{X} \cong \nu^{*}V \oplus \nu^{*}W, \end{align} $$

and the $\mathrm {GL}$ -representation $\rho ' \colon \pi _{1}(X') \to \mathrm {GL}(r\mathord {:}\,\mathbb {C})$ determines the flat subbundle

$$ \begin{align*}\nu^{*}V \subset \nu^{*}T_{X} = T_{X'}.\end{align*} $$

Note that if $X'$ is projective, so is X. Therefore, after replacing X with the finite étale cover $X'$ , we may assume that the image $\mathrm {Im}(\rho) $ is an abelian group.

The first homology group $H_{1}(X, \mathbb {Z})$ can be regarded as the abelianization of $\pi _{1}(X)$ . Since $\dim H^{1}(X, \mathbb {C}) = 0$ holds by the assumption $q(X) = 0$ and the Kählerness of X, the first homology group $H_{1}(X, \mathbb {Z})$ is a torsion group (in particular, it is a finite group). Since $\mathrm {Im}(\rho) $ is an abelian group, the universality of abelianizations shows that $\rho \colon \pi _{1}(X) \to \mathrm {GL}(r\mathord {:}\,\mathbb {C})$ factors through

$$ \begin{align*}\rho \colon \pi_{1}(X) \to H_{1}(X, \mathbb{Z}) \to \mathrm{GL}(r\mathord{:}\,\mathbb{C}).\end{align*} $$

This implies that $\mathrm {Im}(\rho) $ is a finite group, which means that V is étale trivializable (i.e., there exists a finite étale cover $\nu \colon X' \to X$ such that the pull-back $\nu ^{*}V $ is a trivial vector bundle on $X'$ ). Thus, by noting the splitting as in (2.2), we can see that the cover $X'$ admits a nonzero $1$ -form by

$$ \begin{align*}\mathcal{O}_{X'}^{\oplus r} \cong (\nu^{*}V)^{*} \subset \Omega^{1}_{X'}.\end{align*} $$

This contradicts the assumption $\hat {q}(X) = 0$ .

Proof. Note that Z is normal by Zariski’s main theorem. Let z be a smooth point of Z, and take a point $x \in X$ such that $z = g(x)$ . We consider the differential maps $df$ , $dg$ , and $d\sigma $ defined on the tangent bundles around x and $z = g(x)$ . Since $f \colon X \to Y$ is a smooth morphism, the differential map $df_{x} = d\sigma _{z} \circ dg_{x}$ is surjective. In particular, the differential map $d\sigma _{z}$ is surjective, which shows that $\sigma \colon Z \to Y$ is a finite étale cover on the smooth locus $Z_{{\mathrm {reg}}}$ . The purity of the branched locus shows that $\sigma \colon Z \to Y$ is étale by $\mathrm {codim} Z_{{\mathrm {sing}}}\geq 2$ . Therefore, since Y is smooth, the variety Z is also smooth.

Proof of Theorem 1.1.

We first check that it is sufficient to prove both Conclusions (a) and (b) after replacing X with a finite étale cover $\nu \colon X' \to X$ . Indeed, this is obvious for Conclusion (a). Following the argument in [Reference Cao and HöringCH19], we check this for Conclusion (b). Suppose that there exists a finite étale cover $\nu \colon X' \to X$ such that $X'$ admits a locally constant fibration $\phi ' \colon X' \to Y'$ with the properties listed in Theorem 1.1. Let $\mathcal {F} \subset T_{X}$ be the (unique) saturated integrable subsheaf corresponding to an MRC fibration $X \dashrightarrow Y$ of X, so that a general $\mathcal {F}$ -leaf is a fiber of the MRC fibration $X \dashrightarrow Y$ . For the reader’s convenience, we briefly explain that the subsheaf $\mathcal {F} \subset T_{X}$ is independent of the choice of an MRC fibration $X \dashrightarrow Y$ . Let $\mathcal {F}$ and $\mathcal {F}'$ be the foliations induced by MRC fibrations $X \dashrightarrow Y$ and $X \dashrightarrow Y'$ , respectively. After replacing $Y'$ with a bimeromorphic model, we may assume that there exists a bimeromorphic morphism $Y' \to Y$ , which yields a morphism $\mathcal {F}' \to \mathcal {F}$ that is an isomorphism on a Zariski open subset of X. Consequently, the induced morphism $T_X/\mathcal {F}' \to T_X/\mathcal {F}$ has a torsion kernel. Since $T_X/\mathcal {F}'$ is torsion-free, this morphism is an isomorphism, and hence $\mathcal {F}' \to \mathcal {F}$ is an isomorphism.

The pull-back $\nu ^{*}\mathcal {F} \subset T_{X'}$ corresponds to the locally constant MRC fibration $X' \to Y'$ of $X'$ . This implies that $\nu ^{*}\mathcal {F}$ is a locally free sheaf, and thus so is $\mathcal {F}$ . Hence, by [Reference HöringHor07, Corollary 2.11], there exists a smooth RC (rationally connected) fibration $\phi \colon X \to Y$ such that $T_{X/Y} = \mathcal {F}$ . Once we know that $K_{Y}$ is pseudo-effective, we can conclude that $X \to Y$ is the desired fibration by Theorem 2.1.

To check that $K_{Y}$ is pseudo-effective, we consider the Stein factorization $X' \to Z \to Y$ of ${X' \to X \to Y}$ (see the diagram below). Note that Z is smooth and $Z \to Y$ is a finite étale cover by Lemma 3.1.

We now check that $g \colon X' \to Z$ is an RC fibration (i.e., a general fiber is rationally connected). Let $z \in Z$ be a general point. The fiber $X_{y} := \phi ^{-1}(y)$ at $y := \sigma (z)$ is rationally connected, since $y = \sigma (z)$ is also a general point in Y. Since $\sigma \colon Z \to Y$ is étale, we obtain the decomposition into connected components:

$$ \begin{align*}\amalg_{w \in \sigma^{-1}(y)} X^{\prime}_{w} = g^{-1}(\sigma^{-1}(y)) = \nu^{-1}(\phi^{-1}(y)) = \nu^{-1}(X_{y}), \end{align*} $$

where $X^{\prime }_{w}:=g^{-1}(w)$ is the fiber of $g \colon X' \to Z$ at w. This induces the étale cover $\nu |_{X^{\prime }_{w}} \colon X^{\prime }_{w} \to X_{y}$ . The fiber $X_{y}=\phi ^{-1}(y)$ is rationally connected and, in particular, simply connected. Therefore, the étale cover $\nu |_{X^{\prime }_{w}} \colon X^{\prime }_{w} \to X_{y}$ is an isomorphism. In particular, $X^{\prime }_{w}$ is rationally connected (i.e., $g \colon X' \to Z$ is an RC fibration).

By the universality of MRC fibrations (see [Reference CampanaCam92, Reference Kollár, Miyaoka and MoriKoMM92] for details), there exists a dominant rational map $Z \dashrightarrow Y'$ . Since $Y'$ is a finite étale quotient of a torus (in particular, it has no rational curves), this map $Z \dashrightarrow Y'$ is actually an everywhere-defined fibration by [Reference FujikiFuj80, Proposition 1] (see also [Reference Hacon and McKernanHM07, Corollary 1.7]), and thus, it is a bimeromorphic morphism by $\dim Y' = \dim Z$ . By comparing the canonical bundles, we can see that $K_{Z}$ is pseudo-effective since $Y'$ is smooth and $K_{Y'}$ is Hermitian flat (in fact, torsion). This shows that $K_{Y}$ is also a pseudo-effective line bundle since $Z \to Y$ is an étale cover. Thus, the fibration $X \to Y$ is the desired fibration by Theorem 2.1. This completes the proof of the reduction step showing that we may replace X with a finite étale cover.

We now prove all the properties of Conclusion (b) after replacing X with a suitable finite étale cover. Let $\alpha \colon X \to Y := \mathrm {Alb}(X)$ be the Albanese map. By replacing X with a finite étale cover, we may assume that

$$ \begin{align*}\hat{q}(X) = q(X) = \dim Y \end{align*} $$

by Theorem 2.6. Furthermore, as an additional reduction, we may assume that the fiber F of the Albanese map $\alpha \colon X \to Y = \mathrm {Alb}(X)$ satisfies

$$ \begin{align*}\hat{q}(F) = q(F) \end{align*} $$

by replacing X with a finite étale cover again. We will explain why this reduction is possible. Note that all the fibers are isomorphic to each other since $\alpha \colon X \to Y = \mathrm {Alb}(X)$ is a locally constant fibration. The fiber F also admits a Kähler metric with semi-positive holomorphic sectional curvature by Corollary 2.4, and thus there exists a finite étale cover $F' \to F$ such that $q(F') = \hat {q}(F)$ by Theorem 2.6. It is sufficient for this reduction to find a finite étale cover $F" \to F'$ (whose irregularity still attains the augmented irregularity of F) such that it is induced by some finite étale cover $X" \to X$ of X.

By the definition of locally constant fibrations (see [Reference Matsumura and WangMW25a, Definition 2.3]), there exists a representation

$$\begin{align*}\tau\colon\pi_1(Y)\longrightarrow\mathrm{Aut}(F) \text{ such that } X\cong({Y}_{\mathrm{univ}}\times F)/\pi_1(Y), \end{align*}$$

where $\gamma \in \pi _1(Y)$ acts diagonally by

$$\begin{align*}(y,z)\longmapsto \big(\gamma\cdot y,\tau(\gamma)(z)\big). \end{align*}$$

We will construct the desired cover using the fundamental groups of this fibration. Since Y is a torus, we have $\pi _2(Y)=0$ . Hence, the homotopy exact sequence gives

$$\begin{align*}1\longrightarrow\pi_1(F) \longrightarrow\pi_1(X) \longrightarrow\pi_1(Y) \longrightarrow1. \end{align*}$$

After fixing compatible base points in F and $F'$ , let $K\subset \pi _1(F)$ be a subgroup of finite index corresponding to $F'\to F$ , and define the core H of $K \subset \pi _1(X)$ by

$$\begin{align*}H:=\bigcap_{g\in\pi_1(X)}gKg^{-1}. \end{align*}$$

Since $\pi _1(F)$ is normal in $\pi _1(X)$ , every $gKg^{-1}$ is contained in $\pi _1(F)$ and has the same index in $\pi _1(F)$ . Moreover, a finitely generated group has only finitely many subgroups of a fixed finite index (see [Reference RoseRos78, Corollary 4.16]). Hence, we can deduce that

$$\begin{align*}[\pi_1(F):H]<\infty,\qquad H\triangleleft\pi_1(X),\qquad H\subset K. \end{align*}$$

Since $H\subset K \subset \pi _{1}(F)$ , we obtain the finite étale covers

$$ \begin{align*}F":={F}_{\mathrm{univ}}/H\longrightarrow F' ={F}_{\mathrm{univ}}/K \longrightarrow F={F}_{\mathrm{univ}}/\pi_{1}(F).\end{align*} $$

This shows that

$$ \begin{align*}q(F")\geq q(F')=\hat q(F).\end{align*} $$

The converse inequality follows from the definition of the augmented irregularity, and hence we obtain $q(F")=\hat q(F)$ .

We now construct a finite étale cover $X" \to X$ of X inducing $F"\to F$ on the fibers. The group $\pi _1(F)/H$ is finite and satisfies

$$\begin{align*}\frac{\pi_1(X)/H}{\pi_1(F)/H}\cong\pi_1(Y). \end{align*}$$

Since $\pi _1(Y)$ is an abelian group, the commutator subgroup of $\pi _1(X)/H$ is finite. Since $\pi _1(X)/H$ is finitely generated, its center has finite index. Indeed, the centralizer of each element has finite index, and it is enough to intersect the centralizers of a finite set of generators. Thus, the center contains a torsion-free subgroup

$$\begin{align*}A\subset Z\big(\pi_1(X)/H\big) \end{align*}$$

of finite index in $\pi _1(X)/H$ . Since $\pi _1(F)/H$ is finite, we see that

$$\begin{align*}A\cap\big(\pi_1(F)/H\big)=\{1\}. \end{align*}$$

Let $G_X\subset \pi _1(X)$ be the inverse image of A under the quotient map $\pi _1(X)\to \pi _1(X)/H$ , and let $G_Y\subset \pi _1(Y)$ be its image. Then, we obtain the following commutative diagram of exact sequences:

All three vertical inclusions have finite index. Indeed, we have already proved that $ [\pi _1(F):H]<\infty $ . Moreover, since $G_X$ is the inverse image of A, we have $ [\pi _1(X):G_X] = [\pi _1(X)/H:A] <\infty $ . Finally, since $G_Y$ is the image of $G_X$ under the surjection $\pi _1(X)\to \pi _1(Y)$ , we have $ [\pi _1(Y):G_Y] \leq [\pi _1(X):G_X] <\infty $ .

Let

$$\begin{align*}\nu\colon X"\longrightarrow X, \qquad \mu\colon Y"\longrightarrow Y \end{align*}$$

be the finite étale covers corresponding to $G_X$ and $G_Y$ , respectively. Let $p_F\colon {F}_{\mathrm {univ}}\to F$ be the universal covering map. The universal covering map of X factors as

$$\begin{align*}{X}_{\mathrm{univ}}\cong{Y}_{\mathrm{univ}}\times{F}_{\mathrm{univ}} \xrightarrow{\ {\mathrm{id}}_{{Y}_{\mathrm{univ}}}\times p_F\ } {Y}_{\mathrm{univ}}\times F \xrightarrow{ \quad \alpha \quad } X={Y}_{\mathrm{univ}}\times F/\pi_{1}(Y). \end{align*}$$

Consequently, if $g\in G_X$ maps to $\gamma \in G_Y$ , then, by the lifting property, g acts as

$$\begin{align*}g(y,z)= \big(\gamma\cdot y,\widetilde{\tau}_g(z)\big), \end{align*}$$

where $\widetilde {\tau }_g\in \mathrm {Aut}({F}_{\mathrm {univ}})$ satisfies

$$\begin{align*}p_F\circ\widetilde{\tau}_g =\tau(\gamma)\circ p_F. \end{align*}$$

For every $h\in H$ , the automorphism $\widetilde {\tau }_g h\widetilde {\tau }_g^{-1}$ corresponds to $ghg^{-1}\in H$ . Hence, $\widetilde {\tau }_g$ normalizes H and descends to an automorphism of $F"={F}_{\mathrm {univ}}/H$ . If two elements of $G_X$ have the same image in $G_Y$ , they differ by an element of H and hence induce the same automorphism of $F"$ . Therefore, we obtain a representation

$$\begin{align*}\tau"\colon G_Y\longrightarrow\mathrm{Aut}(F") \end{align*}$$

such that

$$\begin{align*}p\circ\tau"(\gamma)=\tau(\gamma)\circ p \quad\text{for every }\gamma\in G_Y\!, \end{align*}$$

where $p\colon F"\to F$ is the covering map. Identifying $G_Y$ with $\pi _1(Y")$ , we obtain $ X"\cong ({Y}_{\mathrm {univ}}\times F")/\pi _1(Y") $ , where $\pi _1(Y")$ acts diagonally through $\tau "$ . Consequently, the induced morphism $ \alpha "\colon X"\longrightarrow Y" $ is a locally constant fibration with fiber $F"$ .

After replacing X with the finite étale cover constructed above, we consider the Albanese map $\alpha \colon X \to Y = \mathrm {Alb}(X)$ with fiber F. We will prove that the irregularity $q(F)$ (and thus the augmented irregularity $\hat {q}(F)$ ) is zero, assuming that

$$ \begin{align*}\hat{q}(F) = q(F) \quad \text{ and }\quad \hat{q}(X) = q(X) = \dim Y. \end{align*} $$

To this end, we consider the relative Albanese map

$$ \begin{align*}\beta \colon X \to Z := \mathrm{Alb}(X/Y)\end{align*} $$

associated with the Albanese map $\alpha \colon X \to Y := \mathrm {Alb}(X)$ (see [Reference CampanaCam85, Reference FujikiFuj83] for relative Albanese maps).

The fibration $\gamma \colon Z \to Y$ is locally trivial by the definition of relative Albanese maps, since the same holds for $\alpha \colon X \to Y$ . For any point $y \in Y$ , the fiber $G_{y} := \gamma ^{-1}(y)$ is the Albanese torus $\mathrm {Alb}(F_{y})$ , where $F_{y} := \alpha ^{-1}(y)$ is the fiber of $\alpha \colon X \to Y$ at y. In particular, the canonical bundle $K_{G_{y}}$ is a trivial line bundle. Furthermore, since Y is a finite étale quotient of a torus, the canonical bundle $K_{Y}$ is a Hermitian flat line bundle. Thus, the direct image sheaf

$$ \begin{align*}\gamma_{*}(\mathcal{O}_{Z}(K_{Z/Y} + \gamma^{*}K_{Y})) = \gamma_{*}(\mathcal{O}_{Z}(K_{Z})) \end{align*} $$

is a line bundle and admits the $L^{2}$ -metric, which is a smooth Hermitian metric with semi-positive curvature by the positivity of direct images. Note that $\gamma \colon Z \to Y$ is not necessarily projective but is a Kähler fibration. Nevertheless, we can apply the positivity of direct images by [Reference WangWan21, Theorem 2.6] (see also [Reference Berndtsson and PăunBP08, Reference CaoCao17, Reference Hacon, Popa and SchnellHPS18, Reference Păun and TakayamaPT18]). By the construction of the $L^{2}$ -metric, the smoothness of the $L^{2}$ -metric follows from the smoothness of $\gamma \colon Z \to Y$ and the Hermitian metric on $K_{Y}$ .

Moreover, since $K_{G_{y}}$ is a trivial line bundle, the evaluation morphism

$$ \begin{align*}\gamma^{*}\gamma_{*}(\mathcal{O}_{Z}(K_{Z})) \to \mathcal{O}_{Z}(K_{Z}) \end{align*} $$

is surjective, and thus it is an isomorphism since $\gamma ^{*}\gamma _{*}(\mathcal {O}_{Z}(K_{Z})) $ is a line bundle. In particular, the canonical bundle $K_{Z}$ is pseudo-effective. Thus, by Theorem 2.1, there exists a finite étale cover $T \to Z$ from a torus.

Consider the fiber product $X' := X \times _Z T$ with the following diagram:

Note that $X' \to X$ is a finite étale cover by construction. If $\dim Z> \dim Y$ , then we obtain

$$ \begin{align*}\hat{q}(X) = q(X') \geq \dim Z> \dim Y = q(X) = \hat{q}(X), \end{align*} $$

which is a contradiction. Hence, we have $\dim Y = \dim Z$ , and thus $\gamma \colon Z \to Y$ is bimeromorphic. The bimeromorphic morphism $\gamma \colon Z \to Y$ is an isomorphism since both $K_{Y}$ and $K_{Z}$ are Hermitian flat line bundles.

Consequently, we can obtain

$$ \begin{align*}\hat{q}(F) = q(F) = 0 \end{align*} $$

from the definition of relative Albanese maps and the choice of F.

Theorem 2.6 shows that F is a rationally connected projective manifold since $\hat {q}(F) = 0$ , which completes the proof of Conclusion (b). Since the rationally connected fiber F is simply connected, we can see that

$$ \begin{align*}\pi_{1}(X) \cong \pi_{1}(Y) \cong \mathbb{Z}^{\oplus 2 \dim Y}\end{align*} $$

by the homotopy exact sequence of the fibration $X \to Y$ . Thus, we can see that $\pi _{1}(X)$ is an abelian group, finishing the proof of Conclusion (a).

Proof of Corollary 1.3.

By Theorem 1.1, there exists an MRC fibration $\phi \colon X \to Y$ such that Y is a finite étale quotient of a torus. Then, by the proof of Corollary 2.4, the tangent bundle $T_X$ splits as in (2.1), and any tangent vectors in $\phi ^{*}T_Y \subset T_X$ are truly flat. This proves the desired inequality.