1 Introduction↩︎

Throughout, we work over the complex numbers; Chow groups and motives are taken with rational coefficients.

1.1 Orlov conjecture for hyper-Kähler varieties↩︎

For a nonsingular projective variety \(X\), the Orlov conjecture [1] predicts a mysterious connection between the bounded derived category of coherent sheaves \(D^b(X)\) and the Chow motive \(h(X)\). We say that nonsingular projective varieties \(X,X'\) are derived equivalent if there is an isomorphism of bounded derived categories of coherent sheaves, \[D^b(X) \simeq D^b(X').\]

In this paper, we focus on hyper-Kähler varieties, for which Fu–Vial [2] proposed a multiplicative version of Conjecture 1.

Assume that \(X,X'\) are of dimension \(2n\). Conjecture 2 predicts the existence of algebraic correspondences \[\label{Orlov95cycle} [Z] \in \mathrm{CH}^{2n}(X \times X'), \quad [Z'] \in \mathrm{CH}^{2n}(X' \times X)\tag{1}\] which are mutually inverse \[[Z']\circ[Z] = [\Delta_X] \in \mathrm{CH}^{2n}(X \times X), \quad [Z]\circ[Z'] = [\Delta_{X'}] \in \mathrm{CH}^{2n}(X'\times X')\] and respect the cup-product. Here \(\Delta_{(-)}\) is the diagonal class, \(\circ\) stands for the composition of correspondences, and the cup-product is induced by the small diagonal class.

When \(X\) is a \(K3\) surface, Conjecture 1 was proven in [4] and Conjecture 2 was proven in [2]. In higher dimension, very little evidence is known for Conjectures 1 and 2. Even the following homological specialization of Conjecture 2 is wide open.

More precisely, Conjecture 3 predicts the existence of algebraic cycles (1 ) which induce mutually inverse cohomological correspondences \[[Z']\circ[Z] = [\Delta_X] \in H^{4n}(X \times X, {\mathbb{Q}}), \quad [Z]\circ[Z'] = [\Delta_{X'}] \in H^{4n}(X'\times X', {\mathbb{Q}})\] respecting the cup-product \[(H^*(X, {\mathbb{Q}}), \cup) \xrightleftharpoons[~~~{[Z']}~~~]{~~~{[Z]}~~~} (H^*(X', {\mathbb{Q}}), \cup).\]

We discuss some evidence for Conjecture 3. Taelman [5] showed that derived equivalent hyper-Kähler varieties \(X,X'\) have isomorphic Hodge structures \[H^i(X, {\mathbb{Q}}) \simeq H^i(X', {\mathbb{Q}});\] therefore, assuming the Hodge conjecture, one obtains the algebraic cycles (1 ) which induce isomorphisms of the homological motives \(h_{\mathrm{hom}}(X)\simeq h_{\mathrm{hom}}(X')\). However, these isomorphisms a priori do not preserve the cup-product. On the other hand, Fu–Vial [2] showed that derived equivalent hyper-Kähler varieties \(X,X'\) admit an isomorphism of cohomology rings with \({\mathbb{C}}\)-coefficients \[(H^*(X,{\mathbb{C}}),\cup) \simeq (H^*(X', {\mathbb{C}}), \cup);\] but it is unclear if such an isomorphism can be lifted to cohomology with \({\mathbb{Q}}\)-coefficients preserving the Hodge structures. Finally, we note that in all the known examples, derived equivalent hyper-Kähler varieties are deformation equivalent; in particular they share isomorphic cohomology rings.

1.2 Hyper-Kähler varieties of \(K3^{[n]}\)-type↩︎

Although the conjectures in Section 1.1 seem to be out of reach in general, we show in this paper that much evidence of these conjectures can be obtained for hyper-Kähler varieties of \(K3^{[n]}\)-type using the projectively hyperholomorphic bundles constructed recently by Markman [6].

Our first result shows that Conjecture 3 concerning homological motives is completely settled for hyper-Kähler varieties of \(K3^{[n]}\)-type.

Next, we consider an interesting class of \(K3^{[n]}\)-type varieties given by the moduli space of stable sheaves on a \(K3\) surface. Let \(S\) be a projective \(K3\) surface whose Mukai lattice is denoted \(\widetilde{H}(S, {\mathbb{Z}})\). We consider a primitive Mukai vector \(v \in \widetilde{H}(S, {\mathbb{Z}})\) with \(v^2\geq 0\). For a \(v\)-generic polarization, the moduli space \(M_{S,v}\) of stable sheaves \({\mathcal{E}}\) on \(S\) with \(v({\mathcal{E}}) = v\) is a hyper-Kähler variety of \(K3^{[n]}\)-type of dimension \(2n = v^2+2\).² The following property of these hyper-Kähler varieties was proven by Beckmann [8].

In view of Theorem 6 and the birational invariance of Chow motives for hyper-Kähler varieties [7], it is natural to ask how the homological and the Chow motives of \(M_{S,v}, M_{S,v'}\) are related. The proof of Theorem 4 also yields the following.

Finally, we discuss evidence for Chow motives. Our main result is that when \(S\) is of Picard rank \(1\), we are able to realize the isomorphism in Theorem 7 via generically defined cycles, which allows us to lift the homological statement to Chow using the (conjectural) Franchetta properties for the self-products of polarized \(K3\) surfaces.

Assume \(g\geq 2\). A \(K3\) surface of genus \(g\) is a primitively polarized \(K3\) surface \((S,H)\) with \(H^2=2g-2\). Let \({\mathcal{S}}_g \to {\mathcal{X}}_g\) be the universal family over the moduli stack of \(K3\) surfaces of genus \(g\). We say that \(K3\) surfaces of genus \(g\) satisfy the \(n\)-th Franchetta property if the generically defined cycles \[\mathrm{GDCH}^*(S^n):= \mathrm{Im}\left( \mathrm{CH}^*({{\mathcal{S}}^n_g}_{/{\mathcal{X}}_g}) \to \mathrm{CH}^*(S^n) \right)\] on the \(n\)-th product of any fiber \(S\subset {\mathcal{S}}_g\) map injectively to the cohomology via the cycle class map, i.e., \[\mathrm{cl}: \mathrm{GDCH}^*(S^n) \hookrightarrow H^{*}(S^n, {\mathbb{Q}}).\] For \(n=1\), the Franchetta property was conjectured by O’Grady [9], known as the generalized Franchetta conjecture for \(K3\) surfaces, and has been verified for low genus cases using the global geometry of the universal family [10]–[12]. The Franchetta property was further conjectured to hold in [13], [14] for any \(n\geq 1\); see [15].

1.3 Idea of the proof↩︎

We first discuss the main difficulty in proving the Orlov conjecture. Assume that \(X, X'\) are derived equivalent varieties of dimension \(d\). The Chern characters of the Fourier–Mukai kernels \({\mathcal{U}}, {\mathcal{U}}^{-1}\) yield correspondences \[\label{corr0} \mathfrak{F} \in \mathrm{CH}^*(X\times X'), \quad \mathfrak{F}^{-1} \in \mathrm{CH}^*(X' \times X)\tag{2}\] satisfying \[\mathfrak{F}^{-1}\circ\mathfrak{F} = [\Delta_X], \quad \mathfrak{F}\circ\mathfrak{F}^{-1} = [\Delta_{X'}].\] The classes (2 ) are a priori of mixed degree in the Chow groups. In order to prove (the homological or the Chow version of) the Orlov conjecture, we need to construct algebraic cycles of pure degree \[\label{cor00} [Z] \in \mathrm{CH}^d(X \times X'), \quad [Z'] \in \mathrm{CH}^d(X' \times X)\tag{3}\] with \[\label{hom-Chow} [Z']\circ[Z] = [\Delta_X] , \quad [Z]\circ[Z'] = [\Delta_{X'}].\tag{4}\] Here the identities (4 ) hold in cohomology or Chow, depending on the version of the Orlov conjecture one would like to prove. In general, it is not clear how to modify the cycles in (2 ) of mixed degree to the desired \(d\)-dimensional cycles in (3 ).

Now we consider hyper-Kähler varieties of \(K3^{[n]}\)-type. We first observe that derived equivalent varieties \(X,X'\) of \(K3^{[n]}\)-type are connected Hodge-theoretically; see Proposition 19. Therefore, both Theorems 4 and 7 are deduced from the following theorem.

The proofs of the main theorems rely on a Fourier transform package discussed in Section 2 for hyper-Kähler varieties of \(K3^{[n]}\)-type, which in cohomology is largely a reformulation of results of Markman [6]. We prove Theorems 4, 7, and 9 in Section 3.

In Section 5, we show that the isomorphism in Theorem 7 can be constructed from generically defined cycles when the \(K3\) surface is of Picard rank \(1\), thereby proving Theorem 8. The key is an algebraic moduli approach to a recent construction of Zhang [16] regarding moduli spaces of stable sheaves, which we discuss in Section 4.

1.4 Acknowledgements↩︎

We are grateful to Eyal Markman and Ruxuan Zhang for helpful discussions.

D.M. was supported by a Simons Investigator Grant. J.S. was supported by the NSF grant DMS-2301474 and a Sloan Research Fellowship.

2 Fourier transforms via hyperholomorphic bundles↩︎

Conjectures of Beauville and Voisin [17], [18] predict that the Chow motive of a hyper-Kähler variety behaves analogously to that of an abelian variety. The purpose of Section 2 is to make this analogy more explicit from the perspective of Fourier transforms. Although most of the discussion in this section is conjectural, the construction introduced here plays a crucial role in the proofs of the main theorems.

2.1 Abelian varieties↩︎

We first briefly review the theory of Fourier transforms in the study of algebraic cycles on abelian varieties [19], [20].

Let \(A\) be a \(g\)-dimensional abelian variety, and let \(A^\vee:= \mathrm{Pic}^0(A)\) be its dual. The normalized Poincaré line bundle \({\mathcal{L}}\) on \(A^\vee\times A\) induces a derived equivalence of coherent categories [21], which further induces algebraic correspondences, known as the Fourier transforms \[\mathfrak{F}:= \mathrm{ch}({\mathcal{L}}) \in \mathrm{CH}^*(A^\vee \times A), \quad \mathfrak{F}^{-1}:= \mathrm{ch}({\mathcal{L}}^\vee) \in \mathrm{CH}^*(A\times A^\vee).\] We set \[\mathfrak{p}_k:= \mathfrak{F}_k\circ \mathfrak{F}^{-1}_{2g-k} \in \mathrm{CH}^g(A\times A), \quad \mathfrak{p}^\vee_k:= \mathfrak{F}^{-1}_k\circ \mathfrak{F}_{2g-k} \in \mathrm{CH}^g(A^\vee\times A^\vee),\quad 0\leq k \leq 2g,\] where \(\mathfrak{F}_i \in\mathrm{CH}^i(A^\vee \times A)\) and \(\mathfrak{F}^{-1}_j\in\mathrm{CH}^j(A \times A^\vee)\) are the degree \(i\) and degree \(j\) parts of \(\mathfrak{F}\) and \(\mathfrak{F}^{-1}\) respectively. Since \(\mathfrak{F}, \mathfrak{F}^{-1}\) are obtained from Fourier–Mukai equivalences, we obtain decompositions of the diagonal classes \[\label{Ch95Ku95Ab0} [\Delta_A] = \sum_{k=0}^{2g} \mathfrak{p}_k \in \mathrm{CH}^g(A \times A), \quad [\Delta_{A^\vee}] = \sum_{k=0}^{2g} \mathfrak{p}^\vee_k \in \mathrm{CH}^g(A^\vee \times A^\vee).\tag{5}\] A key observation of [19], [20] is that the decompositions (5 ) yield Fourier-stable multiplicative Chow–Künneth decompositions \[\begin{align} h(A) &= \bigoplus_{k=0}^{2g} h_k(A),\quad h_k(A)=(A, \mathfrak{p}_k,0),\tag{6}\\ \quad h(A^\vee) &= \bigoplus_{k=0}^{2g} h_k(A^\vee),\quad h_k(A^\vee)=(A^\vee, \mathfrak{p}^\vee_k,0). \tag{7} \end{align}\] More precisely, the decompositions (6 ) and (7 ) respect the cup-product. Furthermore, in the category of Chow motives, the graded algebra object \[\bigoplus_{k=0}^{2g} h_k(A), \quad \cup: h_i(A) \times h_{j}(A) \to h_{i+j}(A)\] given by the cup-product is isomorphic through the Fourier transform \(\mathfrak{F}\) to the graded algebra object (with appropriate Tate-twists) \[\bigoplus_{k=0}^{2g} h_{2g-k}(A^\vee)(g - k), \quad \ast: h_{2g-i}(A^\vee)(g - i) \times h_{2g-j}(A^\vee)(g - j) \to h_{2g-i-j}(A^\vee)(g - i - j)\] given by the convolution product induced by the abelian group structure \(+: A^\vee\times A^\vee \to A^\vee\).

This package relies heavily on special geometric ingredients of abelian varieties, including the normalized Poincaré line bundle and the group structure.

2.2 Hyper-Kähler varieties↩︎

A Fourier transform package similar to the one in Section 2.1 for abelian varieties is expected to hold for hyper-Kähler varieties. We provide here an explicit proposal for varieties of \(K3^{[n]}\)-type. Our proposal is built on the recently constructed projectively hyperholomorphic bundles of Markman [6] which we briefly review as follows; see also [22].

Consider a projective \(K3\) surface \(S_0\) with \(\mathrm{Pic}(S_0)={\mathbb{Z}}H\). Assume that the Mukai vector \[v_0:=(r, mH, s) \in \widetilde{H}(S_0, {\mathbb{Z}}), \quad \mathrm{gcd}(r,s)=1\] is isotropic (i.e. \(v_0^2=0\)), and that all \(H\)-stable sheaves on \(S_0\) with Mukai vector \(v_0\) are stable vector bundles. Let \(M_{S_0,v_0}\) be the moduli space of such stable vector bundles. Then \(M_{S_0,v_0}\) is again a \(K3\) surface with a universal rank \(r\) bundle \({\mathcal{U}}\) on \(M_{S_0,v_0} \times S_0\). Conjugating the Bridgeland–King–Reid (BKR) correspondence [23] yields a projectively hyperholomorphic vector bundle \({\mathcal{U}}^{[n]}\) on \(M_{S_0,v_0}^{[n]} \times S_0^{[n]}\) of rank \[\mathrm{rk}({\mathcal{U}}^{[n]}) = n!r^n,\] which induces a derived equivalence \[\Phi_{{\mathcal{U}}^{[n]}}: D^b(M_{S_0,v_0}^{[n]}) \xrightarrow{\simeq} D^b(S_0^{[n]}).\]

Now let \(X\) be any hyper-Kähler variety of \(K3^{[n]}\)-type. There always exists a variety \(Y\) of \(K3^{[n]}\)-type and a twisted vector bundle \(({\mathcal{E}},\alpha_{\mathcal{E}})\) on \(Y\times X\), such that this twisted vector bundle is deformed from \({\mathcal{U}}^{[n]}\) and induces a (twisted) derived equivalence \[\label{FM} \Phi_{({\mathcal{E}},\alpha_{\mathcal{E}})}: D^b(Y, \alpha_Y) \xrightarrow{\simeq} D^b(X, \alpha_X)\tag{8}\] with \(\alpha_X, \alpha_Y\) certain Brauer classes on \(X, Y\) respectively. We denote by \(({\mathcal{E}}^{-1}, \alpha_{{\mathcal{E}}^{-1}})\) the (shifted) twisted vector bundle given by the kernel of the inverse of (8 ). Note that even for fixed \(X\), \(S_0\), and \(v_0\) (which determines the \(K3\) surface \(M_{S_0,v_0}\) uniquely), the choice of \(Y\) and \(({\mathcal{E}}, \alpha_{\mathcal{E}})\) may not be unique.

We consider the kappa class (see [24] or [6]) associated with the twisted vector bundle \(({\mathcal{E}}, \alpha_{\mathcal{E}})\). For the twisted vector bundle \({\mathcal{E}}\), the vector bundle \[{\mathcal{E}}^{\otimes \mathrm{rank}({\mathcal{E}})} \otimes \mathrm{det}({\mathcal{E}})^{-1}\] is untwisted, and the kappa class \[\kappa({\mathcal{E}}) \in \mathrm{CH}^*(Y \times X)\] is defined to be the \(\mathrm{rank}({\mathcal{E}})\)-th root of the Chern character of \({\mathcal{E}}^{\otimes \mathrm{rank}({\mathcal{E}})} \otimes \mathrm{det}({\mathcal{E}})^{-1}\). We define \(\kappa({\mathcal{E}}^{-1})\) similarly; they yield Fourier transforms \[\mathfrak{F}:= \kappa({\mathcal{E}})\sqrt{\mathrm{td}_{Y\times X}} \in \mathrm{CH}^*(Y \times X),\quad \mathfrak{F}^{-1}:= \kappa({\mathcal{E}}^{-1})\sqrt{\mathrm{td}_{X\times Y}} \in \mathrm{CH}^*(X\times Y).\] Since \(({\mathcal{E}}^{-1}, \alpha_{{\mathcal{E}}^{-1}})\) is the kernel for the Fourier–Mukai inverse of (8 ), the relations \[[\Delta_X] = \mathfrak{F}\circ \mathfrak{F}^{-1} \in \mathrm{CH}^{2n}(X \times X), \quad [\Delta_Y] = \mathfrak{F}^{-1}\circ \mathfrak{F} \in \mathrm{CH}^{2n}(Y \times Y)\] are obtained from the Grothendieck–Riemann–Roch theorem. In other words, the Fourier transforms \(\mathfrak{F}, \mathfrak{F}^{-1}\) induce mutually inverse isomorphisms of ungraded Chow motives \[\label{iso95Chow95XY} h(Y)\xrightleftharpoons[~~~\mathfrak{F}^{-1}~~~]{~~~\mathfrak{F}~~~} h(X), \quad \mathfrak{F}\circ \mathfrak{F}^{-1} = \mathrm{id}_{h(X)}, \quad \mathfrak{F}^{-1}\circ \mathfrak{F} = \mathrm{id}_{h(Y)}.\tag{9}\]

2.3 A Fourier transform package↩︎

Consider any \(X,Y,{\mathcal{E}}, {\mathcal{E}}^{-1}, \mathfrak{F}, \mathfrak{F}^{-1}\) as in Section 2.2. We describe in this section a conjectural Fourier transform package. For integers \(0 \leq i, j \leq 4n\), let \(\mathfrak{F}_i \in\mathrm{CH}^i(Y \times X)\) and \(\mathfrak{F}^{-1}_j\in\mathrm{CH}^j(X \times Y)\) denote the degree \(i\) and degree \(j\) parts of \(\mathfrak{F}\) and \(\mathfrak{F}^{-1}\) respectively.

Exactly as in the abelian variety case, we set \[\label{projectors} \mathfrak{p}_{2k}:= \mathfrak{F}_{2k}\circ \mathfrak{F}^{-1}_{4n-2k} \in \mathrm{CH}^{2n}(X\times X), \quad \mathfrak{p}^\vee_{2k}:= \mathfrak{F}^{-1}_{2k}\circ \mathfrak{F}_{4n-2k} \in \mathrm{CH}^{2n}(Y\times Y),\quad 0\leq k \leq 2n.\tag{10}\] Conjecture 11(a) yields decompositions \[\label{CK} [\Delta_X] = \sum_{k=0}^{2n} \mathfrak{p}_{2k} \in \mathrm{CH}^{2n}(X \times X), \quad [\Delta_Y] = \sum_{k=0}^{2n} \mathfrak{p}^\vee_{2k} \in \mathrm{CH}^{2n}(Y \times Y)\tag{11}\] as in (5 ).

In order to study the interaction between the cup-product and the Fourier transform, we need to normalize \(\mathfrak{F}, \mathfrak{F}^{-1}\) as follows; this is crucial in view of Proposition 17(b) below.

Recall that the rank of \({\mathcal{E}}\) is \(n!r^n\). We set the normalized Fourier transforms as \[\label{normal} \begin{gather} \widetilde{\mathfrak{F}} = \sum_{k} \widetilde{\mathfrak{F}}_{2k}, \quad \widetilde{\mathfrak{F}}_{2k}: = r^{k-n}\mathfrak{F}_{2k}, \\ \widetilde{\mathfrak{F}}^{-1} = \sum_{k} \widetilde{\mathfrak{F}}^{-1}_{2k}, \quad \widetilde{\mathfrak{F}}^{-1}_{2k}:= r^{k-n}\mathfrak{F}_{2k}. \end{gather}\tag{13}\] Clearly, under Conjecture 11(a), they still induce isomorphisms of ungraded Chow motives \[h(Y)\xrightleftharpoons[~~~\widetilde{\mathfrak{F}}^{-1}~~~]{~~~\widetilde{\mathfrak{F}}~~~} h(X),\quad \widetilde{\mathfrak{F}}\circ \widetilde{\mathfrak{F}}^{-1} = \mathrm{id}_{h(X)}, \quad \widetilde{\mathfrak{F}}^{-1}\circ \widetilde{\mathfrak{F}} = \mathrm{id}_{h(Y)},\] and the statements of Proposition 13 hold identically if we replace \(\mathfrak{F},\mathfrak{F}^{-1}\) by \(\widetilde{\mathfrak{F}}, \widetilde{\mathfrak{F}}^{-1}\).

Following [6], we consider the convolution class \[\mathfrak{C}:= \widetilde{\mathfrak{F}}^{-1} \circ [\Delta_X^{\mathrm{sm}}] \circ \left(\widetilde{\mathfrak{F}}\times\widetilde{\mathfrak{F}}\right) \in \mathrm{CH}^*(Y\times Y \times Y),\] and its induced ungraded morphism \[\label{convolution} \ast: h(Y) \times h(Y) \to h(Y).\tag{14}\] Here \([\Delta^{\mathrm{sm}}_X]\) is the class of the small diagonal \(X\subset X \times X \times X\). The class \(\mathfrak{C}\) is a priori of mixed degree and depends on the variety \(X\) and the twisted vector bundle \({\mathcal{E}}\) on \(Y \times X\). We conjecture that it is in fact of pure degree and intrinsic to \(Y\).

2.4 Results in cohomology↩︎

We first note that both conjectures in Section 2.3, which concern algebraic cycles, hold cohomologically by Markman [6].

As a consequence, both Propositions in Section 2.3 hold at the level of homological motives. We thus obtain Künneth decompositions of the homological motives \[h_{\mathrm{hom}}(X) = \bigoplus_{k = 0}^{2n}h_{\mathrm{hom}, 2k}(X), \quad h_{\mathrm{hom}}(Y) = \bigoplus_{k = 0}^{2n}h_{\mathrm{hom}, 2k}(Y),\] and the two graded algebra objects \[\left(\bigoplus_{k = 0}^{2n}h_{\mathrm{hom}, 2k}(X), \cup\right), \quad \left(\bigoplus_{k = 0}^{2n}h_{\mathrm{hom}, 4n-2k}(Y)(2n-2k), \ast\right)\] are isomorphic through the Fourier transform \(\widetilde{{\mathfrak{F}}}\).

Next, we give a description of the kappa class of Section 2.2 in terms of the (rational) extended Mukai lattice. For a hyper-Kähler manifold \(X\), its extended Mukai lattice is the \({\mathbb{Q}}\)-vector space \[\label{extended95Mukai} \widetilde{H}(X,{\mathbb{Q}}) = {\mathbb{Q}}\alpha \oplus H^2(X, {\mathbb{Q}}) \oplus {\mathbb{Q}}\beta, \quad \alpha^2=\beta^2=0, \quad (\alpha,\beta) = -1,\tag{20}\] where the pairing on \(H^2(X,{\mathbb{Q}})\) is induced by the Beauville–Bogomolov–Fujiki (BBF) form, and \(\alpha,\beta\) are orthogonal to \(H^2(X,{\mathbb{Q}})\). By [5], [6], [8], the lattice \(\widetilde{H}(X, {\mathbb{Q}})\) encodes important information of its derived category. Let \(\widetilde{{H}}(X, {\mathbb{C}}):= \widetilde{H}(X, {\mathbb{Q}}) \otimes_{\mathbb{Q}}{\mathbb{C}}\) be the extended Mukai lattice with \({\mathbb{C}}\)-coefficients.

We consider a derived equivalence (8 ) obtained from a \(K3\) surface \(S_0\) of Picard rank \(1\) and an isotropic Mukai vector \(v_0\). We first explain that an isometry \[\label{varphi} \varphi: \widetilde{H}(Y, {\mathbb{C}}) \xrightarrow{\simeq} \widetilde{H}(X, {\mathbb{C}})\tag{21}\] yields an isomorphism of (ungraded) \({\mathbb{C}}\)-vector spaces \[\varphi^H: H^*(Y, {\mathbb{C}}) \xrightarrow{\simeq} H^*(X, {\mathbb{C}}).\] By the construction, \(X,Y\) are deformed from \(S_0^{[n]}, M^{[n]}_{S_0,v_0}\) respectively. We further deform the \(K3\) surface \(S_0\) to a special one \(S'\) such that the corresponding moduli space \(M_{S',v'}\) is isomorphic to \(S'\) itself. We have parallel transports \[\label{PTX} \mathrm{PT}_X: \widetilde{H}(X, {\mathbb{Q}}) \to \widetilde{H}(S'^{[n]}, {\mathbb{Q}}), \quad \mathrm{PT}^H_X: H^*(X, {\mathbb{Q}}) \to H^*(S'^{[n]}, {\mathbb{Q}}),\tag{22}\] and \[\label{PTY39} \mathrm{PT}_Y: \widetilde{H}(Y, {\mathbb{Q}}) \to \widetilde{H}(M^{[n]}_{S',v'}, {\mathbb{Q}}), \quad \mathrm{PT}^H_X: H^*(Y, {\mathbb{Q}}) \to H^*(M^{[n]}_{S',v'}, {\mathbb{Q}}).\tag{23}\] Through a choice of an isomorphism \[\label{iso} M_{S',v'} \simeq S',\tag{24}\] the parallel transports (23 ) can further be expressed as \[\label{PTY} \mathrm{PT}_Y: \widetilde{H}(Y, {\mathbb{Q}}) \to \widetilde{H}(S'^{[n]}, {\mathbb{Q}}), \quad \mathrm{PT}^H_X: H^*(Y, {\mathbb{Q}}) \to H^*(S'^{[n]}, {\mathbb{Q}}).\tag{25}\]

Now, by the parallel transports (22 ) and (25 ), the isometry (21 ) induces an isometry \[\label{isometry5} \mathrm{PT}_Y\circ \varphi \circ \mathrm{PT}^{-1}_X: \widetilde{H}(S'^{[n]}, {\mathbb{Q}}) \to \widetilde{H}(S'^{[n]}, {\mathbb{Q}}).\tag{26}\] Integration of the Looijenga–Lunts–Verbitsky (LLV) algebra provides an action of the special orthogonal group \(\mathrm{SO}(\widetilde{H}(S'^{[n]}, {\mathbb{C}}))\) on the total cohomology \(H^*(S'^{[n]}, {\mathbb{C}})\), \[\label{integ} \mathrm{SO}(\widetilde{H}(S'^{[n]}, {\mathbb{C}})) \to \mathrm{GL}( H^*(S'^{[n]}, {\mathbb{C}})).\tag{27}\] A further choice of a monodromy reflection of \(S'\), which induces a monodromy reflection of \(S'^{[n]}\), upgrades 27 to an action of the full orthogonal group \[\label{integ2} O(\widetilde{H}(S'^{[n]}, {\mathbb{C}})) \to \mathrm{GL}( H^*(S'^{[n]}, {\mathbb{C}}));\tag{28}\] see [6]. By composing with the parallel transports (22 ) and (25 ), we obtain \[\varphi^H : H^*(Y, {\mathbb{C}}) \xrightarrow{\mathrm{PT}^H_Y} H^*(S'^{[n]}, {\mathbb{C}}) \to H^*(S'^{[n]}, {\mathbb{C}}) \xrightarrow{(\mathrm{PT}_X^{H})^{-1}} H^*(X, {\mathbb{C}})\] where the second map is given by the integration map 28 applied to (26 ). This gives the desired (ungraded) assignment \[\left( \varphi: \widetilde{H}(Y, {\mathbb{C}}) \xrightarrow{\simeq} \widetilde{H}(X, {\mathbb{C}}) \right) \mapsto \left( \varphi^H: H^*(Y, {\mathbb{C}}) \xrightarrow{\simeq} H^*(X, {\mathbb{C}}) \right).\]

For the derived equivalence (8 ), recall that we have the Fourier transform \[\mathfrak{F}: H^*(Y, {\mathbb{Q}}) \to H^*(X, {\mathbb{Q}})\] induced by the kappa class of the Fourier–Mukai kernel \({\mathcal{E}}\). By results of Taelman [5], the morphism \(\mathfrak{F}\) induces an isometry of the extended Mukai lattice \[\label{eq:htilde} \widetilde{H}(\mathfrak{F}): \widetilde{H}(Y, {\mathbb{C}}) \to \widetilde{H}(X, {\mathbb{C}})\tag{29}\] whose construction is representation-theoretic; see also [6]. Note that when \(n\) is even, the isometry \(\widetilde{H}(\mathfrak{F})\) is defined up to a sign depending on the orientations of \(X, Y\).³.

The following lemma due to Markman [6] shows that the morphism \(\mathfrak{F}\) is determined up to a sign by the isometry \(\widetilde{H}(\mathfrak{F})\) via the integration map (28 ).

2.5 Relations to other work↩︎

1. Section 2 focuses on generalizing the Fourier transform package from abelian varieties to hyper-Kähler varieties, where the role of the Poincaré line bundle is played by higher-rank hyperholomorphic bundles. In another direction, the Fourier transform package has been considered for abelian fibrations with singular fibers where the Poincaré line bundle is replaced by certain generically rank \(1\) Cohen–Macaulay sheaves; see [25], [26] for more details.

2. Markman’s hyperholomorphic bundles, which are crucial in our construction of Section 2, have also been applied to various other algebro-geometric questions for hyper-Kähler varieties of \(K3^{[n]}\)-type, including algebraic realizations of Hodge isometries [6], symmetries of derived categories [16], [22], [27], and the period–index problem [28], [29].

3. A different Fourier transform was studied in [30] for the self-product of a hyper-Kähler fourfold which is either the Hilbert square of a \(K3\) surface or the Fano variety of lines of a nonsingular cubic fourfold.

3 Proofs of Theorems 4, 7, and 9↩︎

In this section, we complete the proofs of all the theorems concerning homological motives for hyper-Kähler varieties of \(K3^{[n]}\)-type: Theorems 4, 7, and 9. A key step is Proposition 22 which shows the existence of algebraic correspondences conjugating the cup-product and the convolution product for a hyper-Kähler variety of \(K3^{[n]}\)-type.

3.1 Derived equivalences and Hodge isometries↩︎

Let \(X, X'\) be hyper-Kähler varieties of \(K3^{[n]}\)-type. We say that \(X, X'\) are Hodge isometric, if there is a Hodge isometry \[H^2(X, {\mathbb{Q}}) \simeq H^2(X', {\mathbb{Q}})\] with respect to the BBF form. The following proposition shows that derived equivalent varieties of \(K3^{[n]}\)-type must be Hodge isometric.

The following corollary is immediate.

3.2 Fourier autoduality↩︎

Let \(Y\) be the variety of \(K3^{[n]}\)-type introduced in Section 2.2. The goal is to relate the two graded algebra objects (both on the \(Y\) side) \[\label{eq:hommot} \left(\bigoplus_{k = 0}^{2n}h_{\mathrm{hom}, 2k}(Y), \cup\right), \quad \left(\bigoplus_{k = 0}^{2n}h_{\mathrm{hom}, 4n-2k}(Y)(2n-2k), \ast\right)\tag{31}\] in the category of homological motives.

Nevertheless, as we observe in this section, it is always possible to construct a degree-reversing, product-conjugating self-correspondence in \(H^*(Y \times Y, {\mathbb{Q}})\), which is further induced by algebraic cycles. The construction relies on the Lefschetz standard conjecture for varieties of \(K3^{[n]}\)-type established in [6], [31].

Choose an ample class \(L \in H^2(Y, {\mathbb{Q}})\) of BBF norm \((L, L) = 2d\), and consider the associated \(\mathfrak{sl}_2\)-triple \((e_L, f_L, h)\) with \[e_L = {}\cup L: H^*(Y, {\mathbb{Q}}) \to H^{* + 2}(Y, {\mathbb{Q}}), \quad h|_{H^i(Y, {\mathbb{Q}})} = (i - 2n)\mathrm{id}.\] Viewed as a correspondence, the Lefschetz operator \(e_L\) is given by the cycle class \[\label{eq:defL} e_L = \Delta_*L \in H^{4n + 2}(Y \times Y, {\mathbb{Q}})\tag{32}\] where \(\Delta: Y \to Y \times Y\) is the diagonal morphism. By [6], the dual Lefschetz operator \(f_L\) is also algebraic, and is explicitly given by the correspondence \[\label{eq:defLambda} f_L = \mathfrak{F}^{-1} \circ e_{\pm\frac{1}{rd}\widetilde{H}(\mathfrak{F})^{-1}(L)} \circ \mathfrak{F} \in H^{4n - 2}(Y \times Y, {\mathbb{Q}}).\tag{33}\] Here we follow the notation of Section 2: \(\mathfrak{F}\) is the Fourier transform induced by the rank \(n!r^n\) twisted vector bundle on \(Y \times X\), and \(\widetilde{H}(\mathfrak{F})\) is as in 29 which for \(n\) even is defined up to a sign.

Now consider the correspondence \[\mathfrak{F}_L := \exp(e_L) \circ \exp(-f_L) \circ \exp(e_L) \in H^*(Y \times Y, {\mathbb{Q}})\] which we call the cohomological Fourier self-transform associated with \(L\). Note that if we view the \(\mathfrak{sl}_2\)-triple \((e_L, f_L, h)\) as elements of the LLV algebra \(\mathfrak{so}(\widetilde{H}(X, {\mathbb{C}}))\), then \(\mathfrak{F}_L\) naturally lives in the associated Lie group \(\mathrm{SO}(\widetilde{H}(Y, {\mathbb{C}}))\) via the exponential map \[\mathrm{exp}: \mathfrak{so}(\widetilde{H}(X, {\mathbb{C}})) \to \mathrm{SO}(\widetilde{H}(Y, {\mathbb{C}})).\] The latter acts on \(H^*(Y, {\mathbb{C}})\) by integrating the LLV algebra action \[\mathrm{SO}(\widetilde{H}(Y, {\mathbb{C}})) \to \mathrm{GL}( H^*(Y, {\mathbb{C}}) )\] as in (27 ). In other words, the class \(\mathfrak{F}_L \in H^*(Y\times Y, {\mathbb{Q}})\), viewed as an element in \(\mathrm{GL}( {H}^*(Y, {\mathbb{C}}))\), lies in the image of the integration map above. The inverse of \({\mathfrak{F}}_L\) is given by \[{\mathfrak{F}}_L^{-1} = \exp(-e_L) \circ \exp(f_L) \circ \exp(-e_L) \in H^*(Y \times Y, {\mathbb{Q}}).\]

We have by construction \(\mathfrak{F}_L^{-1}h\mathfrak{F}_L = -h \in \mathfrak{so}(\widetilde{H}(X, {\mathbb{C}}))\), and therefore \[{\mathfrak{F}}_L^{-1} \circ h \circ {\mathfrak{F}}_L = -h \in H^*(Y \times Y, {\mathbb{Q}}).\] In particular, \({\mathfrak{F}}_L\) induces a degree-reversing automorphism \[{\mathfrak{F}}_L: H^*(Y, {\mathbb{Q}}) \to H^*(Y, {\mathbb{Q}}), \quad H^i(Y, {\mathbb{Q}}) \to H^{4n - i}(Y, {\mathbb{Q}}).\] Then as in Proposition 17(a), we obtain immediately the Fourier vanishing: that for any integers \(0 \leq i, j < 2n\) we have \[{\mathfrak{F}}_{L, 2i + 1} = 0 \in H^{4i + 2}(Y \times Y, {\mathbb{Q}}), \quad {\mathfrak{F}}^{-1}_{L, 2j + 1} = 0 \in H^{4j + 2}(Y \times Y, {\mathbb{Q}}),\] and for any integers \(0 \leq i, j \leq 2n\), we have \[\mathfrak{F}_{L, 2i} \circ \mathfrak{F}^{-1}_{L, 2j}= 0 \in H^{4i+4j-4n}(Y \times Y, {\mathbb{Q}}), \quad \mathfrak{F}^{-1}_{L, 2i} \circ \mathfrak{F}_{L, 2j}= 0 \in H^{4i+4j-4n}(Y \times Y, {\mathbb{Q}}).\] Here \({\mathfrak{F}}_{L, i}\) and \({\mathfrak{F}}^{-1}_{L, j}\) are the summands of \({\mathfrak{F}}_L, {\mathfrak{F}}_L^{-1}\) in \(H^{2i}(Y \times Y, {\mathbb{Q}})\) and \(H^{2j}(Y \times Y, {\mathbb{Q}})\) respectively.

The normalizations of the Fourier self-transforms \({\mathfrak{F}}_L, {\mathfrak{F}}_L^{-1}\) take the form \[\label{eq:normal2} \begin{gather} \widetilde{\mathfrak{F}}_L = \sum_{k} \widetilde{\mathfrak{F}}_{L, 2k}, \quad \widetilde{\mathfrak{F}}_{L, 2k}: = d^{n - k}\mathfrak{F}_{L, 2k}, \\ \widetilde{\mathfrak{F}}^{-1}_L = \sum_{k} \widetilde{\mathfrak{F}}^{-1}_{L, 2k}, \quad \widetilde{\mathfrak{F}}^{-1}_{L, 2k}:= d^{n - k}\mathfrak{F}_{L, 2k}, \end{gather}\tag{34}\] where \(2d\) is the BBF norm of \(L\).

Recall the two graded algebra objects from 31 . We remark that although the two motivic decompositions of \(h_{\mathrm{hom}}(Y)\) in 31 (the second one with appropriate Tate twists) are obtained via different projectors, they are identical decompositions of homological motives. This is simply because modulo homological equivalence, the Künneth projectors are unique if they exist.

The following result is key to the proofs of the main theorems.

3.3 Proofs of Theorems 7 and 9↩︎

In view of Corollary 20, it remains to prove Theorems 7 and 9. The proof of Theorem 9 relies on the following characterization, due to Markman [6], of Hodge isometric varieties of \(K3^{[n]}\)-type.

We move on to Theorem 7. Let \(S\) be a projective \(K3\) surface, and let \(v \in \widetilde{H}(S, {\mathbb{Z}})\) be a primitive Mukai vector with \(v^2 = 2n-2\geq 0\). The moduli space \(M_{S,v}\) parameterizes \(H\)-stable sheaves on \(S\) with Mukai vector \(v\), where the polarization \(H\) is \(v\)-generic. We now prove that there is an isomorphism of homological motives which respects the cup-product, \[\label{main95thm95sec2} (h_{\mathrm{hom}}(S^{[n]}), \cup) \simeq (h_{\mathrm{hom}}(M_{S,v}), \cup).\tag{37}\] This clearly implies Theorem 7.

Again by [7], it suffices to consider the case when \(M_{S,v}\) is not birational to \(S^{[n]}\). In this case, Zhang proved in [16] that there is a derived equivalence \[\label{Zhang95FM} D^b(S^{[n]}) \simeq D^b(M_{S,v}, \theta_{v}),\tag{38}\] where \(\theta_v\) is a particular Brauer class on \(M_{S,v}\). More precisely, Zhang’s equivalence (38 ) can be realized as the composition of \[\label{Zhang95FM2} \Phi_{({\mathcal{E}},\theta_v)}: D^b(S^{[n]}) \xrightarrow{\simeq} D^b(M'_{S,v}, \theta_v)\tag{39}\] and \[\label{D-equiv} D^b(M'_{S,v}, \theta_{v}) \xrightarrow{\simeq} D^b(M_{S,v}, \theta_v).\tag{40}\] Here \(M'_{S,v}\) is a hyper-Kähler variety birational to \(M_{S,v}\), and we use the same notation \(\theta_v\) to denote the Brauer class on \(M'_{S,v}\) obtained via the birational map. The equivalence (39 ) is a special case of (8 ) induced by a projectively hyperholomorphic bundle \({\mathcal{E}}\) on \(S^{[n]} \times M'_{S,v}\), and (40 ) is given by the proven (twisted) \(D\)-equivalence conjecture [22], [33] for \(K3^{[n]}\)-type.

4 Generically defined kappa classes↩︎

Throughout this section, we assume that \((S,H)\) is a fixed \(K3\) surface of genus \(g \geq 2\) with \(\mathrm{Pic}(S)={\mathbb{Z}}H\). Let \[v = (r, mH, s) \in \widetilde{H}(S, {\mathbb{Z}}), \quad r, m, s \in {\mathbb{Z}}\] be a primitive Mukai vector with \(v^2 = 2n-2\geq 0\). The main purpose of this section is to construct generically defined kappa class; see Theorem 30.

4.1 Moduli of polarized \(K3\) surfaces↩︎

Recall the derived equivalences (39 ) and (40 ) for the fixed polarized \(K3\) surface \((S,H)\). Now we vary \((S,H)\) over the moduli stack \({\mathcal{X}}_g\). We denote \[{\mathcal{X}}^*_g \subset {\mathcal{X}}_g\] the open substack formed by polarized \(K3\) surfaces whose polarization is general with respect to the Mukai vector \(v\). Clearly all polarized \(K3\) surfaces of Picard rank \(1\) lie in \({\mathcal{X}}^*_g\). We assume that the original \(K3\) surface \((S,H)\) is represented by a point \(P \in {\mathcal{X}}^*_g\). The universal family \({\mathcal{S}}_g \to {\mathcal{X}}^*_g\) carries a relative polarization \({\mathcal{H}}\in \mathrm{Pic}({\mathcal{S}}_g)\) of genus \(g\), and we have the proper and smooth families \[\label{families} {\mathcal{S}}_g^{[n]} \to {\mathcal{X}}^*_g, \quad {\mathcal{M}}_{g,v} \to {\mathcal{X}}^*_g\tag{41}\] formed by the Hilbert schemes of points and the moduli of stable sheaves with respect to the Mukai vector \(v\) respectively. Here the Mukai vector and the polarization are defined relatively over \({\mathcal{X}}_g\) by the relative polarization \({\mathcal{H}}\); the general assumption of the polarization ensures that \({\mathcal{M}}_{g,v} \to {\mathcal{X}}^\ast_g\) is smooth with hyper-Kähler fibers. The fibers of (41 ) over \(P \in {\mathcal{X}}^*_g\) recover the hyper-Kähler varieties \(S^{[n]}\) and \(M_{S,v}\) respectively.

We may modify the family \({\mathcal{M}}_{g,v} \to {\mathcal{X}}^*_g\) to include the birational model \(M'_{S,v}\) used in (39 ) and (40 ) as a fiber.

We consider the following family given by (41 ) and Proposition 27, \[\label{family2} \Pi: {\mathcal{S}}_g^{[n]}\times_{{\mathcal{X}}^\circ} {\mathcal{M}}'_{g,v} \to {\mathcal{X}}^\circ_g,\tag{42}\] whose fiber over \(P \in {\mathcal{X}}^\circ_g\) is the product \(S^{[n]} \times M'_{S,v}\). We say that a class \[\gamma \in H^*(S^{[n]} \times M'_{S,v}, {\mathbb{Q}})\] is monodromy invariant, if it is invariant under the geometric monodromy action for the family (42 ).

The Fourier–Mukai kernel of (39 ) induces the (cohomological) kappa class \[\label{kappa} \kappa({\mathcal{E}}) \in H^*(S^{[n]} \times M'_{S,v}, {\mathbb{Q}})\tag{43}\] as in Section 2.2 which, by the Künneth formula, is completely determined by its induced (cohomological) Fourier transform \[\label{Fourier2} \mathfrak{F}:H^*(S^{[n]}, {\mathbb{Q}}) \to H^*(M'_{S,v}, {\mathbb{Q}}).\tag{44}\]

The main result of Section 4 is the following theorem, whose proof will be completed in Section 4.3.

The Fourier–Mukai transform (39 ) is constructed for any \(K3\) surface using transcendental techniques (e.g. twistor lines, hyper-holomorphic bundles, etc.). In order to apply the Franchetta properties over the moduli of polarized \(K3\) surfaces, we want to globalize it over a Zariski open subset of \({\mathcal{X}}_g\). This is achieved by a moduli approach which we discuss in the following.

4.2 Stable vector bundles on gerbes↩︎

The Brauer class \(\theta_v\) in Zhang’s Fourier–Mukai transform (39 ) is given by a class \[\label{theta} \theta_v \in H^2(M'_{S,v}, {\mathbb{Z}}/(2n-2){\mathbb{Z}}).\tag{46}\] Geometrically, this corresponds to a \(\mathbf{\mu}_{2n-2}\)-gerbe \[\mathfrak{M}'_{S,v} \to M'_{S,v}.\]

The class (46 ) is canonical up to a sign (see [16]). Therefore it can be defined as a relative class for the family \(\Pi_M: {\mathcal{M}}'_{g,v} \to {\mathcal{X}}^\circ_g\); in other words, it gives rise to a section \[\label{theta2} [\theta_v] \in H^0( {\mathcal{X}}^\circ_g, R^2{\Pi_M}_*({\mathbb{Z}}/(2n-2){\mathbb{Z}})),\tag{47}\] which yields a \(\mu_{2n-2}\)-gerbe for each fiber of \(\Pi_M: {\mathcal{M}}'_{g,v} \to {\mathcal{X}}^\circ_g\).

The next lemma shows that we can globalize these gerbes by passing to a generically finite cover.

By Lemma 31, we obtain a \(\mu_{2n-2}\)-gerbe \[\label{global95gerbe} \mathfrak{M}'^\natural_{g,v} \to {\mathcal{M}}'^{\natural}_{g,v} \to {\mathcal{X}}_g^\natural\tag{48}\] whose restriction to every fiber recovers the \(\mu_{2n-2}\)-gerbe induced by \([\theta_v]\). We also consider the family obtained from the pullback of the Hilbert schemes, \[{\mathcal{S}}^{[n]}_g \to {\mathcal{X}}^\natural_g,\] and the relative product \[\label{family3} {\mathcal{S}}^{[n]}_g \times_{{\mathcal{X}}^\natural_g} \mathfrak{M}'^\natural_{g,v} \to {{\mathcal{X}}^\natural_g}.\tag{49}\]

Assume that \(Q \in {\mathcal{X}}_g^\natural\) is a point over \(P \in {\mathcal{X}}^\circ_g\), so that the fiber over \(Q\) of the family (49 ) recovers \(S^{[n]} \times \mathfrak{M}'_{S,v}\). The families (48 ) and (49 ) play a key role in the proof of Theorem 30.

4.3 Proof of Theorem 30↩︎

The Fourier–Mukai kernel \(({\mathcal{E}}, \theta_v)\) is a twisted vector bundle on \(S^{[n]} \times M'_{S,v}\). Since the twisted vector bundle \(({\mathcal{E}}, \theta_v)\) is deformed along a twistor path from a slope stable vector bundle on a pair of Hilbert schemes, it is slope stable with respect to an ample class \[A\in \mathrm{Pic}(S^{[n]} \times M'_{S,v})_{\mathbb{Q}}.\] Equivalently, the twisted vector bundle \(({\mathcal{E}}, \theta_v)\) on \(S^{[n]} \times M'_{S,v}\) corresponds to an \(A\)-slope stable vector bundle on the \(\mu_{2n-2}\)-gerbe: \[\label{vector95bundle} {\mathcal{E}}\in \mathrm{Coh}(S^{[n]} \times \mathfrak{M}'_{S,v}).\tag{50}\] By Lemma 28, the polarization \(A\) on \(S^{[n]} \times M'_{S,v}\) can be lifted to a relative polarization \[{\mathcal{A}}\in \mathrm{Pic}\left({\mathcal{S}}^{[n]}_g \times_{{\mathcal{X}}^\natural_g} {\mathcal{M}}'^\natural_{g,v}\right)_{\mathbb{Q}}.\] Let \(\mathfrak{T}^\natural_g \to {\mathcal{X}}^\natural_g\) be the relative moduli stack of vector bundles on the fibers of (49 ) satisfying

1. they are slope stable with respect to the polarization \({\mathcal{A}}\), and

2. the kappa classes of the vector bundles are given by (?? ).

We rigidify the moduli stack with respect to the scaling automorphism \(\mathbb{C}^*\), so that it is Deligne–Mumford. For convenience, from now on we use \(\mathfrak{T}^\natural_g\) to denote this Deligne–Mumford stack. Clearly the vector bundle (50 ) lies in the moduli stack \(\mathfrak{T}^\natural_g\) over the point \(Q \in {\mathcal{X}}^\natural_g\): \[[{\mathcal{E}}] \in \mathfrak{T}^\natural_g \mapsto Q \in {\mathcal{X}}^\natural_g.\] In particular, the moduli stack \(\mathfrak{T}^\natural_g\) is non-empty.

5 Franchetta properties and the proof of Theorem 8↩︎

We complete the proof of Theorem 8 by applying the Fourier transform package of Section 2 to the universal kappa class given by Theorem 30.

5.1 Franchetta properties↩︎

Assume \(g\geq 2\). We consider the universal family \({\mathcal{S}}_g \to {\mathcal{X}}_g\) over the moduli stack of \(K3\) surfaces of genus \(g\). Recall the generically defined cycles \[\mathrm{GDCH}^*(S^n):= \mathrm{Im}\left( \mathrm{CH}^*({{\mathcal{S}}^n_g}_{/{\mathcal{X}}_g}) \to \mathrm{CH}^*(S^n) \right)\] on the \(n\)-th product of any fiber \(S\subset {\mathcal{S}}_g\). In fact, by a standard spreading out argument, a class in \(\mathrm{CH}^*(S^n)\) is generically defined if it is given by the restriction of a class on the \(n\)-th relative product over a Zariski dense open subset \({\mathcal{U}}\subset {\mathcal{X}}_g\). The \(n\)-th Franchetta property holds for \(K3\) surfaces of genus \(g\) if the cycle class map is injective restricting to the subgroup \[\mathrm{GDCH}^*(S^n) \subset \mathrm{CH}^*(S^n).\] The following conjecture, which generalizes O’Grady’s generalized Franchetta conjecture [9], was studied in [13], [14] and stated explicitly in [15].

We can consider similarly generically defined cycles on (products of) nonsingular moduli spaces of stable sheaves on a \(K3\) surface, which are defined relatively over a Zariski open subset of \({\mathcal{X}}_g\). The following result given by [14] reduces the Franchetta properties for moduli spaces of stable sheaves to the Franchetta properties for products of \(K3\) surfaces.

5.2 Generically defined Fourier transforms↩︎

We fix \((S,H)\) to be a \(K3\) surface of genus \(g\) with \(\mathrm{Pic}(S) = {\mathbb{Z}}H\). Under the notation of Section 4.1, the pair \((S,H)\) is represented by a moduli point \(P:= [(S,H)] \in {\mathcal{X}}^\circ_g\).

Recall the hyper-Kähler varieties \(M_{S,v}, M'_{S,v}, S^{[n]}\) with \[\dim M_{S,v} = \dim M'_{S,v} = 2n,\] and the Fourier–Mukai transform (39 ). We define the generically defined kappa class \[\label{GD95kappa} \kappa_v^{\mathrm{GD}} \in \mathrm{GDCH}^*(S^{[n]} \times M'_{S,v})\tag{53}\] as the restriction of the class \(\mathfrak{K}^{\mathrm{univ}}\) in Theorem 30 to the fiber over \(P \in {\mathcal{X}}^\circ_g\). We have \[\label{coh95match} \kappa({\mathcal{E}}) = \kappa_v^{\mathrm{GD}} \in H^*(S^{[n]} \times M'_{S,v}, {\mathbb{Q}}).\tag{54}\] Following Section 2.2, the class (53 ) induces the generically defined Fourier transforms \[\label{Fourier95GD} \mathfrak{F}_v^{\mathrm{GD}} \in \mathrm{GDCH}^*(S^{[n]} \times M'_{S,v}), \quad \mathfrak{F}_v^{\mathrm{GD,-1}} \in \mathrm{GDCH}^*(M'_{S,v} \times S^{[n]})\tag{55}\] where we replace \(\kappa({\mathcal{E}})\) by \(\kappa_v^{\mathrm{GD}}\). Similarly, we define the normalizations \(\widetilde{\mathfrak{F}}_v^{\mathrm{GD}}, {{\widetilde{\mathfrak{F}}_v}^{\mathrm{GD}, -1} }\) and the convolution class \[\mathfrak{C}^{\mathrm{GD}}_v \in \mathrm{GDCH}^*(S^{[n]} \times S^{[n]} \times S^{[n]} )\] following Section 2.3.

The following proposition shows that Conjectures 11 and 14 for the Fourier transform package \((\mathfrak{F}^{\mathrm{GD}}_v, \mathfrak{F}^{\mathrm{GD},-1}_v, \mathfrak{C}^{\mathrm{GD}}_v)\) is a consequence of certain special cases of Conjecture 35.

Assuming Conjecture 35 for \(g\) and \(2n\), Proposition 38(a) (together with Proposition 13) yields generically defined Fourier-stable Chow–Künneth decompositions \[\label{CK1} h(M'_{S,v}) = \bigoplus_{k=0}^{2n} h_{2k}(M'_{S,v}), \quad h(S^{[n]}) = \bigoplus_{i=0}^{2n} h_{2k}(S^{[n]})\tag{56}\] with projectors given by the classes (55 ) as in (10 ). By [15], the existence and the uniqueness of generically defined Chow–Künneth decompositions are known for \(M'_{S,v}\) and \(S^{[n]}\) if we assume the \(2n\)-th Franchetta property. Here the explicit forms of the projectors are needed to deduce the Fourier stability, which is crucial in the proof of Theorem 8.

From now on, when we assume that Conjecture 35 holds for \(g\) and \(2n\), we work with the (canonical) generically defined Chow–Künneth decompositions 56 without specifying the projectors.

Further assuming Conjecture 35 for \(g\) and \(3n\), Proposition 38(b) (together with Proposition 16) shows that the graded algebra object \[\bigoplus_{k=0}^{2n} h_{2k}(M'_{S,v}), \quad \cup: h_{2i}(M'_{S,v}) \times h_{2j}(M'_{S,v}) \to h_{2i+2j}(M'_{S,v})\] is isomorphic through \(\widetilde{{\mathfrak{F}}}_v^{\mathrm{GD}}\) to the graded algebra object \[\begin{gather} \bigoplus_{k=0}^{2n} h_{4n-2k}(S^{[n]})(2n-2k), \\ \ast: h_{4n-2i}(S^{[n]})(2n-2i) \times h_{4n-2j}(S^{[n]})(2n-2j) \to h_{4n-2i-2j}(S^{[n]})(2n-2i-2j) \end{gather}\] with respect to the generically defined convolution class \(\mathfrak{C}^{\mathrm{GD}}_v\).

Next, we consider a generically defined version of the Fourier autoduality in Section 3.2 for the Hilbert scheme \(S^{[n]}\). Any ample class \(L \in \mathrm{CH}^1(S^{[n]})\) of BBF norm \(2d\) is obviously generically defined. Following 32 and 33 , we then define \[e_L^{\mathrm{GD}} \in \mathrm{GDCH}^{2n + 1}(S^{[n]} \times S^{[n]}), \quad f_L^{\mathrm{GD}} \in \mathrm{GDCH}^{2n - 1}(S^{[n]} \times S^{[n]}),\] the latter using the generically defined Fourier transforms \(\mathfrak{F}_v^{\mathrm{GD}}, \mathfrak{F}_v^{\mathrm{GD}, -1}\). These in turn yield the generically defined Fourier self-transforms \[{\mathfrak{F}}_L^{\mathrm{GD}} := \exp(e_L^{\mathrm{GD}})\exp(-f_L^{\mathrm{GD}})\exp(e_L^{\mathrm{GD}}) \in \mathrm{GDCH}^*(S^{[n]} \times S^{[n]}), \quad {\mathfrak{F}}_L^{\mathrm{GD}, -1} \in\mathrm{GDCH}^*(S^{[n]} \times S^{[n]}),\] the normalizations \(\widetilde{{\mathfrak{F}}}_L^{\mathrm{GD}}, \widetilde{{\mathfrak{F}}}_L^{\mathrm{GD}, -1}\), and the convolution class \[\mathfrak{C}_L^{\mathrm{GD}} \in \mathrm{GDCH}^*(S^{[n]} \times S^{[n]} \times S^{[n]}).\]

The next proposition concerning \(({\mathfrak{F}}_L^{\mathrm{GD}}, {\mathfrak{F}}_L^{\mathrm{GD}, -1}, \mathfrak{C}_L^{\mathrm{GD}})\) is parallel to Proposition 38.

Assuming Conjecture 35 for \(g\) and \(2n\), Proposition 39(a) implies that the decomposition of \(h(S^{[n]})\) in 56 is also stable under the generically defined Fourier self-transforms. Further assuming Conjecture 35 for \(g\) and \(3n\), Proposition 39(b) shows that the graded algebra object \[\bigoplus_{k=0}^{2n} h_{2k}(S^{[n]}), \quad \cup: h_{2i}(S^{[n]}) \times h_{2j}(S^{[n]}) \to h_{2i+2j}(S^{[n]})\] is isomorphic through \(\widetilde{{\mathfrak{F}}}_L^{\mathrm{GD}}\) to the graded algebra object \[\begin{gather} \bigoplus_{k=0}^{2n} h_{4n-2k}(S^{[n]})(2n-2k), \\ \ast: h_{4n-2i}(S^{[n]})(2n-2i) \times h_{4n-2j}(S^{[n]})(2n-2j) \to h_{4n-2i-2j}(S^{[n]})(2n-2i-2j) \end{gather}\] with respect to the convolution class \(\mathfrak{C}^{\mathrm{GD}}_L = \mathfrak{C}^{\mathrm{GD}}_v\).

5.3 Proof of Theorem 8↩︎

The proof is completely parallel to the proof of Theorem 7 in Section 3.3, where it suffices to relate the motives of \(S^{[n]}\) and \(M'_{S, v}\).

We first assume that Conjecture 35 holds for \(g\) and \(2n\). Recall the generically defined Chow–Künneth decompositions 56 . We construct an isomorphism of Chow motives \[h_{2k}(S^{[n]}) \simeq h_{2k}(M'_{S,v})\] as follows.

By Propositions 38(a) and 39(a) (together with Proposition 13(b)), we obtain mutually inverse isomorphisms of Chow motives \[\begin{gather} h_{4n-2k}(S^{[n]}) (2n-2k)\xrightleftharpoons[~~~\widetilde{\mathfrak{F}}^{\mathrm{GD},-1}_{v, 4n-2k}~~~]{~~~\widetilde{\mathfrak{F}}^{\mathrm{GD}}_{v, 2k}~~~} h_{2k}(M'_{S,v}), \tag{57} \\ h_{4n-2k}(S^{[n]}) (2n-2k) \xrightleftharpoons[~~~\mathfrak{F}^{\mathrm{GD},-1}_{L, 4n-2k}~~~]{~~~\mathfrak{F}^{\mathrm{GD}}_{L, 2k}~~~} h_{2k}(S^{[n]}).\tag{58} \end{gather}\] Their composition yields mutually inverse isomorphisms of Chow motives \[h_{2k}(S^{[n]}) \xrightleftharpoons[~~~\mathfrak{F}^{\mathrm{GD}}_{L,2k}\circ\widetilde{\mathfrak{F}}^{\mathrm{GD},-1}_{v, 4n-2k}~~~]{ ~~~\widetilde{\mathfrak{F}}_{v, 2k}^{\mathrm{GD}}\circ \mathfrak{F}^{\mathrm{GD},-1}_{L, 4n-2k}~~~} h_{2k}(M'_{S,v})\] which completes the proof of Theorem 8(a).

If we further assume that Conjecture 35 holds for \(g\) and \(3n\), then the desired isomorphism respecting the cup-product is obtained as the composition of the isomorphism \[\left(\bigoplus_{k = 0}^{2n}h_{2k}(M'_{S, v}), \cup\right) \simeq \left(\bigoplus_{k = 0}^{2n}h_{4n-2k}(S^{[n]})(2n-2k), \ast\right)\] shown in Proposition 38(b), and the isomorphism \[\left(\bigoplus_{k = 0}^{2n}h_{2k}(S^{[n]}), \cup\right) \simeq \left(\bigoplus_{k = 0}^{2n}h_{4n-2k}(S^{[n]})(2n-2k), \ast\right)\] shown in Proposition 39(b). The proof of Theorem 8(b) is also complete. 0◻

References↩︎

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1. In fact, if one of the varieties is hyper-Kähler, the derived equivalence condition forces the other to be hyper-Kähler of the same dimension by [3].↩︎

2. For different choices of \(v\)-generic polarizations, we get birational hyper-Kähler varieties. We are mainly concerned with Chow motives in this paper; by a result of Reiss [7] the Chow motive of a hyper-Kähler variety, as an algebra object, is not sensitive to the birational model.↩︎

3. We refer to [6], especially Section 3 and Convention 5.16 therein for a careful analysis of the sign. As the precise sign is irrelevant for our purpose, we write “\(\pm\)” in the sequel to simplify the exposition.↩︎