1 Introduction
For a complex manifold X, let
$S^mT_X^*$
denote the mth symmetric power of the holomorphic cotangent bundle of X. Our main theorem in this article is as follows.
Theorem 1.1. Let
$\mathbb B^n$
be the unit ball in
$\mathbb C^n$
. Let
$\Gamma $
be a torsion-free lattice of
$\text {Aut}(\mathbb {B}^n)$
with only unipotent parabolic automorphisms. Let
$\Sigma = \mathbb {B}^n/ \Gamma $
be a quotient of
$\mathbb B^n$
with finite volume and
$\overline \Sigma $
be its toroidal compactification. Then, for each
$m,r,s \in \mathbb {N} \cup \{0 \},$
there exists a holomorphic vector bundle
$E_{r,m}$
over
$\overline \Sigma $
such that if
$s=0$
, or if
$m \geq n+1-r$
and
$s>0$
, then
$$ \begin{align*} H^{r,s}_{L^2, \bar \partial} ( \Sigma, S^m T_{\Sigma}^*)\cong H^{s} \big( \overline{\Sigma}, E_{r,m} \big), \end{align*} $$
where
$H^{r,s}_{L^2, \bar \partial } ( \Sigma , S^m T_{\Sigma }^*)$
is the
$L^2$
-Dolbeault cohomology group of
$S^mT_\Sigma ^*$
over
$\Sigma $
with respect to the metric induced from the Bergman metric on
$\mathbb B^n$
.
The requirement in Theorem 1.1 that
$\Gamma $
has only unipotent parabolic automorphisms is given for the smoothness of its toroidal compactification. In fact, every torsion-free lattice of
$\text {Aut}(\mathbb {B}^n)$
has a finite index subgroup having only unipotent parabolic automorphisms. Hence, any complex hyperbolic space form with finite volume has a finite covering which has a smooth toroidal compactification.
One motivation for considering the isomorphism in Theorem 1.1 is that it provides a way to show the finiteness of the dimension of the
$L^2$
-Dolbeault cohomology over finite volume ball quotients. For a compact Hermitian manifold X and a holomorphic Hermitian vector bundle E over X, it is well known that the Dolbeault cohomology group
$H^{r,s}(X,E)$
is finite dimensional. In the context of Hodge theory, it has been shown that the set of E-valued harmonic
$(r,s)$
-forms on X is finite dimensional and this set is isomorphic to
$H^{r,s}(X,E)$
. However, when X is non-compact, interesting phenomenon occur. For instance, in 1983, Donnelly–Fefferman [Reference Donnelly and Fefferman5] showed that the dimension of the space of square integrable harmonic
$(r,s)$
-forms vanishes if
$r+s\neq n$
and is infinite if
$r+s=n$
, when X is a strictly pseudococonvex domain in
$\mathbb C^n$
equipped with its Bergman metric. Along similar lines, we refer the readers to [Reference Gromov7], [Reference Ohsawa14].
For sufficiently large m, Theorem 1.1 immediately implies the following result.
Corollary 1.2. If
$s=0$
, or if
$s \in \mathbb {N}$
and
$m \geq n+1-r$
, then
$$ \begin{align*}\text{dim} \, H^{r,s}_{L^2, \bar \partial} (\Sigma, S^m T_{\Sigma}^*) <\infty. \end{align*} $$
For small m, the situation is more delicate and requires additional analysis. A key observation regarding the toroidal compactification
$\overline {\Sigma }$
of
$\Sigma $
is that there exists a d-bounded Kähler potential for
$\omega $
in a neighborhood of the union of boundary divisors of
$\overline {\Sigma }$
. This yields a Donnelly–Fefferman type estimate (4.7), which in turn implies the finite dimensionality of
$H^{r,s}_{L^2,\bar \partial }(\Sigma , S^m T_{\Sigma }^*)$
in certain cases.
Theorem 1.3. Let
$\Gamma $
be a torsion-free lattice of
$\text {Aut}(\mathbb {B}^n)$
with only unipotent parabolic automorphisms and let
$\Sigma =\mathbb {B}^n/\Gamma $
be a ball quotient with finite volume. If
$0 {<} s \leq n-1$
and
$$ \begin{align*} 0 \leq m \leq \bigg\{ \begin{array}{@{}ll} \big[ \frac{(n-s)^2 - {2(r+1)}}{2(n+1)} \big] & \text{when } r \not =0 \\ \big[ \frac{(n-s)^2}{2(n+1)} \big] & \text{when } r =0, \end{array} \end{align*} $$
then
$\text {dim} \, H^{r,s}_{L^2,\bar \partial }(\Sigma , S^m T_{\Sigma }^*) <\infty $
.
To prove the isomorphism part of Theorem 1.1, we establish the
$L^2$
-Dolbeault resolution of the sheaf of holomorphic sections
$\mathcal {O}(E_{r,m})$
on
$\overline {\Sigma }$
. For this, we use the
$L^2$
-Hörmander method for the
$\bar \partial $
-equation to prove the exactness of the sequence near a point of a boundary divisor in the smooth toroidal compactification
$\overline {\Sigma }$
.
On the other hand, in [Reference Fujiki6, Proposition 2.1], Fujiki established a resolution for a compact Kähler manifold X with a simple normal crossing divisor D and a holomorphic line bundle L over X with a smooth Hermitian metric
$h^L$
on
$X-D$
. More precisely, he showed that the
$L^2$
-Dolbeault complex
$(L^{0,*}_{2,L}, \bar \partial )$
is a resolution of a sheaf, provided that
$\omega $
is a Kähler metric of the Poincaré type, ball quotient, or Hilbert Modular type on
$X-D$
, and
$h^L$
on
$X-D$
satisfies certain conditions near D. For the definition of
$L^{0,*}_{2,L}$
, see Section 2.4. He used an
$L^2$
-Dolbeault type lemma generalizing [Reference Zucker21, Proposition 6.4] to the higher-dimensional manifolds under more general setting, without using Hörmander’s
$L^2$
-method. One advantage of their delicate approach is that it only requires control of
$h^L$
and
$\omega $
near the boundary divisor, and they did not need to consider curvatures. Our method, while different in spirit, is particularly well-suited to the present problem, the Hörmander
$L^2$
-method has been extensively employed in this line of research as a standard method to find a resolution of a compact Kähler manifold X with a simple normal crossing divisor D (see [Reference Chen3], [Reference Huang, Liu, Wan and Yang8] and references therein) and the finite dimensionality of
$L^2$
-Dolbeault cohomologies.
By Corollary 1.2, Theorem 1.3, and
$L^2$
Hodge decomposition given in [Reference Chen3], [Reference Kashiwara and Kawai9], we have the following.
Theorem 1.4. For the unit ball
$\mathbb B^n$
in
$\mathbb C^n$
, let
$\Gamma $
be a torsion-free lattice of
$\text {Aut}(\mathbb {B}^n)$
with only unipotent parabolic automorphisms and let
$\Sigma = \mathbb {B}^n/ \Gamma $
be a quotient of
$\mathbb B^n$
with finite volume. Let g be the induced metric on
$\Sigma $
from the Bergman metric on
$\mathbb B^n$
. If one of the following conditions holds:
-
1.
$s = 0;$
-
2.
$s \in \mathbb {N}$
and
$m \geq n+1-r;$
-
3.
$0 < s \leq n-1$
and
$$ \begin{align*} 0 \leq m \leq \bigg\{ \begin{array}{@{}ll} \big[ \frac{(n-s)^2 - {2(r+1)}}{2(n+1)} \big] & \text{when } r \not =0\\ \big[ \frac{(n-s)^2}{2(n+1)} \big] & \text{when } r =0 \end{array} \end{align*} $$
then
-
1. The operator
$\bar \partial \colon L^{r,s}_2(\Sigma ,S^m T_{\Sigma }^*)\to L^{r,s+1}_2(\Sigma ,S^m T_{\Sigma }^*)$
and its Hilbert adjoint
$\bar \partial ^*$
have closed images, and there is an orthogonal decomposition for each s, where
$$ \begin{align*} L^{r,s}_2(\Sigma,S^m T_{\Sigma}^*) = \text{Im}\, \bar\partial \oplus \text{Im}\, \bar\partial^* \oplus \mathcal H^{r,s}_2( \Sigma,S^m T_{\Sigma}^*), \end{align*} $$
$\mathcal H^{r,s}_2(\Sigma ,S^m T_{\Sigma }^*):= \ker \bar \partial \cap \ker \bar \partial ^*$
. As a consequence, there is an isomorphism
$$ \begin{align*}H^{r,s}_{L^2, \bar\partial}(\Sigma,S^m T_{\Sigma}^*)\cong \mathcal H^{r,s}_2(\Sigma,S^m T_{\Sigma}^*). \end{align*} $$
-
2. Denoting H be the projection operator
$H\colon L^{r,s}_2(\Sigma ,S^m T_{\Sigma }^*)\to \mathcal H^{r,s}_2(\Sigma ,S^m T_{\Sigma }^*)$
, then the Green operator
$G:= \left (\Delta _{\bar \partial } |_{\mathcal H^{r,s}_2(\Sigma ,S^m T_{\Sigma }^*)^\perp }\right )^{-1}(I -H)$
is well defined and bounded. Moreover, we have the following identity:
$$ \begin{align*}\Delta_{\bar \partial} \circ G = G \circ \Delta_{\bar \partial } = Id -H,\quad H \circ G=G \circ H=0. \end{align*} $$
This article is organized as follows: in Section 2, we recall the description of the toroidal compactification of the ball quotient with finite volume and
$L^2$
-Dolbeault cohomology for holomorphic vector bundles over complete Kähler manifolds. In Section 2.2, we introduce a quasi-isometric metric on the ball quotient, comparing it to the induced metric from the Bergman metric on the unit ball. In Section 3, we prove Theorems 1.1 and 1.3. Moreover, in Proposition 4.3, we prove the finite dimensionality of
$L^2$
-Dolbeault cohomologies of the canonical line bundle of
$\Sigma $
.
Throughout this article, we will use the multi-index. For example, for
$I=(i_1,\ldots , i_n)$
, we denote
$i_1+\cdots + i_n$
by
$|I|$
and a symmetric product
$e_1^{i_1} \ldots e_n^{i_n}$
by
$e^{I}$
for a local frame
$\{e_1,\ldots , e_n \}$
of
$T_{\Sigma }^*$
. For any real number
$\alpha $
,
$[\alpha ]$
denotes the greatest integer less than or equal to
$\alpha $
.
2 Preliminaries
2.1 Toroidal compactification of finite volume ball quotient
In this section, we will recall the description of the toroidal compactifications for
$\mathbb B^n/\Gamma $
given in [Reference Ash, Mumford, Rapoport and Tai1], [Reference Baily and Borel2], [Reference Mok and Itenberg12], [Reference Wong20].
Let
$\Gamma $
be a torsion-free subgroup of
$\text {Aut}(\mathbb B^n)$
and let
$\Sigma =\mathbb B^n/\Gamma $
. If
$\Sigma $
is of finite volume for the induced Bergman metric on
$\mathbb {B}^n$
, then there exists only finite number of cusps
$b_1, \ldots , b_k$
in
$\partial \mathbb {B}^n$
[Reference Ash, Mumford, Rapoport and Tai1], [Reference Baily and Borel2], [Reference Satake16], [Reference Siu and Yau18] and let
$B:=\{b_1,\ldots ,b_k\}$
. For each
$b\in \partial \mathbb B^n$
, let
$N_b:=\{\varphi \in \text {Aut}(\mathbb B^n): \varphi (b)=b\}$
be the stabilizer of b.
Now fix
$b\in B$
and let
$$ \begin{align*}c\colon \mathbb B^n\to S_n:=\left\{(z',z_n)\in \mathbb C^{n-1}\times \mathbb C: \text{Im}\, z_n> |z'|^2\right\} \end{align*} $$
be a Cayley transformation such that c extends real analytically to
$\overline {\mathbb B^n}-\{b\}$
and
$c|_{\partial \mathbb B^n-\{b\}} \to \partial S_n$
is a real analytic diffeomorphism. For any
$N\geq 0$
, define an open subset of
$S_n$
by
$$ \begin{align*}S^{(N)} := \left\{(z',z_n)\in \mathbb C^{n-1}\times \mathbb C : \text{Im}\,z_n> |z'|^2 +N \right\}. \end{align*} $$
Consider the holomorphic map
$\Psi \colon \mathbb C^{n-1}\times \mathbb C\to \mathbb C^{n-1}\times \mathbb C^* $
given by
$$ \begin{align} \Psi(z',z_n)=(z', e^{\frac{2\pi i z_n}{\tau}}):=(w', w_n) \end{align} $$
for some
${\tau> 0}$
and let
$G := \Psi (S_n)$
,
$G^{(N)}:=\Psi (S^{(N)})$
. Then, G and
$G^{(N)}$
are total spaces of a family of punctured discs over
$\mathbb C^{n-1}$
. Define
$\widehat G$
and
$\widehat G^{(N)}$
by adding
$\mathbb C^{n-1}\times \{0\}$
to G and
$G^{(N)}$
, respectively. Then, we have
$$ \begin{align*} \begin{aligned} \widehat G &= \left\{(w', w_n)\in \mathbb C^{n-1}\times\mathbb C: |w_n|^2 < e^{-\frac{4\pi}{\tau} |w' |^2}\right\},\\ \widehat G^{(N)} &= \left\{(w', w_n)\in \mathbb C^{n-1}\times\mathbb C: |w_n|^2 < e^{-\frac{4\pi N}{\tau}} e^{-\frac{4\pi}{\tau} |w' |^2}\right\}. \end{aligned} \end{align*} $$
Let
$W_b$
be the unipotent radical of
$N_b$
whose elements act on
$S_n$
as affine automorphisms. Let
$U_b:=[W_b, W_b]$
whose elements act on
$S_n$
as translations in
$\mathrm {Re} \; z_n$
direction. Then,
$\Gamma \cap W_b$
acts on
$S_n$
as a discrete group of automorphisms and
$[\Gamma \cap W_b, \Gamma \cap U_b]=0$
, which implies that the action of
$\Gamma \cap W_b$
descends from
$S_n$
to
$S_n/(\Gamma \cap U_b)$
. Since
$U_b$
is one-dimensional,
$U_b\cap \Gamma \cong \mathbb Z$
is generated by some
$\tau>0$
.
This implies that if we choose such
$\tau $
to define the holomorphic map
$\Psi $
, we have
$S_n/(\Gamma \cap U_b)\cong \Psi (S_n)=G$
and
$S^{(N)}/(\Gamma \cap U_b)\cong \Psi (S^{(N)})=G^{(N)}$
. Moreover, there is a group homomorphism
$\pi \colon \Gamma \cap W_b\to \text {Aut}(G)$
such that
$\Psi \circ \varphi = \pi (\varphi )\circ \Psi $
for any
$\varphi \in \Gamma \cap W_b$
. If
$\Gamma $
has only unipotent parabolic automorphisms, then the action of
$\pi (\Gamma \cap W_b)$
on
$\mathbb C^{n-1}\times \{0\}$
is a lattice of translation
$\Lambda _b$
. From now on, we assume that
$\Gamma $
has only unipotent parabolic automorphisms. Define
$D_b:=\mathbb C^{n-1}\times \{0\}/\Lambda _b$
be a torus. The toroidal compactification
$\overline \Sigma $
of
$\Sigma $
is set-theoretically given by
$$ \begin{align*}\overline \Sigma = \Sigma \cup \bigcup_{b\in B}D_b. \end{align*} $$
Define
$$ \begin{align*}\Omega_b^{(N)}:= \widehat G^{(N)}/\pi(\Gamma\cap W_b).\end{align*} $$
Thus, there exists an embedding
$\Omega _b^{(N)}-D_b \hookrightarrow \Sigma $
for sufficiently large N, and we have
$$ \begin{align*} \Omega_b^{(N)} \supset G^{(N)}/\pi(\Gamma\cap W_b)\cong S^{(N)}/(\Gamma\cap W_b). \end{align*} $$
For sufficiently large N,
$\Omega _b^{(N)} -D_b$
do not overlap in
$\Sigma $
for
$b\in B$
.
Consider the trivial line bundle
$$ \begin{align*} \begin{aligned} \mathbb C^{n-1}\times\mathbb C&\to\mathbb C^{n-1}\\ (w', w_n) &\mapsto w' \end{aligned} \end{align*} $$
and define a Hermitian metric
$$ \begin{align} \mu(w,w):= \|w\|^2 := e^{\frac{4\pi}{\tau}|w'|^2} |w_n|^2, \end{align} $$
which has negative curvature. Since
$\widehat G^{(N)}$
can be expressed by
$$ \begin{align*}\left\{w\in\mathbb C^n : \mu(w,w)<e^{-\frac{4\pi N}{\tau}}\right\},\end{align*} $$
$\widehat G^{(N)} $
is a level set of the trivial line bundle under the metric
$\mu $
. Then, the quotient of the trivial line bundle gives a line bundle
$L\to D_b$
with the induced Hermitian metric
$\overline \mu $
of negative curvature. This implies that
$D_b$
is an abelian variety and
$\Omega _b^{(N)}$
is the tubular neighborhood of
$D_b$
in L with
$\overline \mu $
-length
$<e^{\frac {-2\pi }{\tau }N}$
.
Let
$\omega _{S_n}$
be a Kähler form of
$S_n$
given by
$$ \begin{align*} \omega_{S_n} = \sqrt{-1}\partial\bar\partial(-\log(\text{Im}\, z_n-|z'|^2)). \end{align*} $$
Remark that
$\|\partial \left (\log (\text {Im}\, z_n -|z'|^2)\right )\|_{\omega _{S_n}}\equiv 1$
(cf. [Reference Lee and Seo10]). For the local coordinate
$(w', w_n)$
given in (2.1) in
$\Omega _b^{(N)}$
, the divisor
$D_b$
is defined by
$\{w_n=0\}$
. Let
$\omega _{\Omega _{b}^{(N)}}$
denote the induced metric on
$\Omega _{b}^{(N)}$
from
$\omega _{S_n}$
near the cusp b. Using the coordinate change in (2.1), by the relation
$$ \begin{align*} \log\|w\| = \frac{2\pi}{\tau} \left(|z'|^2 - \text{Im}\, z_n\right), \end{align*} $$
we have
$$ \begin{align} \begin{aligned} \omega_{\Omega_{b}^{(N)}} &=\sqrt{-1}\partial \bar \partial (-\log(-\log \| w \|) ) \\ \end{aligned} \end{align} $$
and
$$ \begin{align} \|\partial(-\log(-\log \| w \|) ) \|_{\omega_{\Omega_b^{(N)}}} \equiv 1. \end{align} $$
Denote by
$\rho (w)$
the strictly plurisubharmonic function which is defined near the cusp b induced by the Kähler potential
$-\log (-\log \| w \|) $
. We let
$dV_{\omega _{S_n}}$
be the volume form of
$S_n$
with respect to its Bergman metric. We define the Euclidean volume form as
$dV = \sqrt {-1}\partial \bar \partial |w|^2$
. Then as
$\|w\|\to 0$
, there exist constants
$C_1$
,
$C_2>0$
such that
$$ \begin{align*} \frac{C_1}{\|w\|^2 (-\log\|w\|)^{n+1}} dV \leq dV_{\omega_{S_n}} \leq \frac{C_2}{\|w\|^2 (-\log\|w\|)^{n+1}} dV. \end{align*} $$
2.2 Ball quotient type metric
From now on, we denote
$\mathbb {D}(\epsilon )$
by the complex disc centered at
$0$
in
$\mathbb {C}$
with radius
$\epsilon $
and
$\mathbb {D}^*(\epsilon ) := \mathbb {D} (\epsilon ) - \{0 \}$
. We say that two Riemannian metrics g and
$g'$
are quasi-isometric, if there exist two positive constants
$C_1$
,
$C_2$
such that
$$ \begin{align*}C_1 g \leq g' \leq C_2 g \end{align*} $$
and we denote by
$g \sim g'$
if g and
$g'$
are quasi-isometric. The proof of Lemma 2.1 uses a similar method exploited in [Reference Saper15, Lemma 2.3].
Lemma 2.1. Let
$\Gamma $
be a torsion-free lattice of
$\text {Aut}(\mathbb {B}^n)$
with only unipotent parabolic automorphisms and let
$\Sigma = \mathbb {B}^n/\Gamma $
be a ball quotient with finite volume. Let b be a cusp of
$\Gamma $
. Then, for any point
$p \in D_b$
, there exists a positive constant
$\epsilon $
satisfying
-
1.
$p=(0,\ldots ,0) \in \mathbb {D}^{n} (\epsilon ) \subset \subset \Omega _{b}^{(N)}$
; -
2. the induced Kähler form
$\omega _{\Omega _{b}^{(N)}}$
is quasi-isometric to a Kähler form on
$$ \begin{align*} \widetilde \omega_{\Omega_b^{(N)}} := \sum_{j=1}^{n-1} \frac{dw_j \wedge d \bar w_j}{(-\log \| w \|)} + \frac{dw_n \wedge d \bar w_n}{\|w \|^2 (-\log \| w \|)^2}, \end{align*} $$
$\mathbb {D}^{n-1} (\epsilon ) \times \mathbb {D}^* (\epsilon ) $
, where
$(w_1, \ldots , w_n)$
are the Euclidean coordinates.
Proof.
$$ \begin{align*} \begin{aligned} \omega_{\Omega_b^{(N)}} &= \sqrt{-1} \partial \bar \partial (- \log (-\log \| w \|)) \\ &= \sqrt{-1} \frac{\partial (- \log \| w \|) \wedge \bar \partial (-\log \|w \|)}{(-\log \|w \|)^2} - \sqrt{-1} \frac{\partial \bar \partial (-\log \|w \|)}{(-\log \|w \|)} \\ &=\sqrt{-1} \frac{\big(-\frac{1}{2}\frac{dw_n}{w_n} - \frac{2\pi}{\tau} \sum_{j=1}^{n-1} \bar w_j dw_j \big) \wedge \overline{\big(-\frac{1}{2}\frac{dw_n}{w_n} - \frac{2\pi}{\tau} \sum_{j=1}^{n-1} \bar w_j dw_j \big)}}{(-\log \|w \|)^2} \\ &\quad+ \sqrt{-1} \frac{2\pi}{\tau}\sum_{j=1}^{n-1} \frac{ dw_j \wedge d \bar w_j}{(-\log \|w \|)} \\&= \sqrt{-1} \frac{2\pi}{\tau} \sum_{j=1}^{n-1} \frac{dw_j \wedge d \bar w_j}{(-\log \| w \|)} \bigg( 1+ \frac{2\pi}{\tau} \frac{|w_j|^2}{(-\log \| w \|)} \bigg) \end{aligned} \end{align*} $$
$$ \begin{align*} \begin{aligned} \qquad&\qquad+ \frac{\sqrt{-1}}{4} \frac{dw_n \wedge d \bar w_n}{|w_n|^2(-\log \| w \|)^2} + {\frac{\left(\frac{2\pi}{\tau}\right)^2\sqrt{-1}}{(-\log\|w\|)^2}}{ \sum_{1\leq \ell \neq j\leq n-1} \bar w_\ell w_j dw_\ell \wedge d \bar w_j } \\&\qquad + \sqrt{-1} \frac{\pi}{\tau} \frac{1}{(-\log \| w \|)^2} \sum_{j=1}^{n-1}{\bigg( \frac{w_j}{w_n} dw_n \wedge d \bar w_j + {\frac{ \overline w_j}{ \overline w_n}} dw_j \wedge d \bar w_n \bigg).} \end{aligned} \end{align*} $$
Therefore, under the frame
$$ \begin{align*} d \zeta_j := \frac{\sqrt{\frac{2 \pi}{\tau}} d \omega_j}{\sqrt{-\log \| w \|}} \quad \text{for } j=1,\ldots, n-1, \quad d\zeta_n := \frac{1}{2} \frac{ e^{\frac{2\pi}{\tau} |w'|^2 } d w_n }{ \|w \| (-\log \|w \|)} \end{align*} $$
for
$T_{\Sigma }^*$
on
$\mathbb {D}^{n-1}(\epsilon ) \times \mathbb {D}^* (\epsilon )$
, we have
$$ \begin{align*} \begin{aligned} \omega_{\Omega_b^{(N)}} & = \sqrt{-1} \bigg( \sum_{j=1}^{n-1} \big( 1+ O( |w_j|^2 (-\log \| w \|)^{-1} ) \big) d\zeta_j \wedge d \bar \zeta_j + d \zeta_n \wedge d \bar \zeta_n \\ & \qquad\qquad\,\,+ {\sum_{1 \leq \ell \not = j \leq n-1} O (\bar w_\ell w_j (-\log \|w \|)^{-1} ) d \zeta_\ell \wedge d \bar \zeta_j } \\&\qquad\qquad\,\,+ \sum_{j=1}^{n-1} O({w_j }(-\log \| w \|)^{-\frac{1}{2} }) d\zeta_n \wedge d \bar \zeta_j \\&\qquad\qquad\,\, + \,\, \sum_{j=1}^{n-1} O({\overline{w_j}}(-\log \| w \|)^{-\frac{1}{2} }) d \zeta_j \wedge d \bar \zeta_n \bigg). \end{aligned} \end{align*} $$
Since
$ (-\log \|w \|)^{-1}$
and
$ (-\log \|w \|)^{-\frac {1}{2} }$
converge to zero as
$\| w \| \sim |w_n| \rightarrow 0$
, the lemma is proved.
2.3
$L^2$
-Dolbeault cohomology
In this section, we will review some results for
$L^2$
-Dolbeault cohomologies. For details, see [Reference Demailly4].
Let
$(X,\omega )$
be a Kähler manifold and
$(E,h^{E})$
be a holomorphic vector bundle over X. Let
$|\cdot |^2_{\omega }$
be the induced norm on
$\Lambda ^{r,s}T_{X}^*$
from
$\omega $
. For E-valued
$(r,s)$
forms u and v, we denote by
$\langle u, v \rangle _{h^E, \omega }$
the inner product on
$E \otimes \Lambda ^{r,s} T_{X}^*$
induced from
$h^E$
and
$\omega $
. Denote
$\langle u, u \rangle _{h^E, \omega }$
by
$|u|^2_{h^E, \omega }$
. For any measurable E-valued
$(r,s)$
forms u and v, we define an
$L^2$
-inner product of u and v by
$$ \begin{align*}\langle \langle u , v \rangle \rangle_{h^E, \, \omega} := \int_{X} \langle u, v \rangle_{h^E, \, \omega} dV_{\omega}. \end{align*} $$
Then, the
$L^2$
-norm of u is given by
$$ \begin{align*}\|u \|^2_{h^E, \, \omega} := \int_{X} |u|^2_{h^E, \, \omega} dV_{\omega}. \end{align*} $$
For simplicity, we write
$ \| u \|^2_{h^E, \, \omega }$
as
$\| u \|^2$
if there is no ambiguity. The
$L^2$
-space of E-valued
$(r,s)$
forms
$L^{r,s}_{2} (X,E, h^E, \omega )$
on X is defined by
$$ \begin{align*}\{ f: f \,\, \text{is a measurable section of} \, \, E \otimes \Lambda^{r,s} T_{X}^* \,\, \text{such that} \,\, \| f \|^2 <\infty \}. \end{align*} $$
If there is no danger of confusion, we abbreviate it to
$L_{2}^{r,s}(X, E)$
. Let
$C^\infty _{c,(r,s)}(X, E)$
be the space of compactly supported E-valued
$(r,s)$
forms. We extend the operator
$ \bar \partial : C^{\infty }_{c,(r,s)} (X, E) \rightarrow C^{\infty }_{c,(r,s+1)}(X, E)$
to a closed densely defined linear operator
$$ \begin{align*}\bar \partial : L^{r,s}_{2} (X,E) \rightarrow L^{r,s+1}_{2} (X,E) \end{align*} $$
by taking the maximal closed extension of
$\bar \partial $
. Then, we have the Hilbert adjoint
$$ \begin{align*}\bar \partial^*: L_{2}^{r,s}(X,E) \rightarrow L_{2}^{r,s-1}(X, E) \end{align*} $$
of
$\bar \partial $
, which is also a closed densely defined linear operator. We denote by
$$ \begin{align*}H^{r,s}_{L^2, \bar \partial} (X,E, h^E, \omega) \quad \text{or} \quad H^{r,s}_{L^2, \bar \partial} (X, E) \end{align*} $$
the
$L^2$
-Dolbeault cohomology group of
$(X, E, h^E, \omega )$
.
2.4 Definition of fine sheaves
Let
$(F, h^F)$
be a holomorphic vector bundle over
$\overline {\Sigma }$
which has a smooth Hermitian metric
$h^F$
on
$\Sigma $
. Given any open set
$U \subset \overline {\Sigma }$
, let
$L^{r,*}_{2,loc}(U,F)$
be the space of measurable sections of F-valued
$(r,*)$
forms which is
$L^2$
-integrable on
$K- \bigcup _{b \in B} D_{b} $
for any compact subset K of U with respect to
$dV_{\omega }$
and
$h^F$
. Define sheaves
$L^{r,*}_{2,F}$
by
$$ \begin{align*}L^{r,*}_{2,F} (U) := \{ \nu \in L^{r,*}_{2,loc} (U, F) : \bar \partial \nu\in L^{r, *+1}_{2, loc} (U, F) \} \end{align*} $$
for any open set U on
$\overline {\Sigma }$
. Note that sheaves
$L^{r,*}_{2,F}$
are fine sheaves by the completeness of
$\omega $
on
$\Sigma $
which has finite volume on
$\Sigma $
, and so
$$ \begin{align} H^{s} (\overline \Sigma, L^{r,*}_{2,F})=0 \quad \text{for every } s>0. \end{align} $$
For details of fine sheaves, see [Reference Warner19]. We denote by
$\bar \partial _{(r,s),F}$
the sheaf morphisms
$$ \begin{align*}\bar \partial_{(r,s),F} : L^{r,s}_{2,F} \longrightarrow L^{r,s+1}_{2,F}, \end{align*} $$
which is defined by the maximal closed extension of
$\bar \partial $
.
3 Proof of Theorem 1.1
The proofs in this section are influenced by [Reference Chen3], [Reference Fujiki6]. Throughout the section, we let
$F=(S^m {T_{ \Sigma }^*}, g^{-m})$
, where g is the induced Bergman metric on
$\Sigma $
. Let
$D_b$
be the boundary divisor in
$\overline {\Sigma }$
corresponding to the cusp b.
From now on, we will use the Euclidean coordinates
$(w', w_n):= (w_1, \ldots , w_n)$
as local holomorphic coordinates on
$\Omega _{b}^{(N)}$
which are induced by the uniformization
$\widehat G^{(N)}$
of
$\Omega _b^{(N)} = \widehat G^{(N)} / \pi (\Gamma \cap W_b)$
and we assume that
$\| w \| < 1$
on
$\Omega _{b}^{(N)}$
.
Let
$b_1, \ldots , b_k$
be the cusps of
$\Sigma $
and
$D_j$
be the boundary divisor corresponding to
$b_j$
for each
$j=1, \ldots , k$
in the toroidal compactification
$\overline {\Sigma }$
of
$\Sigma $
. For notation simplicity, from now on, we denote
$\Omega _{b_j}^{(N)}$
by
$\Omega _{j}^{(N)}$
.
Let
$\pi _j : \Omega _{j}^{(N)} \rightarrow D_{j}$
be the canonical projection. Let
$[D_{j}]$
be the associated holomorphic line bundle to
$D_{j}$
and
$T^*_{\Omega _{j}^{(N)} / D_{j}}$
be the relative cotangent bundle for
$\pi _j$
. Define
$$ \begin{align*} \begin{aligned} E_{r,m,j} :=&\ \bigg \{ \Lambda^r \pi_j^* T_{D_j}^* \otimes \bigg( \bigoplus_{\substack{ \ell < n-(m+r) } } \left( S^{m-\ell} \pi_j^* T_{D_j}^* \otimes S^{\ell} T^*_{\Omega_j^{(N)} /D_j} \otimes [\ell D_j] \right) \\ & \quad \quad \oplus \bigoplus_{ \substack{ \ell \geq n-(m+r) } } \left( { S^{m-\ell} \pi_{j}^* T_{D_j}^* \otimes S^{\ell} T^*_{\Omega_j^{(N)} /D_j} } \otimes [(\ell-1) D_j] \right) \bigg)\bigg \} \\ &\oplus \bigg \{ \left( \Lambda^{r-1} \pi_j^* T_{D_j}^* \otimes T_{\Omega_j^{(N)} / D_j}^* \right) \otimes \bigg( \bigoplus_{\substack{ \ell < n-(m+r+1) } } \bigg( S^{m-\ell} \pi_j^* T_{D_j}^* \otimes S^{\ell} T^*_{\Omega_j^{(N)} /D_j} \\ &\,\,\,\,\, \otimes [(\ell+1) D_j] \bigg) \oplus \bigoplus_{\substack{ \ell \geq n-(m+r+1) } } \left( S^{m-\ell} \pi_j^* T_{D_j}^* \otimes S^{\ell} T^*_{\Omega_j^{(N)} /D_j} \otimes [\ell D_j] \right)\bigg) \bigg \} \end{aligned} \end{align*} $$
as a vector bundle over
$\Omega _{j}^{(N)}$
. In the definition of
$E_{r,m,j}$
, we use the canonical identification of
$\Omega _{j}^{(N)}$
and a tubular neighborhood of the zero section of the holomorphic normal bundle
$N_{j}$
of
$D_j$
in
$\overline {\Sigma }$
. For
$r \geq 0$
, we define a holomorphic vector bundle
$E_{r,m}$
as follows:
$$ \begin{align*} E_{r,m} := \bigg \{ \begin{array}{@{}ll} E_{r,m,j} & \text{on } \Omega_{j}^{(N)}\\ \Lambda^{r} T_{\Sigma}^* \otimes S^m T_{\Sigma}^* & \text{ on } \overline{\Sigma} - \bigcup_{j=1}^{k} {\Omega_{j}^{(N)}}. \end{array} \end{align*} $$
It is well defined since the restriction of the line bundle
$[tD_j]$
to
$\Sigma $
is trivial for each j and
$t \in \mathbb N\cup \{0\}$
.
Lemma 3.1. Let
$\Gamma $
be a torsion-free lattice of
$\text {Aut}(\mathbb {B}^n)$
with only unipotent parabolic automorphisms and let
$\Sigma = \mathbb {B}^n/ \Gamma $
be a ball quotient with finite volume. Then, for each
$r, \, m \in \mathbb {N} \cup \{ 0 \}$
, the following sequence
$$ \begin{align} 0 \xrightarrow {} \mathcal{O} ( E_{r,m} ) \xrightarrow { } L^{r,0}_{2,F} \xrightarrow {\bar \partial} L^{r,1}_{2,F} \end{align} $$
is exact.
Proof. At first, we will show that
$$ \begin{align} \mathcal{O}_{} ( E_{r,m} ) \xrightarrow { } L^{r,0}_{2,F} \xrightarrow {\bar \partial} L^{r,1}_{2,F} \end{align} $$
is exact. For any open set in
$\Sigma $
, the exactness of the sequence is well known and hence we only need to consider open sets intersecting with divisors
$D_{j}, j=1, \ldots , k$
in
$\overline {\Sigma }$
. Consider a divisor
$D_{b} \subset \overline {\Sigma }$
and fix a point
$p \in D_{b}$
. Let
$U_{p} := \mathbb {D}^{n-1} (\epsilon ) \times \mathbb {D}^* (\epsilon ),$
where
$\mathbb {D}^{n-1} (\epsilon ) := \{ (w_1, \ldots , w_{n-1}) : w_i \in \mathbb {D}(\epsilon ) , 1 \leq i \leq n-1 \}$
with a sufficiently small
$\epsilon>0$
so that
$p \in U_{p} \subset \subset \Omega _{{b}}^{(N)}$
and
$p=(0,\ldots ,0)$
.
Note that every holomorphic section s in
$L^{r,0}_{2,F}(U_{p})$
is expressed as
$$ \begin{align*}s= \sum_{\substack{|I|=m \\ j_1 < \cdots < j_r}} s_{j_1 \ldots j_r, I} (w', w_n) dw_{j_1} \wedge \cdots \wedge dw_{j_r} \otimes dw^{I} \,\, \text{ with } \,\, s_{j_1 \ldots j_r, I} (w', w_n) \in \mathcal{O}(U_{p}). \end{align*} $$
Since for each fixed
$w'$
,
$s_{j_1 \ldots j_r, I}$
can be regarded as a holomorphic function on
$\mathbb {D}^* (\epsilon )$
, each
$s_{j_1 \ldots j_r, I} (w', w_n)$
has the Laurent expansion:
$$ \begin{align*}s_{j_1 \ldots j_r, I} (w', w_n) = \sum_{k=-\infty}^{\infty} s_{j_1\ldots j_r, I, k}(w',0) w_n^{k}. \end{align*} $$
By Lemma 2.1,
$$ \begin{align*} | dw_k |^2_{\omega^{-1}} \sim \left \{ \begin{array}{@{}ll} -\log \|w \| & \text{if } k=1, \ldots, n-1,\\ \|w\|^2 (-\log\|w\|)^{2} & \text{if } k=n. \end{array} \right. \end{align*} $$
Let
$dV$
be the Lebesgue measure. Then,
$s \in L^{r,0}_{2,F}(U_{p})$
implies that
$$ \begin{align*} \int_{\mathbb D^{n-1}(\epsilon)\times\mathbb{D}^*(\epsilon)} |s_{j_1 \ldots j_r, I} (w', w_n)|^{2} \| w \|^{2(i_n-1)}(-\log \| w \|)^{r+m+i_n - (n+1)} dV < \infty \end{align*} $$
for any
$|I|=m$
when
$j_r \neq n$
, and
$$ \begin{align*} \int_{\mathbb D^{n-1}(\epsilon)\times\mathbb{D}^*(\epsilon)} |s_{j_1 \ldots j_r, I} (w', w_n)|^{2} \| w \|^{2 i_n }(-\log \| w \|)^{{(r+1)}+m+i_n - (n+1)} dV < \infty \end{align*} $$
for any
$|I|=m$
when
$j_r = n$
. Let
$dV_{w_n}$
be the Lebesgue measure of
$\mathbb {C}$
. Since for any
$k,$
the function
$x\mapsto x^2(-\log x)^k$
is increasing on
$0<x<\epsilon $
for sufficiently small
$\epsilon $
, we have
$$ \begin{align} \int_{\mathbb{D}^*(\epsilon)} |s_{j_1 \ldots j_r, I} (w', w_n)|^{2} | w_n|^{2(i_n-1)}(-\log | w_n|)^{r+m+i_n - (n+1)} dV_{w_n} < \infty \end{align} $$
for any
$|I|=m$
when
$j_r \neq n$
, and
$$ \begin{align} \int_{\mathbb{D}^*(\epsilon)} |s_{j_1 \ldots j_r, I} (w', w_n)|^{2} | w_n|^{2 i_n }(-\log | w_n|)^{r+m+i_n -n} dV_{w_n} < \infty \end{align} $$
for any
$|I|=m$
when
$j_r = n$
. Using polar coordinate
$(\rho , \theta )$
, we obtain
$$ \begin{align*} \begin{aligned} ({3.3}) &= 2\pi \sum_{k=-\infty}^{\infty} |s_{j_1 \ldots j_r, I, k} (w', 0)|^2 \bigg( \int_{-\log \epsilon}^{\infty} e^{-(2k+2i_n) u} u^{r+m+i_n-(n+1)}du \bigg) \end{aligned} \end{align*} $$
and similarly, we obtain
$$ \begin{align*} \begin{aligned} ({3.4}) &= 2\pi \sum_{k=-\infty}^{\infty} |s_{j_1 \ldots j_r, I, k} (w', 0)|^2 \bigg( \int_{-\log \epsilon}^{\infty} e^{-(2k+2i_n+2) u} u^{r+m+i_n-n}du \bigg). \end{aligned} \end{align*} $$
Remark that
$$ \begin{align} \begin{aligned} \int_{-\log \epsilon}^{\infty} e^{-(2k+2i_n) u} & u^{r+m+i_n-(n+1)} du < \infty \\ &\Longleftrightarrow \bigg \{ \begin{array}{@{}ll} k \geq -(i_n -1) & \text{if } r+m+i_n \geq n,\\ k \geq -i_n & \text{if } r+m+i_n < n \end{array} \end{aligned} \end{align} $$
and
$$ \begin{align} \begin{aligned} \int_{-\log \epsilon}^{\infty} e^{-(2k+2i_n+2) u} &u^{(r+1)+m+i_n-(n+1)}du < \infty \\ &\Longleftrightarrow \bigg \{ \begin{array}{@{}ll} k \geq -i_n & \text{if } r+m+i_n \geq n-1,\\ k \geq -(i_n+1) & \text{if } r+m+i_n < n-1. \end{array} \end{aligned} \end{align} $$
Therefore, (3.2) is exact. Secondly, we will show that
$$ \begin{align*} 0 \rightarrow \mathcal{O} ({ E_{r,m}}) \xrightarrow {} L^{r,0}_{2,F} \end{align*} $$
is exact, that is, we will prove that if
$s \in \mathcal {O} ( E_{r,m}) (U_{p})$
, then it follows that
$s \in L^{r,0}_{2,F}(U_{p})$
. Note that by the definition of
${E_{r,m}}$
, we can express s as
$$ \begin{align*}s= \sum_{\substack{|I|=m \\ j_1 < \cdots <j_r } } s_{j_1 \ldots j_r, I} (w', w_n) dw_{j_1} \wedge \cdots \wedge dw_{j_r} \otimes dw^{I} \,\, \text{ with } \,\, s_{j_1 \ldots j_r, I} (w', w_n) \in \mathcal{O}(U_{p}). \end{align*} $$
If
$j_r \neq n$
, then
$s_{j_1 \ldots j_r, I}$
may have a pole up to the order
$i_n -1$
if
$i_n \geq n-(m+r)$
and
$i_n$
if
$ i_n < n-(m+r)$
, for each fixed
$w'$
. Similarly, if
$ j_r=n, $
then
$s_{j_1 \ldots j_r, I}$
may have a pole up to the order
$i_n $
if
$i_n \geq n-(m+r+1)$
and
$i_r +1 $
if
$ i_r < n-(m+r+1)$
, for each fixed
$w'$
. Finally, using (3.5) and (3.6), it follows that
$s \in L_{2,F}^{(r,0)}(U_{p})$
and the proof is completed.
The proposition presented below can be considered an
$L^2$
-version of the Dolbeault–Grothendieck lemma.
Proposition 3.2. Let
$\Gamma $
be a torsion-free lattice of
$\text {Aut}(\mathbb {B}^n)$
with only unipotent parabolic automorphisms and
$\Sigma = \mathbb {B}^n/ \Gamma $
be the ball quotient with finite volume. Let
${F=S^mT_{\Sigma }^*}$
. If
$m \geq n+1-r$
, then for any
$\ell , r \in \mathbb {N} \cup \{ 0 \}$
, the following sequence
$$ \begin{align} 0 \rightarrow \ker \bar\partial_{(r,\ell),F} \xrightarrow {} L^{{r,\ell}}_{2,F} \xrightarrow {\bar \partial} \ker \bar \partial_{(r,\ell+1),F} \rightarrow 0 \end{align} $$
is exact.
Proof. For any open set in
$\Sigma $
, the exactness of (3.7) is well known and hence we only need to consider open sets intersecting with the corresponding divisor
$D_{b}$
in
$\overline {\Sigma }$
of a cusp b of
$\Gamma $
. Consider a divisor
$D_{b} \subset \overline \Sigma $
and fix a point
$p \in D_{b}$
. Take a sufficiently small
$\delta>0$
such that for an open set
$U_p = \mathbb {D}^{n-1}(\delta ) \times \mathbb {D}^* (\delta ) \subset \subset \Omega _{b}^{(N)}$
of p, there exists a holomorphic coordinate system
$(w_1,\ldots ,w_n)$
on an open set
$W_p$
containing
$\overline {U_p}$
which satisfies the statements of Lemma 2.1.
Let
$\sigma _r$
be the induced metric on
$\Lambda ^{(r,0)} T^{*}_{U_p - D_b}$
from
$\omega $
. For
$r \geq 0$
and
$J=(j_1, \ldots , j_r)$
satisfying
$ 1 \leq j_1 < j_2 < \cdots < j_r \leq n$
, let
$$ \begin{align*} \tau := \bigg \{ \begin{array}{@{}ll} - r \log( - \log \| w \|) & \text{if } j_r \not = n\\ -(r+1) \log (- \log \| w \|) {-} \log \|w \|^2 & \text{if } j_r = n \end{array} \end{align*} $$
and define
$L_{IJ}$
by the trivial line bundle over
$U_p-D_b$
with a Hermitian metric
$$ \begin{align*}\| w \|^{2i_n}(-\log \| w \|)^{m+i_n} e^{-\tau}. \end{align*} $$
Then, under the local trivialization of
$\Lambda ^{(r,0)} T^*_{U_p - D_b} \otimes F$
over
$U_p - D_b$
via
$(w_1, \ldots , w_n)$
, we have a decomposition
$$ \begin{align*} \begin{aligned} &L^{0,s}_{2}(U_p - D_b, \Lambda^{(r,0)} T_{U_p - D_b}^* \otimes F, \sigma_r h^F, \omega) \\ &= \bigoplus_{|I|=m, |J|=r} L^{0,s}_{2} (U_p - D_b, L_{IJ}, \|w \|^{2i_n} (-\log \|w \|)^{m+i_n} e^{-\tau} , \omega) \end{aligned} \end{align*} $$
by Lemma 2.1. Hence,
$$ \begin{align*}H^{r,\ell}_{L^2, \bar \partial}(U_p - D_b, F, h^F, \omega) \cong H^{0,\ell}_{L^2, \bar \partial}(U_p - D_b, \Lambda^{(r,0)} T_{\Sigma}^* \otimes F, \sigma_r h^F, \omega) \end{align*} $$
is isomorphic to
$$ \begin{align*} \begin{aligned} \bigoplus_{|I|=m, |J|=r} H^{0,\ell}_{L^2, \bar \partial}(U_p - D_b, L_{IJ}, \|w \|^{2i_n} (-\log \|w \|)^{m+i_n} e^{-\tau}, \omega). \end{aligned} \end{align*} $$
Moreover,
$$ \begin{align*} \begin{aligned} H^{0,\ell}_{L^2, \bar \partial}(U_p - D_b, L_{IJ}, \|w \|^{2i_n} (-\log \|w \|)^{m+i_n} e^{-\tau}, \omega) \end{aligned} \end{align*} $$
is isomorphic to
$$ \begin{align} H^{n,\ell}_{L^2, \bar \partial} (U_p - D_b, K_{U_p - D_b}^{-1} \otimes L_{IJ}, e^{-k|w'|^2} \|w \|^{2i_n} (-\log \|w \|)^{m+i_n} e^{-\tau} (\det \omega)^{-1}, \omega), \end{align} $$
since
$|w'|^2$
is bounded on
$U_p - D_b$
. Therefore, to prove that the sequence (3.7) is exact, it suffices to show the vanishing of (3.8) for all
$|I|=m$
and
$|J|=r$
. Now, by (2.3), we have
$$ \begin{align} \begin{aligned} &\sqrt{-1} \Theta(K_{U_p-D_b}^{-1})+ \sqrt{-1} \partial \bar \partial \big( k |w'|^2 - i_n \log \| w \|^2 + \tau \\&\qquad\qquad\qquad\qquad\qquad\qquad+ (m+i_n)(-\log(-\log \|w \|)) \big) \\ &= (m+i_n - (n+1)) \omega + k \sqrt{-1} \partial \bar \partial |w'|^2 + \sqrt{-1} \partial \bar \partial \tau \\ &\geq (m+i_n+r - (n+1)) \omega + \bigg(k - \frac{4\pi (i_n+1)}{\tau} \bigg) \sqrt{-1} \partial \bar \partial |w'|^2 \end{aligned} \end{align} $$
and (3.9) is positive by taking k sufficiently big.
For a sufficiently small
$\epsilon $
, let
$\omega _{\epsilon }$
be a complete Kähler metric defined by
$$ \begin{align*} \omega_{\epsilon} = \omega + \epsilon \sqrt{-1} \partial \bar \partial \bigg( \sum_{i=1}^{n} \frac{1}{\delta^2 - |w_i|^2} \bigg) \quad \text{on} \quad U_p-D_b. \end{align*} $$
Since
$\omega _{\epsilon } \geq \omega $
, by the proof of [Reference Demailly4, Theorem 6.1 in Chapter VIII] and [Reference Demailly4, Lemma 6.3 in Chapter VIII], by taking k sufficiently big, (3.9) implies the vanishing of (3.8). Therefore, the proof is completed.
Proof of Theorem 1.1
First, we prove for the case of
$s=0$
. Notice that the exactness of (3.1) implies that
$$ \begin{align*}0 \xrightarrow { } H^0 (\overline \Sigma, \mathcal{O}(E_{r,m})) \xrightarrow { } H^0 (\overline \Sigma, L^{r,0}_{2,F}) \xrightarrow {\bar \partial} H^0 (\overline \Sigma, L^{r,1}_{2,F}) \end{align*} $$
is exact. Since
$H^0 (\overline \Sigma , L^{r,0}_{2,F}) \cong L^{r,0}_{2} (\Sigma , S^m T_{\Sigma }^*)$
by the definition of sheaves
$L^{r,*}_{2,F}$
, we then obtain the following exact sequence:
$$ \begin{align} 0 \xrightarrow { } H^0 (\overline \Sigma, \mathcal{O}(E_{r,m})) \xrightarrow { } L^{r,0}_{2}(\Sigma, S^m T_{\Sigma}^*) \xrightarrow {\bar \partial} L^{r,1}_{2}(\Sigma, S^m T_{\Sigma}^*). \end{align} $$
Hence, by the exactness of (3.10),
$$ \begin{align*}H^0(\Sigma, S^m T_{\Sigma}^*) \cong H^0(\overline \Sigma, \mathcal{O}(E_{r,m})) \end{align*} $$
and the proof for the case of
$s=0$
is completed. Now, the remaining part is to prove the cases for
$m \geq n+1-r$
and
$s \in \mathbb {N}$
. By Lemma 3.1 and Proposition 3.2, it follows that
$$ \begin{align*} 0 \xrightarrow {} \mathcal{O}(E_{r,m}) \xrightarrow { } L^{r,0}_{2, F} \xrightarrow {\bar \partial} \ker \bar \partial_{(r,1), F} \rightarrow 0 \end{align*} $$
and
$$ \begin{align*} 0 \rightarrow \ker \bar\partial_{(r,s), F } \xrightarrow { } L^{r,s}_{2,F} \xrightarrow {\bar \partial} \ker \bar \partial_{(r,s+1), F} \rightarrow 0 \end{align*} $$
are exact for every
$0 \leq s \leq n$
. It means that the resolution
$$ \begin{align*} 0 \rightarrow \mathcal{O}( E_{r,m}) \rightarrow L^{0,*}_{2, F } \end{align*} $$
is exact. Since
$L^{r,*}_{2,F}$
are fine sheaves by the completeness of
$\omega $
, the theorem is proved.
4 Proof of Theorem 1.3
Let
$\Theta (E)$
be the Chern curvature tensor of
$(E,h^E)$
. Let
$(w_1, \ldots , w_n)$
be a local coordinate system and
$\{ e_k \}$
be a local orthonormal frame for
$(E,h^E)$
. The Chern curvature tensor
$\Theta (E)$
is of the form
$$ \begin{align*}\Theta(E) = \sum_{ \lambda\mu jk} c_{ \lambda \mu jk } (e_\lambda^* \otimes e_\mu )\otimes dw_j \wedge d \overline w_k. \end{align*} $$
We say that E is of Nakano q-positive if, for any non-zero set of complex number
$\tau _{\lambda ,K}$
which is skew-symmetric in the indices of
$K=(i_1,\ldots ,i_q)$
,
$$ \begin{align} \sum_{\lambda,\mu,j,k,|S|=q-1} c_{\lambda\mu jk} \tau_{\lambda, jS } \overline{\tau}_{\mu,kS }>0. \end{align} $$
If E is of Nakano
$1$
-positive, then we say that E is of Nakano positive. Note that by (4.1), if E is of Nakano positive, then E is also of Nakano q-positive for any
$1 \leq q \leq n$
. Since
$T_{\Sigma }^*$
is of Nakano positive, by [Reference Siu17, Section 4.1],
$\Lambda ^{(r,0)} T_{\Sigma }^*$
is of Nakano positive. Furthermore, it is known that
$S^m T_{\Sigma }^*$
is of Nakano positive by [Reference Liu, Sun and Yang11, Proposition 7.4].
Let
$\Lambda _{\omega }$
denote the adjoint of the left multiplication of
$\omega $
.
Lemma 4.1. For any
$u \in C_{(0,s)}^{\infty }(\Sigma , \Lambda ^r T_{\Sigma }^* \otimes S^m T_{\Sigma }^*)$
,
$$ \begin{align*} \langle [\sqrt{-1} \Theta(\Lambda^r T_{\Sigma}^* \otimes S^m T_{\Sigma}^*), \Lambda_{\omega}] u , u \rangle \geq \left\{ \begin{array}{@{}lll} -\big( m(n+1) +(r+1) \big) |u|^2 & \text{if } r \not =0, \\ \\ -m(n+1) |u|^2 & \text{if } r=0. \end{array} \right. \end{align*} $$
Proof. Let
$\{e_1,\ldots , e_n \}$
be a local orthonormal frame of
$T_{\Sigma }^*$
. Note that
$$ \begin{align*}\Theta(T_{\Sigma}^*)(e_j)= \sum_{a} e_a \otimes e_j \wedge \bar e_{a} + \sum_s e_j \otimes e_s \wedge \bar e_s. \end{align*} $$
Hence,
$$ \begin{align} \begin{aligned} &\Theta(S^m T_{\Sigma}^*)(e_1^{i_1} \ldots e_n^{i_n}) \\ &= \sum_{j=1}^{n}i_j e_1^{i_1} \ldots e_j^{i_j-1} \ldots e_n^{i_n} \cdot \Theta(T_{\Sigma}^*)(e_j) \\ &=\sum_{j=1}^{n}i_j e_1^{i_1} \ldots e_j^{i_j-1} \ldots e_n^{i_n} \cdot \bigg(\sum_{a} e_a \otimes e_j \wedge \bar e_a + \sum_{s} e_j \otimes e_s \wedge \bar e_s \bigg)\\ &=\sum_{j=1}^{n} \bigg(i_j e_1^{i_1} \ldots e_{j}^{i_j} \ldots e_n^{i_n} \otimes e_j \wedge \bar e_j + \sum_s i_j e_1^{i_1} \ldots e_{j}^{i_j} \ldots e_n^{i_n} \otimes e_s \wedge \bar e_s \bigg) \\ &\quad + \sum_{j=1}^{n} \sum_{a \not= j} i_j e_1^{i_1} \ldots e_a^{i_a+1} \ldots e_j^{i_j-1} \ldots e_n^{i_n} \otimes e_j \wedge \bar e_a \\ &= \sum_{j=1}^{n} (i_j +m) e_1^{i_1} \ldots e_n^{i_n} \otimes e_j \wedge \bar e_j + \sum_{j=1}^{n} \sum_{a \not = j} i_j e_1^{i_1} \ldots e_a^{i_a+1} \ldots e_j^{i_j-1} \ldots e_n^{i_n} \otimes e_j \wedge \bar e_a\end{aligned} \end{align} $$
and
$$ \begin{align} \begin{aligned} &\Theta(\Lambda^r T_{\Sigma}^*)(e_{i_1} \wedge \cdots \wedge e_{i_r}) \\ &= \sum_{q=1}^{r} e_{i_1} \wedge \cdots \wedge\Theta(T_{\Sigma}^*) (e_{i_q}) \wedge \cdots \wedge e_{i_r} \\ &=\sum_{q=1}^{r} e_{i_1} \wedge \cdots \wedge\bigg(\sum_{a=1}^{n} e_a \otimes e_{i_q} \wedge \bar e_a + \sum_{s=1}^{n} e_{i_q} \otimes e_s \wedge \bar e_s \bigg) \wedge \cdots \wedge e_{i_r} \\ &= \sum_{q=1}^{r} \left( e_{i_1} \wedge \cdots \wedge e_{i_q} \wedge \cdots \wedge e_{i_r} \right) \otimes ( e_{i_q} \wedge \bar e_{i_q}) \\ &\quad+ r \left( e_{i_1} \wedge \cdots \wedge e_{i_q} \wedge \cdots \wedge e_{i_r} \right) \otimes \sum_{s=1}^{n} e_s \wedge \bar e_s \\ & \quad+ \sum_{q=1}^{r} \sum_{a \not= i_q} \left( e_{i_1} \wedge \cdots \wedge e_{i_{q-1}} \wedge e_a \wedge e_{i_{q+1}} \wedge \cdots \wedge e_{i_r} \right) \otimes (e_{i_q} \wedge \bar e_{a}). \end{aligned} \end{align} $$
Write the Chern curvature form of
$S^m T_{\Sigma }^*$
as
$$ \begin{align*}\Theta(S^m T_{\Sigma}^*) = \sum_{ I J k \ell }c_{ IJ k\ell} (e_I^* \otimes e_J) \otimes e_k \wedge {\bar e}_\ell \end{align*} $$
and the Chern curvature form of
$\Lambda ^{r} T_{\Sigma }^*$
as
$$ \begin{align*}\Theta(\Lambda^{r} T_{\Sigma}^*) = \sum_{\substack{k \ell \\ i_1 < \cdots < i_r \\ j_1< \cdots <j_r}} \widetilde c_{(i_1, \ldots, i_r) \,(j_1, \ldots, j_r)\, k\ell} \big( (e_{i_1} \wedge \cdots \wedge e_{i_r})^* \otimes (e_{j_1} \wedge \cdots \wedge e_{j_r}) \big) \otimes e_k \wedge \bar e_\ell.\end{align*} $$
$$ \begin{align} c_{I J k\ell } = \left\{ \begin{array}{@{}lll} i_k+m &\quad \text{if } I=J \text{ and } k=\ell \\ i_{k} &\quad \text{if } (j_1, \ldots, j_n)= (i_1, \ldots, i_{\ell}+1, \ldots, i_{k}-1, \ldots, i_n) \text{ and } k \not=\ell \\ 0 &\quad \text{otherwise} \end{array} \right. \end{align} $$
and
$$ \begin{align} \begin{aligned} &\widetilde c_{(i_1, \ldots, i_r) \,(j_1, \ldots, j_r)\, k\ell} \\ &= \left\{ \begin{array}{@{}lll} r &\quad \text{if } (i_1, \ldots, i_r) = (j_1, \ldots, j_r), \,k=\ell \text{ and } k \not \in \{ i_1, \ldots, i_r \}, \\ r+1 &\quad \text{if } (i_1, \ldots, i_r) = (j_1, \ldots, j_r), \,k=\ell \text{ and } k \in \{ i_1, \ldots, i_r \}, \\ 1 &\quad \text{if } k = i_k, \text{ and } \ell \notin \{i_1 \ldots i_r \}, i_{k-1} < \ell < i_{k+1}, \\ &\,\,\,\quad {(j_1, \ldots, j_r)=(i_1, \ldots, i_{k-1}, \ell , i_{k+1} ,\ldots, i_r)}, \\ {(-1)^s} &\quad \text{if } k = i_k, \text{ and } \ell \notin \{i_1 \ldots i_r \} , i_{k-s-1} < \ell < i_{k-s}, \\ &\,\,\,\quad {(j_1, \ldots, j_r)=(i_1, \ldots, i_{k-s-1}, \ell, \ldots, i_{k-1},i_{k+1},\ldots, i_r)},\\ {(-1)^s} &\quad \text{if } k = i_k, \text{ and } \ell \notin \{i_1 \ldots i_r \}, i_{k+s} < \ell < i_{k+s+1}, \\ &\,\,\,\quad {(j_1, \ldots, j_r)=(i_1, \ldots, i_{k-1},i_{k},\ldots,i_{k+s}, \ell, i_{k+s+1}, \ldots, i_r),} \\ 0 &\quad \text{otherwise.} \end{array} \right. \end{aligned} \end{align} $$
Since
$\Lambda ^r T_{\Sigma }^*$
and
$S^m T_{\Sigma }^*$
are of Nakano positive, by applying calculations given in [Reference Demailly4, Section 7 in Chapter VII] with (4.4) and (4.5), the proof is completed.
Proposition 4.2. Let
$\Gamma $
be a torsion-free lattice of
$\text {Aut}(\mathbb {B}^n)$
with only unipotent parabolic automorphisms and let
$\Sigma =\mathbb {B}^n/\Gamma $
be a ball quotient with finite volume. Let
$r \in \mathbb {N} \cup \{ 0 \}$
. There exists a compact set
$K \subset \Sigma $
such that if
$0 \leq s \leq n-1$
and
$$ \begin{align} 0 \leq m \leq \bigg\{ \begin{array}{@{}ll} \big[ \frac{(n-s)^2 -2 (r+1)}{2(n+1)} \big] & \mathrm{when } r \not =0, \\ \big[ \frac{(n-s)^2}{2(n+1)} \big] & \mathrm{when } r =0, \end{array} \end{align} $$
then there exists a constant
$C>0$
such that for any
$\tau \in \text {Dom }\bar \partial \cap \text {Dom } \bar \partial ^* \cap L^{0,s}_{2}(\Sigma , \Lambda ^{r} T_{\Sigma }^* \otimes S^m T_{\Sigma }^*)$
, the following inequality holds:
$$ \begin{align} \begin{aligned} \int_{\Sigma} |\tau|^2 dV_{\omega} &\leq C_{} \left( \| \bar \partial \tau \|^2 + \| \bar \partial^* \tau \|^2 + \int_{K} |\tau |^2 dV_{\omega} \right). \end{aligned} \end{align} $$
Proof. Throughout the proof, we let
$F:=\Lambda ^{r} T_{\Sigma }^* \otimes S^m T_{\Sigma }^*$
. Since
$\omega $
is a complete Hermitian metric on
$\Sigma $
, it suffices to prove (4.7) for any
$\tau \in C^{\infty }_{c,(0,s)}(\Sigma , F)$
. Let
$\psi $
be a real-valued smooth function on
$\Sigma $
which will be chosen later and let
$\zeta =e^{\psi /2} \tau $
. We denote by
$\bar \partial ^*_{h^F}$
the Hilbert adjoint of
$\bar \partial $
with respect to
$(X,F, h^F,\omega )$
and denote by
$\bar \partial ^*_{h^F,\psi }$
the Hilbert adjoint of
$\bar \partial $
with respect to
$(X,F,h^F e^{-\psi },\omega )$
.
Since
$$ \begin{align*} \begin{aligned} | e^{-\psi/2} \bar \partial (e^{\psi/2} \tau) |^2_{h^{F},\omega} &= \left| \bar \partial \tau + \frac{\bar \partial \psi}{2} \wedge \tau \right|^2_{h^{F}, \omega} \\ &= |\bar \partial \tau |^2_{h^{F}, \omega} + 2 \text{Re} \left\langle \bar \partial \tau, \frac{\bar \partial \psi}{2}\wedge \tau \right\rangle_{h^{F}, \omega} + \bigg | \frac{ \bar \partial \psi}{2} \wedge \tau \bigg |^2_{h^{F},\omega} \\ & \leq \bigg(1+\frac{1}{d} \bigg) |\bar \partial \tau|^2_{h^{F}, \omega}+ \frac{1+d}{4}| \bar \partial \psi|^2_{\omega} | \tau |^2_{h^{F}, \omega} \end{aligned} \end{align*} $$
and
$$ \begin{align*} \begin{aligned} | e^{-\psi/2} \bar \partial^*_{h^{F},\psi} (e^{\psi/2} \tau) |^2_{h^{F},\omega} &= \bigg| \bar \partial^*_{h^{F}} \tau + \frac{\partial \psi}{2} \lrcorner \tau \bigg|^2_{ h^{F}, \omega} \\ &= |\bar \partial^*_{h^{F}} \tau |^2_{h^{F}, \omega} + 2 \text{Re} \left\langle \bar \partial^*_{h^{F}} \tau, \frac{\partial \psi}{2} \lrcorner \tau \right\rangle_{h^{F},\omega} + \bigg| \frac{ \partial \psi}{2} \lrcorner \tau \bigg|^2_{h^{F},\omega} \\ & \leq\bigg(1+\frac{1}{d} \bigg) |\bar \partial^*_{h^{F}} \tau|^2_{h^{F},\omega} + \frac{1+d}{4} | \partial \psi |^2_{\omega} |\tau |^2_{h^{F},\omega,} \end{aligned} \end{align*} $$
we obtain
$$ \begin{align} \begin{aligned} &\int_{\Sigma} | \bar \partial \zeta|^2_{h^{F},\omega} e^{-\psi} dV_{\omega} + \int_{\Sigma} |\bar \partial^*_{h^{F}, \psi} \zeta|^2_{h^{F},\omega} e^{-\psi} dV_{\omega}\\ &\leq \left(1+\frac{1}{d}\right) \bigg( \int_{\Sigma} |\bar \partial \tau|^2_{h^{F}, \omega} + \int_{\Sigma} | \bar \partial^*_{h^{F}} \tau|^2_{h^{F}, \omega} dV_{\omega} \bigg) + \frac{1+d}{2} \int_{\Sigma} | \bar \partial \psi |^2_{\omega} |\tau|^2_{h^{F},\omega} dV_{\omega}. \end{aligned} \end{align} $$
On the other hand, by the basic estimate of
$(\Sigma , F, h^{F} e^{-\psi }, \omega )$
(e.g., see [Reference Demailly4]),
$$ \begin{align} \begin{aligned} \int_{\Sigma} |\bar \partial \zeta |^2_{h^{F}, \omega} e^{-\psi} dV_{\omega} + & \int_{\Sigma} | \bar \partial^*_{h^{F}} \zeta |^2_{h^{F}, \omega} e^{-\psi} dV_{\omega} \\ &\geq \int_{\Sigma} \langle [\sqrt{-1}\Theta(F ) + \sqrt{-1} \partial \bar \partial \psi , \Lambda_{\omega}] \zeta, \zeta \rangle e^{-\psi} dV_{\omega}. \end{aligned} \end{align} $$
Now, we consider a smooth function
$\chi _{b} \in C^{\infty } (\overline {\Sigma })$
for each cusp
$b \in B$
satisfying
$$ \begin{align*} \chi_{b} = \left\{ \begin{array}{@{}ll} 1 & \text{if } z \in \Omega_{b}^{(3N)}, \\ 0 & \text{if } z \in \overline{\Sigma}-\Omega_{b}^{(2N)}. \end{array} \right. \end{align*} $$
Let
$$ \begin{align*}\eta := \sum_{b \in B} \chi_b \cdot \big( -\log(-\log \|w \|^2 ) \big)\end{align*} $$
and
$K:=\overline {\Sigma } - \bigcup _{b \in B} \Omega _{b}^{(4N)}$
. We choose
$\psi = -t \eta $
with a constant
$t>0$
which will be chosen later. Let
$$ \begin{align*} {m_{n,r}} := \bigg \{ \begin{array}{@{}ll} m(n+1)+(r+1) & \text{if } r \not =0, \\ m(n+1) & \text{if } r =0. \end{array} \end{align*} $$
By (2.3) and Lemma 4.1, it follows that
$$ \begin{align*} \begin{aligned} &\int_{\Sigma} \langle [\sqrt{-1} \Theta(F ) +\sqrt{-1} \partial \bar \partial \psi, \Lambda_{\omega}] \zeta, \zeta \rangle e^{-\psi}dV_{\omega} \\ &\geq - m_{n,r} \int_{\Sigma} | \tau |_{h^{F},\omega}^2 dV_{\omega} + {(n-s)}t \int_{\Sigma-K} |\tau|^2_{h^{F},\omega } dV_{\omega} + D_{}\int_{K} |\tau |^2_{h^{F},\omega} dV_{\omega,} \end{aligned} \end{align*} $$
where
$$ \begin{align*}D_{} := \inf_{\int_{K} |\tau|^2_{h^{F}, \omega} dV_{\omega} \not = 0} \frac{\int_{K} \langle [\sqrt{-1} \partial \bar \partial \psi, \Lambda_{\omega}] \tau , \tau \rangle dV_{\omega}}{ \int_{K} |\tau|^2_{h^{F},\omega} dV_{\omega} }. \end{align*} $$
Let
$\lambda := m_{n,r}+ \max \{0, -D_{} \}$
. Then, we have
$$ \begin{align*} \begin{aligned} &\left(1+\frac{1}{d}\right) \bigg( \| \bar \partial \tau \|^2_{h^{F}, \omega} + \| \bar \partial^*_{h^{F}} \tau \|^2_{h^{F}, \omega} \bigg)+ \frac{1+d}{2} t^2 \int_{\Sigma} | \tau |^2_{h^{F}, \omega} dV_{\omega}+ \lambda \int_K | \tau |_{h^{F} , \omega}^2 dV_{\omega} \\ &\geq \big( -m_{n,r} + (n-s)t \big) \int_{\Sigma-K} | \tau |^2_{h^{F},\omega} dV_{\omega}. \end{aligned} \end{align*} $$
By letting
$t = \frac {n-s}{1+d}$
, it follows that
$$ \begin{align*} &\left(1+\frac{1}{d}\right) \bigg( \| \bar \partial \tau \|^2_{h^{F}, \omega} + \| \bar \partial^*_{h^{F}} \tau \|^2_{h^{F}, \omega} \bigg) + \left( \frac{1+d}{2} t^2 + \lambda \right) \int_{K} | \tau |^2_{h^{F},\omega} dV_{\omega} \\ &\geq \left(- m_{n,r} + \bigg((n-s)t -\frac{1+d}{{2}} t^2 \bigg) \right) \int_{\Sigma-K} | \tau |^2_{h^{F},\omega} dV_{\omega} \\ & \geq { \bigg( \frac{(n-s)^2}{2(1+d)} -m_{n,r} \bigg) } \int_{\Sigma-K} |\tau|^2_{h^{F},\omega} dV_{\omega} \end{align*} $$
and by taking
$d>0$
sufficiently small, we have
$$ \begin{align*} \frac{(n-s)^2}{2(1+d)} - m_{n,r}>0 \quad \text{when } m \text{ satisfies (4.6).} \end{align*} $$
Therefore, for
$$ \begin{align*}C_{} := \frac{\max \{ 1+ \frac{1}{d}, \frac{1+d}{2} t^2 + \lambda \}}{\frac{(n-s)^2}{2(1+d)}-m_{n,r}}, \end{align*} $$
we obtain (4.7) for any
$\tau \in C^{\infty }_{c, (0,s)} (\Sigma , F)$
.
Proof of Theorem 1.3
Let
$\sigma _r$
be the induced metric on
$\Lambda ^{(r,0)} T^{*}_{\Sigma }$
from
$\omega $
. By (4.7) and [Reference Ohsawa13, Proposition 1.2 in Chapter 2],
$$ \begin{align*}\text{dim} \, H^{r,s}_{L^2, \bar \partial} (\Sigma, S^m T_{\Sigma}^*, h^{S^m T_{\Sigma}^*}, \omega) = \text{dim} \, H^{0,s}_{L^2, \bar \partial} (\Sigma, \Lambda^r T_{\Sigma}^* \otimes S^m T_{\Sigma}^*, \sigma_r h^{S^m T_{\Sigma^*}}, \omega) <\infty \end{align*} $$
follows. Therefore, the proof is completed.
Proposition 4.3. Let
$\Gamma $
be a torsion-free lattice of
$\text {Aut}(\mathbb {B}^n)$
with only unipotent parabolic automorphisms and let
$\Sigma =\mathbb {B}^n/\Gamma $
be a ball quotient with finite volume. Denote
$K_{\Sigma }$
by the canonical line bundle over
$\Sigma $
. If
$m \geq 1$
, then for every
$r,s \in \mathbb {N} \cup \{ 0 \}$
,
$\text {dim} \, H^{r,s}_{L^2,\bar \partial }(\Sigma , K_{\Sigma }^m) <\infty $
. If
$m = 0$
, then
$\text {dim} \, H^{r,s}_{L^2,\bar \partial }(\Sigma , \mathbb {C}) <\infty $
for any
$r+s \not =n$
.
Proof. Let
$b_1, \ldots , b_k$
be cusps of
$\Sigma $
, and let
$D_j$
be the boundary divisor associated with each
$b_j, j=1,\ldots , k$
. In order to prove
$H^{r,0}_{L^2, \bar \partial }(\Sigma , K_{\Sigma }^m) < \infty $
for any
$m \in \mathbb {N} \cup \{ 0\}$
, using a similar calculation given in the proof of Lemma 3.1, we define a vector bundle over
$\Omega _j^{(N)}$
by
$$ \begin{align*} \begin{aligned}\widetilde E_{r,m,j} :=&\ \bigg \{ \Lambda^r \pi_j^* T_{D_j}^* \otimes \bigg( K_{\Omega_{j}^{(N)}} \otimes [(m-1) D_j] \bigg) \bigg \} \\ &\oplus \bigg \{ \left( \Lambda^{r-1} \pi_j^* T_{D_j}^* \otimes T_{\Omega_j / D_j}^* \right) \otimes \bigg( K_{\Omega_{j}^{(N)}} \otimes [m D_j] \bigg) \bigg \}. \end{aligned} \end{align*} $$
For
$r \geq 0$
, we define a holomorphic vector bundle
$\widetilde E_{r,m}$
over
$\overline \Sigma $
as follows:
$$ \begin{align*} \widetilde E_{r,m} := \bigg \{ \begin{array}{@{}ll} \widetilde E_{r,m,j} & \text{on } \Omega_{j}^{(N)},\\ \Lambda^{r} T_{\Sigma}^* \otimes K_{\Sigma}^m & \text{ on } \overline{\Sigma} - \bigcup_{j=1}^{k} \Omega_{b_j}^{(N)}.\end{array} \end{align*} $$
Then, it is a holomorphic vector bundle over
$\overline \Sigma $
, and we obtain the following exact sequence:
$$ \begin{align*}0 \xrightarrow { } \mathcal{O} ( \widetilde E_{r,m} ) \xrightarrow { } L^{r,0}_{2,K_{\Sigma}^m} \xrightarrow {\bar \partial} L^{r,1}_{2,K_{\Sigma}^m}. \end{align*} $$
By following the proof of Theorem 1.1, we then obtain
$$ \begin{align*}\text{dim} \, H^{r,0}_{L^2, \bar \partial}(\Sigma, K_{\Sigma}^m) <\infty. \end{align*} $$
Now, what remains is to prove the proposition for the cases when
$s>0$
. For
$m=0$
, since
$\omega $
has a d-bounded Kähler potential near the union of the boundary divisors of
$\overline {\Sigma }$
, by a similar argument to the proof of Proposition 4.2 and using [Reference Ohsawa13, Proposition 1.2 in Chapter 2], we have
$$ \begin{align} \text{dim} \, H^{r,s}_{L^2, \bar \partial} (\Sigma , \mathbb{C}) < \infty \quad \text {if } r+s \not =n. \end{align} $$
For
$r=0$
, by (4.10), we realize
$$ \begin{align*}\text{dim} \; H^{0,s}_{L^2, \bar \partial} (\Sigma, K_{\Sigma}) \cong \text{dim} \; H^{n,s}_{L^2, \bar \partial} (\Sigma, \mathbb{C})<\infty \end{align*} $$
and if
$m \geq 2$
, by the completeness of
$\omega ,$
we obtain
$$ \begin{align*}H^{0,s}_{L^2, \bar \partial} (\Sigma, K_{\Sigma}^m) \cong H^{n,s}_{L^2, \bar \partial} (\Sigma, K_{\Sigma}^{m-1}) \cong0. \end{align*} $$
For
$r \geq 1$
, since
$\Lambda ^{(r,0)} T_{\Sigma }^*$
is of Nakano positive if
$m \geq 1$
, then
$\Lambda ^{(r,0)}T_{\Sigma }^* \otimes K_{\Sigma }^{m-1}$
is of Nakano positive. Hence, by the completeness of
$\omega $
, we have
$$ \begin{align*}\, H^{r,s}_{L^2, \bar \partial} (\Sigma, K_{\Sigma}^m) \cong \, H^{n,s}_{L^2, \bar \partial} (\Sigma, \Lambda^{(r,0)} T_{\Sigma}^* \otimes K_{\Sigma}^{m-1}) {\cong 0}. \end{align*} $$
Therefore, the proof is completed.
Acknowledgements
The authors thank the anonymous referee for a careful reading and helpful suggestions that improved the presentation of the article. The first author was partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (no. RS-2024-00339854) and the Institute for Basic Science (IBS-R032-D1). The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2022R1F1A1063038, RS-2025-00561084).
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