Tautological classes on moduli space of hyperk\"ahler manifolds

TL;DR

This paper develops the tautological ring on moduli spaces of hyperk"ahler manifolds, showing most classes are combinations of Noether-Lefschetz classes and proving related conjectures for specific cases.

Contribution

It constructs the tautological ring on moduli spaces of hyperk"ahler manifolds and proves the cohomological tautological and Franchetta conjectures in certain cases.

Findings

Most tautological classes are linear combinations of Noether-Lefschetz classes.

Proved the cohomological tautological conjecture for K3$^{[n]}$-type hyperk"ahler manifolds with n≤2.

Established the cohomological generalized Franchetta conjecture for these moduli spaces.

Abstract

In this paper, we discuss the cycle theory on moduli spaces $\cF_h$ of $h$-polarized hyperk\"ahler manifolds. Firstly, we construct the tautological ring on $\cF_h$ following the work of Marian, Oprea and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces. We study the tautological classes in cohomology groups and prove that most of them are linear combinations of Noether-Lefschetz cycle classes. In particular, we prove the cohomological version of the tautological conjecture on moduli space of K3$^{[n]}$-type hyperk\"ahler manifolds with $n\leq 2$. Secondly, we prove the cohomological generalized Franchetta conjecture on universal family of these hyperk\"ahler manifolds.

Equations

$${\mathcal{N}}^r({\mathcal{K}}_g)\subseteq {\mathcal{K}}_g$$

$$\Phi :H^{\ast }(X,\mathbb{Z})\stackrel{\sim }{\to }R,$$

$$L_n=U^{\oplus 3}\oplus E_8(-1{)}^{\oplus 2}\oplus ⟨-2(n-1)⟩$$

$$L_{K,n}=U^{\oplus 3}\oplus ⟨-2(n+1)⟩.$$

$$\mathrm{Aut}(X)\to \mathrm{GL}(H^{\ast }(X,\mathbb{Z}))$$

$$\mathrm{Aut}(X)\to \mathrm{O}(H^2(X,\mathbb{Z})).$$

$$1\to T\to \mathrm{Mon}(X)\to {\mathrm{Mon}}^2(X)\to 1$$

$${\mathrm{Mon}}^2(X,H)\subseteq {\mathrm{Mon}}^2(X)$$

$${\tilde{G}}_X:=\mathrm{Spin}(H^2(X,\mathbb{Q}))$$

$$\rho :G_X\to \mathrm{Aut}(H^{\mathrm{even}}(X,\mathbb{Q}))$$

$${\mathrm{DCH}}^{\ast }(X)\to H^{\ast }(X,\mathbb{Q})$$

$$D^{\pm }=\{x\in \mathbb{P}({\Lambda }_h\otimes \mathbb{C})|(x,x)=0,(x,\bar{x})>0\},$$

$$N_h=(X,H,\phi )/\sim$$

$${\tilde{\mathcal{P}}}_h:N_h\to D$$

$${\mathcal{P}}_h:F_h\hookrightarrow {\Gamma }_h\backslash D.$$

$${\mathrm{Tei}}_h(M)/\mathrm{MCG}(M,h),$$

$$\mathrm{MCG}(M,h)\to \mathrm{GL}(H_{\mathrm{prim}}^{\ast }(M,\mathbb{Z}/\ell \mathbb{Z}))$$

$$\mathrm{MCG}(M,h)\to \mathrm{O}(H_{\mathrm{prim}}^2(M,\mathbb{Z}/\ell \mathbb{Z})),$$

$$H^{\ast }(X,\mathbb{Z}/\ell \mathbb{Z})\cong H^{\ast }(M,\mathbb{Z})\otimes \mathbb{Z}/\ell \mathbb{Z},$$

$${\mathrm{Mon}}_{\ell }(X,H)\subseteq \mathrm{Mon}(X,H)$$

$${\mathrm{Mon}}_{\ell }(X,H)\cong {\mathrm{Mon}}_{\ell }^2(X,H)$$

$${\mathcal{P}}_{\ell }:F_h^{\ell }\to {\Gamma }_h^{\ell }\backslash D.$$

$$\{g\in \mathrm{SO}(\Lambda )|g(h)=h\}\to \mathrm{SO}({\Lambda }_h).$$

$$\left\{K\backslash \{g\in \mathrm{Isometry}(\Lambda (\hat{\mathbb{Z}}),H^2(X,\hat{\mathbb{Z}})(1))|g(h)=c_1(H)\}\right\}.$$

$$F_{h,K}\hookrightarrow {\Gamma }_{h,K}\backslash D,$$

$${\mathbb{H}}_{\pi }:=R{\pi }_{\ast }\mathbb{Q}$$

$${\pi }_{\ell }:{\mathcal{U}}_h^{\ell }\to {\mathcal{F}}_h^{\ell }$$

$$D\times H^{\ast }(M,\mathbb{Q})$$

$$j:\Sigma \hookrightarrow \Lambda$$

$$\varphi :\Sigma \to \mathrm{NS}(X)\subset H^2(X,\mathbb{Z})$$

Full text

1. Introduction

The tautological ring of the moduli space $\mathcal{M}_{g}$ of genus $g\geq 2$ curves, originally studied by Mumford, is the subring of the Chow ring generated by $\kappa$-classes which appear most naturally in geometry. There have been substantial progress in understanding tautological ring of moduli space of curves in the past decades. In higher dimensional moduli theory, little is known even regarding definitions. Recently, there have been some developments towards the cycle theory on moduli spaces of K3 surfaces. In this paper, we would like to investigate the tautological class problem on moduli spaces of K3 type varieties.

1.1. MOP conjecture and its generalization

Let $\mathcal{K}_{g}$ be the moduli space of primitively polarized K3 surfaces of genus $g$. The cycle theory on $\mathcal{K}_{g}$ appears to be much more complicated than on $\mathcal{M}_{g}$ because there are many more classes in the Chow groups. First of all, there are so called NL-cycles on $\mathcal{K}_{g}$ arising from the Noether-Lefschetz theory. For $1\leq r\leq 19$, the $r$-th higher Noether-Lefschetz locus

[TABLE]

parametrizing the K3 surfaces in $\mathcal{K}_{g}$ with Picard number greater than $r$, is a countable union of subvarieties of codimension $r$. Each irreducible component of $\mathcal{N}^{r}(\mathcal{K}_{g})$ occurs as the moduli space of certain lattice-polarized K3 surfaces in $\mathcal{K}_{g}$ and a NL-cycle of codimension $r$ is defined as some linear combinations of such irreducible subvarieties. They play a very important role in the cycle theory of $\mathcal{K}_{g}$. It has been shown that the Picard group ${\rm Pic}_{\mathbb{Q}}(\mathcal{K}_{g})$ with rational coefficients is generated by NL-divisors, see [6]. This motivates us to study the subring generated by all NL-cycles, denoted by ${\rm NL}^{\ast}(\mathcal{K}_{g})$.

Secondly, similar as the case in $\mathcal{M}_{g}$, one can obtain natural cycle classes from the tautological bundles on the universal family. Let $\mathcal{K}_{\Sigma}\subset\mathcal{K}_{g}$ be the $r$-th higher Noether-Lefschetz locus corresponding to some Picard lattice $\Sigma$ of rank $r+1$. In [34], Marian, Oprea and Pandharipande (MOP) have introduced $\kappa$-classes $\kappa^{\mathcal{B}}_{a_{0},\ldots,a_{r};b}$ on $\mathcal{K}_{\Sigma}$, see Definition 4.2.1. The tautological ring $\mathrm{R}^{\ast}(\mathcal{K}_{g})\subseteq{\rm CH}_{\mathbb{Q}}^{\ast}(\mathcal{K}_{g})$ is the $\mathbb{Q}$-subalgebra generated by the images of the $\kappa$-classes on $\mathcal{K}_{\Sigma}$ via pushforward maps. The tautological conjecture on $\mathcal{K}_{g}$ states:

Conjecture 1** (MOP).**

Let $\mathrm{R}^{\ast}(\mathcal{K}_{g})$ be the subring of ${\rm CH}^{\ast}_{\mathbb{Q}}(\mathcal{K}_{g})$ generated by all the images of the $\kappa$-classes on $\mathcal{K}_{\Lambda}$. Then ${\rm NL}^{\ast}(\mathcal{K}_{g})=\mathrm{R}^{\ast}(\mathcal{K}_{g})$.

More generally, as polarized hyperkähler manifolds are higher-dimensional generalizations of algebraic K3 surfaces and behave like K3 surfaces in many ways (cf. §2), it is natural to study tautological rings on moduli spaces of all polarized hyperkähler manifolds. Guided by the K3 case, we define kappa classes on smooth families of hyperkähler manifolds as pushforwards of “Beauville-Voisin classes” on the generic fiber. These classes are named in reference to the work of Beauville and Voisin [4] on weak splitting properties of algebraic hyperkähler manifolds. In this way, we define tautological rings on moduli spaces of polarized hyperkähler manifolds and formulate a generalization of Conjecture 1 for polarized hyperkähler manifolds. We will refer to it as the generalized tautological conjecture, see Conjecture 4 for the details.

In this paper, we mainly consider the cohomological version of the generalized tautological conjecture and prove that most tautological classes in cohomology groups are in the span of Noether-Lefschetz classes. Our main results are Theorems 4.3.1 and 8.3.1. As a consequence, we obtain

Theorem 1**.**

Let $\mathrm{R}^{\ast}_{\rm hom}(\mathcal{K}_{g})\subseteq H^{\ast}(\mathcal{K}_{g},\mathbb{Q})$ be the image of $\mathrm{R}^{\ast}(\mathcal{K}_{g})$ in the cohomology ring, then ${\rm NL}^{\ast}_{\rm hom}(\mathcal{K}_{g})=\mathrm{R}^{\ast}_{\rm hom}(\mathcal{K}_{g}).$

Our approach is to show that all $\kappa$-classes on $\mathcal{K}_{\Sigma}$ are in the span of Noether-Lefschetz cycles on $\mathcal{K}_{\Sigma}$. However, this actually is not true when $r>17$ (see Remark 8.3.2). Fortunately, this does not cause contradict Theorem 1 because we will prove that $\mathrm{R}^{i}_{\rm hom}(\mathcal{K}_{g})=0$ for $i>17$.

While we were writing this paper, the article [45] appeared on arXiv. There Conjecture 1 is essentially proved. They use Gromov-Witten theory. Beware however that they work with a slightly different (more canonical) definition of kappa classes.

1.2. Generalized Franchetta conjecture

Denote by $\mathcal{K}_{g}^{\circ}$ be the subspace of $\mathcal{K}_{g}$ that consists of K3 surfaces with only trivial automorphisms. It carries a universal family $\pi:\mathcal{U}_{g}^{\circ}\rightarrow\mathcal{K}^{\circ}_{g}$. Recall that, O’Grady’s generalized Franchetta conjecture states

Conjecture 2** (O’Grady).**

For any $\alpha\in{\rm CH}^{2}(\mathcal{U}_{g}^{\circ})$, the restriction $\alpha|_{S}$ to a closed fiber $S$ is a multiple of the Beauville-Voisin class $c_{S}$.

Here $c_{S}$ is a canonical class in ${\rm CH}_{0}(S)$ represented by a point on a rational curve of $S$, which satisfies the following properties:

1. (1)

The intersection of two divisors on $S$ lies in $\mathbb{Z}c_{S}\subset{\rm CH}_{0}(S)$. 2. (2)

The second Chern class $c_{2}(T_{S})$ equals $24c_{S}\in{\rm CH}_{0}(S)$.

Conjecture 2 has been confirmed when $g\leq 12$ (see [46]) but remains widely open for large $g$. Let $\mathcal{T}_{\pi}$ be the relative tangent bundle of $\pi$. According to [58, Theorem 10.19], Conjecture 2 is equivalent to the following

Conjecture 3**.**

For any $\alpha\in{\rm CH}^{2}(\mathcal{U}_{g}^{\circ})$, there exists $m\in\mathbb{Q}$ such that $\alpha-mc_{2}(\mathcal{T}_{\pi})$ supports on a proper subvariety of $\mathcal{K}_{g}^{\circ}$.

Our second result is a cohomological version of the generalized Franchetta conjecture (cf. [60]).

Theorem 1.2.1**.**

For any $\alpha\in{\rm CH}^{2}(\mathcal{U}_{g}^{\circ})$, there exists a rational number $m$ such that the class $[\alpha-mc_{2}(\mathcal{T}_{\pi})]\in H^{4}(\mathcal{U}_{g}^{\circ},\mathbb{Q})$ supports on Noether-Lefschetz divisors. In other words, $\alpha-mc_{2}(\mathcal{T}_{\pi})$ is cohomologically equivalent to zero on $\pi^{-1}(W)$ for some open subset $W\subseteq\mathcal{K}_{g}^{\circ}$.

Here again, Theorem 1.2.1 follows from a more general theorem concerning the cohomological generalized Franchetta conjecture on hyperkähler manifolds (see $\S$8.1).

1.3. Organization of the paper

In Sections 2 and 3, we review the hyperkähler geometry, especially the cycle theory and Torelli theorem. The tautological ring on moduli spaces of polarized hyperkählers is defined in Section 4. In Section 5 and Section 6, we recap the work of [7] and [6] about surjectivity results on special cycles of Shimura varieties of orthogonal type. Following [22] in Section 7 we construct a so-called Funke-Kudla-Millson ring in the space of differential forms with coefficients. Section 8 is devoted to the proof of the cohomological tautological conjecture and the generalized Franchetta conjecture. In the last section, we discuss the properties of the ring generated by special cycles on Shimura varieties.

1.4. Notation and conventions

Throughout this paper, we write $\widehat{\mathbb{Z}}$ for the profinite completion of $\mathbb{Z}$. We denote by $\mathbb{A}$ the adèle ring of $\mathbb{Q}$ and $\mathbb{A}_{f}$ the ring of finite adèles. If $G$ is a semisimple classical group over $\mathbb{Q}$, we let $L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}))$ be the space of square integrable functions on $G(\mathbb{Q})\backslash G(\mathbb{A})$ and denote by $\mathcal{A}(G)$ (resp. $\mathcal{A}_{\rm cusp}(G)$) the set of irreducible square integral (resp. cuspidal) representations occuring discretely in $L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}))$. For a complex variety $X$, from now on we shall use ${\rm CH}^{\ast}(X)$ to denote the Chow ring of $X$ with rational coefficients and denote by $\mathrm{DCH}^{\ast}(X)\subseteq{\rm CH}^{\ast}(X)$ the subring generated by divisor classes.

1.5. Acknowledgements

We are very grateful to Brendan Hassett, Daniel Huybrechts and Eyal Markman for many helpful discussions and useful suggestions. The first named author would like to thank Claire Voisin. She first told him about O’Grady’s generalized Franchetta conjecture and asked if the methods of [6] could be used to prove the cohomological version of it. He would also like to thank her for many explanations related to these topics. The second named author would like to thank Panhdaripande and Yin for a lot of useful conversations during his stay in ETH.

2. Hyperkähler manifolds and moduli

2.1. Basic theory of hyperkähler manifolds

A smooth complex compact $2n$-dimensional manifold $X$ is an irreducible holomorphic symplectic or hyperkähler manifold if it is simply connected and $H^{0}(X,\Omega^{2}_{X})$ is spanned by an everywhere nondegenerate holomorphic 2-form $\omega_{X}$. It carries an integral, primitive quadratic form $q_{X}$ on $H^{2}(X,\mathbb{Z})$, called the Beauville-Bogomolov (BB) form, which satisfies

1. (1)

$q_{X}$ is non-degenerate and of signature $(3,b_{2}(X)-3)$ 2. (2)

There exists a positive rational number $c$, the Fujiki invariant such that $q_{X}^{n}(\alpha)=c\int_{X}\alpha^{2n}$ for all classes $\alpha\in H^{2}(X,\mathbb{Z})$. 3. (3)

The Hodge decomposition $H^{2}(X,\mathbb{C})=H^{2,0}(X)\oplus H^{1,1}(X)\oplus H^{0,2}(X)$ is orthogonal with respect to $q_{X}\otimes\mathbb{C}$.

From $(2)$, the BB-form $q_{X}$ is deformation invariant and we will thus restrict to compact hyperkähler manifolds of a fixed deformation class, the isomorphism type, say $\Lambda$, of the lattice realized by the BB-form is unique. We say an isometry $\varphi:H^{2}(X,\mathbb{Z})\rightarrow\Lambda$ is a marking of $X$. Moreover, if we consider the entire cohomology of $X$, we let $R$ denote an abstract ring such that the cohomology ring $H^{\ast}(X,\mathbb{Z})$ is isomorphic to $R$. Then we say that an isomorphism

[TABLE]

is a full marking of $X$. If $X$ comes with an ample line bundle $H$, we say that $(X,H)$ is a polarized hyperkähler manifold.

Such manifolds are quite rare to construct. The two well-known series of examples, found by Beauville [3], are the Hilbert scheme of points on K3 surfaces and generalized Kummer varieties (See Example 2.1.1). The only other known examples were constructed by O’Grady [41, 42]. See the details as below.

Example 2.1.1**.**

1. (i)

For $n>0$, the length $n$ Hilbert scheme $S^{[n]}$ of a K3 surface $S$ is a hyperkähler manifold of dimension $2n$. The second cohomology $H^{2}(S^{[n]},\mathbb{Z})$ under BB-form is an even lattice of signature $(3,20)$ and it is isomorphic to

[TABLE]

where $U$ is the hyperbolic lattice of rank two and $E_{8}$ is the positive definite lattice associated to the Lie group of the same name. We say that a hyperkähler manifold $X$ is of $\mathrm{K3}^{[n]}$-type if it is deformation equivalent to $S^{[n]}$. In that case the Fujiki invariant is $c=(2n)!/(n!2n)$. 2. (ii)

If $A$ is an abelian surface and $s:A^{[n+1]}\rightarrow A$ is the morphism induced by the additive structure of $A$, then the generalized Kummer variety $K_{n}(A):=s^{-1}(0)$ is a projective hyperkähler manifold of dimension $2n$. We say that $X$ is of generalized Kummer type if it is deformation equivalent to $K_{n}(A)$. In this case, $H^{2}(X,\mathbb{Z})$ is isomorphic to

[TABLE]

while the Fujiki invariant is $c=(n+1)(2n)!/(n!2n)$. 3. (iii)

(OG6) Let $S$ be a K3 surface, and $M$ the moduli space of stable rank 2 vector bundles on $S$, with Chern classes $c_{1}=0$, $c_{2}=4$. It admits a natural compactification $\overline{M}$, obtained by adding classes of semi-stable torsion free sheaves. There is a desingularization of $\overline{M}$ which is a hyperkähler manifold of dimension 10. Its second cohomology is a lattice of signature $(3,21)$. 4. (iv)

(OG10) The similar construction can be done starting from rank 2 bundles with $c_{1}=0$, $c_{2}=2$ on an abelian surface, that gives a hyperkähler manifold of dimension 6 as in (iii). Its second cohomology is a lattice of signature $(3,5)$.

2.2. Automorphisms and monodromy group

Let $X$ be a hyperkähler manifold of type $\Lambda$ and real dimension $2n$. Let ${\rm Aut}(X)$ be the group of automorphisms of $X$. Its action on the cohomology gives a map

[TABLE]

which has finite kernel. We say that $X$ is cohomologically rigidified if ${\rm Aut}(X)$ acts faithfully on $H^{\ast}(X,\mathbb{Z})$, i.e. (2.3) is injective. So far, for all hyperkähler manifolds in Example 2.1.1, they are known to be cohomologically rigidified except the case of OG6 type (cf. [3, 44]). In general, it remains open whether (2.3) is injective for all hyperkähler manifolds.

Moreover, the automorphisms act on $H^{2}(X,\mathbb{Z})$ preserving the BB-form. So projection to the second cohomology yields a map

[TABLE]

with finite kernel. According to [25], this kernel is deformation invariant and it is trivial if $\Lambda$ is of K3*[n]* or OG10 type and nontrivial if $\Lambda$ is of generalized Kummer or OG6 type.

2.3.

The monodromy group ${\rm Mon}(X)\subseteq\mathrm{GL}(H^{\ast}(X,\mathbb{Z}))$ is defined as the subgroup generated by monodromy representations of all connected families containing $X$. Let ${\rm Mon}^{2}(X)\subseteq\mathrm{Aut}(H^{2}(X,\mathbb{Z}))$ be its image under the projection to the second cohomology group. Verbitsky [55] has shown that the group ${\rm Mon}^{2}(X)$ is an arithmetic subgroup of $\mathrm{O}(H^{2}(X,\mathbb{Z}))$ and there is an exact sequence

[TABLE]

with $T$ a finite group. Moreover, $T$ is trivial if (2.4) is injective (cf. [55, Corollary 7.3]).

Furthermore, given a polarized hyperkähler manifold $(X,H)$, the polarized monodromy group

[TABLE]

is defined to be the stabilizer of $c_{1}(H)$. Then ${\rm Mon}^{2}(X,H)$ is isomorphic to an arithmetic subgroup of $\mathrm{O}(H^{2}_{\rm prim}(X,\mathbb{Z}))$ (cf. [55, Theorem 3.4]) via a given marking. It has been shown by Markman [36, Proposition 7.1] that the image is independent of the marking.

2.4. Group actions on cohomology

Let $G_{X}$ be the $\mathbb{Q}$-algebraic group associated to ${\rm SO}(\Lambda)$. It acts naturally on $H^{2}(X,\mathbb{Q})$ through to the standard representation. As $G_{X}$ is a connected component of the Zariski closure of ${\rm Mon}^{2}(X)$ and ${\rm Mon}(X)\rightarrow{\rm Mon}^{2}(X)$ has finite kernel, the monodromy action of ${\rm Mon}(X)$ also gives rise to an action of $G_{X}$ on $H^{\ast}(X)$ of $X$ via automorphisms (cf. [24, Proposition 4.1]).

The $G_{X}$ action also respects the Hodge structure. For instance, the trivial summands in $H^{\ast}(X,\mathbb{Q})$ are Hodge classes. In particular, the Chern classes $c_{i}(T_{X})$ of the tangent bundle ly in a trivial summand of $H^{2i}(X,\mathbb{Q})$ and $c_{i}(T_{X})=0$ is zero when $i$ is odd.

Remark 2.4.1**.**

The theory of Lefschetz modules, developed by Verbitsky [54] and Looijenga-Lunts [32], provides another action of the group

[TABLE]

on $H^{\ast}(X,\mathbb{Q})$. The action also preserves the ring structure and on the even cohomology it factors through a representation

[TABLE]

that coincides with the standard representation on $H^{2}(X,\mathbb{Q})$. Therefore, the two representation of ${\rm Mon}^{2}(X)$ in ${\rm GL}(H^{\ast}(X,\mathbb{Q}))$ agree on a finite index subgroup (cf. [35, §4.6]).

2.5. Beauville-Voisin classes

We now recall Beauville and Voisin’s work on weak splitting property of algebraic hyperkähler manifolds, which is observed from the following result on the Chow ring of K3 surfaces.

Theorem 2.5.1** (Beauville-Voisin [4]).**

Let $X$ be a smooth projective K3 surface. Then there exists a canonical zero cycle $c_{X}\in{\rm CH}_{0}(X)$ of degree one such that

• •

$c_{X}$* is represented by a point on a rational curve on $X$*

• •

for any divisors $D_{1},D_{2}\in{\rm Pic}(X)$, the intersection $D_{1}\cdot D_{2}$ is proportional to $c_{X}$ in ${\rm CH}_{0}(X)$.

• •

$c_{2}(T_{X})=24c_{X}$.

In particular, the natural map

[TABLE]

is injective.

Beauville has pointed out that this can be understood in a broader framework of Bloch-Beilinson-Murre filtration, the so called weak splitting property, and he conjectures that this can be generalized to hyperkähler manifolds, i.e. (2.7) holds when $X$ is hyperkähler. This was later strengthen by Voisin [60] via involving the Chern classes of the tangent bundle $T_{X}$, i.e. replacing $\mathrm{DCH}^{\ast}(X)$ in (2.7) by the subalgebra ${\rm BV}^{\ast}(X)\subseteq{\rm CH}^{\ast}(X)$ generated by divisor classes and $c_{i}(T_{X})$. We will call the elements in ${\rm BV}^{\ast}(X)$ the *Beauville-Voisin (BV) classes * of $X$.

The conjecture has been confirmed by Voisin [59] and Fu [20] when $X$ is a K3 surface, Fano variety of lines of a cubic fourfold, generalized Kummer varieties and $S^{[n]}$ if $n\leq 2b_{2,tr}(S)+4$, where $b_{2,tr}(S)$ is the second Betti number of $S$ minus its Picard number. In these cases, there exists a canonical zero cycle $c_{X}\in{\rm CH}_{0}(X)$ such that ${\rm BV}^{n}(X)$ is spanned by $c_{X}$. In general, the existence of $c_{X}$ remains widely open. Note that if the canonical cycle $c_{X}$ exists, then the top Chern class $c_{2n}(T_{X})$ is a multiple of $c_{X}$.

3. Moduli space of hyperkähler manifolds

3.1. Moduli of polarized hyperkähler manifolds

In the sequel we will always assume that a $2n$-dimensional hyperkähler manifold $X$ of type $\Lambda$ has a primitive polarization $H$, i.e. $c_{1}(H)\in H^{2}(X,\mathbb{Z})$ is primitive. In order to discuss the moduli space of polarized hyperkähler manifolds, we shall choose a polarization type, i.e. an $\mathrm{O}(\Lambda)$-orbit of a primitive vector $h\in\Lambda$. We say that $(X,H)$ is $h$-polarized if $\varphi(c_{1}(H))=h$ for a given marking $\varphi:H^{2}(X,\mathbb{Z})\cong\Lambda$.

Consider the moduli stack of $h$ primitively polarized hyperkähler manifolds of dimension $2n$ and type $\Lambda$ with a given Fujiki invariant, which associates to a scheme $T$ (over $\mathbb{C}$) the set of isomorphism classes of flat families of primitively polarized hyperkähler manifolds over $T$. This stack in general is disconnected with only finitely many connected components. Let $\mathcal{F}_{h}$ be a connected component of this stack. Then a standard result is

Proposition 3.1.1**.**

$\mathcal{F}_{h}$* is a smooth Deligne-Mumford quotient stack and it can be coarsely represented by a quasi-projective variety $F_{h}$.*

Proof.

The proof is the same as in the case of K3 surfaces. The quasi-projectivity of $F_{h}$ follows from the Torelli theorem and the standard result of Baily-Borel [2]. ∎

Remark 3.1.2**.**

Here we work over the base field $\mathbb{C}$ only for simplicity. We also refer to [1] for the results of moduli space of hyperkähler varieties over fields of characteristic [math].

3.2. Period map

Let us recall Verbitsky and Markman’s Hodge theoretic Torelli theorem for hyperkähler manifolds. Given a polarization type vector $h\in\Lambda$, the orthogonal completement $\Lambda_{h}:=h^{\perp}\subseteq\Lambda$ is a lattice of signature $(2,b_{2}(X)-3)$. The period domain $D$ of $h$-polarized hyperkähler manifolds of type $\Lambda$ is a connected component of

[TABLE]

seen as a Hermitian symmetric domain of type IV. Consider the connected component of the coarse moduli space of marked hyperkähler manifolds

[TABLE]

parametrizing marked $h$-polarized hyperkähler manifolds $(X,H,\varphi)$ up to equivalence. Then we have a period map

[TABLE]

that maps $(X,H,\varphi)$ to the line $[\varphi(\omega_{X})]$.

Remark 3.2.1**.**

One can also consider the moduli (analytic) stack $\widetilde{\mathcal{N}}_{h}$ of fully marked $h$-polarized hyperkähler manifolds and denote by $\widetilde{N}_{h}$ its coarse moduli space. This also defines a period map $\widetilde{N}_{h}\rightarrow D$. See §3.3 for more details.

Write $\Gamma_{h}:={\rm Mon}^{2}(X,H)$, then Markman shows that the map (3.2) is a $\Gamma_{h}$-equivariant open immersion (cf. [36]). To forget the marking, we quotient (3.2) on both sides and get

Theorem 3.2.2**.**

[36, Theorem 8.4]** The period map (3.2) induces an open immersion

[TABLE]

The analytic orbifold $\Gamma_{h}\backslash D$ has an algebraic structure and it is a quasiprojective variety. In this paper, we may also use $\mathcal{P}_{h}$ to denote the map from the stack $\mathcal{F}_{h}$ to $\Gamma_{h}\backslash D$ by abuse of notations.

Remark 3.2.3**.**

If we consider the coarse moduli space of quasipolarized hyperkähler manifolds, i.e. the line bundle $H$ is big and nef, the open immersion (3.3) becomes an isomorphism, see [26].

3.3.

There is an analytic construction of $\mathcal{F}_{h}$ via Teichmüller theory. As the hyperkähler manifolds only differ by their complex structures, we can write $X$ as $(M,I)$, where $M$ is a hyperkähler manifold and $I$ is a complex structure on $M$. Then the moduli space $\mathcal{F}_{h}$ can be described as the orbifold

[TABLE]

given as the quotient of a divisor $\mathrm{Tei}_{h}(M)$ in a connected component of the Teichmüller space of $M$ by a subgroup $\mathrm{MCG}(M,h)$ of the mapping class group.

The space $\mathrm{Tei}_{h}(M)$ actually coarsely represents the moduli stack $\widetilde{\mathcal{N}}_{h}$ in Remark 3.2.1 and the image of the $\rm{MCG}(M,h)$ in ${\rm GL}(H^{\ast}(M,\mathbb{Z}))$ is nothing but the polarized monodromy group ${\rm Mon}(X,H)$. In this way, most of the results we discussed in this section can be found in [55] using the language of Teichmüller spaces.

3.4. Level structures

We now introduce the level structures on polarized hyperkähler manifolds to help rigidify our moduli problem. Just as the case for moduli of curves with level structures, this can be viewed as the quotient of $\mathrm{Tei_{h}(M)}$ by a finite index subgroup of $\mathrm{MCG}(M,h)$. For polarized hyperkähler manifolds, there are two natural ways to add level structures by taking the kernel of either the map

[TABLE]

or the map

[TABLE]

for $\ell>0$. In this paper, we mainly concern about the moduli space of polarized hyperkähler manifolds with a level strucutre on the total cohomology, i.e. the quotient of $\mathrm{Tei}_{h}(M)$ by the kernel of (3.4).

In the algebraic setting, we say that a full $\ell$-level structure on a $h$-polarized hyperkähler $(X,H)$ is an isomorphism

[TABLE]

mapping the class $c_{1}(H)$ to $h$. We let $\mathcal{F}^{\ell}_{h}$ be the connected component (associated to $\mathcal{F}_{h}$) of the moduli stack of $h$-polarized hyperkähler manifold with a full $\ell$-level structure. The forgetful map $\mathcal{F}_{h}^{\ell}\rightarrow\mathcal{F}_{h}$ is finite and étale.

Denote by

[TABLE]

the polarized monodromy group with a full $\ell$-level structure, which is the image of the kernel of (3.4) in ${\rm Mon}(X,H)$. If $\ell$ is sufficiently large, the projection to the second cohomology yields an isomorphism

[TABLE]

because of (2.5). We let $\Gamma^{\ell}_{h}\subseteq\Gamma_{h}$ be the image of ${\rm Mon}^{2}_{\ell}(X,H)$ via markings. Then the coarse moduli space $F_{h}^{\ell}$ of $\mathcal{F}_{h}^{\ell}$ is the quotient of the coarse moduli space of fully marked $h$-polarized hyperkähler manifolds by ${\rm Mon}^{2}_{\ell}(X,H)$ and it admits an open immersion

[TABLE]

induced by the period map.

If the kernel of (2.3) is trivial, then $\mathcal{F}_{h}^{\ell}$ is represented by $F_{h}^{\ell}$ for $\ell\gg 1$ because any object in $\mathcal{F}_{h}^{\ell}$ has only trivial automorphism (cf. [48, Lemma 1.5.12]). In this case, $F_{h}^{\ell}$ carries a universal family in the category of varieties. As mentioned in §2.2, this has been confirmed for all known examples except OG6.

3.5.

Let us give a few remarks about the second construction. Rizov [48] has dealt with the case of K3 surfaces in adelic language (see also [38, 33]), but there is no difficulty to extend his construction to hyperkähler manifolds, see e.g. [16, §3.3]. The resulting moduli space better fits with our notions of Shimura varities defined in Section 5.

Write $G={\rm SO}(\Lambda_{h})$ and let $K\subseteq G(\mathbb{A}_{f})$ be an open compact subgroup. Recall that there is an injective morphism

[TABLE]

Following [48, 2.2], we say that $K$ is admissible if every element of $K$ can be viewed as an isometry of $\Lambda(\mathbb{A}_{f})$ fixing $h$ and stabilizing $\Lambda_{h}(\widehat{\mathbb{Z}})$. In this case, we define a $K$-level structure on a $h$ primitively polarized hyperkähler $(X,H)$ as an element of the set

[TABLE]

As in [48, Definition 1.5.16], one can consider the moduli stack $\mathcal{F}_{h,K}$ of primitively $h$-polarized hyperkähler manifolds with a $K$-level structure and let $F_{h,K}$ be the corresponding coarse moduli space. The period map gives an open immersion

[TABLE]

where $\Gamma_{h,K}=\Gamma_{h}\cap K$ (cf. [48, Proposition 3.2.1]). Again, this follows from Verbitsky-Markman’s Hodge theoretic Torelli theorem on hyperkähler manifolds.

3.6. Descent of local systems

Let $\pi:\mathcal{U}_{h}\rightarrow\mathcal{F}_{h}$ be the universal family of $h$-polarized hyperkähler manifolds over $\mathcal{F}_{h}$. We denote by

[TABLE]

the local system that fiberwise corresponds to the cohomology algebra of the fiber. As we explained in §2.3, the speical orthogonal group ${\rm SO}(\Lambda_{h})$ (up to a finite covering) acts fiberwisely on $\mathbb{H}_{\pi}$. As ${\rm SO}(\Lambda_{h})$ acts naturally on $\Gamma_{h}\backslash D$, our goal is to show that the local system $\mathbb{H}_{\pi}$ descends to an automorphic local system on $\Gamma_{h}\backslash D$ after taking a finite cover of $\mathcal{F}_{h}$. We prove it by adding level structures defined in §3.4.

Lemma 3.6.1**.**

Consider the universal family

[TABLE]

of h-polarized hyperkähler manifolds with a full $\ell$-level strucutre. When $\ell$ is sufficiently large, the local system $\mathbb{H}_{\pi_{\ell}}:=R(\pi_{\ell})_{\ast}\mathbb{Q}$ descends to an automorphic local system $\mathbf{H}^{\bullet}$ on $\Gamma_{h}^{\ell}\backslash D$ via the period map (3.7).

Proof.

We can consider the universal fibration $\widetilde{\pi}:U\rightarrow\widetilde{\mathcal{N}}_{h}$ over $\widetilde{\mathcal{N}}_{h}$. Let $\mathbb{H}_{\widetilde{\pi}}$ be the associated local system. One can easily see that the local system $\mathbb{H}_{\widetilde{\pi}}$ is a constant system on $\widetilde{\mathcal{N}}_{h}$ and it descends to the constant system

[TABLE]

on $D$ via the period map (3.2) (cf. [37]).

For $\ell$ sufficiently large, the action of $\Gamma_{h}^{\ell}\cong{\rm Mon}_{\ell}(X,H)$ on $\widetilde{\mathcal{N}}_{h}$ naturally lifts to an action on $\mathbb{H}_{\widetilde{\pi}}$. After taking the quotient by $\Gamma_{h}^{\ell}$, it gives the locally constant system $\mathbb{H}_{\pi}$ on $\mathcal{F}^{\ell}_{h}$. On the other hand, the map (3.7) is obtained by passing to the quotient on both sides of (3.2). It follows that $\mathbb{H}_{\pi}$ descends to the automorphic bundle $\mathbf{H}^{\bullet}:=\Gamma_{\ell}\backslash(D\times H^{\ast}(M,\mathbb{Q}))$ via (3.7). ∎

3.7. Moduli of lattice polarized hyperkähler

We have to introduce the lattice polarized hyperkähler manifolds for later use. The moduli space of lattice polarized hyperkähler manifolds are briefly studied by Camere in [15, 14], which generalize the work of Dolgachev [19] on K3 surfaces. Here we review the basic notions with certain modifications. Let

[TABLE]

be a fixed primitive embedding of a lattice $\Sigma$ with signature $(1,r)$. An ample $\Sigma$-polarized hyperkähler manifold (with respect to $j$) is a pair $(X,\phi)$, where $X$ is a hyperkähler manifold and

[TABLE]

is a primitive lattice embedding satisfying

• •

$\phi(\Sigma)$ contains an ample divisor class of $X$,

• •

there exists a marking $\varphi$ such that $\varphi\circ\phi=j$.

If we fix an ample class $h$, we can define the so called $h$-ample $\Sigma$-polarized hyperkahler manifold (with respect to $j$) as a triple $(X,H,\phi)$ satisfying that $(X,H)$ is $h$-polarized and $\phi(\Sigma)$ contains $c_{1}(H)$. Denote by $\mathcal{F}_{\Sigma,h}$ a connected component of the moduli stack of $h$-ample $\Sigma$-polarized hyperkähler manifolds. It comes with the natural forgetful map $\mathcal{F}_{\Sigma,h}\rightarrow\mathcal{F}_{h}$.

There is also a Hodge theoretic description of $\mathcal{F}_{\Sigma,h}$ by Torelli theorem. Regard $\Sigma$ as a sublattice of $\Lambda$ and we consider the restricted period domain $D_{\Sigma}\subseteq D$ as a connected component of

[TABLE]

Let $\Gamma_{\Sigma}$ be the stabilizer of $\phi(\Sigma)$ in the monodromy group ${\rm Mon}^{2}(X)$. It is an arithmetic subgroup of $\mathrm{O}(\Sigma^{\perp})$. As before, let $F_{\Sigma,h}$ be the corresponding coarse moduli space. Then the restriction of period map on moduli space of marked polarized hyperkähler manifolds defines a map

[TABLE]

which is an open immersion and there is a commutative diagram

[TABLE]

Remark 3.7.1**.**

In general, the definition of $\mathcal{F}_{\Sigma,h}$ depends on the primitive embedding of $\Lambda$. In some cases, such as $\Lambda$ is unimodular, the primitive embedding of $\Sigma$ to $\Lambda$ is unique up to isometry and $\mathcal{F}_{\Sigma,h}$ is thus independent of the choice of $j$.

Finally, one can easily see that the results in §3.4 and §3.6 naturally extend to lattice polarized hyperkähler manifolds. More precisely, concerning the full $\ell$-level structures on lattice polarized hyperkähler manifolds, we let $\mathcal{F}_{\Sigma,h}^{\ell}$ (resp. $F_{\Sigma,h}^{\ell}$) be the (coarse) moduli space of $h$-ample $\Sigma$-polarized hyperkähler manifolds with a full $\ell$-level structure. Then there is an open immersion

[TABLE]

for some finite index subgroup $\Gamma_{\Sigma}^{\ell}\subseteq\Gamma_{\Sigma}$ and the associated local systems on $F^{\ell}_{\Sigma,h}$ descend via $\mathcal{P}^{\ell}_{\Sigma}$.

4. Cycle classes on moduli of hyperkähler manifolds

Analogous to the case of K3 surfaces, there exist natural cycle classes on $\mathcal{F}_{h}$ and its universal family, which comes from either Hodge theory or the geometry of the generic fiber. In this section, we discuss the relations between these cycles and define the tautological ring on the moduli space. Natural conjectures are made motived from MOP conjecture and generalized Franchetta conjecture.

4.1. Noether-Lefschetz cycles

We first recall the Noether-Lefschetz theory for hyperkähler manifolds. Let $\pi:\mathcal{U}\rightarrow\mathcal{F}$ be a family of projective hyperkähler manifolds over $\mathcal{F}$. When $\mathcal{F}$ is a quasiprojective variety, we define the $r$-th Noether-Lefschetz loci on $\mathcal{F}$ (with respect to $\pi$) as

[TABLE]

where $\rho$ is the Picard number of the generic fiber of $\pi$. We can also define the Noether-Lefschetz loci with respect to the universal family $\pi:\mathcal{U}_{h}\rightarrow\mathcal{F}_{h}$ over the moduli stack $\mathcal{F}_{h}$, where the irreducible component of the Noether-Lefschetz loci on $\mathcal{F}_{h}$ will occur as the proper map

[TABLE]

for some Lorentian lattice $\Sigma\hookrightarrow\Lambda$ containing $h$. In either cases, we define

[TABLE]

as the subalgebra generated by Noether-Lefschetz loci on $\mathcal{F}$ (cf. [45]). We may omit the subscript $\pi$ when $\mathcal{F}$ is the moduli stack.

Proposition 4.1.1** (see [6]).**

The first Chern class of the Hodge line bundle is in the span of NL-divisors. Moreover, let $m_{\Lambda}={\rm rank}~{}\Lambda$ be the second Betti number of hyperkähler manifolds in $\mathcal{F}_{\Sigma,h}$ and assume that $m_{\Lambda}-{\rm rank}(\Sigma)\geq 5$, then

[TABLE]

In particular, we have an inclusion $\mathrm{DCH}^{\ast}(\mathcal{F}_{\Sigma,h})\subseteq{\rm NL}^{\ast}(\mathcal{F}_{\Sigma,h})$.

Let us roughly explain the proof. It actually relies on the study of Picard groups on the coarse moduli space $F_{\Sigma,h}$ as we have ${\rm Pic}(\mathcal{F}_{\Sigma,h})\cong{\rm Pic}(F_{\Sigma,h})$. In [6, Theorem 3.8], we deal with orthogonal Shimura varieties whereas $F_{\Sigma,h}$ is only an open subset of $\Gamma_{\Sigma}\backslash D_{\Sigma}$. However the map

[TABLE]

is onto and we are easily reduced to $\Gamma_{\Sigma}\backslash D_{\Sigma}$, see Proposition 5.3.4 below for details. Then we can conclude the assertion by applying [6, Theorem 3.8]. Here, the assumption $m_{\Lambda}-{\rm rank}(\Sigma)\geq 5$ or equivalently $\dim D_{\Sigma}\geq 3$ is a necessary condition for us to apply [6, Theorem 3.8].

4.2. Kappa classes and tautological ring

In this subsection, we construct the $\kappa$-classes on $\mathcal{F}_{\Sigma,h}$ by taking the pushforward of BV classes on the generic fiber of universal families of hyperkähler manifolds of (complex) dimension $n$. This idea is initiated in [34] for the case of K3 surfaces. To work in greater generality, we first define $\kappa$-classes with respect to an arbitrary smooth family.

Definition 4.2.1**.**

Let $\pi:\mathcal{U}\rightarrow\mathcal{F}$ be a smooth projective family of hyperkähler manifolds over a smooth Deligne-Mumford stack. Let

[TABLE]

be a collection of line bundles whose image in ${\rm Pic}(\mathcal{U}/\mathcal{F})$ form a basis. Then we define the $\kappa$-classes

[TABLE]

where $m=\sum\limits_{i=0}^{r}a_{i}+\sum\limits_{j=1}^{2n}jb_{j}-2n$. For simplicity, we may write

[TABLE]

and refer to these as the special $\kappa$-classes.

In particular, we can define $\kappa$-classes and also special $\kappa$-classes on $\mathcal{F}_{\Sigma,h}$ as the $\kappa$-classes with respect to the universal family $\pi_{\Sigma}:\mathcal{U}_{\Sigma}\rightarrow\mathcal{F}_{\Sigma,h}$. This yields

Definition 4.2.2**.**

The tautological ring $\mathrm{R}^{\ast}(\mathcal{F}_{h})\subseteq{\rm CH}^{\ast}(\mathcal{F}_{h})$ of $\mathcal{F}_{h}$ is defined to be the subalgebra generated by the images of all $\kappa$-classes on $\mathcal{F}_{\Sigma,h}$ via the pushforward maps

[TABLE]

for all $\Sigma$. Instead, if we only use the special $\kappa$-classes on $\mathcal{F}_{\Sigma,h}$ via (4.6), we obtain a subring $\mathrm{DR}^{\ast}(\mathcal{F}_{h})\subseteq\mathrm{R}^{\ast}(\mathcal{F}_{h})$.

Clearly, we have natural inclusions

[TABLE]

from the definition. Note that we can vary the choices $\mathcal{B}$ of universal line bundles on $\mathcal{U}_{\Sigma}$. (These choices differ by the pullback of line bundles on $\mathcal{F}_{\Sigma,h}$.) It follows that $\mathrm{DR}^{\ast}(\mathcal{F}_{h})$ and hence $\mathrm{R}^{\ast}(\mathcal{F}_{h})$ contains all classes in $\mathrm{DCH}^{\ast}(\mathcal{F}_{\Sigma,h})$ via (4.6). Then we propose the following generalization of MOP conjecture:

Conjecture 4** (Generalized Tautological Conjecture).**

[TABLE]

4.3. Cohomological tautological conjecture

As a weaker version of the generalized tautological conjecture, we mainly consider the cohomological tautological conjecture as follows. Let $H^{\ast}(\mathcal{F}_{h},\mathbb{Q})$ denote the singular cohomology of $\mathcal{F}_{h}$ which is isomorphic to the singular cohomology of the coarse moduli space $F_{h}$ (cf. [5]). There is a cycle class map

[TABLE]

and we add subscript hom to denote the image of the corresponding ring in $H^{\ast}(\mathcal{F}_{h},\mathbb{Q})$ via the cycle class map. Then we have the cohomological tautological conjecture (CTC).

Conjecture 5**.**

${\rm NL}^{\ast}_{\rm hom}(\mathcal{F}_{h})=\mathrm{R}^{\ast}_{\rm hom}(\mathcal{F}_{h}).$**

One of our main results is

Theorem 4.3.1**.**

Assume that the lattice $\Lambda$ has rank $m_{\Lambda}\geq 6$, or equivalently $\dim\mathcal{F}_{h}\geq 3$. Then we have

[TABLE]

Moreover, if $n<\frac{m_{\Lambda}-3}{8}$, then

[TABLE]

In particular, Conjecture 5 holds when $\Lambda$ is of $\mathrm{K3}^{[n]}$-type with $n\leq 2$.

We prove Theorem 4.3.1 in Section 8. Indeed, we will prove a more general result concerning $\kappa$-classes with respect to general families of hyperkähler manifolds. Unfortunately, our results can not imply the full Conjecture 5 for hyperkähler manifolds of generalized Kummer type, OG6, OG10, or K3*[n]*-type with large $n$ as their second Betti number is relative small. Instead, we show that a large part of the tautological classes are lying in R${}^{\ast}_{\hom}(\mathcal{F}_{h})$.

4.4. Examples

As a completement to Theorem 4.3.1, we follow [53] to give some examples of $\kappa$-classes on $\mathcal{F}_{h}$ with $a_{i}=0$ and $b_{j}\neq 0$ lying in ${\rm NL}^{\ast}(\mathcal{F}_{h})$. For simplicity, we may write

[TABLE]

Let $\Omega_{\mathcal{U}_{h}/\mathcal{F}_{h}}^{i}$ be the relative sheaf of differential $i$-forms and we denote by $\lambda=c_{1}(R^{0}\pi_{\ast}\Omega^{2}_{\mathcal{U}_{h}/\mathcal{F}_{h}})$ the first Chern class of the line bundle on $\mathcal{F}_{h}$ that fiberwise corresponds to the vector space spanned by the non-degenerate holomorphic 2-form of the fiber. By [6, Corollary 8.5], we know that $\lambda\in{\rm NL}^{\ast}(\mathcal{F}_{h})$. Easily one can get

[TABLE]

Furthermore, note that there is an isomorphism $\wedge^{i}(R^{2}\pi_{\ast}\mathcal{O}_{\mathcal{U}_{h}})\cong R^{2i}\pi_{\ast}\mathcal{O}_{\mathcal{U}_{h}}$ via cup product and $R^{2n}\pi_{\ast}\mathcal{O}_{\mathcal{U}_{h}}\cong(R^{0}\pi_{\ast}\Omega_{\pi})^{\vee}$, this yields

[TABLE]

Using Grothendieck-Riemann-Roch Theorem, we have

[TABLE]

Let ${\rm td}_{i}$ be the $i$-th Todd class of $(\Omega^{1}_{\mathcal{U}_{h}/\mathcal{F}_{h}})^{\vee}=T_{\mathcal{U}_{h}/\mathcal{F}_{h}}$. By definition, the class

[TABLE]

is a linear combination of $\tilde{\kappa}_{b_{1}\ldots,b_{n}}$ and we find

[TABLE]

by comparing terms in (4.11) of degree $i$ at both sides. The first Chern class $c_{1}(\Omega^{1}_{\mathcal{U}_{h}/\mathcal{F}_{h}})$ is the pull back of a divisor class on $\mathcal{F}_{h}$. By projection formula, this gives

[TABLE]

if $\sum\limits_{j\geq 2}jb_{j}<2n$ or $\sum\limits_{j\geq 2}jb_{j}=2n$ with some $b_{2k+1}\neq 0$. When $n=1$, the relations in (4.12) and (4.13) suffice to show that all classes $\tilde{\kappa}_{b_{1}\ldots,b_{2n}}$ are lying in the subalgebra $\mathbb{Q}\left<\lambda\right>$ of ${\rm CH}^{\ast}(\mathcal{F}_{h})$ generated by $\lambda$ (cf. [53]). However, it seems more complicated as $n$ gets larger. For instance, let us compute the first few terms of $\tilde{\kappa}_{b_{1},\ldots,b_{4}}$ (i.e. $n=2$). Using (4.12), we have

• •

$i=0$, $\frac{1}{720}(3\kappa_{0,2,0,0}-\kappa_{0,0,0,1})=3;$

• •

$i=1$, $c_{1}(\Omega^{1}_{\mathcal{U}_{h}/\mathcal{F}_{h}})=2\pi^{\ast}(\lambda)$;

• •

$i=2,~{}\frac{1}{60480}(\tilde{\kappa}_{0,3,0,0}-9\tilde{\kappa}_{0,1,0,1}-5\tilde{\kappa}_{2,0,0,1}+11\tilde{\kappa}_{2,2,0,0})=\frac{5\lambda^{2}}{2}$

$\vdots$

Let $\chi=\chi(X)$ be the Euler characteristic of $X\in\mathcal{F}_{h}$, e.g. $\chi(S^{[2]})=324$. It follows that

[TABLE]

and

[TABLE]

It would be interesting to know if all such classes are lying in $\mathbb{Q}\left<\lambda\right>$.

4.5. Generalized Franchetta conjecture for hyperkählers

Assume that very general members in $\mathcal{F}_{h}$ have only the trivial automorphism. Let $\mathcal{F}_{h}^{\circ}\subseteq\mathcal{F}_{h}$ be the open subset parametrizing the hyperkähler manifolds with only trivial automorphisms and let

[TABLE]

be the associated universal family. The generalized Franchetta conjecture for $\mathcal{U}_{h}^{\circ}$ is

Conjecture 6**.**

Let $\mathcal{T}_{\pi^{\circ}}$ be the relative tangent bundle of $\pi^{\circ}:\mathcal{U}^{\circ}_{h}\rightarrow\mathcal{F}_{h}^{\circ}$. Then for any $\alpha\in{\rm CH}^{2n}(\mathcal{U}_{h}^{\circ})$, there exists $m\in\mathbb{Q}$ such that $\alpha-mc_{2n}(T_{\pi^{\circ}})$ supports on a proper subset of $\mathcal{F}^{\circ}_{h}$.

The conjecture can be stated for the moduli stack and the assumption of trivial automorphism group on the generic fiber of the universal family can be removed. In §8.1, we prove the cohomological version of this conjecture for hyperkähler manifolds whose second Betti number is sufficiently large (with respect to the dimension).

5. Shimura varieties of orthogonal type

In this section, we discuss Kudla-Millson’s construction of special cycles on Shimura varieties of orthogonal type as well as its generalization by Funke and Millson to special cycles with coefficients. We also discuss the connection with Noether-Lefschetz cycles on moduli spaces of polarized hyperkähler manifolds.

5.1. Shimura varieties of orthogonal type

Let $V$ be a non-degenerate quadratic space over $\mathbb{Q}$ of signature $(2,b)$ and let $G={\rm SO}(V)$ the corresponding special orthogonal group. The group $G(\mathbb{R})$ of real points of $G$ is isomorphic to ${\rm SO}(2,b)$. We denote by $G(\mathbb{R})_{+}\cong{\rm SO}_{0}(2,b)$ the component of the identity; recall that it is precisely the kernel of the spinor norm. Let $K_{\mathbb{R}}\cong{\rm SO}(2)\times{\rm SO}(b)$ be a maximal compact subgroup of $G(\mathbb{R})_{+}$ and let $\widehat{D}=G(\mathbb{R})/K_{\mathbb{R}}$. We denote by $D$ the connected component of $\widehat{D}$ associated to $G(\mathbb{R})_{+}$. We have $D\cong{\rm SO}_{0}(2,b)/{\rm SO}(2)\times{\rm SO}(b)$, it is a Hermitian symmetric domain of dimension $b$.

Let $\widetilde{G}$ be the general spin group ${\rm GSpin}(V)$ associated to $V$. For any compact open subgroup $K\subseteq G(\mathbb{A}_{f})$, we set $\widetilde{K}$ to be its preimage in $\widetilde{G}(\mathbb{A}_{f})$ and denote by $X_{K}$ the double coset

[TABLE]

Let $\widetilde{G}(\mathbb{Q})_{+}\subseteq\widetilde{G}(\mathbb{Q})$ be the subgroup consisting of elements with totally positive spinor norm, which can be viewed as the subgroup of $\widetilde{G}(\mathbb{Q})$ lying in the identity component of the adjoint group of $\widetilde{G}(\mathbb{R})$. Write

[TABLE]

The decomposition of $X_{K}$ into connected components is

[TABLE]

where $\Gamma_{g_{j}}$ is the image of $\widetilde{G}(\mathbb{Q})_{+}\cap g_{j}\tilde{K}g_{j}^{-1}$ in ${\rm SO}_{0}(2,b)$. When $g_{j}=1$, we denote by $\Gamma_{K}$ the arithmetic group $\Gamma_{1}=K\cap G(\mathbb{Q})$ and $Y_{K}=\Gamma_{K}\backslash D$ the connected component of $X_{K}$.

5.2. Connected cycles with coefficients

Let $U$ be a $\mathbb{Q}$-subspace of $V$ with $\dim U=r$ and such that the quadratic form is totally negative definite when restricted to $U$. Let $\widehat{D}_{U}\subset\widehat{D}$ be the subset consisting of the 2-planes that are perpendicular to $U$ and let $D_{U}=\widehat{D}_{U}\cap D$.

For a fixed compact open subgroup $K\subseteq G(\mathbb{A}_{f})$ and an element $g\in\widetilde{G}(\mathbb{A}_{f})$ we let $\Gamma_{g}$ be the image of $\widetilde{G}(\mathbb{Q})_{+}\cap g\tilde{K}g^{-1}$ in ${\rm SO}_{0}(2,b)$. Denote by $\Gamma_{g,U}$ the image in ${\rm SO}_{0}(2,b)$ of the pointwise stabilizer of $U$ in $\widetilde{G}(\mathbb{Q})_{+}\cap g\tilde{K}g^{-1}$. We then have a map

[TABLE]

Following Kudla [28] we will denote this connected cycle (with trivial coefficients) by $c(U,g,K)$; it is of codimension $r$. We let $\mathrm{SC}^{r}(Y_{K})\subseteq{\rm CH}^{r}(Y_{K})$ be the subgroup spanned by connected cycles $c(U,1.K)$ of codimension $r$ and set

[TABLE]

to be the subalgebra generated by connected cycles of all codimensions. We list some simple properties of connected cycles.

• •

Let $K^{\prime}\subseteq K$ be an open compact subgroup and let $f:Y_{K^{\prime}}\rightarrow Y_{K}$ be the covering map. Then

[TABLE]

• •

For the map $\iota:\Gamma_{1,U}\backslash D_{U}\rightarrow Y_{K}$ in (5.1), we have

[TABLE]

Here we write ${\rm SC}_{k}(X):={\rm SC}^{\dim X-k}(X)$.

5.3.

Now we give some description of the ring ${\rm SC}^{\ast}(Y_{K})$. Firstly, besides the connected cycles, there is Hodge line bundle on $Y_{K}$ and we use $\lambda$ to denote its first Chern class in ${\rm CH}^{1}(Y_{K})$. It is preserved under the pullback via (5.1) and there is a very useful result

Lemma 5.3.1**.**

[6, Corollary 8.7]** The class $\lambda\in{\rm SC}^{1}(Y_{K}).$

This yields the following

Theorem 5.3.2**.**

$\mathrm{SC}^{*}(Y_{K})=\bigoplus\limits_{r}\mathrm{SC}^{r}(Y_{K}).$**

Proof.

We need to show that the image of the intersection produt

[TABLE]

lies in ${\rm SC}^{r_{1}+r_{2}}(Y_{K})$. Let us first consider the case when $r_{1}=1$. For any two connected cycles $\alpha=c(U_{1},1,K)\in{\rm SC}^{r_{1}}(Y_{K})$ and $\beta=c(U_{2},1,K)\in{\rm SC}^{r_{2}}(Y_{K})$, there are two possibilities:

1. (1)

If $\alpha$ and $\beta$ intersect properly, then $\alpha\cdot\beta$ is a linear combination of connected cycles $c(U,1,K)$ where $U$ is spanned by $U_{2}$ and an element in the $\Gamma_{K}$-orbit of $U_{1}$. 2. (2)

If $\beta$ is contained in $\alpha$, then

[TABLE]

by O’Grady (cf. [43, Lemma 1.2]). By Lemma 5.3.1 and (5.3), we know that

[TABLE]

It follows that $\alpha\cdot\beta\in{\rm SC}^{1+r_{2}}(Y_{K})$.

This further gives

[TABLE]

for any $\alpha_{1},\ldots,\alpha_{r}\in{\rm SC}^{1}(Y_{K})$ and $\gamma\in{\rm SC}^{r_{2}}(Y_{K})$.

When $r_{1}>1$, given a connected cycle $c(U_{1},1,K)$ of codimension $r_{1}$ on $Y_{K}$, an easy inductive argument shows that there exists an open compact subgroup $K^{\prime}\subseteq K$ such that the connected cycle

[TABLE]

can be written as an intersection of $r_{1}$ connected cycles of codimension one, i.e.

[TABLE]

for some subspace $W_{j}$ with $\dim W_{j}=1$. Consider the covering map $f:Y_{K^{\prime}}\rightarrow Y_{K}$, we have

[TABLE]

by (5.2) and (5.4). As $f(c(U_{1},1,K^{\prime}))=c(U_{1},1,K)$, the projection formula gives $c(U_{1},1,K)\cdot\beta\in{\rm SC}^{r_{1}+r_{2}}(Y_{k})$.

∎

Remark 5.3.3**.**

Similar questions have been asked by Kudla in [28], where one consider the ring generated by special cycles instead of connected cycles. We believe our argument might still work.

To conclude this paragraph we relate connected cycles to Noether-Lefschetz cycles: we have a period map $\mathcal{P}_{h}:\mathcal{F}_{h}\rightarrow\Gamma_{h}\backslash D$. The following proposition is then straightforward.

Proposition 5.3.4**.**

The irreducible Noether-Lefschetz cycles are the restriction of connected cycles on $\Gamma_{h}\backslash D$ to $\mathcal{F}_{h}$. In particular,

[TABLE]

More generally, one can easily see that (5.7) holds for the Noether-Lefschetz ring on the moduli space of lattice polarized hyperkähler manifolds with level structures.

5.4.

Now following Funke and Millson [22] we promote the connected cycles $c(U,g,K)$ to cycles with non-trivial coefficients. First recall that given any partition $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})$, with $k=\left[\frac{b+2}{2}\right]$, the harmonic Schur functor associates to the quadratic space $V$ a finite dimensional $\mathrm{O}(V)$-module $\SS_{[\lambda]}(V)$ which is irreducible of highest weight $\lambda$. As such it defines a fiber bundle

[TABLE]

We will use $\mathbf{E}$ to denote the associated local system.

Now let $\mathbf{x}=(x_{1},\ldots,x_{r})$ be a rational basis of $U$. The vectors $x_{1},\ldots,x_{n}$ are all fixed by $\Gamma_{g,U}$. Hence any tensor word in the $x_{j}$’s is also fixed by $\Gamma_{g,U}$. Given a tableau $T$ on $\lambda$ we denote by $\mathbf{x}_{T}$ the corresponding harmonic tensor in $\SS_{[\lambda]}(V)$, see [21].111Beware that a tableau is called a semi standard filling in [22]. It is fixed by $\Gamma_{g,U}$ and therefore gives rise to a parallel section of the restriction of $\mathbf{E}$ over the connected cycle $c(U,g,K)$. We define connected cycles with coefficients in $\mathbf{E}$ by setting

[TABLE]

The scalar product on $V$ induces an inner product on $\SS_{[\lambda]}(V)$. Composing the wedge product with this inner product and integrating the result over $Y_{K}$ we define a pairing

[TABLE]

where $H_{c}^{*}(-)$ denotes the de Rham cohomology with compact support. It follows from Poincaré duality that the pairing (5.9) is perfect. Now if $\eta$ is a compactly supported $\SS_{[\lambda]}(V)$-valued closed $2(b-r)$-form on $Y_{K}$ we can form the period

[TABLE]

It thus corresponds to the connected cycle with coefficient $c(\mathbf{x},g,K)_{T}$ a linear form on $H_{c}^{2(b-r)}(Y_{K},\mathbf{E})$. Since the pairing (5.9) is perfect this linear form in turn defines a class $[c(\mathbf{x},g,K)_{T}]$ in $H^{2r}(Y_{K},\mathbf{E})$. We shall refer to the corresponding map as the cycle class map (with coefficients). We denote by $\mathrm{SC}^{\ast}_{\rm hom}(Y_{K},\mathbf{E})$ the subring of $H^{\ast}(Y_{K},\mathbf{E})$ generated by the cycle classes $[c(\mathbf{x},g,K)_{T}]$ (as $\mathbf{x}$ and $T$ vary).

Remark 5.4.1**.**

When $\lambda=0$ the representation $\SS_{[\lambda]}(V)$ is trivial and the cycle class map is obtained by composing the inclusion map $\mathrm{SC}^{r}(Y_{K})\subseteq{\rm CH}^{r}(Y_{K})$ with the usual cycle class map ${\rm CH}^{r}(Y_{K})\to H^{2r}(Y_{K})$.

If $K^{\prime}\subseteq K$ is a compact open normal subgroup, the finite covering group $\Gamma_{K}/\Gamma_{K^{\prime}}$ that acts on $H^{2r}(Y_{K^{\prime}},\mathbf{E})$ preserves the image $\mathrm{SC}^{r}_{\rm hom}(Y_{K^{\prime}},\mathbf{E})$ of the cycle class map and, since the covering projection map $Y_{K^{\prime}}\to Y_{K}$ maps connected cycles to connected cycles, we have:

[TABLE]

5.5. Zucker’s conjecture and Hodge theory

As a key ingredient, we shall explain the connection between the ordinary cohomology and $L^{2}$-cohomology of $Y_{K}$. As we will see later, the second one is well understood as relative Lie algebra cohomology which naturally links to representation theory.

We assume that $Y_{K}$ is smooth and fix a local system $\mathbf{E}$ as in the preceding paragraph. Let $H^{i}_{(2)}(Y_{K},\mathbf{E})$ be the $i$-th $L^{2}$-cohomology of $Y_{K}$ with coefficients in $\mathbf{E}$. By Hodge theory, the group $H^{i}_{(2)}(Y_{K},\mathbf{E})$ is isomorphic to the space of $L^{2}$-harmonic $i$-forms, which is a finite dimensional vector space with a natural Hodge structure (see [9]). Let $\overline{Y}_{K}^{bb}$ be the Baily-Borel-Satake compactification of $Y_{K}$. Looijenga [31], Saper and Stern [50] have shown that

Theorem 5.5.1** (Zucker’s conjecture).**

There is an isomorphism

[TABLE]

where $IH^{\bullet}(\overline{Y}_{K}^{bb},\mathbf{E})$ is the intersection cohomology with middle perversity on $\overline{Y}_{K}^{bb}$.

For our purpose, we shall also consider the Hodge, or mixed Hodge, structures on these cohomology groups. Here, as the local system $\mathbf{E}$ underlies a natural variation of Hodge structure over $Y_{K}$ (cf. [62]), $IH^{k}(\overline{Y}_{K}^{bb},\mathbf{E})$ carries a mixed Hodge structure by Saito’s [49] mixed Hodge module theory. A priori the two Hodge structures need not correspond under the isomorphism but Harris and Zucker [23, Theorem 5.4] nevertheless prove:

Theorem 5.5.2**.**

The composition

[TABLE]

is a morphism of mixed Hodge structures.

The image of $\xi_{k}$ is the lowest non-zero weight in the mixed Hodge structure of $H^{k}(Y_{K},\mathbf{E})$ (see Remark 5.5 of [23]). In particular, the subspace $\mathrm{SC}^{r}_{\rm hom}(Y_{K},\mathbf{E})$ lies in the image of $\xi_{2r}$.

Corollary 5.5.3**.**

We have

[TABLE]

as a morphism of Hodge structure for all $k<b-1$.

Proof.

The isomorphism (5.13) holds because the boundary of $\overline{Y}_{K}^{bb}$ has dimension at most one. See [23, Remark 5.5] for the second statement. ∎

6. A surjectivity result for theta lifting

In this section, we recap the work in [6] on the surjectivity of global theta lifting for cohomological automorphic representations with trivial coefficients of orthogonal groups. We also explain how to extend to the case of nontrivial coefficients.

6.1. Global theta correspondence

Here we briefly review Howe’s global theta correspondence. Let us first fix some notations. Let $V$ be a non-degenerate quadratic space over $\mathbb{Q}$ of signature $(2,b)$ and let $W$ be a symplectic space over $\mathbb{Q}$ of dimension $2r$. We denote by $G^{\prime}={\rm Mp}_{2r}(W)$ the symplectic group ${\rm Sp}(W)$ if $b$ is even and the metaplectic double cover of ${\rm Sp}(W)$ if $b$ is odd.

Fix a nontrivial additive character $\psi$ of $\mathbb{A}/\mathbb{Q}$. We denote by $\omega_{\psi}$ the (automorphic) Weil representation of $\mathrm{O}(V)(\mathbb{A})\times G^{\prime}(\mathbb{A})$ realized in the space $\mathcal{S}(V^{r}(\mathbb{A}))$ of Schwartz-Bruhat functions on $V^{r}(\mathbb{A})$ — the so-called Schr$\ddot{\hbox{o}}$dinger model of the Weil representation. For each $\phi\in\mathcal{S}(V^{r}(\mathbb{A}))$ we form the theta function on $\mathrm{O}(V)(\mathbb{A})\times G^{\prime}(\mathbb{A})$:

[TABLE]

Given an irreducible cuspidal automorphic representation $(\tau,H_{\tau})$ of $G^{\prime}(\mathbb{A})$ which occurs as an irreducible subspace of $L^{2}(G^{\prime}\mathbb{Q})\backslash G^{\prime}\mathbb{A}))$ and given an element $f$ in that subspace, we can form the theta integral

[TABLE]

It is absolutely convergent and defines an automorphic function on $\mathrm{O}(V)(\mathbb{A})$, called the global theta lift of $f$. We shall denote by $\theta_{\psi,V}(\tau)$ the space of the automorphic representation generated by all the global theta lifts $\theta_{\psi,\phi}^{f}$ as $f$ and $\phi$ vary. We shall refer to the corresponding automorphic representation of $\mathrm{O}(V)(\mathbb{A})$ as the global $\psi$-theta lift of $\tau$ to $\mathrm{O}(V)$.

Definition 6.1.1**.**

We say that a representation $\pi\in\mathcal{A}(G)$ is in the image of the cuspidal $\psi$-theta correspondence from a smaller symplectic group if there exists a symplectic space $W$ with $\dim W\leq 2\left[\frac{b+2}{2}\right]$ and an extension $\widetilde{\pi}$ of $\pi$ to $\mathrm{O}(V)$ such that $\widetilde{\pi}$ is the global $\psi$-theta lift of an irreducible cuspidal automorphic representation of $G^{\prime}={\rm Mp}_{2r}(W)$, i.e. there exists $\tau\in\mathcal{A}_{\rm cusp}(G^{\prime})$ such that $\tilde{\pi}\hookrightarrow\theta_{\psi,V}(\tau)$.

6.2. Cohomological representations

Throughout this section, we let $E$ be a finite dimensional irreducible representation of $G(\mathbb{R})$. Let $K_{\mathbb{R}}\cong{\rm SO}(2)\times{\rm SO}(b)$ be as above. Denote by $\theta$ the corresponding Cartan involution and let $\mathfrak{g}_{0}=\mathfrak{k}_{0}\oplus\mathfrak{p}_{0}$ be the associated Cartan decomposition of the Lie algebra $\mathfrak{g}_{0}$ of $G(\mathbb{R})$. We fix a Cartan subalgebra $\mathfrak{t}_{0}\subset\mathfrak{k}_{0}$. We shall denote by $\mathfrak{a}$ the complexification of a real Lie algebra $\mathfrak{a}_{0}$.

A unitary representation $\pi_{\mathbb{R}}$ of $G(\mathbb{R})$ is cohomological (with respect to $E$) if its associated $(\mathfrak{g},K_{\mathbb{R}})$-module $\pi_{\mathbb{R}}^{\infty}$ has nonzero relative Lie algebra cohomology, i.e.

[TABLE]

The unitary $(\mathfrak{g},K_{\mathbb{R}})$-modules with nonzero cohomology have been classified by Vogan and Zuckerman [57]. They are determined by $\theta$-stable parabolic subalgebras $\mathfrak{q}\subset\mathfrak{g}$: $\mathfrak{q}=\mathfrak{l}\oplus\mathfrak{u}$, where $\mathfrak{l}$ is the centralizer of an element $X\in i\mathfrak{t_{0}}$ and $\mathfrak{u}$ is the span of the positive roots of $X$ in $\mathfrak{g}$. Then the Lie algebra $\mathfrak{l}$ is the complexification of $\mathfrak{l}_{0}=\mathfrak{l}\cap\mathfrak{g}_{0}$ and we let $L$ be the connected subgroup of $G(\mathbb{R})$ with Lie algebra $\mathfrak{l}_{0}$. Associated to $\mathfrak{q}$, there is a well-defined, irreducible representation $V(\mathfrak{q})$ of $K_{\mathbb{R}}$ that occurs with multiplicity one in $\wedge^{R}\mathfrak{p}$ where $R=\dim(\mathfrak{u}\cap\mathfrak{p})$. Now if (6.3) holds there exists some $\mathfrak{q}$ such that the Cartan product of $V(\mathfrak{q})$ and $E^{*}$ occurs with multiplicity one in $\wedge^{R}\mathfrak{p}\otimes E^{*}$ and the $(\mathfrak{g},K_{\mathbb{R}})$-module $\pi_{\mathbb{R}}^{\infty}$ is isomorphic to some irreducible $(\mathfrak{g},K_{\mathbb{R}})$-module $A_{\mathfrak{q}}(E)$ characterized by the following two properties:

1. (1)

$A_{\mathfrak{q}}(E)$ is unitary with the same infinitesimal character as $E$. 2. (2)

The Cartan product of $V(\mathfrak{q})$ and $E^{*}$ occurs (with multiplicity one) in $A_{\mathfrak{q}}(E)$.

In our case $K_{\mathbb{R}}={\rm SO}(2)\times{\rm SO}(b)$ acts on $\mathfrak{p}=(\mathbb{C}^{2})^{*}\otimes\mathbb{C}^{b}$ through the standard representation of ${\rm SO}(2)$ on $\mathbb{C}^{2}$ and the standard representation of ${\rm SO}(b)$ on $\mathbb{C}^{b}$. We denote by $\mathbb{C}^{+}$ and $\mathbb{C}^{-}$ the $\mathbb{C}$-span of the vectors $e_{1}+ie_{2}$ and $e_{1}-ie_{2}$ in $\mathbb{C}^{2}$. The two lines $\mathbb{C}^{+}$ and $\mathbb{C}^{-}$ are left stable by ${\rm SO}(2)$. This yields a decomposition $\mathfrak{p}=\mathfrak{p}^{+}\oplus\mathfrak{p}^{-}$ which corresponds to the decomposition given by the natural complex structure on $\mathfrak{p}_{0}$. For each non-negative integer $p$ the $K_{\mathbb{R}}$-representation $\wedge^{p}\mathfrak{p}=\wedge^{p}(\mathfrak{p}^{+}\oplus\mathfrak{p}^{-})$ decomposes as the sum:

[TABLE]

The $K_{\mathbb{R}}$-representations $\wedge^{r}\mathfrak{p}^{+}\otimes\wedge^{s}\mathfrak{p}^{-}$ are not irreducible in general: there is at least a further splitting given by the Lefschetz decomposition:

[TABLE]

One can check that for $2(r+s)<b$ each $K_{\mathbb{R}}$-representation $\tau_{r,s}$ is irreducible. Moreover in the range $2(r+s)<b$ only those with $r=s$ can occur as a $K_{\mathbb{R}}$-type $V(\mathfrak{q})$ associated to a cohomological module. In the special case $r=r$ one can moreover check that each $\tau_{r,r}$ is irreducible as long as $r<b$; it is isomorphic to some $V(\mathfrak{q})$ where the Levi subgroup $L$ associated to $\mathfrak{q}$ is isomorphic to $C\times{\rm SO}_{0}(2,b-2r)$ with $C\subset K_{\mathbb{R}}$.

It follows in particular that if $\pi_{\mathbb{R}}$ is an irreducible unitary representation of $G(\mathbb{R})$ that satisfies

[TABLE]

for some $r<b$ then $\pi_{\mathbb{R}}^{\infty}$ is isomorphic to the unique unitary $(\mathfrak{g},K_{\mathbb{R}})$-module that has the same infinitesimal character as $E$ and contains the Cartan product of $\tau_{r,r}$ and $E^{*}$. We shall denote by $A_{r,r}(E)$ this $(\mathfrak{g},K_{\mathbb{R}})$-module. We have:

[TABLE]

We refer to [7, §5.2 and 5.4] for more details.

Note that $A_{r,r}(E)$ can only contribute to even degree cohomology. The following vanishing result therefore follows from the above classification (and Matsushima’s formula (6.4) below).

Proposition 6.2.1**.**

For any odd degree $i<b/2$ and any local system $\mathbf{E}$ we have

[TABLE]

Proof.

Since $i<b/2$ implies $i<b-1$ we have $H^{i}(Y_{K},\mathbf{E})\cong H_{(2)}^{i}(Y_{K},\mathbf{E})$. The proposition then follows from the fact that in the range $r+s<b/2$ the only $K_{\mathbb{R}}$-types $\tau_{r,s}$ than can occur in a cohomological representation are the ones with $r=s$. ∎

6.3. Surjectivity of the theta lift

The following theorem is proved in [6, Theorem 7.7] when $E$ is the trivial representation. It was proved in [7] for general $E$ but under the hypothesis that $\pi$ is cuspidal. To be able to deal with residual representations as well was the main input of [6]. Though the results of [6] only addresses the case where $E$ is the trivial representation, the proofs (see especially the key Proposition 6.2) only make use of the fact that cohomological representations have integral regular infinitesimal character which is still true for representations that are cohomological with respect to $E$. We shall therefore not repeat the proof and simply refer to [6] for the proof of:

Theorem 6.3.1**.**

Let $\pi=\otimes\pi_{v}\in\mathcal{A}(G)$ be a square integrable automorphic representation of $G$. Suppose that the $(\mathfrak{g},K_{\mathbb{R}})$-module $\pi_{\mathbb{R}}^{\infty}$ of the local Archimedean component of $\pi$ is isomorphic to some cohomological module $A_{r,r}(E)$ with $3r<b+1$. Then there exists a cuspidal representation $\tau$ of ${\rm Mp}_{2r}(\mathbb{A})$ such that $\pi$ (up to a twist by a quadratic character) is in the image of the theta lift of $\tau$.

6.4. Theta classes in cohomology groups

We keep all notations as in §5.1. We construct some special cohomology classes (w.r.t the local system $E$) on $Y_{K}$ from the relative Lie algebra cohomology of $(\mathfrak{g},K_{\mathbb{R}})$-modules. We write

[TABLE]

and let $\pi^{K}_{f}$ be the finite dimensional subspaces of $K$-invariant vectors in $\pi_{f}$. According to Matsushima’s formula and Langlands spectral decomposition (see [8], [6, §7.8]), we have

[TABLE]

where $\pi$ occurs discretely in $L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}))$ with multiplicity $m(\pi)$. Following [6], we define the space of theta classes

[TABLE]

as the subspace generated by the image of $H^{i}(\mathfrak{g},K_{\mathbb{R}};\pi_{\mathbb{R}}^{\infty}\otimes E)$ via (6.4) and (5.12), where $\pi$ varies among the irreducible representations in $\mathcal{A}(G)$ which are in the image of the $\psi$-cuspidal theta correspondence from a smaller symplectic group. The space $H^{i}_{\theta}(Y_{K},\mathbf{E})$ of theta classes on $Y_{K}$ are naturally defined by restriction. The main result in [6] can be reformulated as:

Theorem 6.4.1**.**

There is an inclusion

[TABLE]

Moreover, $H^{2r}_{\theta}(Y_{K},\mathbf{E})=H^{r,r}(Y_{K},\mathbf{E})$ when $r<\frac{b+1}{3}$ or $r>\frac{2b-1}{3}$.

Proof.

The inclusion (6.5) follows from (the proof of) [7, Proposition 11.3].222Beware that our space $\mathrm{SC}^{\ast}$ is a priori larger than the one defined in [7]. The last assertion follows from Theorem 6.5, the decomposition (6.4) and the fact that if $\pi_{\mathbb{R}}$ is an irreducible unitary representation of $G(\mathbb{R})$ that satisfies

[TABLE]

for some $r<b$ then $\pi_{\mathbb{R}}^{\infty}$ is isomorphic to $A_{r,r}(E)$. Note that the fact that $H^{2r}_{\theta}(Y_{K},\mathbf{E})$ is contained in $H^{r,r}(Y_{K},\mathbf{E})$ obviously follows from the inclusion (6.5). It also follows from the fact that the only cohomological representations that are in the image of the local Archimedean theta correspondence are the $A_{r,r}(E)$, see [6, §7.2] and [30]. ∎

Let $H^{2k}_{\rm alg}(Y_{K},\mathbb{C})$ be the image of ${\rm CH}^{k}(Y_{K})\otimes\mathbb{C}$ in $H^{2k}(Y_{K},\mathbb{C})$ via the cycle class map. Following [47], one can use Zucker’s conjecture and the hard Lefschetz theorem on intersection cohomology to get the following result

Corollary 6.4.2**.**

When $b>3$, we have

[TABLE]

Proof.

Let us quickly sketch the proof. Let $\lambda$ be the Hodge line bundle on $Y_{K}$. By [6] and Zucker’s conjecture, we have that

[TABLE]

is generated by connected cycles of codimension one. Next, as $\lambda$ is ample, the Hard Lefschetz theorem for intersection cohomology implies that the image of the map

[TABLE]

is spanned by the class of $\lambda^{b-2}\cdot c(U,g,K)$. By [53], the class $\lambda^{b-2}\cdot c(U,g,K)$ is zero in $H^{2b-2}(Y_{K},\mathbb{C})$ when $\dim c(U,g,K)\geq 3$. Since $H^{2b-2}_{alg}(Y_{K},\mathbb{C})$ lies in the image of $\xi_{2b-2}$, this proves the assertion. ∎

Let us finally define theta classes in the cohomology groups of the moduli spaces discussed in Section 3. Assume that $E$ is a finite dimensional representation of ${\rm SO}(\Lambda_{\mathbb{R}})$. Recall that there is a period map $\mathcal{P}_{\Sigma}:\mathcal{F}_{\Sigma,h}\rightarrow\Gamma_{\Sigma}\backslash D_{\Sigma}$. Let $\mathbb{E}=\mathcal{P}_{\Sigma}^{\ast}(\mathbf{E})$ be the pullback of $\mathbf{E}$ to $\mathcal{F}_{\Sigma,h}$. Then we define

[TABLE]

as the subspace of theta classes on $\mathcal{F}_{\Sigma,h}$. It follows from Theorem 6.4.1 that:

Corollary 6.4.3**.**

There is an inclusion

[TABLE]

for all $r$.

7. The Funke-Kudla-Millson ring

In this section, we introduce special Schwartz forms at the Archimedean place and Kudla-Millson’s special theta lift. Following [7] and [6], this establishes a connection between special cycle classes and theta classes as defined in §5.3. More importantly for us, we show that this yields a so called Funke-Kudla-Millson ring, or just FKM ring, in the space of differential forms with coefficients, which plays the key role in our study of the cohomological tautological conjecture.

7.1. Special Schwartz forms

In this subsection we let $V$ be a real quadratic space of signature $(b,2)$ of dimension $m=b+2$. We use signature $(b,2)$ rather than signature $(2,b)$ in order to follow the notations of the works of Kudla-Millson and Funke-Millson; when applied to our geometric situation we will reverse the variables.

Pick an oriented orthogonal basis $\{v_{i}\}$ of $V$ such that $(v_{\alpha},v_{\alpha})=1$ for $\alpha=1,\ldots,b$ and $(v_{\mu},v_{\mu})=-1$ for $\mu=b+1,b+2$. We shall use the notations of [7].

Kudla and Millson [29] and then Funke and Millson [22] have constructed special Schwartz forms $\varphi$ in

[TABLE]

By abuse of notations we also denote by $T^{\ell}(V)$ the local system on $D$ associated to $T^{\ell}(V)$. If $\mathbf{x}\in V^{r}$ we then have $\varphi(\mathbf{x})\in A^{2r}(D,T^{\ell}(V))$.

We shall rather work with the polynomial Fock model for the dual pair $\mathrm{O}(b,2)\times\mathrm{Sp}_{2r}(\mathbb{R})$, see [7, Section 7]. We denote by $\iota$ the intertwining operator from the Schödinger model to the Fock model, see [29, Section 6]. The map $\iota$ maps the vectors of $\mathcal{S}(V^{r})$ that are finite under the action of a maximal compact subgroup $\mathrm{U}(2rm)$ of the symplectic group $\mathrm{Sp}_{2rm}$ containing the dual pair onto the space of polynomials

[TABLE]

Here if $\mathbf{x}=(x_{1},\ldots,x_{r})\in V^{r}$ we have:

[TABLE]

We will use the notation $\mathcal{P}(\mathbb{C}^{mr})_{+}$ to denote the polynomial in the “positive” variables $z_{\alpha,j}$, $1\leq\alpha\leq b$, $1\leq j\leq r$. We have an isomorphism of $K_{\infty}\times K_{\infty}^{\prime}$-modules

[TABLE]

Here $K_{\infty}\cong\mathrm{O}(b)\times\mathrm{O}(2)$ is a maximal compact subgroup of $\mathrm{O}(b,2)$, the groups $K_{\infty}^{\prime}$ is a maximal compact subgroup of $\mathrm{Sp}_{2r}(\mathbb{R})$ and the action of the group $\mathrm{O}(b)$ on $\mathrm{Pol}(M_{b,r}(\mathbb{C}))$ is induced by the natural action on the columns (i.e., from the left) of the matrices.

First consider the case $r=1$. Then we simply set $z_{\alpha}=z_{\alpha,1}$. Note that $\mathfrak{p}\cong\mathbb{C}^{b}\otimes\mathbb{C}^{2}$; we let $\omega_{\alpha,\mu}$ be the linear form which maps an element of $\mathfrak{p}$ to its $(\alpha,\mu)$-coordinate. For any multi-indices $\underline{\alpha}=(\alpha_{1},\alpha_{2})$ we write $\omega_{\underline{\alpha}}=\omega_{\alpha_{1},b+1}\wedge\omega_{\alpha_{2},b+2}$ and $z_{\underline{\alpha}}=z_{\alpha_{1}}z_{\alpha_{2}}$. The form

[TABLE]

is precisely the image $\iota(\varphi_{1,0})$ of the Schwartz form $\varphi_{1,0}\in\mathrm{Hom}_{K_{\infty}}(\wedge^{2}\mathfrak{p},\mathcal{S}(V))$, constructed by Kudla and Millson, under the intertwining operator $\iota$. The natural product

[TABLE]

induces a natural map

[TABLE]

which maps

[TABLE]

onto $\iota(\varphi_{r,0})$ the image of the Kudla-Millson form $\varphi_{r,0}\in\mathrm{Hom}_{K_{\infty}}(\wedge^{2r}\mathfrak{p},\mathcal{S}(V^{r}))$. We shall abusively denote it $\varphi_{r,0}$ as well.

We now describe Funke-Millson forms. For any multi-indices $\underline{\alpha}=(\alpha_{1},\ldots,\alpha_{\ell})$ we let

[TABLE]

Given any $\ell$-tuple $I=(i_{1},\ldots,i_{\ell})$ of integers in $[1,b]$ we define

[TABLE]

by

[TABLE]

Using the natural product on $\mathcal{P}(\mathbb{C}^{mr})_{+}$, we have

[TABLE]

We shall denote $\Phi_{T^{\ell}(V)}$ the subspace of $\mathrm{Hom}_{K_{\infty}}(\wedge^{2r}\mathfrak{p},\mathcal{P}(\mathbb{C}^{mr})\otimes T^{\ell}(V))$ spanned by the Schwarz forms $\varphi_{r,\ell}^{I}$.

Now fix a partition $\lambda=(\lambda_{1},\ldots,\lambda_{k})$, with $k=\left[\frac{m}{2}\right]$, and let $\ell=\lambda_{1}+\ldots+\lambda_{k}$. The finite dimensional $\mathrm{O}(V)$-module $\mathbb{S}_{[\lambda]}(V)$ can be obtained as the image of the classical Schur functor $\mathbb{S}_{\lambda}(V)\subset T^{\ell}(V)$ under the $\mathrm{O}(V)$-equivariant projection of $T^{\ell}(V)$ onto the harmonic tensors. We denote by $\pi_{[\lambda]}$ the corresponding $\mathrm{O}(V)$-equivariant projection of $T^{\ell}(V)$ onto $\mathbb{S}_{[\lambda]}(V)$. We set

[TABLE]

We shall denote $\Phi_{[\lambda]}^{2r}$ the subspace of $\mathrm{Hom}_{K_{\infty}}(\wedge^{2r}\mathfrak{p},\mathcal{P}(\mathbb{C}^{mr})\otimes T^{\ell}(V))$ spanned by the Schwarz forms $\varphi_{r,[\lambda]}^{I}$.

More generally if $E$ is any finite dimensional representation, it decomposes into a finite direct sum (with multiplicities) of $\mathrm{O}(V)$-modules $\mathbb{S}_{[\lambda]}(V)$ and we shall denote $\Phi_{E}^{2r}$ the finite direct sum of the corresponding $\Phi_{[\lambda]}^{2r}$’s identified as a subspace of $\mathrm{Hom}_{K_{\infty}}(\wedge^{2r}\mathfrak{p},\mathcal{P}(\mathbb{C}^{mr})\otimes E)$. We will in fact abusively identify $\Phi_{E}^{2r}$ with a subspace

[TABLE]

using the intertwining operator $\iota$.

7.2. The Funke-Kudla-Millson ring

Consider a finite graded ring of finite dimensional representations $\mathbf{H}^{\ast}$ of $\mathrm{O}(V)$. We use the above special Schwartz forms to construct a subring of the bi-graded complex

[TABLE]

Indeed: the wedge product

[TABLE]

composed with the product $\mathbf{H}^{j}\otimes\mathbf{H}^{j^{\prime}}\to\mathbf{H}^{j+j^{\prime}}$ yields a ring structure on (7.1) and since the latter obviously maps $\Phi_{\mathbf{H}^{j}}^{2i}\times\Phi_{\mathbf{H}^{j^{\prime}}}^{2i^{\prime}}$ into $\Phi_{\mathbf{H}^{j+j^{\prime}}}^{2(i+i^{\prime})}$ we conclude:

Proposition 7.2.1**.**

The subspace

[TABLE]

is a subring.

Consider now a (global) arithmetic quotient $Y_{K}$ with $K\subset\mathrm{O}(V)(\mathbb{A}_{f})$ an open compact subgroup. Let $\varphi\in\mathcal{S}(V(\mathbb{A}_{f})^{i})$ be a $K$-invariant Schwartz function for some $i$. Then for any $\varphi_{\mathbb{R}}\in\Phi_{\mathbf{H}^{j}}^{2i}$, we define a global Schwartz form

[TABLE]

Applying the theta distribution (6.1) to $\phi$ yields a theta function $\theta_{\psi,\phi}(g,g^{\prime})$ which, as a function of $g\in\mathrm{O}(V)$, defines a differential form in $A^{2i}(X_{K},\mathbf{H}^{j})$. Varying $\varphi$ and $g^{\prime}$ we obtain a subspace

[TABLE]

It follows from Proposition 7.2.1 that

[TABLE]

is a subring of the graded ring $\bigoplus_{i,j}A^{2i}(X_{K},\mathbf{H}^{j})$; we shall refer to it as the Funke-Kudla-Millson ring of $X_{K}$. Restricting to the connected component $Y_{K}\subset X_{K}$ yields the Funke-Kudla-Millson ring of $Y_{K}$.

7.3. Relation with $\theta$-classes

By construction, the cohomology class of a differential form in $A_{\rm special}^{2i}(Y_{K},\mathbf{H}^{j})$ is a $\theta$-class in

[TABLE]

It turns out that the converse is also true:

Proposition 7.3.1** (see Theorem 11.2 of [7]).**

Any class in $H_{\theta}^{i,i}(Y_{K},\mathbf{H}^{j})$ can be represented by a differential form in $A_{\rm special}^{2i}(Y_{K},\mathbf{H}^{j})$.

8. Universal family of polarized hyperkähler manifolds

In this section, we study the cohomology groups of universal family of polarized hyperkähler manifolds via Deligne’s decomposition theorem and prove the cohomological Franchetta conjecture. Furthermore, we prove the cohomological tautological conjecture using the FKM ring. Throughout this section, we concern with hyperkähler manifolds with type $\Lambda$ and ${\rm rank}~{}\Lambda=3+b$.

8.1. Cohomology of universal families of hyperkähler manifolds

Let $\pi:\mathcal{U}\rightarrow\mathcal{F}$ be a smooth family of polarized hyperkähler manifolds over a variety $\mathcal{F}$. Let $\mathbb{H}_{\pi}^{i}$ denote the local system on $\mathcal{F}$ whose fiber at a point $p\in\mathcal{F}$ is $H^{i}(\mathcal{U}_{p},\mathbb{Q})$, with $\mathcal{U}_{p}=\pi^{-1}(p)$. According to the decomposition theorem of Deligne [18], we have

[TABLE]

compatible with the mixed Hodge structure on both sides (cf. [17, $\S$4.3]).

For our purpose, we shall consider the universal family (as stacks)

[TABLE]

of polarized hyperkähler manifolds of type $\Lambda$ and dimension $2n$ with a full $\ell$-level structure defined in §3.4. The decomposition theorem still applies in this case. Our first result is a cohomological version of O’Grady’s generalized Franchetta conjecture for hyperkähler manifolds. Here we work with stack cohomology.

Theorem 8.1.1**.**

Let $m_{\Lambda}$ be the second Betti number of hyperkähler manifolds in $\mathcal{F}_{h}^{\ell}$. Assume that $n<\frac{m_{\Lambda}-3}{8}$. Then, for any class $\alpha\in{\rm CH}^{2n}(\mathcal{U}_{h}^{\ell},\mathbb{Q})$, there exists $m\in\mathbb{Q}$ such that $[\alpha-mc_{2n}(\mathcal{T}_{\pi_{\ell}})]\in H^{4n}(\mathcal{U}^{\ell}_{h},\mathbb{Q})$ supports on the Noether-Lefschetz locus of $\mathcal{F}^{\ell}_{h}$. In particular, Theorem 1.2.1 holds.

Proof.

Obvisouly, we can assume $\ell$ is sufficiently large. The local system $\mathbb{H}^{i}_{\pi_{\ell}}=R^{i}({\pi_{\ell}})_{\ast}\mathbb{Q}$ on $\mathcal{F}_{h}^{\ell}$ descends to its coarse moduli space $F_{h}^{\ell}$ and we write it by $\mathbb{H}^{i}_{\pi_{\ell}}$ by abuse of notations. There is a natural isomorphism

[TABLE]

and we may regard $F_{h}^{\ell}$ as an open subset of $\Gamma_{h}^{\ell}\backslash D$.

Next, under our hypothesis, Proposition 6.2.1 implies that, in the degrees $i\leq 4n$ we are concerned with, only even degree contribute non-trivially to the algebraic part of the decomposition (8.1). Since moreover $H^{0}(\mathcal{F}_{h}^{\ell},\mathbb{H}^{4n}_{\pi})\cong\mathbb{Q}$, it follows that there exists $m\in\mathbb{Q}$ such that

[TABLE]

Denote by $\gamma_{i}\in H^{2i}(\mathcal{F}_{h}^{\ell},\mathbb{H}^{2(2n-i)}_{\pi_{\ell}})$ the $i$-th component of (8.3). Since $[\alpha-mc_{2n}(\mathcal{T}_{\pi})]\in H^{4n}(\mathcal{U}_{h}^{\ell},\mathbb{Q})$ is algebraic, it lies in the lowest weight subspace of $H^{4n}(\mathcal{U}_{h}^{\ell},\mathbb{Q})$. As (8.1) is compatible with the mixed Hodge structure on both sides, each $\gamma_{i}$ lies in the lowest weight subspace of $H^{2i}(\mathcal{F}_{h}^{\ell},\mathbb{H}^{2(2n-i)}_{\pi_{\ell}})$.

Recall that Lemma 3.6.1 implies that there exists an automorphic local system

[TABLE]

on $\Gamma_{h}^{\ell}\backslash D$ such that $\mathbb{H}^{j}_{\pi_{\ell}}=R^{j}({\pi_{\ell}})_{\ast}\mathbb{Q}$ on $\mathcal{F}_{h}^{\ell}$ is the pullback $(\mathcal{P}^{\ell}_{h})^{\ast}\mathbf{H}^{j}$. So we have a natural map

[TABLE]

induced by $\mathcal{P}^{\ell}_{h}$, which is a mixed Hodge structure morphism, and it is surjective onto the non-zero lowest weight part. We can therefore lift $\gamma_{i}$ to an element

[TABLE]

with lowest weight. Now it suffices to show

[TABLE]

since elements in $\mathrm{SC}^{i}_{\rm hom}(\Gamma_{h}^{\ell}\backslash D,\mathbf{H}^{2(2n-i)})$ obviously support on proper Shimura subvarieties.

However, as the local system $\mathbf{H}^{\ast}$ arises from a finite dimensional representation of $\mathrm{O}(V)$, Theorem 6.4.1 can be applied and this directly implies (8.4). In particular, the conditions in Theorem 8.1.1 hold for K3*[n]*-type hyperkähler manifolds with $n\leq 2$ and hence Theorem 1.2.1 holds. ∎

8.2. Leray spectral sequence and cup product

There are natural cup products on both sides of (8.1). In general these coincide only on the associated graded rings, see e.g. [10]. However Voisin [60] has proved that these two cup products are compatible for K3 surfaces after shrinking to a smaller open subset. In what follows we partially extend her results to families of hyperkähler manifolds.

Let us first make some conventions. In this subsection, we will deal with the universal family of lattice polarized hyperkähler manifolds with level structures (See $\S$3.3 for all notions). For simplicity of the notations, we will use $\pi:\mathcal{U}\rightarrow\mathcal{F}$ to denote the universal family $\mathcal{U}_{\Sigma,h}^{\ell}\rightarrow\mathcal{F}_{\Sigma,h}^{\ell}$ of $h$-ample $\Sigma$-polarized hyperkähler manifolds with a full $\ell$-level structure and let $Y=\Gamma_{\Sigma}^{\ell}\backslash D_{\Sigma}$. We denote by

[TABLE]

the period map. Moreover, we choose $\ell$ sufficiently large so that $Y$ is smooth and Lemma 3.6.1 applies.

Consider on $\mathcal{U}$ the short exact sequence of vector bundles

[TABLE]

that defines the fiber bundle of relative differential forms $\Omega_{\mathcal{U}/\mathcal{F}}$. This gives a decreasing filtration $L^{*}\Omega^{q}_{\mathcal{U}}$ of the fiber bundle $\Omega^{q}_{\mathcal{U}}$ defined by

[TABLE]

The associated graded vector bundle is $\mathrm{Gr}^{p}_{L}\Omega^{q}_{\mathcal{U}}=\pi^{*}\Omega^{p}_{\mathcal{F}}\otimes\Omega^{q-p}_{\mathcal{U}}$. This yields a filtration on the complex $A^{q}(\mathcal{U})$ which is the space of smooth sections of $\Omega^{q}_{\mathcal{U}}$. We therefore have a corresponding (Leray) spectral sequence that computes the cohomology of $\mathcal{U}$ from that of the base $\mathcal{F}$ and the fiber. The $E_{1}$ term is

[TABLE]

Note that the filtration is compatible with the cup-product meaning that if $\alpha\in L^{p_{1}}A^{k_{1}}(\mathcal{U})$ and $\beta\in L^{p_{2}}A^{k_{2}}(\mathcal{U})$ then $\alpha\wedge\beta\in L^{p_{1}+p_{2}}A^{k_{1}+k_{2}}(\mathcal{U})$. The cup-pruduct on $A^{\ast}(\mathcal{U})$ induces the natural structure on $E_{1}$. Recall that $R\pi_{\ast}\mathbb{R}$ is the pullback of a local system $\mathbf{H}^{\ast}$ on $Y$ that comes from a finite dimensional representation of $\mathrm{O}(\Sigma^{\perp}(\mathbb{R}))$. The pullback $\mathcal{P}^{*}\Phi_{\mathbf{H}^{\ast}}$ of the FKM ring on $Y$ associated to $\Phi_{\mathbf{H}^{\ast}}$ defines a subring of $E_{1}$; we will abusively refer to it as the FKM subring of the $E_{1}$ term of the Leray spectral sequence that computes the cohomology of $\mathcal{U}$ from that of the base $\mathcal{F}$ and the fiber.

Theorem 8.2.1**.**

With the notations as above, for any two cycles $\alpha_{1},\alpha_{2}\in{\rm CH}^{\ast}(\mathcal{U})$ of codimension $<\frac{m_{\Lambda}-3}{4}$, the two cup-products of $[\alpha_{1}]$ and $[\alpha_{2}]$ in $H^{\ast}(\mathcal{U},\mathbb{C})$ associated to the two sides of (8.1) differ by a class supported on the Noether-Lefschetz locus of $\mathcal{F}$.

Proof.

Let $2k_{\ell}$ be the degree of $\alpha_{\ell}$ ($\ell=1,2$). Denote by $\alpha_{\ell}^{(i)}$ the $2i$-th component of $\alpha_{\ell}$ in the decomposition (8.1). Since $k_{\ell}<\frac{m_{\Lambda}-3}{4}$, the argument in the proof of Theorem 8.1.1 implies that each $\alpha_{\ell}^{(i)}$ can be lifted as a $\theta$-class in $H_{\theta}^{2i}(Y,\mathbf{H}^{2(k_{\ell}-i)})$. Now Proposition 7.3.1 implies that such a $\theta$-class can be represented by a differential form in the FKM ring. It follows that both $\alpha_{1}$ and $\alpha_{2}$ can be represented by elements of the FKM subring of the $E_{1}$ term of the Leray spectral sequence that computes the cohomology of $\mathcal{U}$ from that of the base $\mathcal{F}$ and the fiber. Now being a subring FKM is stable by cup-products. It follows that $\alpha_{1}\wedge\alpha_{2}$ belongs to the FKM ring and that

[TABLE]

is supported on the Noether-Lefschetz locus of $\mathcal{F}$ by Corollary 6.4.3. ∎

8.3. Towards the Cohomological Tautological Conjecture

Let $\mathcal{U}\rightarrow\mathcal{F}$ be a smooth connected family of projective hyperkähler manifolds. We denote by $R^{\ast}_{\pi}(\mathcal{F})\subseteq{\rm CH}^{\ast}(\mathcal{F})$ the subring generated by all $\kappa$-classes. An easy fact is that inclusions

[TABLE]

are preserved under the pullback by morphisms preversing the relative Picard group (modulo $\rm{DCH}^{\ast}(\mathcal{F})$). Here, a morphism between two families $\pi:\mathcal{U}\rightarrow\mathcal{F}$ and $\pi^{\prime}:\mathcal{U}^{\prime}\rightarrow\mathcal{F}^{\prime}$ preserving the relative Picard group is a commutative diagram

[TABLE]

with $\bar{f}^{\ast}{\rm Pic}(\mathcal{U}/\mathcal{F})\cong{\rm Pic}(\mathcal{U}^{\prime}/\mathcal{F}^{\prime})$. This is because $f^{\ast}({\rm NL}_{\pi}^{\ast}(\mathcal{F}))\subseteq{\rm NL}^{\ast}_{\pi^{\prime}}(\mathcal{F}^{\prime})$ and $f^{\ast}R_{\pi}^{\ast}(\mathcal{F})=R_{\pi^{\prime}}^{\ast}(\mathcal{F}^{\prime})$ modulo $\rm{DCH}^{\ast}(\mathcal{F}^{\prime})$.

Using this fact, we can work on $\kappa$-classes on a general family of hyperkähler manifolds and Theorem 4.3.1 will be deduced from the following result as a direct consequence.

Theorem 8.3.1**.**

Let $\pi:\mathcal{U}\rightarrow\mathcal{F}$ be a smooth family of $h$-polarized hyperkähler manifolds of type $\Lambda$ over an irreducible quasi-projective variety $\mathcal{F}$ of dimension $b\geq 3$. Let $r+1$ be Picard number of the generic fiber of $\pi$ and let

[TABLE]

be a collection of line bundles whose images in $\mathrm{Pic}_{\mathbb{Q}}(\mathcal{U}/\mathcal{F})$ form a basis. If $b_{j}=0$ for $j\geq\frac{1}{4}(m_{\Lambda}-3)$, the $\kappa$-class $[\kappa_{a_{0},\ldots,a_{r},b_{1},\ldots b_{2n}}^{\mathcal{B}}]\in H^{\ast}(\mathcal{F},\mathbb{Q})$ is lying in ${\rm NL}^{\ast}_{\rm hom}(\mathcal{F})$.

Proof.

According to the discussion above, it suffices to show that the assertion holds when $\pi:\mathcal{U}\rightarrow\mathcal{F}$ is a connected component of the universal family (as stacks) of lattice-polarized hyperkähler manifolds. Because of (5.10), we can add level structures and further be reduced to the case where $\pi:\mathcal{U}\rightarrow\mathcal{F}$ is the universal family $\mathcal{U}_{\Sigma,h}^{\ell}\rightarrow\mathcal{F}_{\Sigma,h}^{\ell}$ in §8.2.

To prove the assertion, we only need to show that the cup product

[TABLE]

lies in the FKM-ring because the pushward of classes in FKM-ring are lying in $H^{\ast}_{\theta}(\mathcal{F},\mathbb{C})$. Then the argument in the proof of Theorem 8.2.1 implies that as long as $b>2$, it suffices to show $c_{1}(\mathcal{L}_{i})$ and $c_{j}(\mathcal{T}_{\pi})$ are lying in the FKM-ring on $\mathcal{U}$.

Using the notations in $\S$8.2, the cycle classes of $c_{1}(\mathcal{L}_{i})$ and $c_{j}(\mathcal{T}_{\pi})$ can be lifted to classes in $\bigoplus\limits_{p+q=2}H^{p}(Y,\mathbf{H}^{q})$ and $\bigoplus\limits_{p+q=2j}H^{p}(Y,\mathbf{H}^{q})$ respectively via the period map $\mathcal{P}:\mathcal{F}\rightarrow Y$ (See the proof of Theorem 8.1.1).

Next, by our assumption $b\geq 3$, Theorem 6.4.1 implies that the FKM-ring on $Y$ contains $\bigoplus\limits_{p+q=2}H^{p}(Y,\mathbf{H}^{q})$ and $\bigoplus\limits_{p+q=2j}H^{p}(Y,\mathbf{H}^{q})$ when $j<\frac{m_{\Lambda}-3}{4}$. This proves our assertion.

Last, let us explain how we get the bound of $j$. Note that Theorem 6.4.1 actually only applies when $j<\frac{\dim Y}{4}$. However, recall that the moduli space $\mathcal{F}_{h}^{\ell}$ has dimension $m_{\Lambda}-3$, and $c_{j}(\mathcal{T}_{\pi})$ can be lifted to the universal family $\mathcal{U}_{h}^{\ell}\rightarrow\mathcal{F}_{h}^{\ell}$ as $\mathcal{T}_{\pi}$ descends. The lifted class lies in the FKM-ring of $\mathcal{U}_{h}^{\ell}$ by Theorem 6.4.1 and the same argument as above. Then our claim follows from the fact the pullback of FKM-ring on $\mathcal{U}_{h}^{\ell}$ is contained in the FKM-ring of $\mathcal{U}$. ∎

Remark 8.3.2**.**

Theorem 8.3.1 is no longer true when $\dim\mathcal{F}\leq 2$. For instance, there exist divisors on Hilbert modular surfaces which are not in the span of modular curves. (cf. [52])

8.4.

As one can see from the proof, the only obstruction for proving the cohomological tautological conjecture of all hyperkähler is that we do not know whether the $i$-th component of $c_{k}(\mathcal{T}_{\pi})$ in the decomposition (8.1) are theta classes when $\frac{m_{\Lambda}-3}{2}\leq i\leq\min\{2k,\frac{3(m_{\Lambda}-3)}{2}\}$.

As pointed out in [60] and [56, 51], the Beauville-Voisin conjecture actually predicts that all $i$-th component of $c_{k}(\mathcal{T}_{\pi})$ lies in the last component (after possibly shrinking to an open subset of $\mathcal{F}_{h}$). This can be deduced from the existence of so called “Chow-Kunneth” decomposition (cf. [39]). Therefore, the Beauville-Voisin conjecture implies that Conjecture 5 is true at least after shrinking to an open subset of $\mathcal{F}_{h}$.

8.5. Further remarks

For polarized hyperähler manifolds of generalized Kummer type, we know that the moduli space $\mathcal{F}_{h}$ is 4-dimensional. In this case, the $\kappa$-classes on moduli space of lattice-polarized hyperkähler manifolds automatically maps to ${\rm NL}^{\ast}_{\hom}(\mathcal{F}_{h})$ via the pushforward map. This is because ${\rm CH}^{1}(\mathcal{F}_{h})$ is spanned by NL-divisors and $R^{3}_{\hom}(\mathcal{F}_{h})=R^{4}_{\hom}(\mathcal{F}_{h})=0$. So we have

[TABLE]

Then it suffices to check whether the $\kappa$-class

[TABLE]

(i.e. $a+\sum\limits_{j=1}^{2n}jb_{j}-2n=2$) is lying in ${\rm NL}^{2}_{\hom}(\mathcal{F}_{h})$. By Theorem 8.3.1, we know this is true when $b_{i}=0$ for $i>1$. When $b_{i}\neq 0$ for some $i>1$, one can see some examples computed in $\S$4.11. In the case of $n=2$, combining these results together, we know that the conjecture holds if and only if

[TABLE]

are contained in ${\rm NL}^{2}_{\hom}(\mathcal{F}_{h})$.

9. Ring of Special cycles

In Kudla’s program, it is more natural to consider the weighted cycles on Shimura varieties instead of the connected cycles. These are so called special cycles. In this section, we discuss the properties of the special cycles and their applications.

9.1. Kudla’s special cycles

Keep the same notations as in §5.1. We let $\lambda$ be the first Chern class of Hodge line bundle on $Y_{K}$. For any $\beta\in\mathrm{Sym}^{r}(\mathbb{Q})$, we set

[TABLE]

and the special cycles on $Y_{\Gamma}$ of codimension $r$ are defined as

[TABLE]

with $\varphi$ being a $K$-invariant Schwartz function on $V^{r}(\mathbb{A}_{f})$. Then $Z(\beta,\varphi,K)$ can be viewed as a cycle class in ${\rm CH}^{r}(Y_{K})$ if ${\rm rank}\beta=r$. As before, we let $\widetilde{{\rm SC}}^{r}(Y_{K})$ be the subspace of ${\rm CH}^{r}(Y_{K})$ spanned by $Z(\beta,\varphi,K)$ and

[TABLE]

the subring generated by all special cycles. The special cycles have many nice properties, e.g. it behaves well under pullback (cf. [28]). More importantly, although in general the Chow ring ${\rm CH}^{\ast}(Y_{K})$ is not finitely generated, we have

Theorem 9.1.1**.**

The ring $\widetilde{{\rm SC}}^{\ast}(Y_{K})$ is finitely generated and its image in $H^{\ast}(Y_{K},\mathbb{Q})$ contains $H_{\theta}^{\ast}(Y_{K},\mathbb{Q})$.

The first statement follows from the work of Zhang [61] and a recent result of Bruinier and Westerholt-Raum [13], while the second assertion is proved in [6]. Clearly, there is an inclusion

[TABLE]

Moreover, we believe that the following statement should be true.

Conjecture 7**.**

${\rm SC}^{\ast}(Y_{K})=\widetilde{{\rm SC}}^{\ast}(Y_{K})$.

We shall remark that this conjecture can be easily checked in some simple cases such as the moduli space of polarized K3 surfaces, or the moduli space of some polarized $K3^{[n]}$-type hyperkähler manifolds (See Proposition 9.1.2). Recall that the second cohomology of a K3 surface is isomorphic to the even unimodular lattice

[TABLE]

and the primitive part of a polarized K3 surface $(X,H)$ in $\mathcal{K}_{g}$ is isomorphic to

[TABLE]

where $v\in L_{K3}$ represents the class $c_{1}(H)$. The moduli space $\mathcal{K}_{g}$ of polarized K3 surfaces of genus $g$ is an open subset of the arithmetic quotient $Y_{L_{g}}=\Gamma_{L_{g}}\backslash D$, where $\Gamma_{L_{g}}$ is the associated polarized monodromy group. In this situation, we have

Proposition 9.1.2**.**

${\rm SC}^{r}(Y_{L_{g}})=\widetilde{{\rm SC}}^{r}(Y_{L_{g}})$* for all $r$.*

Proof.

The arithmetic subgroup $\Gamma_{L_{g}}$ is nothing but the collection of isometries of $L_{K_{3}}$ preserving $v$. Every connected cycle $c(U)$ in $Y_{L_{g}}$ of codimension $r$ can be described as the map

[TABLE]

where $\Sigma=\mathrm{span}\{U,v\}\cap L_{K3}$ is a primitive sublattice of $L_{K3}$ and $D_{\Sigma}$ and $\Gamma_{\Sigma}$ are defined as in §3.7. We write $c(\Sigma)=c(U)$ for this connected cycle.

As $L_{K_{3}}$ is unimodular, every even lattice of signature $(1,r)$ ($r\leq 19$) admits a unique (up to isometry) primitive embedding to $L_{K_{3}}$. Let $\Sigma$ be an even lattice of signature $(1,r)$. The special cycle on $Y_{L_{g}}$ is of the form

[TABLE]

where the sum runs over all embeddings $\phi:\Sigma\hookrightarrow L_{K3}$ modulo the isometries in $\Gamma_{L_{g}}$ and $\widehat{\phi(\Sigma)}$ is the saturation of $\phi(\Sigma)$ in $L_{K3}$. We can write

[TABLE]

for some integers $m(\Sigma^{\prime})>0$. Then it is easy to see that $c(\Sigma)$ is in the span of $Z(\Sigma^{\prime})$ and this proves the assertion.

∎

9.2. Dimension of $\widetilde{{\rm SC}}^{r}(Y_{K})$

Let us give some examples to explain how to use Zhang-Bruinier-Raum’s result to construct relations between special cycles and estimate the dimension of $\widetilde{{\rm SC}}^{i}(Y_{K})$. Let $\Lambda$ be an even lattice of signature $(2,b)$. For simplicity, we restrict ourselves to the case $Y_{K}$ is the arithmetic quotient $Y_{\Lambda}:=\Gamma_{\Lambda}\backslash D$, where $\Gamma_{\Lambda}$ is the arithmetic subgroup

[TABLE]

Let $\mathbb{H}_{r}$ be the Siegel upper half-plane of genus $r$. The metaplectic double cover ${\rm Mp}_{2r}(\mathbb{Z})$ of ${\rm Sp}_{2r}(\mathbb{Z})$ consists of pairs $\left(M,\phi(\tau)\right)$, where

[TABLE]

Let $\rho_{\Lambda}^{(r)}$ be the Weil-representation of ${\rm Mp}_{2r}(\mathbb{Z})$ on $\mathbb{C}[d(\Lambda)^{r}]$. For any $k\in\frac{1}{2}\mathbb{Z}$, a vector-valued Siegel modular form $f(\tau)$ of weight $k$ and type $\rho_{\Lambda}^{(r)}$ is a $\mathbb{C}[d(\Lambda)^{r}]$-valued holomorphic function on $\mathbb{H}_{r}$, such that

[TABLE]

and it is a Siegel cusp form if $f(\tau)$ is holomorphic at all cusps. We denote by $M^{r}_{k,\Lambda}(\rho_{\Lambda})$ (resp. $S^{r}_{k,\Lambda}(\rho_{\Lambda})$) the space of $\mathbb{C}[d(\Lambda)^{r}]$-valued modular (resp. cusp) forms of weight $k$ and type $\rho_{\Lambda}^{(r)}$. Then

Proposition 9.2.1**.**

For $1\leq r\leq b$, we have

[TABLE]

Proof.

The proof is similar as the case $r=1$ proved in [11, §5]. The work of Zhang-Bruinier-Raum says that the generating series

[TABLE]

is a $\mathbb{C}[d(\Lambda)^{r}]$-valued Siegel modular form of weight $\frac{b+2}{2}$ and type $\rho_{\Lambda}$, where $q^{\beta}=\exp(2\pi{\rm Tr}(\beta\tau))$ (cf. [13, Theorem 5.2]). Using the Petersson inner product

[TABLE]

we obtain a natural map

[TABLE]

by tensoring with $\Phi(\tau)$. The assertion then follows directly from the surjectivity of (9.5). ∎

For general $Y_{K}$, the statement is still true once we replace $S_{\frac{b+2}{2},M}^{r}(\rho_{\Lambda}^{\vee})$ by the space of cuspidale vector-valued modular forms with respect to certain congruence subgroup of ${\rm Mp}_{2r}(\mathbb{Z})$. This also follows from the modularity result of Zhang-Bruinier-Raum.

Remark 9.2.2**.**

The inequality (9.3) is in general strict (see [12, §6.1]). A very interesting question to investigate is when the equality holds (see [12, Theorem 2]). This would enable us to compute the Betti numbers of $Y_{K}$ and also the dimension of $\widetilde{{\rm SC}}^{r}(Y_{K})$.

9.3. Further questions

Let us step back to ${\rm SC}^{r}(Y_{K})$. Similar as the result in [53], when $b\geq 3$, one can show that

[TABLE]

while $\lambda^{b-1}=\lambda^{b}=0$ in ${\rm CH}^{\ast}(Y_{K})$. Furthermore, we have

Proposition 9.3.1**.**

For $i>b-2$ and $j>1$, the subgroup

[TABLE]

is zero.

Proof.

The proof is the same as in [53], where one can show that the self intersection of Hodge line bundle of codimension $\geq b-1$ on a Shimura variety of orthogonal type with dimension $b\geq 3$ is rationally equivalently to zero. ∎

Recall that ${\rm SC}^{r}_{\hom}(Y_{K})=0$ for $r>b-2$ by Corollary 6.4.2, we wonder if the questions below have a positive answer.

Question 9.4**.**

Assume that $\dim Y_{K}\geq 3$. Do the following hold?

1. (1)

$\mathrm{SC}^{r}(Y_{K})=0$ for $r>b-2$. 2. (2)

The cycle class map induces an isomorphism $cl:\mathrm{SC}^{\ast}(Y_{K})\rightarrow\mathrm{SC}^{\ast}_{\rm hom}(Y_{K})$.

Let us come back to $\mathcal{K}_{g}$, where the first part of Question 9.4 indicates that ${\rm NL}^{r}(\mathcal{K}_{g})=0$ for $r>17$. This can be viewed as an analogous of the Faber’s conjecture on $\mathcal{M}_{g}$. The second part is then equivalent to saying that there is an isomorphism

[TABLE]

For small $g$, one can get partial results towards these questions via the geometric construction of $\mathcal{K}_{g}$.

Proposition 9.4.1**.**

When $g\leq 14$ or $g\in\{16,18,20\}$, ${\rm NL}^{19}(\mathcal{K}_{g})=0$.

Proof.

The first assertion follows from the unirationality of $\mathcal{K}_{g}$ for small $g$. Indeed, Mukai has proved that $\mathcal{K}_{g}$ is unirational when $g\leq 13$ or $g\in\{16,18,20\}$ and Nuer [40] recently proved that $\mathcal{K}_{14}$ is also unirational. ∎

Remark 9.4.2**.**

In this case, as the Betti number of $\mathcal{K}_{2}$ has been computed in [27], one can make explicity computation to verify Question 9.4 (2) via Proposition 9.2.1.