1. Introduction
Let
${{\mathcal{F}_{g,n}}}$
be the moduli space of n-pointed K3 surfaces of genus
$g>2$
, i.e., primitively polarised of degree
$2g-2$
. It is a quasi-projective variety of dimension
$19+2n$
with a natural morphism
${{\mathcal{F}_{g,n}}}\to {{\mathcal{F}_{g}}}$
to the moduli space
${{\mathcal{F}_{g}}}$
of K3 surfaces of genus g, which is generically a
$K3^{n}$
-fibration. In this paper we study holomorphic differential k-forms on a smooth projective model of
${{\mathcal{F}_{g,n}}}$
. They do not depend on the choice of a smooth projective model, and thus are fundamental birational invariants of
${{\mathcal{F}_{g,n}}}$
. We prove a vanishing result for about half of the values of the degree k, and for the remaining degrees give a correspondence with modular forms on the period domain.
Our main result is stated as follows.
Theorem 1·1. Let
${{\bar{\mathcal{F}}_{g,n}}}$
be a smooth projective model of
${{\mathcal{F}_{g,n}}}$
with
$g>2$
. Then we have a natural isomorphism:
\begin{equation} H^{0}({{\bar{\mathcal{F}}_{g,n}}}, \Omega^{k}) \simeq\begin{cases}0 & \: \: 0<k\leq 9 \\ M_{\wedge^{k},k}({{\Gamma_{g}}}) & \: \: 10 \leq k \leq 18 \\ 0 & \: \: k>19, \: k\in 2{{\mathbb{Z}}} \\ S_{19+m}({{\Gamma_{g}}}, \det)\otimes {{\mathbb{C}}}\mathcal{S}_{n,m} & \: \: k=19+2m, \: 0\leq m \leq n\end{cases}. \end{equation}
Here
${{\Gamma_{g}}}$
is the modular group for K3 surfaces of genus g, which is defined as the kernel of
$\mathrm{O}^{+}(L_{g})\to \mathrm{O}(L_{g}^{\vee}/L_{g})$
where
$L_{g}=2U\oplus 2E_{8}\oplus \langle 2-2g \rangle$
is the period lattice of K3 surfaces of genus g. In the second case,
$M_{\wedge^{k},k}({{\Gamma_{g}}})$
stands for the space of vector-valued modular forms of weight
$(\wedge^{k},k)$
for
${{\Gamma_{g}}}$
(see [
Reference Ma4
]). In the last case,
$S_{19+m}({{\Gamma_{g}}}, \det)$
stands for the space of scalar-valued cusp forms of weight
$19+m$
and determinant character for
${{\Gamma_{g}}}$
, and
$\mathcal{S}_{n,m}$
stands for the right quotient
$\mathfrak{S}_{n}/(\mathfrak{S}_{m}\times \mathfrak{S}_{n-m})$
, which is a left
$\mathfrak{S}_{n}$
-set. Theorem 1·1 is actually formulated and proved in the generality of lattice-polarisation (Theorem 2·6).
In the case of the top degree
$k=19+2n$
, namely for canonical forms, the isomorphism (1·1) is proved in [
Reference Ma2
]. Theorem 1·1 is the extension of this result to all degrees
$k<19+2n$
. The spaces in the right-hand side of (1·1) can also be geometrically explained as follows. In the case
$k\leq 18$
,
$M_{\wedge^{k},k}({{\Gamma_{g}}})$
is identified with the space of holomorphic k-forms on a smooth projective model of
${{\mathcal{F}_{g}}}$
, pulled back by
${{\mathcal{F}_{g,n}}}\to {{\mathcal{F}_{g}}}$
. In the case
$k=19+2m$
,
$S_{19+m}({{\Gamma_{g}}}, \det)$
is identified with the space of canonical forms on
$\bar{\mathcal{F}}_{g,m}$
, and the tensor product
$S_{19+m}({{\Gamma_{g}}}, \det)\otimes {{\mathbb{C}}}\mathcal{S}_{n,m}$
is the direct sum of pullback of such canonical forms by various projections
${{\mathcal{F}_{g,n}}}\to \mathcal{F}_{g,m}$
. Therefore Theorem 1·1 can be understood as a kind of classification result which says that except for canonical forms, there are essentially no new differential forms on the tower
$({{\mathcal{F}_{g,n}}})_{n}$
of moduli spaces. In fact, this is how the proof proceeds.
The space
$S_{l}({{\Gamma_{g}}}, \det)$
is nonzero for every sufficiently large l, so the space
$H^{0}({{\bar{\mathcal{F}}_{g,n}}}, \Omega^{k})$
for odd
$k\geq 19$
is typically nonzero (at least when k is large). On the other hand, it is not clear at present whether
$M_{\wedge^{k},k}({{\Gamma_{g}}})\ne 0$
or not in the range
$10\leq k \leq 18$
. This is a subject of study in the theory of vector-valued orthogonal modular forms.
The isomorphism (1·1) in the case
$k=19+2m$
is an
$\mathfrak{S}_{n}$
-equivariant isomorphism, where
$\mathfrak{S}_{n}$
acts on
$H^{0}({{\bar{\mathcal{F}}_{g,n}}}, \Omega^{k})$
by its permutation action on
${{\mathcal{F}_{g,n}}}$
, while it acts on
$S_{19+m}({{\Gamma_{g}}}, \det)\otimes {{\mathbb{C}}}\mathcal{S}_{n,m}$
by its natural left action on
$\mathcal{S}_{n,m}$
. Therefore, taking the
$\mathfrak{S}_{n}$
-invariant part, we obtain the following simpler result for the unordered pointed moduli space
${{\mathcal{F}_{g,n}}}/\mathfrak{S}_{n}$
, which is birationally a
$K3^{[n]}$
-fibration over
${{\mathcal{F}_{g}}}$
.
Corollary 1·2. Let
$\overline{{{\mathcal{F}_{g,n}}}/\mathfrak{S}_{n}}$
be a smooth projective model of
${{\mathcal{F}_{g,n}}}/\mathfrak{S}_{n}$
. Then we have a natural isomorphism:
\begin{equation*}H^{0}(\overline{{{\mathcal{F}_{g,n}}}/\mathfrak{S}_{n}}, \Omega^{k}) \simeq\begin{cases}0 & \: \: 0<k\leq 9 \\ M_{\wedge^{k},k}({{\Gamma_{g}}}) & \: \: 10\leq k \leq 18 \\ 0 & \: \: k>19, \: k\in 2{{\mathbb{Z}}} \\ S_{19+m}({{\Gamma_{g}}}, \det) & \: \: k=19+2m, \: 0 \leq m \leq n\end{cases}.\end{equation*}
The universal K3 surface
$\mathcal{F}_{g,1}$
is an analogue of elliptic modular surfaces ([
Reference Shioda6
]), and the moduli spaces
${{\mathcal{F}_{g,n}}}$
for general n are analogues of the so-called Kuga varieties over modular curves ([
Reference Shokurov7
]). Starting with the case of elliptic modular surfaces [
Reference Shioda6
], holomorphic differential forms on the Kuga varieties have been described in terms of elliptic modular forms: [
Reference Shokurov7
] for canonical forms, and [
Reference Gordon1
] for the case of lower degrees (somewhat implicitly). Theorem 1·1 can be regarded as a K3 version of these results.
As a final remark, in view of the analogy between universal K3 surfaces and elliptic modular surfaces, invoking the classical fact that elliptic modular surfaces have maximal Picard number ([
Reference Shioda6
]) now raises the question if
$H^{k,0}({{\bar{\mathcal{F}}_{g,n}}})\oplus H^{0,k}({{\bar{\mathcal{F}}_{g,n}}})$
is a sub
${{\mathbb{Q}}}$
-Hodge structure of
$H^{k}({{\bar{\mathcal{F}}_{g,n}}}, {{\mathbb{C}}})$
. This is independent of the choice of a smooth projective model
${{\bar{\mathcal{F}}_{g,n}}}$
.
The rest of this paper is devoted to the proof of Theorem 1·1. In Section 2·1 we compute a part of the holomorphic Leray spectral sequence associated to a certain type of
$K3^{n}$
-fibration. This is the main step of the proof. In Section 2·2 we study differential forms on a compactification of such a fibration. In Section 2·3 we deduce (a generalised version of) Theorem 1·1 by combining the result of Section 2·2 with some results from [
Reference Ma2–Reference Pommerening5
]. Sometimes we drop the subscript X from the notation
$\Omega_{X}^{k}$
when the variety X is clear from the context.
2. Proof
2·1. Holomorphic Leray spectral sequence
Let
$\pi\colon X\to B$
be a smooth family of K3 surfaces over a smooth connected base B. In this subsection X and B may be analytic. We put the following assumption:
Condition 2·1. In a neighbourhood of every point of B, the period map is an embedding.
This is equivalent to the condition that the differential of the period map
\begin{equation*}T_{b}B \to \mathrm{Hom}(H^{2,0}(X_{b}), H^{1,1}(X_{b}))\end{equation*}
is injective for every
$b\in B$
, where
$X_{b}$
is the fiber of
$\pi$
over b.
For a natural number
$n>0$
we denote by
$X_{n}=X\times_{B}\cdots \times_{B}X$
the n-fold fiber product of X over B, and let
$\pi_{n}\colon X_{n}\to B$
be the projection. We denote by
$\Omega_{\pi_{n}}$
the relative cotangent bundle of
$\pi_{n}$
, and
$\Omega_{\pi_{n}}^{p}=\wedge^{p}\Omega_{\pi_{n}}$
for
$p\geq 0$
as usual.
Proposition 2·2. Let
$\pi\colon X\to B$
be a K3 fibration satisfying Condition 2·1. Then we have a natural isomorphism:
\begin{equation*}(\pi_{n})_{\ast}\Omega_{X_{n}}^{k} \simeq\begin{cases}\Omega_{B}^{k} & \; \; k\leq \dim B \\ 0 & \; \; k>\dim B, \: \: k\not \equiv \dim B \: \: \mathrm{mod} \; 2 \\ K_{B}\otimes (\pi_{n})_{\ast}\Omega_{\pi_{n}}^{2m} & \; \; k=\dim B+2m, \: 0\leq m \leq n\end{cases}.\end{equation*}
This assertion amounts to a partial degeneration of the holomorphic Leray spectral sequence. Recall ([
Reference Voisin8
, section 5·2]) that
$\Omega_{X_{n}}^k$
has the holomorphic Leray filtration
$L^{\bullet}\Omega_{X_{n}}^k$
defined by
\begin{equation*}L^{l}\Omega_{X_{n}}^k= \pi_{n}^{\ast}\Omega_{B}^{l}\wedge \Omega_{X_{n}}^{k-l},\end{equation*}
whose graded quotients are naturally isomorphic to
\begin{equation*}\mathrm{Gr}_{L}^{l}\Omega_{X_{n}}^{k} = \pi_{n}^{\ast}\Omega_{B}^{l} \otimes \Omega_{\pi_{n}}^{k-l}.\end{equation*}
This filtration induces the holomorphic Leray spectral sequence
\begin{equation*}(E_{r}^{l,q}, d_{r}) \: \: \: \Rightarrow \: \: \: E_{\infty}^{l+q} = R^{l+q}(\pi_{n})_{\ast}\Omega_{X_{n}}^{k}\end{equation*}
which converges to the filtration
\begin{equation*}L^{l}R^{l+q}(\pi_{n})_{\ast}\Omega_{X_{n}}^{k} \: = \:\mathrm{Im}(R^{l+q}(\pi_{n})_{\ast}L^{l}\Omega_{X_{n}}^{k} \to R^{l+q}(\pi_{n})_{\ast}\Omega_{X_{n}}^{k}).\end{equation*}
By [
Reference Voisin8
, proposition 5·9], the
$E_{1}$
page coincides with the collection of the Koszul complexes associated to the variation of Hodge structures for
$\pi_{n}$
:
\begin{equation}(E_{1}^{l,q}, d_{1}) = (\mathcal{H}^{k-l,l+q}\otimes \Omega_{B}^{l}, \bar{\nabla}).\end{equation}
Here
$\mathcal{H}^{\ast, \ast}$
are the Hodge bundles associated to the fibration
$\pi_{n}\colon X_{n}\to B$
, and
\begin{equation*}\bar{\nabla} : \mathcal{H}^{\ast, \ast}\otimes \Omega_{B}^{\ast} \to \mathcal{H}^{\ast-1, \ast+1}\otimes \Omega_{B}^{\ast+1}\end{equation*}
are the differentials in the Koszul complexes (see [
Reference Voisin8
, section 5·1·3]). For degree reasons, the range of (l, q) in the
$E_{1}$
page satisfies the inequalities
\begin{equation*}0\leq l \leq \dim B, \quad 0\leq k-l \leq 2n, \quad 0\leq l+q \leq 2n.\end{equation*}
The first two can be unified:
\begin{equation}\max\!(0, k-2n) \leq l \leq \min\!(\!\dim B, k), \quad 0\leq l+q \leq 2n.\end{equation}
We calculate the
$E_{1}$
to
$E_{2}$
pages on the edge line
$l+q=0$
.
Lemma 2·3. The following holds:
-
(1)
$E_{1}^{l,-l}=0$
when
$l\leq \min\!(\!\dim B, k)$
with
$l\not\equiv k$
mod 2; -
(2)
$E_{2}^{l,-l}=0$
when
$l< \min\!(\!\dim B, k)$
; -
(3) For
$l_{0} = \min\!(\!\dim B, k)$
we have
$E_{1}^{l_{0},-l_{0}}=E_{2}^{l_{0},-l_{0}}= \cdots = E_{\infty}^{l_{0},-l_{0}}$
.
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
\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
\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
\begin{equation}H^{2m,0}(X_{b}^{n}) = \bigoplus_{|\sigma|=m} H^{2,0}(X_{b})^{\otimes \sigma},\end{equation}
\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
\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
\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
\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.
We can now complete the proof of Proposition 2·2.
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).
In the case
$k=\dim B + 2m$
, the vector bundle
$\mathcal{H}^{2m,0}\otimes K_{B}=(\pi_{n})_{\ast}\Omega_{\pi_{n}}^{2m}\otimes K_{B}$
can be written more specifically as follows. For a subset
$\sigma$
of
$\{ 1, \cdots, n \}$
with cardinality
$| \sigma |=m$
, we denote by
$X_{\sigma}\simeq X_{m}$
the fiber product of the ith factors
$X\to B$
of
$X_{n}\to B$
over all
$i\in \sigma$
. We denote by
\begin{equation*}X_{n} \stackrel{\pi_{\sigma}}{\to} X_{\sigma} \stackrel{\pi^{\sigma}}{\to} B\end{equation*}
the natural projections. The Künneth formula (2·5) says that
\begin{equation*}(\pi_{n})_{\ast}\Omega_{\pi_{n}}^{2m} \simeq\bigoplus_{|\sigma|=m} \pi^{\sigma}_{\ast}K_{\pi^{\sigma}}.\end{equation*}
Combining this with the isomorphism
\begin{equation}\pi^{\sigma}_{\ast}K_{X_{\sigma}}\simeq K_{B}\otimes \pi^{\sigma}_{\ast}K_{\pi^{\sigma}}\end{equation}
for each
$X_{\sigma}$
, we can rewrite the isomorphism in the last case of Proposition 2·2 as
\begin{equation}(\pi_{n})_{\ast}\Omega_{X_{n}}^{\dim B+2m} \simeq\bigoplus_{|\sigma|=m} \pi^{\sigma}_{\ast}K_{X_{\sigma}}.\end{equation}
2·2. Extension over compactification
Let
$\pi\colon X\to B$
be a K3 fibration as in Section 2·1. We now assume that X, B are quasi-projective and
$\pi$
is a morphism of algebraic varieties. We take smooth projective compactifications of
$X_{n}, X_{\sigma}, B$
and denote them by
$\bar{X}_{n}, \bar{X}_{\sigma}, \bar{B}$
respectively.
Proposition 2·4. We have
\begin{equation*}H^{0}(\bar{X}_{n}, \Omega^{k}) \simeq\begin{cases}H^{0}(\bar{B}, \Omega^{k}) & \: \: k\leq \dim B \\ 0 & \: \: k>\dim B, \: \: k\not\equiv \dim B \: \: \mathrm{mod} \; 2 \\ \oplus_{\sigma}H^{0}(\bar{X}_{\sigma}, K_{\bar{X}_{\sigma}}) & \: \: k=\dim B+2m, \: 0\leq m \leq n\end{cases}\end{equation*}
In the last case,
$\sigma$
ranges over all subsets of
$\{ 1, \cdots, n \}$
with
$|\sigma|=m$
. The isomorphism in the first case is given by the pullback by
$\pi_{n}\colon X_{n}\to B$
, and the isomorphism in the last case is given by the direct sum of the pullbacks by
$\pi_{\sigma}\colon X_{n}\to X_{\sigma}$
for all
$\sigma$
.
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
$X_{\sigma}$
.
Claim 2.5. For each
$\sigma$
,
$\omega_{\sigma}$
is holomorphic over
$\bar{X}_{\sigma}$
.
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.
The above argument will be clear if we consider over the generic point
$\eta$
of B: we restrict
$\omega$
to the fiber of
$(X_{\eta})^{n}\to (X_{\eta})^{\tau}$
over the geometric point
$\tilde{B}$
of
$(X_{\eta})^{\tau}$
over
$\eta$
.
By Claim 2.5, the pullback
\begin{equation*}(\pi_{\sigma}^{\ast})_{\sigma} :\bigoplus_{|\sigma|=m} H^{0}(\bar{X}_{\sigma}, K_{\bar{X}_{\sigma}}) \to H^{0}(\bar{X}_{n}, \Omega^{\dim B + 2m})\end{equation*}
is surjective. It is also injective as implied by (2·11). This proves Proposition 2·4.
2·3. Universal K3 surface
Now we prove Theorem 1·1, in the generality of lattice-polarisation. Let L be an even lattice of signature (2, d) which can be embedded as a primitive sublattice of the K3 lattice
$3U\oplus 2E_{8}$
. We denote by
\begin{equation*}\mathcal{D} = \{ \: {{\mathbb{C}}} \omega \in {{{\mathbb P}}}L_{{{\mathbb{C}}}} \: | \: (\omega, \omega)=0, (\omega, \bar{\omega})>0 \: \}^{+}\end{equation*}
the Hermitian symmetric domain associated to L, where
$+$
means a connected component.
Let
$\pi\colon X\to B$
be a smooth projective family of K3 surfaces over a smooth quasi-projective connected base B. We say ([
Reference Ma3
]) that the family
$\pi\colon X\to B$
is lattice-polarised with period lattice L if there exists a sub local system
$\Lambda$
of
$R^{2}\pi_{\ast}{{\mathbb{Z}}}$
such that each fiber
$\Lambda_{b}$
is a hyperbolic sublattice of the Néron-Severi lattice
$NS(X_{b})$
and the fibers of the orthogonal complement
$\Lambda^{\perp}$
are isometric to L. Then we have a period map
\begin{equation*}\mathcal{P} \,:\, B \to {{\Gamma}}\backslash \mathcal{D}\end{equation*}
for some finite-index subgroup
${{\Gamma}}$
of
$\mathrm{O}^{+}(L)$
. By Borel’s extension theorem,
$\mathcal{P}$
is a morphism of algebraic varieties.
Let us put the assumption
\begin{equation}\mathcal{P} \; \mathrm{is\ birational\ and} \: -\mathrm{id}\not\in {{\Gamma}}.\end{equation}
For such a family
$\pi\colon X\to B$
, if we shrink B as necessary, then
$\mathcal{P}$
is an open immersion and Condition 2·1 is satisfied. For example, the universal K3 surface
$\mathcal{F}_{g,1}\to {{\mathcal{F}_{g}}}$
for
$g>2$
restricted over a Zariski open set of
${{\mathcal{F}_{g}}}$
satisfies this assumption with
$L=L_{g}$
and
${{\Gamma}}={{\Gamma_{g}}}$
(see Section 1 for these notations).
As in Section 1, we denote by
$M_{\wedge^{k},k}({{\Gamma}})$
the space of vector-valued modular forms of weight
$(\wedge^{k},k)$
for
${{\Gamma}}$
,
$S_{l}({{\Gamma}}, \det)$
the space of scalar-valued cusp forms of weight l and character
$\det$
for
${{\Gamma}}$
, and
$\mathcal{S}_{n,m}=\mathfrak{S}_{n}/(\mathfrak{S}_{m}\times \mathfrak{S}_{n-m})$
.
Theorem 2·6. Let
$\pi\colon X\to B$
be a lattice-polarised K3 family with period lattice L of signature (2, d) with
$d\geq 3$
and monodromy group
${{\Gamma}}$
satisfying (2·12). Then we have an
$\mathfrak{S}_{n}$
-equivariant isomorphism
\begin{equation*}H^{0}(\bar{X}_{n}, \Omega^{k}) \simeq\begin{cases}0 & \: \: 0<k< d/2 \\ M_{\wedge^{k},k}({{\Gamma}}) & \: \: d/2 \leq k < d \\ 0 & \: \: k>d, \: k-d\not\in 2{{\mathbb{Z}}} \\ S_{d+m}({{\Gamma}}, \det)\otimes {{\mathbb{C}}}\mathcal{S}_{n,m} & \: \: k=d+2m, \: 0\leq m \leq n\end{cases}.\end{equation*}
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].
Remark 2·7. The case
$k\geq d$
of Theorem 2·6 holds also when
$d=1, 2$
. We put the assumption
$d\geq 3$
for the requirement of the Koecher principle from [
Reference Pommerening5
]. Therefore, in fact, only the case
$(d, k)=(2, 1)$
with Witt index 2 is not covered.
Open access
