2.1 Kähler–Ricci flow

Hamilton’s Ricci flow has proved itself to be a natural candidate for a geometric flow which deforms a fixed Kähler metric to a canonical metric. Indeed, the Ricci flow is known to preserve the Kähler property of the metric; the Ricci flow starting from a Kähler metric $\omega _0$ is therefore referred to as the (normalized) Kähler–Ricci flow:

(2.1) $$ \begin{align} \frac{\partial \omega_t}{\partial t} &= - \text{Ric}(\omega_t) - \omega_t, \hspace{1cm} \omega_t \vert_{t=0} = \omega_0. \end{align} $$

Cao [Reference Cao5] showed that starting with any initial reference Kähler metric $\omega _0$ on a Calabi–Yau manifold, the Kähler–Ricci flow converges smoothly to the Ricci-flat Kähler metric in the polarization $[\omega _0]$ . Similar results hold for the Kähler–Ricci flow on canonically polarized manifolds, i.e., compact Kähler manifolds with ample canonical bundle. For Fano manifolds which admit a Kähler–Einstein metric, or more generally, admit a Kähler–Ricci soliton, the Kähler–Ricci flow converges exponentially fast to the Kähler–Einstein metric or Kähler–Ricci soliton. From the work of Tian and Zhang [Reference Tian and Zhang32], it is known that the maximal existence time T for the Kähler–Ricci flow on a compact Kähler manifold X is determined by cohomological data:

(2.2) $$ \begin{align} T \ = \ \sup \{ t \in \mathbb{R} : [\omega_0] + t [K_X ]> 0 \}. \end{align} $$

In particular, this gives a sharp local existence theorem for the Kähler–Ricci flow, and the Kähler–Ricci flow encounters finite-time singularities if and only if the flow intersects the boundary of the Kähler cone in finite time. If the canonical bundle is nef, then it follows from (2.2) that the Kähler–Ricci flow exists for all time. The resulting solutions, in this case, are referred to as longtime solutions of the Kähler–Ricci flow.

In [Reference Song and Tian28, Reference Song and Tian29], Song and Tian outlined a program for the study of the longtime solutions of the Kähler–Ricci flow on compact Kähler manifolds with semi-ample canonical bundle. The additional assumption of $K_X$ being semi-ample is fruitful since it endows X with the structure of the total space of a Calabi–Yau fiber space. As before, a fiber space is understood to mean a surjective holomorphic map $f : X \longrightarrow Y$ with connected fibers from a compact Kähler manifold X onto a normal irreducible, reduced, projective variety Y. Such a map is said to be a Calabi–Yau fiber space if the smooth fibers $X_y:= f^{-1}(y)$ are Calabi–Yau manifolds in the sense that $K_{X_y}$ is holomorphically torsion.

2.2 Sequences of Ricci-flat Kähler metrics

In [Reference Tosatti35], Tosatti laid the foundational framework, building on [Reference Song and Tian28], for the study of noncollapsed and collapsed limits of Ricci-flat Kähler metrics. Here, let X be a Calabi–Yau manifold and denote by $\mathcal {K}$ the Kähler cone of X. The Kähler cone is an open salient convex cone in the finite-dimensional vector space $H^{1,1}(X, \mathbb {R})$ . With respect to the metric topology induced by any norm on $\mathcal {K}$ , we let $\overline {\mathcal {K}}$ denote the closure of the Kähler cone in $H^{1,1}(X, \mathbb {R})$ and denote by $\partial \mathcal {K}$ its boundary. Fix a nonzero class $\alpha _0$ on the boundary of the Kähler cone,Footnote ¹ i.e., a nef class, which we may assume to have the form $\alpha _0 = f^{\ast }[\omega _Y]$ for some Kähler metric $\omega _Y$ on Y. Given a polarization $[\omega _0]$ on M, we may consider the path

(2.3) $$ \begin{align} \alpha_t = \alpha_0 + t[\omega_0], \end{align} $$

for $0 < t \leq 1$ . Yau’s solution of the Calabi conjecture furnishes a bijection between the Ricci-flat Kähler metrics and the points of $\mathcal {K}$ . Hence, given that for each $t> 0$ , the class $\alpha _t$ is Kähler, Yau’s theorem endows $\alpha _t$ with a unique Ricci-flat Kähler representative $\omega _t$ . To understand the behavior of these metrics, therefore, we can vary the complex structure, keeping the cohomological (or symplectic) data fixed, or keep the complex structure fixed and vary the cohomological data; of course, one can also vary both pieces of data, but we will not discuss that here. The former leads to the study of large complex structure limits, which is important in mirror symmetry, but here we will treat only the latter, given its intimate link with the Kähler–Ricci flow and canonical metrics. Therefore, we focus on the problem of understanding the behavior of the metrics $\omega _t$ as the cohomology class $\alpha _t$ degenerates to the boundary of the Kähler cone.

The metrics $\omega _t$ are given by solutions to the Monge–Ampère equation

(2.4) $$ \begin{align} \omega_t^m &= (f^{\ast} \omega_Y + t \omega_X + \sqrt{-1} \partial \overline{\partial} \varphi_t)^m \ = \ c_t t^{m-n} \omega_X^m, \hspace{1cm} \sup_X \varphi_t =0, \end{align} $$

where the constants $c_t$ are bounded away from $0$ and $+\infty $ , and converge as $t\longrightarrow 0$ .

Some remarks concerning the choice of path (2.3) are in order: the reason for considering such a path was initially provided by the fact that this was the path chosen by Gross and Wilson [Reference Gross and Wilson15] in their study of elliptically fibered K3 surfaces with $\text {I}_1$ -singular fibers. However, the motivation for sticking with such a path, however, is the formal analog with the Kähler–Ricci flow (cf. (2.2)). Much of the behavior of the Kähler–Ricci flow can be understood from the study of these sequences of Ricci-flat Kähler metrics (and vice versa). Hence, it has become standard practice to treat both contexts simultaneously, and this practice will be maintained here.

2.3 Twisted Kähler–Einstein metrics

The first systematic approach to the study of these collapsed limits of Ricci-flat Kähler metrics (and the Kähler–Ricci flow in this setting) was given by Song and Tian [Reference Song and Tian28]. They showed that the metrics $\omega _t$ converge to the pullback of a metric $\omega _{\text {can}}$ on the base of the Calabi–Yau fiber space, which satisfies an elliptic equation away from the discriminant locus:

(2.5) $$ \begin{align} \text{Ric}(\omega_{\text{can}}) &= \lambda \omega_{\text{can}} + \omega_{\text{WP}}, \end{align} $$

where $\omega _{\text {WP}}$ is the Weil–Petersson metric measuring the variation in the complex structure of the smooth fibers. A precise description of $\omega _{\text {WP}}$ was given by Tian [Reference Tian31], where he showed that $\omega _{\text {WP}}$ is the curvature form of the Hodge metric on $f_{\ast } \Omega _{M/N}$ . Let us note that in (2.5), $\lambda =0$ for sequences of Ricci-flat Kähler metrics, and $\lambda =-1$ for the longtime solution of the Kähler–Ricci flow (see [Reference Song and Tian28, Reference Song and Tian29, Reference Tosatti35]). In [Reference Tosatti35], it was shown that $\omega _t \to f^{\ast } \omega $ in the $\mathcal {C}_{\text {loc}}^{1,\gamma }(M^{\circ })$ topology of Kähler potentials for all $0 < \gamma < 1$ . In [Reference Tosatti, Weinkove and Yang38], this convergence was improved to local uniform convergence and to $\mathcal {C}_{\text {loc}}^{\alpha }(M^{\circ })$ in [Reference Hein and Tosatti18]. If the generic fiber is a (finite quotient) of a torus, or if the family is isotrivial in the sense that all smooth fibers are biholomorphic, the convergence can be improved to the $\mathcal {C}_{\text {loc}}^{\infty }(M^{\circ })$ topology (see [Reference Gross, Tosatti and Zhang12, Reference Hein and Tosatti17, Reference Tosatti and Zhang40] and [Reference Hein and Tosatti18], respectively).

If $\kappa = n$ , then $K_M$ is nef and big. In this case, it is known [Reference Tian and Zhang32, Reference Tsuji42] that the Kähler–Ricci flow converges weakly to the canonical metric on N, which is smooth on the regular part of Y. Furthermore, in [Reference Song27], it is shown that the metric completion of the Kähler–Einstein metric on $N^{\circ }$ is homeomorphic to N. If $0 < \kappa < n$ , then $K_M$ is no longer big, and much less is known; it is this case that we consider here. It is expected that the Ricci curvature of the metrics along the flow remains uniformly bounded on compact subsets of $M^{\circ }$ (see, e.g., [Reference Tian and Zhang33]). This is known to be the case when the generic fiber is a (finite quotient of a) torus. Recently, in [Reference Fong and Lee10], the higher-order estimates in [Reference Hein and Tosatti18] were used to obtain the uniform bound on $\text {Ric}(\omega _t)$ when the smooth fibers are biholomorphic. Assuming this uniform bound on the Ricci curvature of the metrics along the flow, one can formulate Conjecture 1.1 for the Kähler–Ricci flow.

In [Reference Tosatti and Zhang41], Tosatti and Zhang initiated a program to attack Conjecture 1.1 by understanding the nature of the singularities of the twisted Kähler–Einstein metric $\omega _{\text {can}}$ near the discriminant locus. They showed that if the canonical metric afforded the following $\mathcal {C}^2$ -estimate:

(2.6) $$ \begin{align} \pi^{\ast} \omega_{\text{can}} & \leq C \left( 1 - \sum_{i=1}^{\mu} \log | s_i |_{h_i}^2 \right)^d \omega_{\text{cone}}, \end{align} $$

then parts (i) and (ii) of Conjecture 1.1 are true. That is, to prove the conjecture, it suffices to show that (modulo some logarithmic poles) the canonical metric is at worst conical near the discriminant locus. For part (iii), Tosatti and Zhang required the additional assumption that the base of the Calabi–Yau fiber space is smooth and the divisorial component of the discriminant locus has simple normal crossings (in particular, they require the resolution $\pi $ to be the identity).

Proposition 3.1

[Reference Gross, Tosatti and Zhang14, Theorem 2.3]. There is a constant $C>0$ , positive integers $d \in \mathbb {N}$ , $0 \leq p \leq \mu $ , and rational numbers $\beta _i> 0$ ( $1 \leq i \leq p$ ), $0 < \alpha _i \leq 1$ ( $p+1 \leq i \leq \mu $ ), such that on $Y^{\circ }$ ,

$$ \begin{align*} C^{-1} \prod_{j=1}^p | s_j |_{h_j}^{2\beta_j} \omega_{\text{cone}}^n \ \leq \ \pi^{\ast} \omega_{\text{can}}^n \ \leq \ C \prod_{j=1}^p | s_j |_{h_j}^{2\beta_j} \left( 1 - \sum_{i=1}^{\mu} \log | s_i |_{h_i}^2 \right)^d \omega_{\text{cone}}^n, \end{align*} $$

where $\omega _{\text {cone}}$ is a conical Kähler metric with cone angle $2\pi \alpha _i$ along components $\mathcal {E}_i$ with $p+1 \leq i \leq \mu $ .

Consider the smooth nonnegative function

$$ \begin{align*}H \ : = \ \prod_{j=1}^p | s_j |_{h_j}^{2\beta_j},\end{align*} $$

where the product is over those $\pi $ -exceptional components (which we denote by $\mathcal {F}$ ) on which the volume form $\pi ^{\ast }\omega _{\text {can}}^{n}$ degenerates. Away from the exceptional divisor $\mathcal {F}$ ,

$$ \begin{align*} \sqrt{-1} \partial \overline{\partial} \log(H) &= -\sum_{j=1}^p \beta_j \Theta^{(\mathcal{F}_j,h_j)}, \end{align*} $$

where $\Theta ^{(\mathcal {F}_j,h_j)}$ is the curvature form of the Hermitian metric $h_j$ on $\mathcal {F}_j$ . Pulling back the Monge–Ampère equation to Y via $\pi $ , we get the degenerate, singular Monge–Ampère equation:

(3.1) $$ \begin{align} \pi^{\ast} \omega_{\text{can}}^n &= (\pi^{\ast} \omega_N + \sqrt{-1} \partial \overline{\partial}(\pi^{\ast}\varphi))^{n} \ \ = \ \ H \psi \frac{\omega_Y^n}{\prod_{i=p+1}^{\mu} | s_i |_{h_i}^{2(1-\alpha_i)}}, \end{align} $$

where $\omega _N$ is the reference form on N and $\omega _Y$ is a reference form on Y. From Proposition 3.1, the function

$$ \begin{align*}\psi \ : = \ \frac{\pi^{\ast} \omega_{\text{can}}^{n} \prod_i | s_i |_{h_i}^{2(1-\alpha_i)}}{H \omega_{Y}^{n}},\end{align*} $$

which is smooth on $Y^{\circ } \ = \ Y \backslash E$ , has the following growth control:

(3.2) $$ \begin{align} C^{-1} \ \leq \ \psi \ \leq \ C \left( 1 - \sum_{i=1}^{\mu} \log | s_i |_{h_i}^2 \right)^d. \end{align} $$

Away from the divisor $\mathcal {E}$ , compute

(3.3) $$ \begin{align} \sqrt{-1} \partial \overline{\partial} \log(1/\psi) &= \text{Ric}(\pi^{\ast} \omega_{\text{can}}) - \text{Ric}(\omega_{Y}) + \sum_i (1-\alpha_i) \Theta^{(\mathcal{E}_i,h_i)} + \sqrt{-1} \partial \overline{\partial} \log(H) \nonumber \\ & \geq - \text{Ric}(\omega_{Y}) + \sum_i (1-\alpha_i) \Theta^{(\mathcal{E}_i,h_i)} - \sum_j \beta_j \Theta^{(\mathcal{F}_j,h_j)}, \end{align} $$

where the right-hand side of (3.3) is smooth on all of Y. In light of (3.2), an old result of Grauert and Remmert [Reference Grauert and Remmert11] tells us that $\log (1/\psi )$ extends to a quasi-plurisubharmonic function with vanishing Lelong numbers on all of Y, satisfying (3.3) everywhere in the sense of currents. Apply Demailly’s regularization technique [Reference Demailly6], such that we can approach $\log (1/\psi )$ from above by a decreasing sequence of smooth functions $u_j$ which afford the lower bound

$$ \begin{align*} \sqrt{-1} \partial \overline{\partial} u_j & \geq - \text{Ric}(\omega_{Y}) + \sum_i (1-\alpha_i) \Theta^{(\mathcal{E}_i,h_i)} - \sum_j \beta_j \Theta^{(\mathcal{F}_j,h_j)}- \frac{1}{j} \omega_{Y}, \end{align*} $$

on all of Y. Similarly, since $\sqrt {-1} \partial \overline {\partial } \log (H) \geq - C \omega _{Y}$ (in the sense of currents) on Y, we may regularize $\log (H)$ by smooth functions $v_k$ with $\sqrt {-1} \partial \overline {\partial } v_k \geq -C \omega _{Y}$ and $v_k \leq C$ on Y. Out of this, we obtain a regularization of (3.1):

$$ \begin{align*}\omega_{j,k}^{n} \ = \ c_{j,k} e^{v_k -u_j} \frac{\omega_{Y}^{n}}{\prod_i | s_i |_{h_i}^{2(1-\alpha_i)}},\end{align*} $$

where

$$ \begin{align*}\omega_{j,k} \ = \ \pi^{\ast} \omega_N + \frac{1}{j} \omega_{Y} + \sqrt{-1} \partial \overline{\partial} \varphi_{j,k}\end{align*} $$

is a cone metric (which exists by the main theorems of [Reference Guenancia and Paun16, Reference Jeffres, Mazzeo and Rubinstein19]) with the same cone angles as $\omega _{\text {cone}}$ . In particular, the metrics $\omega _{j,k}$ are smooth on $Y^{\circ }$ and there is a uniform (independent of j) bound on the $L^p$ -norm of $\omega _{j,k}^{n}/\omega _{Y}^{n}$ , for some $p>1$ . By [Reference Eyssidieux, Guedj and Zeriahi9], it follows that

$$ \begin{align*}\sup_Y | \varphi_{j,k} | \ \leq \ C,\end{align*} $$

and since $e^{v_k -u_j} \longrightarrow H \psi $ , we conclude from the stability theorem in [Reference Eyssidieux, Guedj and Zeriahi9] that as $j,k \longrightarrow \infty $ , we have $c_{j,k} \longrightarrow 1$ , $\varphi _{j,k} \longrightarrow \pi ^{\ast } \varphi $ in $\mathcal {C}^0(Y)$ , where $\pi ^{\ast } \varphi $ solves (3.1). Fix a reference metric $\omega _Y$ on Y and consider partial regularizations

(3.4) $$ \begin{align} \omega_j = \pi^{\ast} \omega_N + \frac{1}{j} \omega_Y + \sqrt{-1} \partial \overline{\partial} \varphi_j \end{align} $$

given by solutions of the Monge–Ampère equation

(3.5) $$ \begin{align} \omega_j^n &= c_j H e^{-u_j} \frac{\omega_Y^n}{\prod_{i=1}^{\mu} | s_i |_{h_i}^{2(1-\alpha_i)}}. \end{align} $$

Proof of Theorem 1.3

For positive constants $\delta , \delta _0, \Lambda _{\delta , j}, \varepsilon , \varepsilon _0, \varepsilon _1, b, C_1>0$ , we will apply the maximum principle to the quantity

$$ \begin{align*} \mathcal{Q} & : = \log \text{tr}_{\omega_j}(\omega_{\text{cone}}) + \delta \log(H) + ( (\Lambda_{\delta,j} + \varepsilon_0) b - C_1 \delta) \eta - (\Lambda_{\delta,j} + \varepsilon_1) \varphi_j \\ & \quad - \delta_0(\Lambda_{\delta,j} + \varepsilon_0) \log | s_{\mathcal{F}} |^2 + \varepsilon \log | s_{\mathcal{E}} |^2. \end{align*} $$

Here, we fix a conical Kähler metric $\omega _{\text {cone}}$ with cone angle $2\pi \alpha _i$ along $\mathcal {E}_i$ (an irreducible component of $\mathcal {E})$ . For a reference Kähler metric $\omega _Y$ , we set $\omega _{\text {cone}} = \omega _Y + \sqrt {-1} \partial \bar {\partial } \eta $ , where $\eta : = C^{-1} \sum _i \left ( | s_i |_{h_i}^2 \right )^{\alpha _i}$ for some sufficiently large $C>0$ . The constants will be described throughout the proof. We start with the following observation: from [Reference Gross, Tosatti and Zhang14, Proposition 5.1], there is a j-dependent constant $C_j$ such that $\text {tr}_{\omega _{\text {cone}}}(\omega _j) \leq C_j$ . From the complex Monge–Ampère equation, this implies that for any $\delta \in (0,1)$ ,

$$ \begin{align*} \text{tr}_{\omega_j}(\omega_{\text{cone}}) & \leq \frac{C_j^{n-1}}{c_j H e^{-u_j}} \ \leq \ \frac{C_j^{n-1}}{c_j H^{\delta} e^{-u_j}}. \end{align*} $$

In particular, there is a j-dependent constant (abusively denoted by the same symbol) $C_j$ such that

$$ \begin{align*}H^{\delta} e^{-u_j} \text{tr}_{\omega_j}(\omega_{\text{cone}}) \ \leq \ C_j.\end{align*} $$

This will allow us to apply the maximum principle to terms involving $\text {tr}_{\omega _j}(\omega _{\text {cone}})$ so long as we scale $\text {tr}_{\omega _j}(\omega _{\text {cone}})$ by $H^{\delta } e^{-u_j}$ and add a term which decays to $-\infty $ along the divisor $\mathcal {E}$ (at an arbitrary rate). This latter term is given by $\varepsilon \log | s_{\mathcal {E}} |^2$ , where ${0 < \varepsilon \leq \frac {1}{j}}$ . To compute $\Delta _{\omega _j} \log \text {tr}_{\omega _j}(\omega _{\text {cone}})$ , we observe that the complex Monge–Ampère equation gives the following control of the Ricci curvature of $\omega _j$ :

$$ \begin{align*} \text{Ric}(\omega_j) & = - \sqrt{-1} \partial \bar{\partial} \log(H)+ \sqrt{-1} \partial \bar{\partial}u_j + \text{Ric}(\omega_Y) + \sqrt{-1} \partial \bar{\partial} \log \prod |s_i |_{h_i}^{2(1-\alpha_i)} \\ & \geq - \frac{1}{j} \omega_Y \ \geq \ - \frac{C}{j} \omega_{\text{cone}}. \end{align*} $$

Let $g,h$ be metrics underlying $\omega _j$ and $\omega _{\text {cone}}$ , respectively. Then, for any holomorphic map $f : (Y^{\circ }, \omega _j) \to (Y^{\circ }, \omega _{\text {cone}})$ , with $(f_i^{\alpha })$ denoting the local expression for $\partial f$ , we have (see, e.g., [Reference Broder2, Reference Broder3, Reference Yang and Zheng44])

$$ \begin{align*} \Delta_{\omega_j} | \partial f |^2 & = | \nabla \partial f |^2 + \text{Ric}_{k \overline{\ell}}^g g^{k \overline{q}} g^{p \overline{\ell}} h_{\alpha \overline{\beta}} f_p^{\alpha} \overline{f_q^{\beta}} - R^h_{\alpha \overline{\beta} \gamma \overline{\delta}} g^{i \overline{j}} f_i^{\alpha} \overline{f_j^{\beta}} g^{p \overline{q}} f_p^{\gamma} \overline{f_q^{\delta}}. \end{align*} $$

The lower bound on the Ricci curvature of g implies that

$$ \begin{align*} \text{Ric}_{k \overline{\ell}}^g g^{k \overline{q}} g^{p \overline{\ell}} h_{\alpha \overline{\beta}} f_p^{\alpha} \overline{f_q^{\beta}} & \geq - \frac{C}{j} h_{\gamma \overline{\delta}} g^{k \overline{q}} g^{p \overline{\ell}} h_{\alpha \overline{\beta}} f_p^{\alpha} \overline{f_q^{\beta}}. \end{align*} $$

Choose the frame such that at the point where we are computing, we have $f_i^{\alpha } = \lambda _i \delta _i^{\alpha }$ , $g_{i\overline {j}} = \delta _{ij}$ and $h_{\alpha \overline {\beta }} = \delta _{\alpha \overline {\beta }}$ . Then

$$ \begin{align*} - \frac{C}{j} h_{\gamma \overline{\delta}} f_k^{\gamma} \overline{f_{\ell}^{\delta}} g^{k \overline{q}} g^{p \overline{\ell}} h_{\alpha \overline{\beta}} f_p^{\alpha} \overline{f_q^{\beta}} &= - \frac{C}{j} \delta_{\gamma \delta} \lambda_k \delta_k^{\gamma} \lambda_{\ell} \delta_{\ell}^{\delta} \delta^{kq} \delta^{p\ell} \delta_{\alpha \beta} \lambda_p \delta_p^{\alpha} \lambda_q \delta_q^{\beta} \ = \ - \frac{C}{j} \lambda_{\alpha}^4. \end{align*} $$

Since $\sum _{\alpha } \lambda _{\alpha }^4 \leq \left ( \sum _{\alpha } \lambda _{\alpha }^2 \right )^2 = | \partial f |^4$ , we deduce that

$$ \begin{align*} \text{Ric}_{k \overline{\ell}}^g g^{k \overline{q}} g^{p \overline{\ell}} h_{\alpha \overline{\beta}} f_p^{\alpha} \overline{f_q^{\beta}} & \geq - \frac{C}{j} \text{tr}_{\omega_j}(\omega_{\text{cone}})^2. \end{align*} $$

From [Reference Jeffres, Mazzeo and Rubinstein19, Reference Lin and Shen22], we know that there is a uniform constant $K>0$ such that $\text {HBC}(\omega _{\text {cone}}) \leq K$ . This, in particular, implies that the holomorphic sectional curvature of the cone metric $\omega _{\text {cone}}$ affords the (positive) upper bound ${\text {HSC}(\omega _{\text {cone}}) \leq K}$ for $K>0$ . Royden’s polarization argument [Reference Broder3, Reference Broder4, Reference Lu23, Reference Royden25] implies that

$$ \begin{align*} R^h_{\alpha \overline{\beta} \gamma \overline{\delta}} g^{i \overline{j}} f_i^{\alpha} \overline{f_j^{\beta}} g^{p \overline{q}} f_p^{\gamma} \overline{f_q^{\delta}} & \leq K \text{tr}_{\omega_j}(\omega_{\text{cone}})^2. \end{align*} $$

Hence, by Royden’s Schwarz lemma [Reference Broder4, Reference Lu23, Reference Royden25], we see that

$$ \begin{align*} \Delta_{\omega_j} \log \text{tr}_{\omega_j}(\omega_{\text{cone}}) &\geq - \left( \frac{C}{j} + K \right) \text{tr}_{\omega_j}(\omega_{\text{cone}}), \end{align*} $$

at any point away from the divisor $\mathcal {E}$ . Therefore,

$$ \begin{align*}& \Delta_{\omega_j} (\log \text{tr}_{\omega_j}(\omega_{\text{cone}}) + \delta \log(H) - u_j ) \geq - \left( \frac{C}{j} + K \right) \text{tr}_{\omega_j}(\omega_{\text{cone}}) \\ & \quad+ \delta \text{tr}_{\omega_j}(\sqrt{-1} \partial \bar{\partial} \log(H)) - \Delta_{\omega_j} u_j, \end{align*} $$

at any point away from the divisor $\mathcal {E}$ . Since $\sqrt {-1} \partial \bar {\partial } \log (H) \geq - C_1 \omega _Y$ , and adding $u_j$ to the quantity being differentiated (which will not affect the maximum principle), we see that

$$ \begin{align*} \Delta_{\omega_j} (\log \text{tr}_{\omega_j}(\omega_{\text{cone}}) + \delta \log(H)) & \geq - \left( \frac{C}{j} + K \right) \text{tr}_{\omega_j}(\omega_{\text{cone}}) - \delta C_1 \text{tr}_{\omega_j}(\omega_Y), \end{align*} $$

at any point away from $\mathcal {E}$ . We know that the conical metric is given by $\omega _{\text {cone}} = \omega _Y + \sqrt {-1} \partial \bar {\partial } \eta $ . Hence,

$$ \begin{align*} \Delta_{\omega_j}(\log \text{tr}_{\omega_j}(\omega_{\text{cone}}) +\delta \log(H) - \delta C_1 \eta + \varepsilon \log | s_{\mathcal{E}} |^2) & \geq -\kern-1pt \left( \frac{C}{j} + K + C_1 \delta \right) \kern-1pt\text{tr}_{\omega_j}(\omega_{\text{cone}}), \end{align*} $$

and we adopt the notation $\Lambda _{\delta ,j} : = \frac {C}{j} + K + C_1 \delta $ . To ensure the positivity of the coefficient of $\text {tr}_{\omega _j}(\omega _{\text {cone}})$ , let $\varepsilon _1>0$ be a positive constant to be determined later. Then

$$ \begin{align*} &\Delta_{\omega_j}( \log \text{tr}_{\omega_j}(\omega_{\text{cone}}) + \delta \log(H) - C_1 \delta \eta + \varepsilon \log | s_{\mathcal{E}} |^2 - (\Lambda_{\delta,j} + \varepsilon_1) \varphi_j) \\ &\quad\geq - \Lambda_{\delta,j} \text{tr}_{\omega_j}(\omega_{\text{cone}}) - (\Lambda_{\delta,j} + \varepsilon_1)n + (\Lambda_{\delta,j} +\varepsilon_1) \text{tr}_{\omega_j}(\pi^{\ast} \omega). \end{align*} $$

The pullback $\pi ^{\ast } \omega $ is degenerate in the directions tangent to the divisor. Let $\delta _0>0$ be the suitably small constant such that $\pi ^{\ast } \omega - \delta _0 \Theta ^{(\mathcal {F},h)}$ is positive-definite. Furthermore, we let $b>0$ be the suitably small constant such that $\pi ^{\ast } \omega - \delta _0 \Theta ^{(\mathcal {F},h)} + b \sqrt {-1} \partial \bar {\partial } \eta $ has the same cone angles as $\omega _{\text {cone}}$ . Hence, we can find a constant $c_0>0$ such that

$$ \begin{align*}\pi^{\ast} \omega - \delta_0 \Theta^{(\mathcal{F},h)} + b \sqrt{-1} \partial \bar{\partial} \eta \ \geq \ c_0 \omega_{\text{cone}}.\end{align*} $$

Hence, with $\mathcal {Q}$ defined above, we see that

$$ \begin{align*} \Delta_{\omega_j} \mathcal{Q} & \geq - \Lambda_{\delta, j} \text{tr}_{\omega_j}(\omega_{\text{cone}}) \kern1.3pt{-}\kern1.3pt (\Lambda_{\delta,j} + \varepsilon_1) n + (\Lambda_{\delta,j} + \varepsilon_1) \text{tr}_{\omega_j}(\pi^{\ast} \omega - \delta_0 \Theta^{(\mathcal{F},h)} + b \sqrt{-1} \partial \bar{\partial} \eta) \\ & \geq (c_0(\Lambda_{\delta,j} + \varepsilon_1) - \Lambda_{\delta,j}) \text{tr}_{\omega_j}(\omega_{\text{cone}}) - (\Lambda_{\delta,j} + \varepsilon_1) n. \end{align*} $$

If $c_0 \geq 1$ , then $c_0 (\Lambda _{\delta ,j} + \varepsilon _1) - \Lambda _{\delta ,j} \geq c_0 \varepsilon _0$ , and we take $\varepsilon _1>0$ arbitrary. For instance, if we take $\varepsilon _1 = c_0^{-1}$ , then

$$ \begin{align*} \Delta_{\omega_j} \mathcal{Q} & \geq \text{tr}_{\omega_j}(\omega_{\text{cone}}) - (\Lambda_{\delta,j} + c_0^{-1}) n. \end{align*} $$

By the maximum principle, at the point where $\mathcal {Q}$ achieves its maximum, we have

$$ \begin{align*}\text{tr}_{\omega_j}(\omega_{\text{cone}}) \ \leq \ n(\Lambda_{\delta,j} + c_0^{-1}) n \ \leq \ C,\end{align*} $$

for some uniform constant $C>0$ . On the other hand, if $0 < c_0 < 1$ , then we choose $\varepsilon _1 = c_0^{-1}(1 + \Lambda _{\delta ,j}(1-c_0))$ . In this case, we see that

$$ \begin{align*} \Delta_{\omega_j}(\mathcal{Q}) & \geq \text{tr}_{\omega_j}(\omega_{\text{cone}}) - c_0^{-1}n(1+\Lambda_{\delta,j}), \end{align*} $$

and by the maximum principle, at the point where $\mathcal {Q}$ achieves its maximum,

$$ \begin{align*}\text{tr}_{\omega_j}(\omega_{\text{cone}}) \ \leq \ c_0^{-1} n(1+\Lambda_{\delta,j}).\end{align*} $$

Both estimates imply that $\mathcal {Q} \leq C$ for some uniform constant $C>0$ . Hence, we have the estimate

$$ \begin{align*} \text{tr}_{\omega_j}(\omega_{\text{cone}}) & \leq \frac{C}{H^{\delta} | s_{\mathcal{F}} |^{2\delta_0(\Lambda_{\delta,j}+\varepsilon_1)}} \end{align*} $$

everywhere on $Y^{\circ }$ . From the complex Monge–Ampère equation, we see that

$$ \begin{align*} \text{tr}_{\omega_{\text{cone}}}(\omega_j) & \leq \text{tr}_{\omega_j}(\omega_{\text{cone}})^{n-1} \frac{\omega_j^n}{\omega_{\text{cone}}^n} \ \leq \ \frac{CH e^{u_j}}{| s_{\mathcal{F}} |^{2\delta_0(\Lambda_{\delta,j} + \varepsilon_1)(n-1)} H^{\delta(n-1)}}. \end{align*} $$

Choosing $\delta < \frac {1}{n-1}$ , we see that

$$ \begin{align*} \text{tr}_{\omega_{\text{cone}}}(\omega_j) & \leq \frac{Ce^{-u_j}}{| s_{\mathcal{F}} |^{2\delta_0(\Lambda_{\delta,j} + \varepsilon_1)(n-1)}}, \end{align*} $$

and letting $j \to \infty $ , we obtain the partial second-order estimate

$$ \begin{align*} \pi^{\ast} \omega_{\text{can}} & \leq \frac{C}{| s_{\mathcal{F}} |^{2\delta_0(\Lambda_{\delta,j} + \varepsilon_1)(n-1)}} \left( 1 - \sum_{i=1}^{\mu} \log | s_i |_{h_i}^2 \right)^d \omega_{\text{cone}}. \end{align*} $$

Proof of Corollary 1.4

Suppose the holomorphic sectional curvature of $\omega _{\text {cone}}$ is almost nonpositive, i.e., for any $\varepsilon _0>0$ , we have $\text {HSC}(\omega _{\text {cone}}) \leq \varepsilon _0$ , then

$$ \begin{align*} 2\delta_0(\Lambda_{\delta,j} + \varepsilon_1)(n-1) & \leq 2\delta_0 \left( \frac{C}{j} + \varepsilon_0 + C_1 \delta \right)(n-1). \end{align*} $$

The constant $\delta \in (0, \frac {1}{n-1})$ can be made arbitrarily small, and therefore, as $j \to \infty $ , we have

$$ \begin{align*} 2\delta_0 \left( \frac{C}{j} + \varepsilon_0 + C_1 \delta \right) (n-1) \to 2 \delta_0(n-1)(\varepsilon_0 + C_1 \delta). \end{align*} $$

In particular, if the holomorphic sectional curvature of the conical Kähler metric is bounded above by an arbitrarily small positive constant, then the multiplicity of the divisorial pole can be made arbitrarily small.

Remark 3.2

The assumption that the holomorphic sectional curvature of the conical metric is arbitrarily small is not as restrictive as one may expect. We suspect that by appropriately localizing the function $\mathcal {Q}$ used in the maximum principle, one should be able to ensure that the holomorphic sectional curvature of $\omega _{\text {cone}}$ is almost nonpositive at the point where $\mathcal {Q}$ achieves its maximum. To see this, consider the case where the divisor $\mathcal {E}$ consists of a single component, locally described by $\{ z_1 =0 \}$ . Then the computation in [Reference Jeffres, Mazzeo and Rubinstein19, Appendix] shows that

$$ \begin{align*}R_{1 \overline{1} 1 \overline{1}} \sim -\alpha^2(\alpha-1)^2 | z |^{-2(2-\alpha)}.\end{align*} $$

In other words, if the function $\mathcal {Q}$ can be tweaked such that the maximum occurs sufficiently close to $\mathcal {E}$ , but not on $\mathcal {E}$ , then the holomorphic sectional curvature of $\omega _{\text {cone}}$ should be negative at this point, and the argument goes through.

3.1 The holomorphic sectional curvature

The main theorem, although primarily concerned with Conjecture 1.1, fits into a more general program the author has initiated. There are the three primary aspects of the holomorphic sectional curvature:

1. (i) Symmetries—The holomorphic sectional curvature of a Kähler metric (or more generally, Kähler-like metric) determines the curvature tensor entirely.

2. (ii) Value distribution of curves—A compact Kähler manifold with negative holomorphic sectional curvature is Kobayashi hyperbolic (every entire curve $\mathbb {C} \to X$ is constant). On the other hand, a compact Kähler manifold with positive holomorphic sectional curvature is rationally connected (any two points lie in the image of a rational curve $\mathbb {P}^1 \to X$ ) [Reference Yang43].

3. (iii) Singularities—The holomorphic sectional curvature influences the singularities of the geometry.

The first facet of the holomorphic sectional curvature is well known to all complex geometers. The assertions in statement (ii) are also well known, but it is worth emphasizing a particular subtlety concerning (ii): let $(X, \omega )$ be a Hermitian manifold with a Hermitian metric of negative (Chern) holomorphic sectional curvature ${{}^c \text {HSC}(\omega ) \leq - \kappa <0}$ . Then X is Brody hyperbolic in the sense that every entire curve $\mathbb {C} \to X$ is constant. The proof is an elementary application of the Schwarz lemma. Suppose that there is a nonconstant map $f : \mathbb {C} \to X$ . Then $\Delta _{\omega _{\mathbb {C}}} | \partial f |^2 \geq - {}^c R(\partial f, \bar {\partial f}, \partial f, \bar {\partial f})$ , and by the maximum principle, f is constant. The key point here, however, is that for holomorphic curves, there is only one direction to consider. For holomorphic maps of higher rank (e.g., $f : \mathbb {C}^2 \to X$ ), it is not clear whether the holomorphic sectional curvature of an arbitrary Hermitian metric gives any control. This is the reason for the introduction of the real bisectional curvature [Reference Lee and Streets20, Reference Yang and Zheng44] and the Schwarz bisectional curvatures [Reference Broder2, Reference Broder3].

At this point, the reader may be perplexed, since we used the Schwarz lemma in the proof of the main theorem and this involves a map of rank $>1$ . But here, we note that Royden’s polarization argument [Reference Broder3, Reference Broder4, Reference Royden25] is used. In particular, for holomorphic maps into Kähler manifolds, the holomorphic sectional curvature gives suitable control. In other words, we see the Schwarz lemma as an incarnation of aspects (i) and (ii).

The third aspect is far less understood (emphasized by the vague nature of the statement in (iii)). The main theorem of the present manuscript provides evidence for the relationship, but it is certainly not the only evidence. Recall that Demailly [Reference Demailly7] constructed an example of a projective Kobayashi hyperbolic surface that does not admit a Hermitian metric of negative Chern holomorphic sectional curvature. The construction relies on the following algebraic hyperbolicity criterion [Reference Demailly7].

Theorem 3.3

(Demailly). Let $(X, \omega )$ be a compact Hermitian manifold with ${}^c \text {HSC}_{\omega } \leq \kappa _0$ for some $\kappa _0 \in \mathbb {R}$ . If $f : \mathcal {C} \to X$ is a nonconstant holomorphic map from a compact Riemann surface $\mathcal {C}$ of genus g, then

$$ \begin{align*} 2g-2 & \geq -\frac{\kappa_0}{2\pi} \deg_{\omega}(\mathcal{C}) - \sum_{p \in \mathcal{C}} (m_p-1). \end{align*} $$

We note that, in [Reference Demailly7] (see also [Reference Diverio8]), the constant is assumed to be nonpositive, but the formula holds more generally (see, e.g., [Reference Broder4]). The example is constructed as a fibration (with hyperbolic base and fiber) with a fiber sufficiently singular to violate the above algebraic hyperbolicity criterion.