Proof. The problem being local, we can reduce to the following situation

where $f(t)=t^{\alpha }\varphi (t)$ with $\varphi (0)\neq 0$ , $g(u)=u^{\beta }\psi (u)$ with $\psi (0)\neq 0$, and $\rho (u)=u^{r}$, $\pi (w)=w^{p}$. In particular, we have $\alpha r=\beta p$.

A section $\omega$ of $\Omega _{\pi ,\Delta }$ is locally of the form

\[ \omega = w^{{-}p(1-1/m)}\pi^{{\ast}}\, d z = pw^{(p/m)-1}\, d w, \]

with $1\leqslant m\leqslant \infty$. One infers that $g^{\ast }\omega$ vanishes at order $\beta ((p/m)-1)+(\beta -1)=(\alpha /m) r-1$. Therefore $g^{\ast }\omega /\rho ^{\ast }\, d t$ is holomorphic if and only if $\alpha \geqslant m$. This proves the equivalence of (i) and (ii).

Now we prove the equivalence between (i) and (iii) for a fixed diagram (⋆). Recall that by definition, $j_{k}^{\star }(g)$ belongs to $J_{k}(\pi ,\Delta )$ if and only if $\omega (j_{k}^{\star }(g))$ is holomorphic for all jet differentials $\omega$. Such a jet differential being locally of the form

\[ \sum a_{\alpha} \biggl(\frac{\pi^{{\ast}}\, d^{1}z}{w^{p(1-1/m)^{+}}} \biggr)^{\alpha_{1}} \dotsm \biggl( \frac{\pi^{{\ast}}\, d^{k}z}{w^{p(1-k/m)^{+}}} \biggr)^{\alpha_{k}}, \]

it is necessary and sufficient to check the holomorphicity of

\[ \omega_{j}(j_{k}^{{\star}}(g)) = \biggl( \frac{\pi^{{\ast}}\, d^{j}z}{w^{p(1-j/m)^{+}}} \biggr) (g,f'\circ\rho,\dotsc,f^{(k)}\circ\rho). \]

When $m=\infty$, one has a standard logarithmic derivative:

\[ \omega_{j}(j_{k}^{{\star}}(g)) = \frac{f^{(j)}\circ\rho}{g^{p}} = \frac{f^{(j)}}{f}\circ\rho. \]

It coincides with $g^{\ast }\omega _{1}/\rho ^{\ast }\, d t$ for $j=1$. The vanishing order $r((\alpha -j)-\alpha )$ is indeed nonnegative if and only if $\alpha =\infty$.

When $m$ is finite, if $j\geqslant m$, there is nothing to check. Else, a straightforward computation shows that

\[ \omega_{j}(j_{k}^{{\star}}(g)) = \frac{f^{(j)}\circ\rho}{g^{p(1-j/m)}} = \frac{p!}{j!(p-j)!p^{j}} \ \biggl( \frac{(\rho')^{j}f^{(j)}\circ\rho}{(g')^{j} \pi^{(j)}\circ g} \biggr) \ \biggl(\frac{g^{{\ast}}\omega_{1}}{\rho^{{\ast}}\, d t}\biggr)^{j} \]

(note that $j\leqslant m < p$). In particular, for $j=1$, by commutativity of (⋆) one has

\[ \omega_{1}(j_{k}^{{\star}}(g)) = g^{{\ast}}\omega_{1}/\rho^{{\ast}}\, d t. \]

More generally, since $(\rho ')^{j}f^{(j)}\circ \rho$ and $(g')^{j} \pi ^{(j)}\circ g$ appear both in the development of the $j$th derivative of $f\circ \rho =\pi \circ g$, these have the same vanishing order. Therefore $\omega _{j}(j_{k}^{\star }(g))$ is holomorphic if and only if $(g^{\ast }\omega _{1}/\rho ^{\ast }\, d t)$ is holomorphic.

Proof. The morphism $\varphi \circ f\colon \mathbb {D}\to (X',\Delta ')$ is orbifold.

Proof. We proceed by induction on the length of tensor products in the summand: we will prove that there is a natural filtration of $E_{k,N}\Omega _{\pi ,\Delta }$ induced by the weighted degrees $\lVert \cdot \rVert _{k-1}\dotsc ,\lVert \cdot \rVert _{p}$, with associated graded bundle

\[ \mathrm{Gr}_{{\bullet}}^{p}E_{k,N}\Omega_{\pi,\Delta} = \bigoplus_{p\ell_{p}+\dotsb+k\ell_{k}\leqslant N} E_{p,N-p\ell_{p}-\dotsb-k\ell_{k}}\Omega_{\pi,\Delta} \otimes S^{\ell_{p}}\Omega_{\pi,\Delta^{(p)}} \otimes \dotsb \otimes S^{\ell_{k}}\Omega_{\pi,\Delta^{(k)}}. \]

Since $E_{1,\ell }\Omega _{\pi ,\Delta }=S^{\ell }\Omega _{\pi ,\Delta }$, the sought statement corresponds indeed to the case $p=1$.

The weighted degree $\lVert \cdot \rVert _{k-1}$ induces a descending filtration by subbundles

\[ E_{k-1,N}\Omega_{\pi,\Delta}\cong F_{k-1}^{N}\subset\dotsb\subset F_{k-1}^{w+1}\subset F_{k-1}^{w}\subset\dotsb\subset F_{k-1}^{0}=E_{k,N}\Omega_{\pi,\Delta} \]

of $E_{k,N}\Omega _{\pi ,\Delta }$, with

\[ F_{k-1}^{w} := \{{ \omega\in E_{k,N}\Omega_{\pi,\Delta} \, / \, \lVert \omega \rVert_{k-1}\geqslant w }\}. \]

Note that these are indeed subbundles because in a coordinate change, the weighted degree $\lVert \cdot \rVert _{k-1}$ can only increase. This is an easy corollary of the upper-triangularity of the Faà di Bruno formula.

We claim that the graded pieces are

\[ F_{k-1}^{w}/F_{k-1}^{w+1} \cong \begin{cases} 0 & \text{if }k\nmid N-w,\\ E_{k-1,w}\Omega_{\pi,\Delta}\otimes S^{\ell_{k}}\Omega_{\pi,\Delta^{(k)}} & \text{if }w=N-k\ell_{k}, \end{cases} \]

and the announced result follows. The proof of the claim is standard. Let us simply point out that it relies on the simple observation that if one mods out by jet-coordinates of order less than $k$, $d^{k}(\phi \circ z)=(\phi '\circ z) d^{k}z$; hence, in the filtration, polynomials in jet-coordinates of order $k$ behave under coordinates changes $\phi$ in the exact same way as symmetric differential forms (i.e. polynomials in jet-coordinates of order $1$) do. Here the slight subtlety is that we consider orbifold jet differentials: the pole order of a $k$th jet coordinate $w_{i}^{-p_{i}(1-k/m_{i})^{+}}\pi ^{\ast }\, d^{k}z_{i}$ for the pair $(X,\Delta )$ is not the same as the pole order of the $1$st jet-coordinate $w_{i}^{-p_{i}(1-1/m_{i})}\pi ^{\ast }\, d z_{i}$ for the pair $(X,\Delta )$ but rather the same as the pole order of the $1$st jet-coordinate $w_{i}^{-p_{i}(1-k/m_{i})^{+}}\pi ^{\ast }\, d z_{i}$ for the pair $(X,\Delta ^{(k)})$ (cf. Definition 2.1).

Proof. We follow in spirit Green and Griffiths [Reference Green and GriffithsGG80, Proposition 1.10]. By Proposition 2.14

\[ \operatorname{ch} E_{k,N}\Omega_{\pi,\Delta} = \sum_{\lVert \ell \rVert=N} \operatorname{ch}(S^{\ell_{1}}\Omega_{\pi,\Delta^{(1)}}) \operatorname{ch}(S^{\ell_{2}}\Omega_{\pi,\Delta^{(2)}}) \dotsm \operatorname{ch}(S^{\ell_{k}}\Omega_{\pi,\Delta^{(k)}}). \]

For $i=1,\dotsc ,k$, the orbifold cotangent bundle $\Omega _{\pi ,\Delta ^{(i)}}$ is a vector bundle of rank $n$. Let $\lambda _{1}^{(i)}$, …, $\lambda _{n}^{(i)}$ be a set of Chern roots for it. In terms of these Chern roots, we get

\[ \operatorname{ch} E_{k,N}\Omega_{\pi,\Delta} = \sum_{\sum_{i=1}^{k}\sum_{j=1}^{n}ix_{i,j}=N} \exp\biggl(\sum_{i=1}^{k}\sum_{j=1}^{n}x_{i,j}\lambda_{j}^{(i)}\biggr). \]

Using the sum-integral formula yields

\[ \operatorname{ch} E_{k,N}\Omega_{\pi,\Delta} = N^{kn-1} \int_{\sum_{i=1}^{k}\sum_{j=1}^{n}ix_{i,j}=1} \exp\biggl(\sum_{i=1}^{k}\sum_{j=1}^{n}Nx_{i,j}\lambda_{j}^{(i)}\biggr) \,d\omega + O(N^{kn-2}). \]

Expanding the exponential

\[ \operatorname{ch} E_{k,N}\Omega_{\pi,\Delta} = N^{(k+1)n-1} \int_{\sum_{i=1}^{k}\sum_{j=1}^{n}ix_{i,j}=1} \frac{\bigl(\sum_{i=1}^{k}\sum_{j=1}^{n}x_{i,j}\lambda_{j}^{(i)}\bigr)^{n}}{n!} \,d\omega + O(N^{(k+1)n-2}). \]

Rescaling

\[ \operatorname{ch} E_{k,N}\Omega_{\pi,\Delta} = N^{(k+1)n-1} \int_{\sum_{i=1}^{k}\sum_{j=1}^{n}x_{i,j}=1} \frac{\bigl(\sum_{i=1}^{k}\sum_{j=1}^{n}x_{i,j}{\lambda_{j}^{(i)}}/{i}\bigr)^{n}}{n!} \frac{ d\omega}{(k!)^{n}} + O(N^{(k+1)n-2}). \]

Using multinomial formula, one gets

\begin{align*} & \operatorname{ch} E_{k,m}\Omega_{\pi,\Delta} \\ &\quad = N^{(k+1)n-1} \sum_{\sum q_{i,j}=n} \frac{(\lambda_{1}^{(1)})^{q_{1,1}}\dotsm(\lambda_{n}^{(k)})^{q_{k,n}}} {1^{\sum q_{1,j}}\dotsm k^{\sum q_{k,j}}} \int_{\sum_{i=1}^{k}\sum_{j=1}^{n}x_{i,j}=1} \frac{x_{1,1}^{q_{1,1}}}{q_{1,1}!} \dotsm \frac{x_{k,n}^{q_{k,n}}}{q_{k,n}!} \frac{ d\omega}{(k!)^{n}} \\ &\qquad + O(N^{(k+1)n-2}). \end{align*}

For any $q$ values with $\sum q_{i,j}=n$, by calculus

\[ \int_{\sum_{i=1}^{k}\sum_{j=1}^{n}x_{i,j}=1} \frac{x_{1,1}^{q_{1,1}}}{q_{1,1}!} \dotsm \frac{x_{k,n}^{q_{k,n}}}{q_{k,n}!} \, d\omega = \frac{1}{((k+1)n-1)!}. \]

Factorizing

\[ \operatorname{ch} E_{k,N}\Omega_{\pi,\Delta} = \frac{N^{(k+1)n-1}}{(k!)^{n}((k+1)n-1)!} \sum_{\sum q_{i,j}=n} \frac{(\lambda_{1}^{(1)})^{q_{1,1}}\dotsm(\lambda_{n}^{(k)})^{q_{k,n}}} {1^{\sum q_{1,j}}\dotsm k^{\sum q_{k,j}}} + O(N^{(k+1)n-2}). \]

By plain linear algebra manipulations, one then gets

\[ \operatorname{ch} E_{k,N}\Omega_{\pi,\Delta} = \frac{N^{(k+1)n-1}}{(k!)^{n}((k+1)n-1)!} \sum_{\sum q_{i}=n} \frac{h_{q_{1}}(\lambda^{(1)})\dotsm h_{q_{k}}(\lambda^{(k)})} {1^{q_{1}}\dotsm k^{q_{k}}} + O(N^{(k+1)n-2}), \]

where $h_{q}$ is the $q$th complete symmetric function. It remains to note that a definition of Segre classes is

\[ s_{q}(\Omega_{\pi,\Delta^{(i)}}) = ({-}1)^{q}h_{q}(\lambda^{(i)}). \]

This proves the sought formula for the asymptotic Euler characteristic, by the Riemann–Roch theorem.

Proof. We follow again Green and Griffiths [Reference Green and GriffithsGG80], with some slight modifications. Recall that one can fix some $i$ such that $\Delta ^{(p)}$ coincides with $\Delta ^{(\infty )}$ for $p>i$. Then

\begin{align*} & \sum_{q_{1}+\dotsb+q_{k}=n} \frac{ s_{q_{1}}(\Omega_{\pi,\Delta^{(1)}}) \dotsm s_{q_{k}}(\Omega_{\pi,\Delta^{(k)}}) } {1^{q_{1}}\dotsm k^{q_{k}}}\\ &\quad = \sum_{q_{1}+\dotsb+q_{i}+q=n} \biggl( \frac{ s_{q_{1}}(\Omega_{\pi,\Delta^{(1)}}) \dotsm s_{q_{i}}(\Omega_{\pi,\Delta^{(i)}}) } {1^{q_{1}}\dotsm i^{q_{i}}} \sum_{q_{i+1}+\dotsb+q_{k}=q} \frac{ s_{q_{i+1}}(\Omega_{\pi,\Delta^{(\infty)}}) \dotsm s_{q_{k}}(\Omega_{\pi,\Delta^{(\infty)}}) } {(i+1)^{q_{i+1}}\dotsm k^{q_{k}}} \biggr). \end{align*}

Reasoning in the exact same way as in [Reference Green and GriffithsGG80], we have

\begin{align*} & \sum_{q_{1}+\dotsb+q_{k}=n} \frac{ s_{q_{1}}(\Omega_{\pi,\Delta^{(1)}}) \dotsm s_{q_{k}}(\Omega_{\pi,\Delta^{(k)}}) } {1^{q_{1}}\dotsm k^{q_{k}}}\\ &\quad = \sum_{q_{1}+\dotsb+q_{i}+q=n} \underbrace{ \frac{ s_{q_{1}}(\Omega_{\pi,\Delta^{(1)}}) \dotsm s_{q_{i}}(\Omega_{\pi,\Delta^{(i)}}) } {1^{q_{1}}\dotsm i^{q_{i}}} }_{O(1)} \biggl( \frac{(\log k)^{q}}{q!}s_{1}(\Omega_{\pi,\Delta^{(\infty)}})^{q} + O((\log k)^{q-1}) \biggr). \end{align*}

Hence, keeping only the term in $(\log k)^{n}$ (for which $q_{1}=\dotsb =q_{i}=0$),

\[ ({-}1)^{n} \sum_{q_{1}+\dotsb+q_{k}=n} \frac{ s_{q_{1}}(\Omega_{\pi,\Delta^{(1)}}) \dotsm s_{q_{k}}(\Omega_{\pi,\Delta^{(k)}}) } {1^{q_{1}}\dotsm k^{q_{k}}} = \frac{(\log k)^{n}}{n!}c_{1}(\Omega_{\pi,\Delta^{(\infty)}})^{n} + O((\log k)^{n-1}). \]

This finishes the proof.

Theorem 3.1 (First main theorem) Assume that $g(V)\not \subset \mathrm {Supp}\,D$. One has

\[ T_{g}(r, \mathcal{O}(D)) = N(r,g^{{\ast}}D) + m_{g}(r,D) + O(1). \]

Theorem 3.2 [Nog85, Yam04a] Let $\xi$ be a meromorphic function on $V$ considered as a holomorphic function $V \to \mathbb {P}^{1}$. Then for any $\ell \geqslant 1$, one has

\[ \frac{1}{2\pi\deg\rho} \int_{\partial V(r)} \operatorname{log}^{+} \left \lvert\frac{({\partial^{\ell}}/{\partial t^{\ell}})\xi} {\xi}\right \rvert \cdot \rho^{{\ast}}\, d t \leqslant O(\operatorname{log}^{+} T_{\xi}(r,[\infty])) + O(\log r) \hspace{.5cm}\Vert. \]

The symbol $\Vert$ means that the inequality holds for $r\geqslant 0$ outside a set of finite linear measure and $\operatorname {log^{+}} x= \max \{ \log x,0 \}$.

Theorem 3.3 (Tautological inequality) For an ample line bundle $A\to Y$, one has

\[ T_{g_{[1]}}(r, \mathcal{O}_{ \mathbb{P}(\Omega_{Y})}(1)) \leqslant N(r,\mathrm{Ram}(\rho)) + O(\operatorname{log^{+}} T_{g}(r,A)) + O(\log r) \hspace{.5cm}\Vert. \]

Proof. We refer to the proof of Theorem A7.5.4 in [Reference RuRu01], which can easily be adapted. In order to use Theorem 3.2, recall that the orbifold jet coordinates of $g$ are obtained by applying ${\partial ^{\ell }}/{\partial t^{\ell }}$ to $\pi \circ g$ coordinatewise.

Theorem 3.5 (Orbifold tautological inequality) Let $g\colon V\to Y$ be the holomorphic lifting of an orbifold entire curve $f\colon \mathbb {C}\to (X,\Delta )$. For an ample line bundle $A\to X$, one has

\[ T_{g_{[1]}}(r, \mathcal{O}_{ \mathbb{P}(\Omega_{\pi,\Delta})}(1)) \leqslant O(\operatorname{log^{+}} T_{g}(r,\pi^{{\ast}}A)) + O(\log r) \hspace{.5cm}\Vert. \]

Proof. We follow the approach used by Vojta [Reference VojtaVoj11], to which we refer for the geometric interpretation of the proof. The rough idea is to see the integral in the lemma on logarithmic derivatives for jet differentials as a proximity function to infinity, in an appropriate compactification. Let $S$ be the total space of ${\Omega^{\vee}_{\pi}}, {\Delta }$ and let $\overline {S}= \mathbb {P}(\Omega _{\pi ,\Delta }\oplus \mathcal {O}_{Y})$. Let $[\infty ]$ denote the divisor $\overline {S}\setminus S$. Let $p\colon P\to \overline {S}$ be the blow-up of $\overline {S}$ along the image $[0]$ of the zero section, let $E$ denote its exceptional divisor and let $q\colon P\to \mathbb {P}(\Omega _{\pi ,\Delta })$. There is a lifting $g_{[1]}^{\diamond }$ of $g$ in $\mathbb {P}(\Omega _{\pi ,\Delta }\oplus \mathcal {O}_{Y})$ and a lifting $\phi$ to $P$. To sum up, one has the following commutative diagram.

One then has (cf. [Reference VojtaVoj11] for more details)

\[ p^{{\ast}} \mathcal{O}_{ \mathbb{P}(\Omega_{\pi,\Delta}\oplus \mathcal{O}_{Y})}(1) \cong q^{{\ast}} \mathcal{O}_{ \mathbb{P}(\Omega_{\pi,\Delta})}(1)\otimes \mathcal{O}(E) \cong q^{{\ast}} \mathcal{O}([\infty])\otimes \mathcal{O}(E). \]

Hence

\[ T_{g_{[1]}}(r, \mathcal{O}_{ \mathbb{P}(\Omega_{\pi,\Delta})}(1)) = T_{\phi}(r,q^{{\ast}} \mathcal{O}_{ \mathbb{P}(\Omega_{\pi,\Delta})}(1))+O(1) \\ = T_{g_{[1]}^{{\diamond}}}(r,[\infty])-T_{\phi}(r,E)+O(1). \]

Now, since $g$ is nonconstant, $\phi (V)\not \subset E$, and $T_{\phi }(r,E)$ is bounded from below. It remains to control $T_{g_{[1]}^{\diamond }}(r,[\infty ])$, using the first main theorem. By the lemma on logarithmic derivatives, $m_{g_{[1]}^{\diamond }}(r,[\infty ])$ is bounded from above by $O(\operatorname {log^{+}} T_{g}(r,\pi ^{\ast }A))+O(\log r)$. Lastly, since $g$ is the holomorphic lifting of an orbifold curve, the map $g_{[1]}^{\diamond }$ is holomorphic (cf. Proposition 2.3), and therefore $N_{g_{[1]}^{\diamond }}(r,[\infty ])=0$. This ends the proof.

Proof. For curves, the projection $p\colon \mathbb {P}(\Omega _{\pi ,\Delta })\to Y$ is an isomorphism and $\mathcal {O}(1)\cong p^{*}\Omega _{\pi ,\Delta }\cong p^{\ast }\pi ^{\ast }(K_{X}+\Delta )$. Therefore by Theorem 3.5 one has

\[ T_{f}(r,K_{X}+\Delta) \leqslant O(\operatorname{log^{+}} T_{f}(r,A)) + O(\log r) \hspace{.5cm}\Vert. \]

Therefore $f$ extends to an orbifold morphism $\bar {f}\colon ( \mathbb {P}^{1},D)\to (X,\Delta )$ where $D$ is necessarily supported at infinity. We have that $\deg (K_{ \mathbb {P}^{1}}+D)<0$ and thus $\bar {f}$ has to be constant by the Riemann–Hurwitz formula.

Proof. Considering the projectivization $p\colon \mathbb {P}(\Omega _{\pi ,\Delta })\to Y$, the symmetric differential $P$ can be seen as a global section $\tilde {P}\in H^{0}( \mathbb {P}(\Omega _{\pi ,\Delta }),\mathcal {L})$, where $\mathcal {L} := \mathcal {O}_{ \mathbb {P}(\Omega _{\pi ,\Delta })}(\ell )\otimes p^{\ast }(\pi ^{\ast }A^{\vee })$. Let $g_{[1]}$ be the lifting to $\mathbb {P}(\Omega _{\pi ,\Delta })$ of $g$ (such that $g=p\circ g_{[1]}$). Should $g^{\ast }P=\tilde {P}(g_{[1]})$ not vanish, then, by the boundedness, additivity and functoriality properties of the height function, one would get that

\[ T_{g_{[1]}}(r,\mathcal{L}) = \ell\cdot T_{g_{[1]}}(r, \mathcal{O}(1)) - T_{g}(r,\pi^{{\ast}}A) \]

is bounded from below. Theorem 3.5 then implies that $T_{f}(r,A)=T_{g}(r,\pi ^{\ast }A)=O(\log r)$. Therefore $f$ extends to an orbifold morphism $\bar {f}\colon ( \mathbb {P}^{1},D)\to (X,\Delta )$ where $D$ is necessarily supported at infinity. Since $\deg (K_{ \mathbb {P}^{1}}+D)<0$, $g^{\ast }P$ has to vanish, a contradiction.

Proof. Recall the graduation obtained from the Green–Griffiths filtration

\[ \operatorname{Grad_{{\bullet}}} E_{k,N}\Omega_{\pi,\Delta} \otimes \pi^{{\ast}}A^{{\vee}} = \bigoplus_{\lVert \ell \rVert=N} S^{\ell_{1}}\Omega_{\pi,\Delta} \otimes S^{\ell_{2}}\Omega_{\pi,\Delta^{(2)}} \otimes \dotsb \otimes S^{\ell_{k}}\Omega_{\pi,\Delta^{(k)}} \otimes \pi^{{\ast}}A^{{\vee}}, \]

and remark that for $j=1,\dotsc ,k$, one has $S^{\ell _{j}}\Omega _{\pi ,\Delta ^{(j)}} \subseteq S^{\ell _{j}} \Omega _{\pi ,\Delta } \subseteq (\Omega _{\pi ,\Delta })^{\otimes \ell _{j}}$. Recall also from [Reference Campana and PăunCP15] that if for some integer $q>0$ and some ample line bundle $A$, the vector bundle $(\Omega _{\pi ,\Delta })^{\otimes q}\otimes \pi ^{\ast }A^{\vee }$ has a nonzero global section, then the pair $(X,\Delta )$ is of general type. One infers that under the assumption of the lemma, for any $\ell$, the graded bundle $\operatorname {Grad_{\bullet }} E_{k,N}\Omega _{\pi ,\Delta }\otimes \pi ^{\ast }A^{\vee }$ has no global section. This fact holds a fortiori for the bundle $E_{k,N}\Omega _{\pi ,\Delta }\otimes \pi ^{\ast }A^{\vee }$ itself.

Proof. From [Reference Campana and WinkelmannCW09] we have that $(X, \Delta )$ contains an orbifold entire curve $f\colon \mathbb {C}\to (X,\Delta )$ if and only if $\deg (K_{X}+\Delta )\leqslant 0$. Therefore, if an orbifold curve $(X,\Delta )$ admits a nonconstant orbifold entire curve $f\colon \mathbb {C}\to (X,\Delta )$ then $(X,\Delta )$ is not of general type and the previous lemma gives $H^{0}(Y,E_{k,N}\Omega _{\pi ,\Delta }\otimes \pi ^{\ast }A^{\vee }) = \{ 0 \}$.

Theorem 3.10 Let $(X,\Delta )$ be a smooth orbifold pair, and let $\pi \colon Y\to X$ be an adapted covering. If $P\in H^{0}(Y,E_{k,N}\Omega _{\pi ,\Delta }\otimes \pi ^{\ast }A^{\vee })$ is a global orbifold jet differential vanishing on an ample divisor $A\to X$, then for any holomorphic lifting $g\colon V\to Y$ of an orbifold entire curve, one has $g^{\ast }P\equiv 0$.

Proof. We follow the classical proof (see for example [Reference RuRu01, Theorem A7.5.5]). Let us show that $f$ extends to a rational curve. Then, one gets an orbifold morphism $\bar {f}\colon ( \mathbb {P}^{1},D)\to (X,\Delta )$, where $D$ is necessarily supported at infinity, together with a holomorphic lifting $\bar {g}$. According to Remark 2.5, the jet differential $P$ then pullbacks to a jet differential on $( \mathbb {P}^{1},D)$. Now, by construction, $( \mathbb {P}^{1},D)$ admits an (orbifold) entire curve. By Corollary 3.9, it follows that the pullback of $P$ (and therefore $g^{\ast }P$) vanishes identically.

To show that $f$ extends to a rational curve, by a classical result, it suffices to establish that $T_{f}(r,A) = O(\log r)$, or equivalently that $T_{g}(r,\pi ^{\ast }A)=O(\log r)$.

Since $P$ vanishes on $A$, viewing $g^{\ast }P$ as a holomorphic function $V\to \mathbb {P}^{1}$, one has

\[ T_{g}(r,\pi^{{\ast}}A) \leqslant O(T_{g^{{\ast}}P}(r,[\infty])). \]

Now, recall from Definition 2.6 that the function $g^{\ast }P\colon V\to \mathbb {C}$ is holomorphic. Hence, one has $N_{g^{\ast }P,[\infty ]}\equiv 0$. Furthermore, applying Corollary 3.4, one obtains that the proximity function to infinity of $g^{\ast }P$ satisfies

\[ m_{g^{{\ast}}P}(r,[\infty]) = O(\operatorname{log^{+}} T_{g}(r,\pi^{{\ast}}A)) + O(\log r) \hspace{.5cm}\Vert. \]

Therefore, one has

\[ T_{g}(r,\pi^{{\ast}}A) \leqslant O(\operatorname{log^{+}} T_{g}(r,\pi^{{\ast}}A)) + O(\log r). \]

It follows that $T_{g}(r,\pi ^{\ast }A) = O(\log r)$, which ends the proof.

Proof. It follows at once from Remark 2.7 and from Theorem 3.10.

Theorem 3.12 Let $(X,\Delta )$ be a smooth orbifold surface of general type with a holomorphic foliation $\mathcal {F}$. Any orbifold entire curve tangent to $\mathcal {F}$ is algebraically degenerate.

Proof. We suppose that $f\colon \mathbb {C}\to (X, \Delta )$ is a Zariski-dense orbifold curve. Let us prove that

\[ T(K_{X}+\Delta) \leqslant 0, \]

thus contradicting that $(X, \Delta )$ is of general type.

Let $S \subset \mathbb {P}(\Omega _{X}(\log D))$ be the surface induced by the foliation $\mathcal {F}$ and let $\pi \colon S\to X$ be the projection. We know that $S$ contains $f_{[1]}( \mathbb {C})$ and, supposing that $S$ dominates $X$, $S$ is equipped with a foliation $\mathcal {F}_{0}$. After some blow ups, we obtain a foliated smooth surface $(S_{m}, D_{m}, \mathcal {F}_{m}) \to (S, \pi ^{-1}(D), \mathcal {F}_{0})$, i.e. $S_{m}$ is smooth, $D_{m}$ is normal crossing and $\mathcal {F}_{m}$ has reduced singularities. Let $D_{m}=C+B$ where $C$ is the invariant part of $D_{m}$ by $\mathcal {F}_{m}$. We have an exact sequence

\[ 0 \to \mathcal{N}^{*}(C) \to T^{*}_{S_{m}}(\log D_{m}) \to K_{ \mathcal{F}_{m}}(B).\mathcal{I}_{Z} \to 0, \]

where $\mathcal {I}_{Z}$ is an ideal supported on the singularity set $Z$ of $\mathcal {F}_{m}$.

Now, we apply the logarithmic tautological inequality (4) which gives

\[ T_{{f_{m}}_{[1]}}(r,L) \leqslant N_{1}(r,f_{m}^{{\ast}}D_{m}) + O(\operatorname{log^{+}} T_{f}(r,A)+ \log r) \hspace{.5cm}\Vert, \]

where $L=\mathcal {O}_{ \mathbb {P}(\Omega _{\tilde {S}}(\log D_{m}))}(1)$, $f_{m}$ and ${f_{m}}_{[1]}$ are the lifts of $f$.

We have

\[ L_{|Y}=p^{*}K_{{ \mathcal{F}_{m}}}(B) \otimes\mathcal{O}({-}E_{m}), \]

where $L_{|Y}$ denotes the restriction of $L$ to the graph $Y$ of the foliation, $p\colon Y\to S_{m}$ the projection and $E_{m}$ is the exceptional divisor.

Therefore we obtain

\[ T_{{f}, K_{X}+D}(r) \leqslant T_{{f_{m}}, K_{S_{m}}+D_{m}}(r); \]

hence

\[ T_{{f}, K_{X}+D}(r) \leqslant N_{1}(r,f^{{\ast}}D) + T_{{f_{m}}_{[1]}}(r,E_{m}) + T_{f_{m}}(r,\mathcal{N}^{*}(C)) + O(\operatorname{log^{+}} T_{f}(r,A)+ \log r) \hspace{.5cm}\Vert. \]

Since $f$ is an orbifold curve, we have

\[ m_{i}N_{1}(r,f^{{\ast}}\Delta_{i}) \leqslant N(r,f^{{\ast}}\Delta_{i}) \leqslant T_{f}(r,\Delta_{i}). \]

This gives

\[ T(K_{X}+\Delta) \leqslant T'_{m}(E_{m}) + T_{m}(\mathcal{N}^{*}(C)), \]

where $T'_{m}$ is the current associated to ${f_{m}}_{[1]}$.

To finish the proof, we shall now use the two following results of Brunella [Reference BrunellaBru99] (and McQuillan [Reference McQuillanMcQ98]): $T_{m}(\mathcal {N}^{*}(C)) \leqslant 0$ and $T_{m}(E_{m}) \to 0$ as $m \to \infty$ i.e. performing infinitely many blow ups.

Theorem 3.13 Let $(X, \Delta )$ be a smooth orbifold surface of general type with a $\Delta$-holomorphic foliation $\mathcal {F}$ with reduced singularities, then any (orbifold or not) entire curve tangent to $\mathcal {F}$ is algebraically degenerate.

Proof. We suppose that $f\colon \mathbb {C}\to X$ is a Zariski-dense curve tangent to $\mathcal {F}$. We have the exact sequence $0 \to \mathcal {F} \to T_{X} \to \mathcal {N}$. We have $T(K_{ \mathcal {F}}) \leqslant 0$ by a result of McQuillan (see [Reference BrunellaBru99]). We also have $T(N^{*}(\Delta )) \leqslant T(N^{*}(\lceil \Delta \rceil ))\leqslant 0$ by the already mentioned result of Brunella. Therefore we obtain, $T(K_{X}+\Delta )=T(K_ \mathcal {F}+N^{*}(\Delta )) \leqslant 0$, giving a contradiction.

Proof. By Seidenberg's theorem we can do some blow ups such that on $\tilde {X}$ the induced foliation $\tilde { \mathcal {F}}$ has only reduced singularities. Let us denote by $\tilde {\Delta }$ the strict transform of $\Delta$. Then $(\tilde {X}, \tilde {\Delta })$ is a smooth orbifold of general type thanks to the hypothesis that $(X, \Delta )$ is canonical. Therefore we can apply Theorem 3.13 to finish the proof.

Theorem 4.1 Let $(X,\Delta )$ be a smooth orbifold surface of general type. If one has

\[ H^{0}\big(X,{\bigoplus_{N\geqslant 1}}S^{N}\Omega_{X,\Delta}\otimes L^{{\vee}}\big) \neq \{ 0 \}, \]

for some ample line bundle $L$ on $X$, then there exists a proper closed subvariety $Z\subsetneq X$ such that every nonconstant orbifold entire curve $f\colon \mathbb {C}\to (X,\Delta )$ satisfies $f( \mathbb {C})\subseteq Z$.

Proof. Suppose there is a nontrivial section $s\in H^{0}\big (X,{ {\bigoplus} _{N\geqslant 1}}S^{N}\Omega _{X,\Delta }\otimes L^{\vee }\big )$. Then by Corollary 3.11, any entire orbifold curve $f\colon \mathbb {C}\to (X,\Delta )$ satisfies $f^{\ast }s\equiv 0$. In other words, $f$ is tangent to the (multi-)foliation defined by $s$. Then the same proof as in Theorem 3.12 implies that $f$ is algebraically degenerate and $f$ extends to a morphism $f\colon ( \mathbb {P}^{1}, \Delta ') \to (X,\Delta )$, such that $\deg (K_{( \mathbb {P}^{1}, \Delta ')})\leqslant 0$. Theorem 6.6 in [Reference RousseauRou10] gives that there are only finitely many such curves in $X$, since $(X,\Delta )$ is of general type. This finite set defines a proper algebraic subset $Z\subsetneq X$.

Theorem 4.2 A smooth orbifold surface of general type $(X,\Delta )$ such that

\[ \chi_{1}(\pi,\Delta) = (c_{1}(\Omega_{\pi,\Delta})^{2}-c_{2}(\Omega_{\pi,\Delta})) > 0 \]

satisfies the orbifold Green–Griffiths–Lang Conjecture A.

Proof. By Riemann–Roch, if $\chi _{1}(\pi ,\Delta ) = (c_{1}(\Omega _{\pi ,\Delta })^{2}-c_{2}(\Omega _{\pi ,\Delta })) > 0$ then

\[ h^{0}(Y,S^{m}\Omega_{\pi,\Delta})+h^{2}(Y,S^{m}\Omega_{\pi,\Delta}) \geqslant \frac{m^{3}}{6}(c_{1}(\Omega_{\pi,\Delta})^{2}-c_{2}(\Omega_{\pi,\Delta}))+O(m^{2}). \]

Moreover, by duality,

\[ H^{2}(Y,S^{m}\Omega_{\pi,\Delta})=H^{0}(Y,K_{Y}\otimes S^{m}\Omega_{\pi,\Delta} \otimes \mathcal{O}({-}m \cdot \pi^{*}(K_{X}+\Delta)). \]

Since $K_{X}+\Delta$ is big, for sufficiently large $m$, $m\cdot \pi ^{\ast }(K_{X}+\Delta )-K_{Y}$ is effective. Then

\[ H^{2}(Y,S^{m}\Omega_{\pi,\Delta}) \hookrightarrow H^{0}(Y,S^{m}\Omega_{\pi,\Delta}). \]

This implies that

\[ h^{0}(Y,S^{m}\Omega_{\pi,\Delta}) \geqslant \frac{m^{3}}{12}(c_{1}(\Omega_{\pi,\Delta})^{2}-c_{2}(\Omega_{\pi,\Delta}))+O(m^{2}). \]

Therefore the orbifold cotangent bundle $\Omega _{\pi ,\Delta }$ is big and $(X,\Delta )$ satisfies the hypothesis of Theorem 4.1.

Proof. Considering the conjecture and the definition of orbifold curves, one can always remove some components (i.e. take $m_{i}=1$), and one can always assume that all remaining orbifold multiplicities are equal to $2$. Let us thus consider an orbifold divisor with $c$ components, of respective degrees $d_{1},\dotsc ,d_{c}$, having all orbifold multiplicity $2$. By Theorem 4.2, it is then sufficient to prove that the orbifold pairs under consideration are of general type and satisfy $\chi _{1}=s_{2}(\Omega _{\pi ,\Delta })>0$. Namely, these have to satisfy $d_{1}+\dotsb +d_{c}>6$ and

\[ \chi_{1} = \deg(\pi) \biggl( 6 -3\frac{\sum_{1\leqslant i\leqslant c}d_{i}}{2} + \frac{\sum_{1\leqslant i < j\leqslant c}d_{i}d_{j}-\sum_{1\leqslant i\leqslant c}d_{i}^{2}}{4} \biggr) > 0. \]

The first condition is clearly satisfied. The partial second derivative with respect to $d_{i}$ of the second expression is $(-1/2)$, whence it is a concave function. Let $d_{m}$ be the minimum of the $d_{i}$ and $d_{M}$ their maximum. On the convex set $\{ d_{m}\leqslant d_{i}\leqslant d_{M}\forall i \}\subseteq \mathbb {R}^{c}$, the minimum of the concave function under consideration is attained in an extremal point. At this point, $c_{m}$ of the $d_{i}$ have the value $d_{m}$ and the others have the value $d_{M}$. The minimum value is then

\[ 6 - 3c_{m} \frac{d_{m}}{2} - 3(c-c_{m}) \frac{d_{M}}{2} + c_{m}(c_{m}-3) \frac{d_{m}^{2}}{8} + c_{m}(c-c_{m}) \frac{d_{m}d_{M}}{4} + (c-c_{m})(c-c_{m}-3) \frac{d_{M}^{2}}{8}. \]

Moreover, the derivative of this value with respect to $d_{M}$ must be nonnegative, and the derivative with respect to $d_{m}$ must be nonpositive, namely

\[ (c-c_{m}) \biggl( \frac{-3}{2} + c_{m} \frac{d_{m}}{4} + (c-c_{m}-3) \frac{d_{M}}{4} \biggr) \geqslant 0 \]

and

\[ c_{m} \biggl( \frac{-3}{2} + (c_{m}-3) \frac{d_{m}}{4} + (c-c_{m}) \frac{d_{M}}{4} \biggr) \leqslant 0. \]

One infers that if $c_{m}\not \in \{ 0,c \}$ then

\[ \frac{3}{4}(d_{m}-d_{M}) = \biggl( \frac{-3}{2} + c_{m} \frac{d_{m}}{4} + (c-c_{m}-3) \frac{d_{M}}{4} \biggr) - \biggl( \frac{-3}{2} + (c_{m}-3) \frac{d_{m}}{4} + (c-c_{m}) \frac{d_{M}}{4} \biggr) \geqslant 0. \]

Therefore $d_{m}=d_{M}$. Hence, in any case, the minimum is attained in a point where all degrees are equal. We can thus assume that all degrees are $d$. Then

\[ \chi_{1} = \deg(\pi) \biggl( 6 -\frac{3c}{2}d +\frac{c(c-3)}{8}d^{2} \biggr). \]

It remains to check that this polynomial in $d$ has a positive leading coefficients for $c\geqslant 4$, that its discriminant is negative for $c>12$, and to compute the largest root for $4\leqslant c\leqslant 12$. These are easy computations.

Proof. By Corollary 4.4, all orbifold entire curves $\mathbb {C}\to ( \mathbb {P}^{2},\Delta )$ are contained in algebraic curves. Therefore, by Theorem 1.4, these are constant.

Theorem 4.6 [GP16] Consider an adapted covering $\pi \colon Y\to (X,\Delta )$ of a smooth orbifold pair with $K_{X}+\Delta$ ample. For all $r>s$ one has

\[ H^{0} (Y,\, ({\Omega^{\vee}_{\pi}}, {\Delta})^{{\otimes} r} \otimes (\Omega_{\pi,\Delta})^{{\otimes} s}) = \{ 0 \}. \]

Proof. Since we are in the surface case, it is sufficient to prove that for large $N$,

\[ H^{2}(Y, E_{k,N}\Omega_{\pi,\Delta}) = \{ 0 \}. \]

We use the graduation induced by the Green–Griffiths filtration

\[ \operatorname{Grad_{{\bullet}}} E_{k,N}\Omega_{\pi,\Delta} = \bigoplus_{\lVert \ell \rVert=N} S^{\ell_{1}}\Omega_{\pi,\Delta^{(1)}} \otimes S^{\ell_{2}}\Omega_{\pi,\Delta^{(2)}} \otimes \dotsb \otimes S^{\ell_{k}}\Omega_{\pi,\Delta^{(k)}}. \]

This shows that it is actually sufficient to prove that for all $\ell \in \mathbb {N}^{k}$ with $\lVert \ell \rVert =N$

\[ H^{2} (Y, S^{\ell_{1}}\Omega_{\pi,\Delta^{(1)}} \otimes S^{\ell_{2}}\Omega_{\pi,\Delta^{(2)}} \otimes \dotsb \otimes S^{\ell_{k}}\Omega_{\pi,\Delta^{(k)}} ) = \{ 0 \}. \]

Using Serre duality, this is equivalent to

\[ H^{0} (Y, S^{\ell_{1}}{\Omega^{\vee}_\pi}, {\Delta^{(1)}} \otimes S^{\ell_{2}}{\Omega^{\vee}_\pi},{\Delta^{(2)}} \otimes \dotsb \otimes S^{\ell_{k}}{\Omega^{\vee}_\pi,}{\Delta^{(k)}} \otimes \mathcal{O}(K_{Y}) ) = \{ 0 \}. \]

Now, we remark that we have an injection

\[ S^{\ell_{1}}{\Omega^{\vee}_\pi},{\Delta^{(1)}} \otimes S^{\ell_{2}}{\Omega^{\vee}_\pi},{\Delta^{(2)}} \otimes \dotsb \otimes S^{\ell_{k}}{\Omega^{\vee}_\pi},{\Delta^{(k)}} \hookrightarrow ({\Omega^{\vee}_\pi},{\Delta^{(k)}})^{{\otimes}\lvert \ell \rvert}. \]

On the other hand, choosing $p$ such that $p\cdot \pi ^{\ast }(K_{X}+\Delta ^{(k)})-K_{Y}>0$, we obtain

\[ \mathcal{O}(K_{Y}) \hookrightarrow \mathcal{O}(p\cdot\pi^{{\ast}}(K_{X}+\Delta^{(k)})) \hookrightarrow (\Omega_{\pi,\Delta^{(k)}})^{{\otimes} 2p}. \]

From Theorem 4.6, we see that

\[ S^{\ell_{1}}{\Omega^{\vee}_\pi},{\Delta^{(1)}} \otimes S^{\ell_{2}}{\Omega^{\vee}_\pi},{\Delta^{(2)}} \otimes \dotsb \otimes S^{\ell_{k}}{\Omega^{\vee}_\pi},{\Delta^{(k)}} \otimes \mathcal{O}(K_{Y}) \hookrightarrow ({\Omega^{\vee}_\pi},{\Delta^{(k)}})^{{\otimes}\lvert \ell \rvert} \otimes (\Omega_{\pi,\Delta^{(k)}})^{{\otimes} 2p} \]

has no global sections as soon as $\lvert \ell \rvert >2p$. Since $\lvert \ell \rvert \geqslant {N}/{k}$, this is achieved as soon as $N$ is large enough.

Proof. If $a > 2d/(d-3)$ then $K_{ \mathbb {P}^{2}}+\Delta ^{(2)}>0$, which allows us to apply Corollary 4.7. Now, for $k=2$, $a\geqslant 2$, Proposition 2.16 yields

\[ \chi(E_{2,N}\Omega_{\pi,\Delta}) = \frac{N^{5}}{1920} \frac{\deg(\pi)}{a^{2}} \bigl( (48-27d+2d^{2})a^{2} -12(d-3)d a +12d^{2} \bigr) +O(N^{4}). \]

The result follows.

Theorem 4.11 If $(X,\Delta )$ is a smooth orbifold surface with $K_{X}\equiv 0$, $\Delta$ ample and $\chi _{k}(\pi ,\Delta )>0$, then for any ample line bundle $L\to X$,

\[ H^{0}\biggl(\displaystyle \mathop{\bigoplus}\limits_{N\geqslant 1}E_{k,N}\Omega_{\pi,\Delta}\otimes L^{{\vee}}\biggr) \neq \{ 0 \}. \]

Proof. The case $k=1$ follows at once from Proposition 2.16 and Corollary 4.7.

Assume now that $\chi _{k}(\pi ,\Delta )>0$ for $k>1$. Since

\[ H^{0}(\oplus_{N\geqslant 1}E_{k-1,N}\Omega_{\pi,\Delta}\otimes L^{{\vee}}) \hookrightarrow H^{0}(\oplus_{N\geqslant 1}E_{k,N}\Omega_{\pi,\Delta}\otimes L^{{\vee}}), \]

reasoning by induction, one can moreover assume that $\chi _{k-1}(\pi ,\Delta )\leqslant 0$. We then claim that $K_{X}+\Delta ^{(k)}>0$, and the result follows by Corollary 4.7.

Indeed, if not, then $K_{X}+\Delta ^{(k)}\equiv 0$, i.e. $\Delta ^{(k)}=\varnothing$ and

\[ \chi_{k}(\pi,\Delta) = \chi_{k-1}(\pi,\Delta) + \sum_{i=1}^{k-1} \frac{s_{1}(\Omega_{\pi,\Delta^{(i)}})}{i} \frac{s_{1}(\Omega_{\pi,\Delta^{(k)}})}{k} + \frac{s_{2}(\Omega_{\pi,\Delta^{(k)}})}{k^{2}} = \chi_{k-1}(\pi,\Delta) + \frac{s_{2}(\Omega_{X})}{k^{2}}. \]

But by the classification of surfaces with trivial canonical bundle, $s_{2}(\Omega _{X})=-c_{2}(X)\leqslant 0$ and this yields a contradiction, since then $0< \chi _{k}(\pi ,\Delta ) \leqslant \chi _{k-1}(\pi ,\Delta ) \leqslant 0$.

Proof. Recall that for $k$ big enough, the positivity of the Euler characteristic is given by the positivity of the coefficient

\[ \chi_{k}(\pi,\Delta) := ({-}1)^{n} \sum_{q\in\mathbb{N}^{k}\colon\lvert q \rvert=n} \frac{s_{q_{1}}(\Omega_{\pi,\Delta^{(1)}})}{1^{q_{1}}} \dotsm \frac{s_{q_{k}}(\Omega_{\pi,\Delta^{(k)}})}{k^{q_{k}}}. \]

Now, from the residue short exact sequence

\[ s(\Omega_{\pi,\Delta}) = s(\Omega_{X}) \prod_{i} \frac{(1-c_{1}(D_{i}))}{(1-c_{1}(D_{i})/m_{i})}. \]

If $X$ is a surface with trivial canonical bundle, a formal computation yields that for $k\geqslant m_{i}$, for all $i$:

\begin{align*} \chi_{k}(\pi,\Delta) &={-}\sum_{1\leqslant j\leqslant k} \biggl(\frac{1}{j^{2}}\biggr) c_{2}(X) \\ & \quad+ \sum_{i_{1} < i_{2}} \biggl( \sum_{2\leqslant j_{1}\leqslant m_{i_{1}}} \frac{1}{j_{1}} \sum_{2\leqslant j_{2}\leqslant m_{i_{2}}} \frac{1}{j_{2}} \biggr) c_{1}(D_{i_{1}})c_{1}(D_{i_{2}}) \\ &\quad+ \sum_{i} \biggl( \sum_{2\leqslant j_{1} < j_{2}\leqslant m_{i}} \frac{1}{j_{1}j_{2}} -\frac{(m_{i}-1)}{2m_{i}} \biggr) c_{1}(D_{i})^{2}. \end{align*}

Recall that $c_{2}(X)\geqslant 0$. In the one-component case one gets

\[ \chi_{k}(\pi,\Delta) \geqslant \biggl( \sum_{2\leqslant j_{1} < j_{2}\leqslant m} \frac{1}{j_{1}j_{2}} -\frac{(m-1)}{2m} \biggr) c_{1}(D)^{2} - \frac{\pi^{2}}{6} c_{2}(X). \]

A numerical exploration shows that the coefficient $c_{m}$ of $c_{1}(D)^{2}$ becomes positive for $m\geqslant 5$ and that then $\pi ^{2}/(6c_{m})\leqslant 10$.

Proof. Suppose that for some $k$ and $N$, $H^{0}(Y,E_{k,N}\Omega _{\pi ,\Delta })\neq 0$. Then one infers from Proposition 2.14 that for some $\ell _{1},\dotsc ,\ell _{k}$ with $\lVert \ell \rVert =N$

\[ S^{\ell_{1}}\Omega_{\pi,\Delta^{(1)}} \otimes S^{\ell_{2}}\Omega_{\pi,\Delta^{(2)}} \otimes \dotsb \otimes S^{\ell_{k}}\Omega_{\pi,\Delta^{(k)}} \]

has some nonzero global sections. Note that $\Omega _{\pi ,\Delta ^{(\infty )}}=\pi ^{\ast }\Omega _{ \mathbb {P}^{n}}$. Since ${\Omega^{\vee}}_{ \mathbb {P}^{n}}$ is globally generated, one obtains nonzero global sections of

\[ S^{\ell_{1}}\Omega_{\pi,\Delta^{(1)}} \otimes \dotsb \otimes S^{\ell_{p}}\Omega_{\pi,\Delta^{(p)}} \]

for the largest $p\leqslant k$ such that $p < m$ (i.e. for which $\Delta ^{(p)}>\Delta ^{(\infty )}=\varnothing$).

We remark that a nonzero section $\sigma$ of $E_{k,N}\Omega _{\pi ,\Delta }$ can be made invariant to yield a nonzero section of $E_{k,gN}\Omega _{X,\Delta }$, where $g$ is the order of the Galois group of the covering $\pi \colon Y\to (X,\Delta )$. It is obtained by taking the pushforward along $\pi$ of the product of the Galois conjugates of $\sigma$, which are all nonzero. Applying this result for $k=1$, one deduces the existence of some nonzero global sections of

\[ S^{g\ell_{1}}\Omega_{X,\Delta^{(1)}} \otimes \dotsb \otimes S^{g\ell_{p}}\Omega_{X,\Delta^{(p)}} \subseteq S^{g\ell_{1}}\Omega_{ \mathbb{P}^{n}}(\log H) \otimes \dotsb \otimes S^{g\ell_{p}}\Omega_{ \mathbb{P}^{n}}(\log H). \]

The bundle on the right is a product of symmetric powers of the logarithmic cotangent bundle of a hypersurface in $\mathbb {P}^{n}$, with less than $n$ factors. By the Pieri rule, all partitions in its direct sum decomposition into Schur powers have therefore less than $n$ parts. This yields the sought contradiction, since these Schur powers have no sections, by the vanishing theorem of Brückmann–Rackwitz [Reference Brückmann and RackwitzBR90] (see [Reference DiverioDiv08, Reference DiverioDiv09]).

Proof. Let us first observe that $\Omega _{A}(\log D)$ is nef. Since $\Omega _{A}$ is globally generated, one is reduced to verify the nefness over $D$. On $D$, one has the following short exact sequence:

\[ 0 \to \Omega_{D} \to \Omega_{A}(\log D)\rvert_{D} \to \mathcal{O}_{D} \to 0. \]

Here, as a quotient of $\Omega _{A}\rvert _{D}$, the vector bundle $\Omega _{D}$ is nef. Thus, as an extension of nef vector bundles, $\Omega _{A}(\log D)\rvert _{D}$ is nef.

Consider a partition $\lambda$, and recall (e.g. [Reference DemaillyDem88, Reference ManivelMan94]) that the Schur bundle $S_{\lambda }(\Omega _{A}(\log D))$ is then the direct image of a nef line bundle $\mathscr {L}$ on the flag bundle associated to $\lambda$. Namely, let $1\leqslant j_{1} < j_{2} < \dotsb < j_{m}\leqslant n$ be the jumps of $\lambda$, for a certain $m\leqslant n$ (i.e. $\lambda _{i} > \lambda _{i+1}\iff i\in \{ j_{1},\dotsc ,j_{m} \}$), and let $F$ be the bundle of flags of subspaces with codimension $j_{1},\dotsc ,j_{m}$ in the fibers of $\Omega _{A}(\log D)$. Let $U^{j_{0}},\dotsc ,U^{j_{m+1}}$ be the universal subbundles of codimension $j_{0} < j_{1} < \dotsc < j_{m}\leqslant j_{m+1}$ on $F$, where by convention $j_{0} := 0$ and $j_{m+1} := n$. Then

\[ \mathscr{L} := \mathop{\bigotimes}\limits_{p=1}^{m} \det(U^{j_{p-1}}/U^{j_{p}})^{{\otimes} \lambda_{j_{p}}}. \]

We will now study the bigness of $\mathscr {L}$. To prove that $\mathscr {L}$ is not big, it is sufficient to observe that the Segre number $s_{n}(\mathscr {L})$ is zero. Using the Gysin formula from [Reference Darondeau and PragaczDP17, Proposition 1.2] for the flag bundle $F\to A$ (we transform a little bit), one gets the following expression for $s_{n}(\mathscr {L})$:

\begin{align*} &({-}1)^{n} [t_{1}^{n}\dotsm t_{n}^{1}] \\ &\quad \times\biggl( (\lambda_{1}t_{1}+\dotsb+\lambda_{n}t_{n})^{n} \mathop{\prod}_{p=1}^{m}(t_{j_{p}+1}\dotsm t_{j_{p+1}})^{{-}j_{p}} \mathop{\prod}_{1\leqslant i < j\leqslant n}(t_{i}-t_{j}) \mathop{\prod}_{1\leqslant i\leqslant n}t_{i}s_{{1}/{t_{i}}}(\Omega_{A}(\log D)) \biggr), \end{align*}

where for a monomial $m$ and a Laurent series $P$ in the formal variables $t_{1},\dotsc ,t_{n}$, $[m](P)$ means the coefficient of $m$ in $P$.

Now, the residue exact sequence on $A$ reads as follows:

\[ 0 \to \Omega_{A} \to \Omega_{A}(\log D) \to \mathcal{O}_{D} \to 0. \]

Therefore, by the Whitney sum formula, we obtain the equality of total Segre classes:

\[ s(\Omega_{A}(\log D))=s(\Omega_{A})\cdot s(\mathcal{O}_{D})=s(\Omega_{A})\cdot c( \mathcal{O}({-}D)). \]

The last equality follows again from the Whitney sum formula applied on the short exact sequence $0\to \mathcal {O}_{A}(-D)\to \mathcal {O}_{A}\to \mathcal {O}_{D}\to 0$. The bundle $\Omega _{A}$ being trivial we obtain $s(\Omega _{A}(\log D))=1-c_{1}(D)$. Replacing in the above expression, the number $s_{n}(\mathscr {L})$ becomes

\[ ({-}1)^{n} [t_{1}^{n}\dotsm t_{n}^{1}] \biggl( (\lambda_{1}t_{1}+\dotsb+\lambda_{n}t_{n})^{n} \mathop{\prod}_{p=1}^{m}(t_{j_{p}+1}\dotsm t_{j_{p+1}})^{{-}j_{p}} \mathop{\prod}_{1\leqslant i < j\leqslant n}(t_{i}-t_{j}) \mathop{\prod}_{1\leqslant i\leqslant n}(t_{i}-c_{1}(D)) \biggr). \]

This coefficient is clearly a linear combination of $1,\dotsc ,c_{1}(D)^{n}$ but, for dimensional reasons, the only such number that is nonzero on $A$ is $c_{1}(D)^{n}$. In other words

\[ s_{n}(\mathscr{L}) = [t_{1}^{n}\dotsm t_{n}^{1}] \biggl( (\lambda_{1}t_{1}+\dotsb+\lambda_{n}t_{n})^{n} \mathop{\prod}_{p=1}^{m}(t_{j_{p}+1}\dotsm t_{j_{p+1}})^{{-}j_{p}} \mathop{\prod}_{1\leqslant i < j\leqslant n}(t_{i}-t_{j}) \biggr) c_{1}(D)^{n}. \]

The degree of the polynomial under consideration is $n(n+1)/2-\sum _{p=1}^{m}(j_{p+1}-j_{p})j_{p}$. As a consequence, if $\sum _{p=1}^{m}(j_{p+1}-j_{p})j_{p}>0$, the coefficient of $t_{1}^{n}\dotsm t_{n}^{1}$ is $0$. To conclude, it remains to observe that $\sum _{p=1}^{m}(j_{p+1}-j_{p})j_{p}=0$ if and only if $S_{\lambda }(\Omega _{A}\log D)$ is a tensor power of the canonical bundle (i.e. $j_{1}=n=j_{m+1}$).

Now, since $\mathscr {L}$ is relatively ample and $p_{\ast }\mathscr {L}=S^{\lambda }\Omega _{A}(\log D)$, where $p\colon F\to A$,

\[ \exists L\text{ ample, } H^{0}(A,S^{\lambda}\Omega_{A}(\log D)\otimes L^{{\vee}})\neq 0 \implies \mathscr{L}\text{ big}. \]

The only if direction follows directly from the fact that $(A,D)$ is of log general type.

Proof. By Theorem 1.5, $(A,\Delta ) := (A, (1- {1}/{m})D)$ satisfies Conjecture A.

Let $\pi \colon Y\to (A,\Delta )$ be an adapted covering for the pair $(A,\Delta )$. Suppose that for some $k$ and $N$, $H^{0}(Y,E_{k,N}\Omega _{\pi ,\Delta }\otimes \pi ^{\ast }L^{\vee })\neq 0$ for some ample line bundle $L$. Then, since $m\leqslant n$, one infers from Proposition 2.14 and from the triviality of $\Omega _{\pi ,\Delta ^{(n)}}=\pi ^{\ast }\Omega _{A}$ that for some $\ell _{1},\dotsc ,\ell _{n-1}$

\[ S^{\ell_{1}}\Omega_{\pi,\Delta^{(1)}} \otimes \dotsb \otimes S^{\ell_{n-1}}\Omega_{\pi,\Delta^{(n-1)}} \otimes \pi^{{\ast}}L^{{\vee}} \]

has some nonzero global sections. This would imply that

\[ S^{g\ell_{1}}\Omega_{A}(\log D) \otimes \dotsb \otimes S^{g\ell_{n-1}}\Omega_{A}(\log D) \otimes (L^{g^{n-1}})^{{\vee}} \]

has nonzero global sections ($g := \lvert \operatorname {Aut}(\pi ) \rvert$)). Combined with the previous proposition, this yields a contradiction because according to the Pieri rule $S^{g\ell _{1}}\Omega _{A}(\log D) \otimes \dotsb \otimes S^{g\ell _{n-1}}\Omega _{A}(\log D) \otimes (L^{g^{n-1}})^{\vee }$ is the direct sum of some Schur powers $S^{\lambda }(\Omega _{1}(\log D))\otimes (L^{g^{n-1}})^{\vee }$ for partitions $\lambda$ with at most $n-1$ parts.

Proof. The natural map

\[ p^{{\ast}} \colon H^{0}(Y,E_{k,N}\Omega_{\pi,\Delta}\otimes B_{Y}^{{-}1}) \to H^{0}(Y',E_{k,N}\Omega_{\pi',\Delta'}\otimes p^{*}(B_{Y}^{{-}1})) \]

is obviously injective, and by Hartogs' theorem, the natural map

\[ \alpha^{{\ast}} \colon H^{0}(Y'_{0},E_{k,N}\Omega_{\pi'_{0},\Delta'_{0}}\otimes {B'}_{0}^{{-}1}) \to H^{0}(Y',E_{k,N}\Omega_{\pi',\Delta'}\otimes \alpha^{*}({B'}_{0}^{{-}1})) \]

is isomorphic, for any ample line bundle $B'_{0}$ on $A_{0}$ (its inverse image on $Y'_{0}$ being written in the same way). From the (proof of the) preceding Corollary 5.5, we know that $H^{0}(Y'_{0},E_{k,N}\Omega _{\pi '_{0},\Delta '_{0}}\otimes {B'}_{0}^{-1})=\{0\}$. This implies the claimed vanishing, since $k.\alpha ^{*}({B'}_{0})-p^{*}(B)$ is effective, for $k$ big enough.