Proof. By (2·1), we have $E_{1}^{l,-l}=\mathcal{H}^{k-l,0}\otimes \Omega_{B}^{l}$ . By the Künneth formula, the fiber of $\mathcal{H}^{k-l,0}$ over a point $b\in B$ is identified with

(2·3) \begin{equation}H^{k-l,0}(X_{b}^{n}) = \bigoplus_{(p_{1}, \cdots, p_{n})} H^{p_{1},0}(X_{b}) \otimes \cdots \otimes H^{p_{n},0}(X_{b}),\end{equation}

where $(p_{1}, \cdots, p_{n})$ ranges over all indices with $\sum_{i}p_{i}=k-l$ and $0\leq p_{i} \leq 2$ .

(1) When $k-l$ is odd, every index $(p_{1}, \cdots, p_{n})$ in (2·3) must contain a component $p_{i}=1$ . Since $H^{1,0}(X_b)=0$ , we see that $H^{k-l,0}(X_{b}^{n})=0$ . Therefore $\mathcal{H}^{k-l,0}=0$ when $k-l$ is odd.

(3) Let $l_{0} = \min\!(\!\dim B, k)$ . By the range (2·2) of (l, q), we see that for every $r\geq 1$ the source of $d_{r}$ that hits $E_{r}^{l_{0}, -l_{0}}$ is zero, and the target of $d_{r}$ that starts from $E_{r}^{l_{0}, -l_{0}}$ is also zero. This proves our assertion.

(2) Let $l < \min\!(\!\dim B, k)$ . In view of (1), we may assume that $l=k-2m$ for some $m>0$ . By (2·2), the source of $d_{1}$ that hits $E_{1}^{l,-l}$ is zero. We shall show that $d_{1}\colon E_{1}^{l,-l}\to E_{1}^{l+1,-l}$ is injective. By (2·1), this morphism is identified with

(2·4) \begin{equation}\bar{\nabla} : \mathcal{H}^{2m,0}\otimes \Omega_{B}^{l} \to \mathcal{H}^{2m-1,1}\otimes \Omega_{B}^{l+1}.\end{equation}

By the Künneth formula as in (2·3), the fibers of the Hodge bundles $\mathcal{H}^{2m,0}$ , $\mathcal{H}^{2m-1,1}$ over $b\in B$ are respectively identified with

(2·5) \begin{equation}H^{2m,0}(X_{b}^{n}) = \bigoplus_{|\sigma|=m} H^{2,0}(X_{b})^{\otimes \sigma},\end{equation}

(2·6) \begin{eqnarray}H^{2m-1,1}(X_{b}^{n})& = & \bigoplus_{|\sigma'|=m-1} \bigoplus_{i\not\in \sigma'} H^{2,0}(X_{b})^{\otimes \sigma'}\otimes H^{1,1}(X_{b}) \\ & = & \bigoplus_{|\sigma|=m} \bigoplus_{i\in \sigma} H^{2,0}(X_{b})^{\otimes \sigma-\{ i \}}\otimes H^{1,1}(X_{b}). \nonumber\end{eqnarray}

In (2·5), $\sigma$ ranges over all subsets of $\{ 1, \cdots, n\}$ consisting of m elements, and $H^{2,0}(X_{b})^{\otimes \sigma}$ stands for the tensor product of $H^{2,0}(X_{b})$ for the jth factors $X_{b}$ of $X_{b}^{n}$ over all $j\in \sigma$ . The notations $\sigma', \sigma$ in (2·6) are similar, and $H^{1,1}(X_{b})$ in (2·6) is the $H^{1,1}$ of the ith factor $X_{b}$ of $X_{b}^{n}$ .

Let us write $V=H^{2,0}(X_{b})$ and $W=(T_{b}B)^{\vee}$ for simplicity. The homomorphism (2·4) over $b\in B$ is written as

(2·7) \begin{equation}\bigoplus_{|\sigma|=m} \left( V^{\otimes \sigma}\otimes \wedge^{l}W \to\bigoplus_{i\in \sigma} V^{\otimes \sigma - \{ i\}}\otimes H^{1,1}(X_{b})\otimes \wedge^{l+1}W \right).\end{equation}

By [ Reference Voisin8 , lemma 5·8], the $(\sigma, i)$ -component

(2·8) \begin{equation}V^{\otimes \sigma}\otimes \wedge^{l}W \to V^{\otimes \sigma - \{ i\}}\otimes H^{1,1}(X_{b})\otimes \wedge^{l+1}W\end{equation}

factorises as

\begin{eqnarray*}V^{\otimes \sigma}\otimes \wedge^{l}W & \to &V^{\otimes \sigma - \{ i\}}\otimes H^{1,1}(X_{b}) \otimes W \otimes \wedge^{l}W \\ & \to & V^{\otimes \sigma - \{ i\}}\otimes H^{1,1}(X_{b})\otimes \wedge^{l+1}W,\end{eqnarray*}

where the first map is induced by the adjunction $V\to H^{1,1}(X_b)\otimes W$ of the differential of the period map for the ith factor $X_{b}$ , and the second map is induced by the wedge product $W \otimes \wedge^{l}W \to \wedge^{l+1}W$ . By linear algebra, this composition can also be decomposed as

(2·9) \begin{eqnarray}V^{\otimes \sigma}\otimes \wedge^{l}W & \to &V^{\otimes \sigma - \{ i\}}\otimes V \otimes W^{\vee} \otimes \wedge^{l+1}W \\ & \to & V^{\otimes \sigma - \{ i\}}\otimes H^{1,1}(X_{b})\otimes \wedge^{l+1}W, \nonumber\end{eqnarray}

where the first map is induced by the adjunction $\wedge^{l}W \to W^{\vee} \otimes \wedge^{l+1}W$ of the wedge product, and the second map is induced by the adjunction $V\otimes W^{\vee}\to H^{1,1}(X_{b})$ of the differential of the period map. By our initial Condition 2·1, the second map of (2·9) is injective. Moreover, since $l+1\leq \dim W$ by our assumption, the wedge product $\wedge^{l}W\times W \to \wedge^{l+1}W$ is nondegenerate, so its adjunction $\wedge^{l}W \to W^{\vee} \otimes \wedge^{l+1}W$ is injective. Thus the first map of (2·9) is also injective. It follows that (2·8) is injective. Since the map (2·7) is the direct sum of its $(\sigma, i)$ -components, it is injective. This finishes the proof of Lemma 2·3.

Proof of Proposition 2·2. By Lemma 2·3 (2), we have $E_{\infty}^{l,-l}=0$ when $l<l_{0}=\min\!(\!\dim B, k)$ . Together with Lemma 2·3 (3), we obtain

\begin{equation*}(\pi_{n})_{\ast}\Omega_{X_{n}}^{k} = E_{\infty}^{0} = E_{\infty}^{l_{0}, -l_{0}} = E_{1}^{l_{0}, -l_{0}}.\end{equation*}

When $k\leq \dim B$ , we have $l_{0}=k$ , and $E_{1}^{l_{0}, -l_{0}}=\Omega_{B}^{k}$ by (2·1). When $k> \dim B$ , we have $l_{0}= \dim B$ , and $E_{1}^{l_{0}, -l_{0}}=\mathcal{H}^{k-\dim B, 0}\otimes K_{B}$ by (2·1). When $k-\dim B$ is odd, this vanishes by Lemma 2·3 (1).

Proof. The assertion in the case $k>\dim B$ with $k\not\equiv \dim B$ mod 2 follows directly from the second case of Proposition 2·2. Next we consider the case $k\leq \dim B$ . We may assume that $\pi_{n}\colon X_{n}\to B$ extends to a surjective morphism $\bar{X}_{n}\to \bar{B}$ . Let $\omega$ be a holomorphic k-form on $\bar{X}_{n}$ . By the first case of Proposition 2·2, we have $\omega|_{X_{n}}=\pi_{n}^{\ast}\omega_{B}$ for a holomorphic k-form $\omega_{B}$ on B. Since $\omega$ is holomorphic over $\bar{X}_{n}$ , $\omega_{B}$ is holomorphic over $\bar{B}$ as well by a standard property of holomorphic differential forms. (Otherwise $\omega$ must have pole at the divisors of $\bar{X}_{n}$ dominating the divisors of $\bar{B}$ where $\omega_{B}$ has pole.) Therefore the pullback $H^{0}(\bar{B}, \Omega^{k})\to H^{0}(\bar{X}_{n}, \Omega^{k})$ is surjective.

Finally, we consider the case $k=\dim B+2m$ , $0\leq m \leq n$ . Let $\omega$ be a holomorphic k-form on $\bar{X}_{n}$ . By (2·11), we can uniquely write $\omega|_{X_{n}}=\sum_{\sigma}\pi_{\sigma}^{\ast}\omega_{\sigma}$ for some canonical forms $\omega_{\sigma}$ on $X_{\sigma}$ .

Proof. We identify $X_{n}$ with the fiber product $X_{\sigma}\times_{B}X_{\tau}$ where $\tau=\{ 1, \cdots, n\} - \sigma$ is the complement of $\sigma$ . We may assume that this fiber product diagram extends to a commutative diagram of surjective morphisms

between smooth projective models. We take an irreducible subvariety $\tilde{B}\subset \bar{X}_{\tau}$ such that $\tilde{B}\to \bar{B}$ is surjective and generically finite. Then $\pi_{\tau}^{-1}(\tilde{B})\subset \bar{X}_{n}$ has a unique irreducible component dominating $\tilde{B}$ . We take its desingularisation and denote it by Y. By construction $\pi_{\sigma}|_{Y} \colon Y\to \bar{X}_{\sigma}$ is dominant (and so surjective) and generically finite. On the other hand, for any $\sigma'\ne \sigma$ with $|\sigma'|=m$ , the projection $\pi_{\sigma'}|_{Y} \colon Y\dashrightarrow X_{\sigma'}$ is not dominant. Indeed, such $\sigma'$ contains at least one component $i\in \tau$ , so if $Y\dashrightarrow X_{\sigma'}$ was dominant, then the ith projection $Y\dashrightarrow X$ would be also dominant, which is absurd because it factorises as $Y\to \tilde{B}\subset \bar{X}_{\tau}\dashrightarrow X$ .

We pullback the differential form $\omega=\pi_{\sigma}^{\ast}\omega_{\sigma}+\sum_{\sigma'\ne \sigma}\pi_{\sigma'}^{\ast}\omega_{\sigma'}$ to Y and denote it by $\omega|_{Y}$ . Since $\omega$ is holomorphic over $\bar{X}_{n}$ , $\omega|_{Y}$ is holomorphic over Y. Since $\pi_{\sigma'}^{\ast}\omega_{\sigma'}|_{Y}$ is the pullback of the canonical form $\omega_{\sigma'}$ on $X_{\sigma'}$ by the non-dominant map $Y \dashrightarrow X_{\sigma'}$ , it vanishes identically. Hence $\pi_{\sigma}^{\ast}\omega_{\sigma}|_{Y}=\omega|_{Y}$ is holomorphic over Y. Since $\pi_{\sigma}|_{Y}\colon Y \to \bar{X}_{\sigma}$ is surjective, this implies that $\omega_{\sigma}$ is holomorphic over $\bar{X}_{\sigma}$ as before.

Proof. When $k\leq d$ , we have $H^{0}(\bar{X}_{n}, \Omega^{k}) \simeq H^{0}(\bar{B}, \Omega^{k})$ by Proposition 2·4. Then $\bar{B}$ is a smooth projective model of the modular variety ${{\Gamma}}\backslash \mathcal{D}$ . By a theorem of Pommerening [ Reference Pommerening5 ], the space $H^{0}(\bar{B}, \Omega^{k})$ for $k<d$ is isomorphic to the space of ${{\Gamma}}$ -invariant holomorphic k-forms on $\mathcal{D}$ , which in turn is identified with the space $M_{\wedge^{k},k}({{\Gamma}})$ of vector-valued modular forms of weight $(\wedge^{k},k)$ for ${{\Gamma}}$ (see [ Reference Ma4 ]). The vanishing of this space in $0<k<d/2$ is proved in [ Reference Ma4 , theorem 1·2] in the case when L has Witt index 2, and in [ Reference Ma4 , theorem 1·5 (1)] in the case when L has Witt index $\leq 1$ .

The vanishing in the case $k>d$ with $k\not\equiv d$ mod 2 follows from Proposition 2·4. Finally, we consider the case $k=d+2m$ , $0\leq m \leq n$ . By Proposition 2·4, we have a natural $\mathfrak{S}_{n}$ -equivariant isomorphism

\begin{equation*}H^{0}(\bar{X}_{n}, \Omega^{d+2m}) \simeq\bigoplus_{|\sigma|=m} H^{0}(\bar{X}_{\sigma}, K_{\bar{X}_{\sigma}}),\end{equation*}

where $\mathfrak{S}_{n}$ permutes the subsets $\sigma$ of $\{ 1, \cdots, n \}$ . Here note that the stabiliser of each $\sigma$ acts on $H^{0}(\bar{X}_{\sigma}, K_{\bar{X}_{\sigma}})$ trivially by (2·10). Therefore, as an $\mathfrak{S}_{n}$ -representation, the right-hand side can be written as

\begin{equation*}H^{0}(\bar{X}_{m}, K_{\bar{X}_{m}}) \otimes \left( \bigoplus_{|\sigma|=m}{{\mathbb{C}}}\sigma \right)\simeq H^{0}(\bar{X}_{m}, K_{\bar{X}_{m}}) \otimes {{\mathbb{C}}}\mathcal{S}_{n,m}.\end{equation*}

Finally, we have $H^{0}(\bar{X}_{m}, K_{\bar{X}_{m}})\simeq S_{d+m}({{\Gamma}}, \det)$ by [ Reference Ma3 , theorem 3·1].