1. Introduction
Let ${{\mathcal{F}_{g,n}}}$ be the moduli space of npointed K3 surfaces of genus $g>2$ , i.e., primitively polarised of degree $2g2$ . It is a quasiprojective 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 kforms 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:
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 $L_{g}=2U\oplus 2E_{8}\oplus \langle 22g \rangle$ 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 vectorvalued 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 scalarvalued cusp forms of weight $19+m$ and determinant character for ${{\Gamma_{g}}}$ , and $\mathcal{S}_{n,m}$ stands for the right quotient $\mathfrak{S}_{n}/(\mathfrak{S}_{m}\times \mathfrak{S}_{nm})$ , which is a left $\mathfrak{S}_{n}$ set. Theorem 1·1 is actually formulated and proved in the generality of latticepolarisation (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 righthand 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 kforms 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 vectorvalued 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:
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 socalled 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
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 nfold 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:
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
whose graded quotients are naturally isomorphic to
This filtration induces the holomorphic Leray spectral sequence
which converges to the filtration
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 $\pi_{n}$ :
Here $\mathcal{H}^{\ast, \ast}$ are the Hodge bundles associated to the fibration $\pi_{n}\colon X_{n}\to B$ , and
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
The first two can be unified:
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 $E_{1}^{l,l}=\mathcal{H}^{kl,0}\otimes \Omega_{B}^{l}$ . By the Künneth formula, the fiber of $\mathcal{H}^{kl,0}$ over a point $b\in B$ is identified with
where $(p_{1}, \cdots, p_{n})$ ranges over all indices with $\sum_{i}p_{i}=kl$ and $0\leq p_{i} \leq 2$ .
(1) When $kl$ 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 $H^{kl,0}(X_{b}^{n})=0$ . Therefore $\mathcal{H}^{kl,0}=0$ when $kl$ 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 $l=k2m$ 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
By the Künneth formula as in (2·3), the fibers of the Hodge bundles $\mathcal{H}^{2m,0}$ , $\mathcal{H}^{2m1,1}$ over $b\in B$ are respectively identified with
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
By [ Reference Voisin8 , lemma 5·8], the $(\sigma, i)$ component
factorises as
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
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
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
the natural projections. The Künneth formula (2·5) says that
Combining this with the isomorphism
for each $X_{\sigma}$ , we can rewrite the isomorphism in the last case of Proposition 2·2 as
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 quasiprojective 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
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 kform 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 kform $\omega_{B}$ 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 kform 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 nondominant 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
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 latticepolarisation. 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
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 quasiprojective connected base B. We say ([ Reference Ma3 ]) that the family $\pi\colon X\to B$ is latticepolarised 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 NéronSeveri lattice $NS(X_{b})$ and the fibers of the orthogonal complement $\Lambda^{\perp}$ are isometric to L. Then we have a period map
for some finiteindex subgroup ${{\Gamma}}$ of $\mathrm{O}^{+}(L)$ . By Borel’s extension theorem, $\mathcal{P}$ is a morphism of algebraic varieties.
Let us put the assumption
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 vectorvalued modular forms of weight $(\wedge^{k},k)$ for ${{\Gamma}}$ , $S_{l}({{\Gamma}}, \det)$ the space of scalarvalued cusp forms of weight l and character $\det$ for ${{\Gamma}}$ , and $\mathcal{S}_{n,m}=\mathfrak{S}_{n}/(\mathfrak{S}_{m}\times \mathfrak{S}_{nm})$ .
Theorem 2·6. Let $\pi\colon X\to B$ be a latticepolarised 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
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 kforms on $\mathcal{D}$ , which in turn is identified with the space $M_{\wedge^{k},k}({{\Gamma}})$ of vectorvalued 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
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 righthand side can be written as
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.