Hyper-Kähler Manifolds

Abstract

The aim of this introductory survey is to acquaint the reader with important objects in complex algebraic geometry: K3 surfaces and their higher-dimensional analogs, hyper-Kähler manifolds. These manifolds are interesting from several points of view: dynamical (some have interesting automorphism groups), arithmetical (although we will not say anything on this aspect of the theory), and geometric. It is also one of those rare cases where the Torelli theorem allows for a powerful link between the geometry of these manifolds and lattice theory. We do not prove all the results that we state. Our aim is more to provide, for specific families of hyper-Kähler manifolds (which are projective deformations of punctual Hilbert schemes of K3 surfaces), a panorama of results about projective embeddings, automorphisms, moduli spaces, period maps and domains, rather than a complete reference guide. These results are mostly not new, except perhaps those of Appendix B (written with E. Macrì), where we give in Theorem B.1 an explicit description of the image of the period map for these polarized manifolds.

Appendices

Pell-Type Equations

We state or prove a few elementary results on some diophantine equations.

Given nonzero integers e and t with $e>0$, we denote by ${\mathscr {P}}_e(t)$ the Pell-type equation

$$\begin{aligned} a^2-eb^2=t, \end{aligned}$$

(31)

where a and b are integers (the usual Pell equation is the case $t=1$). A solution (a, b) of this equation is called positive if $a>0$ and $b>0$. If e is not a perfect square, (a, b) is a solution if and only if the norm of $a+b\sqrt{e}$ in the quadratic number field $\textbf{Q}(\sqrt{e})$ is t.

A positive solution with minimal a is called the minimal solution; it is also the positive solution (a, b) for which the “slope” $b/a=\sqrt{\frac{1}{e }-\frac{t}{ea^2}}$ is minimal when $t>0$, maximal when $t<0$. Since the function $x\mapsto x+\frac{t}{x}$ is increasing on the interval $(\sqrt{|t|},+\infty )$, the minimal solution is also the one for which the real number $a+b\sqrt{e}$ is $>\sqrt{|t|}$ and minimal.

Assume that e is not a perfect square. There is always a minimal solution $(a_1,b_1)$ to the Pell equation ${\mathscr {P}}_e(1)$ and if $x_1:=a_1+b_1\sqrt{e}$, all the solutions of the equation ${\mathscr {P}}_e(1)$ correspond to the “powers” $\pm x_1^n$, for $n\in \textbf{Z}$, in $\textbf{Z}[\sqrt{e}]$.

If an equation ${\mathscr {P}}_e(t)$ has a solution (a, b), the elements $\pm (a+b\sqrt{e})x_1^n$ of $\textbf{Z}[\sqrt{e}]$, for $n\in \textbf{Z}$, all give rise to solutions of ${\mathscr {P}}_e(t)$ which are said to be associated with (a, b). The set of all solutions of ${\mathscr {P}}_e(t)$ associated with each other form a class of solutions. A class and its conjugate (generated by $(a,-b)$) may be distinct or equal.

Assume that t is positive but not a perfect square. Let (a, b) be a solution to the equation ${\mathscr {P}}_e(t)$ and set $x:=a+b\sqrt{e}$. If $x=\sqrt{t}$, we have ${\bar{x}}=t/x=x$, hence $b=0$; this contradicts our hypothesis that t is not a perfect square, hence $x\ne \sqrt{t}$.

A class of solutions to the equation ${\mathscr {P}}_e(t)$ and its conjugate give rise to real numbers which are ordered as follows⁵⁵

$$\begin{aligned} \cdots<x_tx_1^{-2}\le {\bar{x}}_tx_1^{-1}<x_tx_1^{-1}\le {\bar{x}}_t<\sqrt{t}<x_t\le {\bar{x}}_tx_1<x_tx_1\le {\bar{x}}_tx_1^2<x_tx_1^2<\cdots \nonumber \\ \end{aligned}$$

(32)

where $x_t=a_t+b_t\sqrt{e}$ corresponds to a solution which is minimal in its class and ${\bar{x}}_t$ is its conjugate. We have $x_t= {\bar{x}}_tx_1$ if and only if the class of the solution $(a_t,b_t)$ is associated with its conjugate. The inequality $x_t\le {\bar{x}}_tx_1$ implies $a_t\le a_ta_1-eb_tb_1$, hence

$$\begin{aligned} \frac{b_t}{a_t}\le \frac{a_1-1}{eb_1}=\frac{a_1^2-a_1}{ea_1b_1}=\frac{b_1}{a_1}-\frac{a_1-1}{ea_1b_1}< \frac{b_1}{a_1}. \end{aligned}$$

(33)

This inequality between slopes also holds for the solution $\bar{x}_tx_1$ in the conjugate class (because $({\bar{x}}_tx_1)x_1^{-1}=\bar{x}_t<\sqrt{t}$) but for no other positive solutions in these two classes.

We will need the following variation on this theme. We still assume that t is positive and not a perfect square. Let $(a'_t,b'_t)$ be the minimal positive solution to the equation ${\mathscr {P}}_{4e}(t)$ (if it exists) and set $x'_t:= a'_t+b'_t\sqrt{4e}$.⁵⁶

If $b_1$ is even, $(a_1,b_1/2)$ is the minimal solution to the equation ${\mathscr {P}}_{4e}(1)$ and we obtain from (33) the inequality

$$\begin{aligned} \frac{b'_t}{a'_t} < \frac{b_1}{2a_1}. \end{aligned}$$

(34)

As above, the only other solution (among the class of $(a'_t,b'_t)$ and its conjugate) for which this inequality between slopes also holds is the “next” solution, which corresponds to ${\bar{x}}'_tx_1$.

If $b_1$ is odd, $(a'_1,b'_1)= (2eb_1^2+1,a_1b_1)$ is the minimal solution to the equation ${\mathscr {P}}_{4e}(1)$, so that $x'_1=x_1^2$. The solutions associated with $(a'_t,b'_t)$ correspond to the $\pm x'_tx_1^{2n}$, $n\in \textbf{Z}$. We still get from (33) the inequality

$$\begin{aligned} \frac{b'_t}{a'_t} \le \frac{a'_1-1}{4eb'_1}=\frac{(2eb_1^2+1)-1}{4ea_1b_1}= \frac{b_1}{2a_1} \end{aligned}$$

(35)

which is in fact strict.⁵⁷ No other solution associated with $(a'_t,b'_t)$ or its conjugate satisfies this inequality between slopes.

The following criterion [88, Theorem 110] ensures that in some cases, any two classes of solutions are conjugate, so that the discussion above applies to all solutions.

Lemma A.1

Let u be a positive integer which is either prime or equal to 1, let e be a positive integer which is not a perfect square, and let $\varepsilon \in \{-1,1\}$. If the equation ${\mathscr {P}}_e(\varepsilon u)$ is solvable, it has one or two classes of solutions according to whether u divides 2e or not; if there are two classes, they are conjugate.

We now extend slightly the class of equations that we are considering: if $e_1$ and $e_2$ are positive integers, we denote by ${\mathscr {P}}_{e_1,e_2}(t)$ the equation

$$\begin{aligned} e_1 a^2 - e_2 b^2 =t. \end{aligned}$$

Given an integral solution (a, b) to ${\mathscr {P}}_{e_1,e_2}(t)$, we obtain a solution $(e_1a,b)$ to ${\mathscr {P}}_{e_1e_2}(e_1t)$. If $e_1$ is square free, all the solutions to ${\mathscr {P}}_{e_1e_2}(e_1t)$ arise in this way; in general, all the solutions whose first argument is divisible by $e_1$ arise. A positive solution (a, b) to ${\mathscr {P}}_{e_1,e_2}(t)$ is called minimal if a is minimal. If $e_1e_2$ is not a perfect square, we say that the solutions (a, b) and $(a',b')$ of ${\mathscr {P}}_{e_1,e_2}(t)$ are associated if $(e_1a,b)$ and $(e_1a',b')$ are associated solutions of ${\mathscr {P}}_{e_1e_2}(e_1t)$.⁵⁸

Let $\varepsilon \in \{-1,1\}$; assume that the equation ${\mathscr {P}}_{e_1,e_2}(\varepsilon )$ has a solution $(a_\varepsilon ,b_\varepsilon )$ and set $x_\varepsilon := e_1a_\varepsilon +b_\varepsilon \sqrt{e_1e_2}$. Let $(a,b)\in \textbf{Z}^2$ and set $x:=e_1a+b\sqrt{e_1e_2}$. We have

$$\begin{aligned} xx_\varepsilon =e_1(e_1aa_\varepsilon +e_2bb_\varepsilon +(a_\varepsilon b+ab_\varepsilon )\sqrt{e_1e_2} )=:e_1y, \end{aligned}$$

where $x{\bar{x}}=e_1\varepsilon y{\bar{y}}$. In particular, x is a solution to the equation ${\mathscr {P}}_{e_1 e_2}(e_1t)$ if and only if y is a solution to the equation ${\mathscr {P}}_{e_1 e_2}( \varepsilon t)$. This defines a bijection between the set of solutions to the equation ${\mathscr {P}}_{e_1, e_2}(t)$ and the set of solutions to the equation ${\mathscr {P}}_{e_1 e_2}(\varepsilon t)$ (the inverse bijection is given by $y\mapsto x=\varepsilon {\bar{x}}_\varepsilon y$).

The proof of the following lemma is left to the reader.

Lemma A.2

Let $e_1$ and $e_2$ be positive integers. Assume that for some $\varepsilon \in \{-1,1\}$, the equation ${\mathscr {P}}_{e_1,e_2}(\varepsilon )$ is solvable and let $(a_\varepsilon ,b_\varepsilon )$ be its minimal solution. Then, $e_1e_2$ is not a perfect square and the minimal solution of the equation ${\mathscr {P}}_{e_1e_2}(1)$ is $(e_1a_\varepsilon ^{2}+e_2b_\varepsilon ^2,2a_\varepsilon b_\varepsilon )$, unless $e_1= \varepsilon = 1$ or $e_2=-\varepsilon =1$,

We now assume $t<0$ (the discussion is entirely analogous when $t>0$ and leads to the reverse inequality in (38)) and $-t$ is not a perfect square. Let $(a_t,b_t)$ be the minimal solution to the equation ${\mathscr {P}}_{e_1,e_2}(t )$ and set $x_t:= e_1a_t+b_t\sqrt{e_1e_2}$, solution to the equation ${\mathscr {P}}_{e_1 e_2}(e_1t)$ which is minimal among all solutions whose first argument is divisible by $e_1$. We have as in (33) the inequalities

$$\begin{aligned} \cdots<x_tx_1^{-1}\le -{\bar{x}}_t<\sqrt{-e_1t}<x_t\le -\bar{x}_tx_1<x_tx_1\le \cdots \end{aligned}$$

(36)

with $x_1:= \frac{1}{e_1}x_\varepsilon ^2$ by Lemma A.2. The increasing correspondence with the solutions to the equation ${\mathscr {P}}_{e_1 e_2}(\varepsilon t)$ that we described above maps $x_tx_1^{-1}$ to $\frac{1}{e_1}x_tx_1^{-1}x_\varepsilon =x_tx_\varepsilon ^{-1}$ and $-{\bar{x}}_t$ to $-\frac{1}{e_1}{\bar{x}}_tx_\varepsilon$. Since the product of these positive numbers is $-t$, we have

$$\begin{aligned} \cdots \le x_tx_\varepsilon ^{-1}<\sqrt{-t}< -\frac{1}{e_1}\bar{x}_tx_\varepsilon \le \cdots \end{aligned}$$

(37)

In particular, $-\frac{1}{e_1}{\bar{x}}_tx_\varepsilon$ corresponds to a positive solution, hence

$$\begin{aligned} \frac{a_t}{b_t} < \frac{a_\varepsilon }{b_\varepsilon }. \end{aligned}$$

(38)

Moreover, $(a_t,b_t)$ is the only positive solution to the equation ${\mathscr {P}}_{e_1,e_2}(t )$ among those appearing in (36) that satisfies this inequality. When |t| is prime, Lemma A.1 implies that any two classes of solutions are conjugate, hence all positive solutions appear in (36), and (38) holds for only one positive solution to the equation ${\mathscr {P}}_{e_1,e_2}(t )$.

Finally, one small variation: we assume that $(a_\varepsilon ,b_\varepsilon )$ is the minimal solution to the equation ${\mathscr {P}}_{e_1,e_2}(\varepsilon )$ but that $(a_t,b_t)$ is the minimal solution to the equation ${\mathscr {P}}_{e_1,4e_2}(t )$. Then we have

$$\begin{aligned} \frac{a_t}{2b_t} < \frac{a_\varepsilon }{b_\varepsilon } \end{aligned}$$

(39)

and, when |t| is prime, $(a_t,b_t)$ is the only solution that satisfies that inequality. The proof is exactly the same: since $2a_\varepsilon b_\varepsilon$ is even, $x_1$ still corresponds to the minimal solution to the equation ${\mathscr {P}}_{4e_1 e_2}(1)$; in (36), we have solutions to the equation ${\mathscr {P}}_{4e_1 e_2}(e_1t)$ and in (37), we have solutions to the equation ${\mathscr {P}}_{e_1 e_2}(\varepsilon t)$, but this does not change the reasoning.

The Image of the Period Map (with E. Macrì)

In this second appendix, we generalize Theorem 3.32 in all dimensions. Recall the set up: a polarization type $\tau$ is the $O(\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}})$-orbit of a primitive element $h_\tau$ of $\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}$ with positive square and ${}^m\!{\mathscr {M}}_\tau$ is the moduli space for hyper-Kähler manifolds of type ${{\,\mathrm{{K3}}\,}}^{[m]}$ with a polarization of type $\tau$. There is a period map

$$\begin{aligned} \wp _\tau :{}^m\!\!{\mathscr {M}}_\tau \longrightarrow {\mathscr {P}}_\tau ={\widehat{O}}({\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}} ,h_\tau )\backslash \Omega _{h_\tau }. \end{aligned}$$

The goal of this appendix is to prove the following result.

Theorem B.1

(Bayer, Debarre–Macrì, Amerik–Verbitsky) Assume $m\ge 2$. Let $\tau$ be a polarization type. The image of the restriction of the period map $\wp _{\tau }$ to any component of the moduli space ${}^m\!{\mathscr {M}}_\tau$ is the complement of a finite union of explicit Heegner divisors.

The procedure for listing these divisors will be explained in Remark B.6. The proof of the theorem is based on the description of the nef cone for hyper-Kähler manifolds of type ${{\,\mathrm{{K3}}\,}}^{[m]}$ in [9, 78] (another proof follows from [4]). We start by revisiting these results. Recall (see (12)) that the (unimodular) extended K3 lattice is

$$\begin{aligned} {\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}:= U^{\oplus 4}\oplus E_8(-1)^{\oplus 2}. \end{aligned}$$

Let X be a projective hyper-Kähler manifold of type $K3^{[m]}$. We defined in Sect. 3.7.1 its Markman–Mukai lattice ${{\widetilde{\Lambda }}}_X$, isometric to ${\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}$ and endowed with a weight-2 Hodge structure. It comes with a canonical morphism of Hodge structures $\theta _X:H^2(X,\textbf{Z}) \hookrightarrow {\widetilde{\Lambda }}_{X}$ and a generator ${{\textbf{v}}}_X$ (of type (1, 1)) of $\theta _X(H^2(X,\textbf{Z}))^\bot$ that satisfies ${{\textbf{v}}}_X^2=2m-2$. We set

$$\begin{aligned} {\widetilde{\Lambda }}_{\text {alg},X} := {{\widetilde{\Lambda }}}_X \cap {{\widetilde{\Lambda }}}_X^{ 1,1}, \end{aligned}$$

so that ${{\,\textrm{NS}\,}}(X)=\theta _X^{-1}({\widetilde{\Lambda }}_{\text {alg},X})$. Finally, we set

$$\begin{aligned} {\mathscr {S}}_X := \{ {{\textbf{s}}}\in {\widetilde{\Lambda }}_{\text {alg},X}\mid {{\textbf{s}}}^2\ge -2 \text { and } 0\le {{\textbf{s}}}\cdot {{\textbf{v}}}_X\le {{\textbf{v}}}_X^2/2 \}\smallsetminus \{0\}. \end{aligned}$$

The hyperplanes $\theta _X^{-1}({{\textbf{s}}}^\perp )\subset {{\,\textrm{NS}\,}}(X)\otimes \textbf{R}$, for ${{\textbf{s}}}\in {\mathscr {S}}_X$, are locally finite in the positive cone ${{\,\textrm{Pos}\,}}(X)$. The dual statement of [9, Theorem 1] is then the following (see also [11, Theorem 12.1]).

Theorem B.2

Let X be a projective hyper-Kähler manifold of type $K3^{[m]}$. The ample cone of X is the connected component of

$$\begin{aligned} {{\,\textrm{Pos}\,}}(X) \smallsetminus \bigcup _{{{\textbf{s}}}\in {\mathscr {S}}_X} \theta _X^{-1}({{\textbf{s}}}^\perp ) \end{aligned}$$

(40)

that contains the class of an ample divisor.

Note that changing ${{\textbf{v}}}_X$ into $-{{\textbf{v}}}_X$ changes ${\mathscr {S}}_X$ into $-{\mathscr {S}}_X$, but the set in (40) remains the same.

Example B.3

Let (S, L) be a polarized K3 surface of degree 2e and let r be a positive integer. Recall from Remark 4.5 that the moduli space $X:={\mathscr {M}}(r,L,r)$, when smooth, is a hyper-Kähler manifold of type ${{\,\mathrm{{K3}}\,}}^{[m]}$, where $m:=1-r^2+e$, and carries a class H of square 2. In fact, the vector ${{\textbf{v}}}_X$ above is $(r,-r,L)\in U\oplus {\Lambda }_{{{\,\mathrm{{K3}}\,}}}={\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}$ and H is $\theta _X^{-1}((1,1,0))$. Consider the vector ${{\textbf{s}}}:= (-1,1,0)\in {\widetilde{\Lambda }}_{\text {alg},X}$. We have ${{\textbf{s}}}^2=-2$, ${{\textbf{s}}}\cdot {{\textbf{v}}}_X=2r$, and $H\in \theta _X^{-1}({{\textbf{s}}}^\perp )$. When $2r\le {{\textbf{v}}}_X^2/2= m-1$, that is, when $e\ge r^2+2r$, the class ${{\textbf{s}}}$ is in ${\mathscr {S}}_X$, hence, by the theorem, H is not ample on X.

Assume now ${{\,\textrm{Pic}\,}}(S)=\textbf{Z}L$, so that the Picard number of X is 2. When $r^2<e\le r^2+2r-1$, the class H is ample on X ([93, Corollary 4.15] and Remark 4.5).

It can actually be shown that for all $e>r^2$, the class H is nef on X.⁵⁹ However, we do not know the nef cone except when $r=1$, where $X=S^{[e]}$ and everything is described in Example 4.11 (see also Example 4.6 for the case $e=r^2+1$). One can only say that

• when $r^2<e\le r^2+2r-1$, the manifold X carries a biregular nontrivial involution (Remark 4.5) that interchanges the two extremal rays of ${{\,\textrm{Nef}\,}}(X)$;⁶⁰

• when $e\ge r^2+2r$, one ray is spanned by H, the other is rational by [99, Theorem 1.3(1)].

We rewrite Theorem B.2 in terms of the existence of certain rank-2 lattices in the Néron–Severi group as follows.

Proposition B.4

Let $m\ge 2$. Let X be a projective hyper-Kähler manifold of type $K3^{[m]}$ and let $h_X\in {{\,\textrm{NS}\,}}(X)$ be a primitive class such that $h_X^2>0$. The following conditions are equivalent:

• (i) there exist a projective hyper-Kähler manifold Y of type $K3^{[m]}$, an ample primitive class $h_Y\in {{\,\textrm{NS}\,}}(Y)$, and a Hodge isometry $g:H^2(X,\textbf{Z}) {{\,\mathrm{{\mathop {\rightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}H^2(Y,\textbf{Z})$ such that $[\theta _X]=[\theta _Y\circ g]$ and $g(h_X)=h_Y$;

• (ii) there are no rank-2 sublattices $L_X\subset {{\,\textrm{NS}\,}}(X)$ such that

  • $h_X\in L_X$, and

  • there exist integers $0\le k \le m-1$ and $a\ge -1$, and $\kappa _X\in L_X$, such that

    $$\begin{aligned} \kappa _X^2 = 2(m-1) ( 4 (m-1) a -k^2 ), \qquad \kappa _X\cdot h_X =0, \qquad \frac{\theta _X(\kappa _X) + k {{\textbf{v}}}_X}{2(m-1)} \in {\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}. \end{aligned}$$

Again, if one changes ${{\textbf{v}}}_X$ into $-{{\textbf{v}}}_X$, one needs to take $-\kappa _X$ instead of $\kappa _X$.

Proof

Assume that a pair $(Y,h_Y)$ satisfies (i) but that there is a lattice $L_X$ as in (ii). The rank-2 sublattice $L_Y:=g(L_X)\subset H^2(Y,\textbf{Z})$ is contained in ${{\,\textrm{NS}\,}}(Y)$. We also let $\kappa _Y:=g(\kappa _X)\in L_Y$ and choose the generator ${{\textbf{v}}}_Y:={{\textbf{v}}}_X\in {\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}$ of $H^2(Y,\textbf{Z})^\perp$. We then have

$$\begin{aligned} \kappa _Y^2 = 2(m-1) ( 4 (m-1) a -k^2 ), \qquad \kappa _Y\cdot h_Y =0, \qquad {{\textbf{s}}}_Y:=\frac{\theta _Y(\kappa _Y) + k {{\textbf{v}}}_Y}{2(m-1)} \in {\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}} \end{aligned}$$

and

$$\begin{aligned} {{\textbf{s}}}_Y^2=2a\ge -2,\qquad 0\le {{\textbf{s}}}_Y\cdot {{\textbf{v}}}_Y =k\le m-1 =\frac{{{\textbf{v}}}_Y^2}{2},\qquad {{\textbf{s}}}_Y\cdot \theta _Y(h_Y) =0. \end{aligned}$$

Moreover, we have ${{\textbf{s}}}_Y\in {\widetilde{\Lambda }}_{\text {alg},Y}$. By Theorem B.2, $h_Y$ cannot be ample on Y, a contradiction.

Conversely, assume that there exist no lattices $L_X$ as in (ii). By [72, Lemma 6.22], there exists a Hodge isometry $g':H^2(X,\textbf{Z}){{\,\mathrm{{\mathop {\rightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}H^2(X,\textbf{Z})$ such that $[\theta _X\circ g']=[\theta _X]$ and $g'(h_X)\in \overline{{{\,\textrm{Mov}\,}}}(X)$. Since $g'(h_X)\in {{\,\textrm{NS}\,}}(X)$ and $g'(h_X)^2>0$, by [72, Theorem 6.17 and Lemma 6.22] and [41, Theorem 7], there exist a projective hyper-Kähler manifold Y of type $K3^{[m]}$, a nef divisor class $h_Y\in {{\,\textrm{Nef}\,}}(Y)$, and a Hodge isometry $g'':H^2(X,\textbf{Z}) {{\,\mathrm{{\mathop {\rightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}H^2(Y,\textbf{Z})$ such that $[\theta _X]=[\theta _Y\circ g'']$ and $g''(g'(h_X))=h_Y$. Assume that $h_Y$ is not ample. Since $h_Y^2>0$ and $h_Y$ is nef, there exists by Theorem B.2 a class ${{\textbf{s}}}_Y\in {\mathscr {S}}_Y$ such that ${{\textbf{s}}}_Y\cdot \theta _Y(h_Y)=0$. We set $a:=\frac{1}{2}{{\textbf{s}}}_Y^2$, $k:= {{\textbf{s}}}_Y\cdot {{\textbf{v}}}_Y$, and

$$\begin{aligned} \kappa _Y:= \theta _Y^{-1}\left( 2(m-1) {{\textbf{s}}}_Y - k {{\textbf{v}}}_Y \right) \in \theta _Y^{-1} ( {\widetilde{\Lambda }}_{\text {alg},Y} ) = {{\,\textrm{NS}\,}}(Y).\end{aligned}$$

Let $L_Y$ be the sublattice of $H^2(Y,\textbf{Z})$ generated by $h_Y$ and $\kappa _Y$. Then $L_X:=g^{-1}(L_Y)$ satisfies the conditions in (ii), a contradiction. $\square$

When $m-1$ is a prime number, this can be written only in terms of $H^2(X,\textbf{Z})$.

Proposition B.5

Assume that $p:=m-1$ is either 1 or a prime number. Let X be a projective hyper-Kähler manifold of type $K3^{[m]}$ and let $h_X\in {{\,\textrm{NS}\,}}(X)$ be a primitive class such that $h_X^2>0$. The following conditions are equivalent:

• (i) there exist a projective hyper-Kähler manifold Y of type $K3^{[m]}$, an ample primitive class $h_Y\in {{\,\textrm{NS}\,}}(Y)$, and a Hodge isometry $g:H^2(X,\textbf{Z}) {{\,\mathrm{{\mathop {\rightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}H^2(Y,\textbf{Z})$ such that $g(h_X)=h_Y$;

• (ii) there are no rank-2 sublattices $L_X\subset {{\,\textrm{NS}\,}}(X)$ such that

  • $h_X\in L_X$, and

  • there exist integers $0\le k \le p$ and $a\ge -1$, and $\kappa _X\in L_X$, such that

    $$\begin{aligned} \kappa _X^2 = 2p ( 4 p a -k^2 ), \qquad \kappa _X\cdot h_X =0, \qquad 2p \mid \mathop {\textrm{div}}\nolimits _{H^2(X,\textbf{Z})}(\kappa _X). \end{aligned}$$

Proof

As explained in [72, Sects. 9.1.1 and 9.1.2], under our assumption on m, there is a unique $O({\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}})$-orbit of primitive isometric embeddings $\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}\hookrightarrow {\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}$. This implies that any Hodge isometry $H^2(X,\textbf{Z}) {{\,\mathrm{{\mathop {\rightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}H^2(Y,\textbf{Z})$ commutes with the orbits $[\theta _X]$ and $[\theta _Y]$.

We can choose the embedding $\theta :\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}\hookrightarrow {\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}$ as follows. Let us fix a canonical basis (u, v) of a hyperbolic plane U in ${\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}$. We define $\theta$ by mapping the generator $\ell$ of $I_1(-2p)$ to $u-pv$. We also choose ${{\textbf{v}}}:=u+pv$ as the generator of $\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}^\perp$.

By Proposition B.4, we only have to show that given a class $\kappa \in \Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}$ with divisibility 2p and such that $\kappa ^2 = 2p ( 4p a -k^2 )$, either $\theta (\kappa )+ k{{\textbf{v}}}$ or $\theta (-\kappa )+ k{{\textbf{v}}}$ is divisible by 2p in ${\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}$. Since $\mathop {\textrm{div}}\nolimits _{H^2(X,\textbf{Z})}(\kappa )$ is divisible by 2p, we can write $\kappa =2pw+r\ell$, with $w\in U^{\oplus 3}\oplus E_8(-1)^{\oplus 2}$ and $r\in \textbf{Z}$. By computing $\kappa ^2$, we obtain the equality

$$\begin{aligned} r^2-k^2 = 2p(w^2-2a). \end{aligned}$$

In particular, $r^2-k^2$ is even, hence so are $r + k$ and $-r + k$. Moreover, p divides $r^2-k^2$, hence also $\varepsilon r + k$, for some $\varepsilon \in \{-1,1\}$. Then

$$\begin{aligned} \theta (\varepsilon \kappa ) + k{{\textbf{v}}}= \varepsilon 2p\theta (w) + (\varepsilon r + k)u + (-\varepsilon r + k)p v \end{aligned}$$

is divisible by 2p, as we wanted. $\square$

Before proving the theorem, we briefly review polarized marked hyper-Kähler manifolds of type ${{\,\mathrm{{K3}}\,}}^{[m]}$, their moduli spaces, and their periods, following the presentation in [72, Sect. 7] (see also [6, Sect. 1]).

Let ${}^m{\mathfrak {M}}$ be the (smooth, nonHausdorff, 21-dimensional) coarse moduli space of marked hyper-Kähler manifolds of type ${{\,\mathrm{{K3}}\,}}^{[m]}$ (see [45, Sect. 6.3.3]), consisting of isomorphisms classes of pairs $(X,\eta )$ such that X is a compact complex (not necessarily projective) hyper-Kähler manifold of type ${{\,\mathrm{{K3}}\,}}^{[m]}$ and $\eta :H^2(X,\textbf{Z}){{\,\mathrm{{\mathop {\rightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}$ is an isometry. We let $({}^m{\mathfrak {M}}^{\,t})_{t\in {\mathfrak {t}}}$ be the family of connected components of ${}^m{\mathfrak {M}}$. By [72, Lemma 7.5], the set ${\mathfrak {t}}$ is finite and is acted on transitively by the group $O(\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}})$.

Let $h\in \Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}$ be a class with $h^2>0$ and let $t\in {\mathfrak {t}}$. Let ${}^m{\mathfrak {M}}^{\,t,+}_{h^\perp }\subset {}^m{\mathfrak {M}}^{\,t}$ be the subset parametrizing the pairs $(X,\eta )$ for which the class $\eta ^{-1}(h)$ is of Hodge type (1, 1) and belongs to the positive cone of X, and let ${}^m{\mathfrak {M}}^{\,t,a}_{h^\perp }\subset {}^m{\mathfrak {M}}^{\,t,+}_{h^\perp }$ be the open subset where $\eta ^{-1}(h)$ is ample on X. By [72, Corollary 7.3], ${}^m{\mathfrak {M}}^{\,t,a}_{h^\perp }$ is connected, Hausdorff, and 20-dimensional.

The relation with our moduli spaces ${}^m\!{\mathscr {M}}_\tau$ is as follows. Let $\tau$ be a polarization type and let M be an irreducible component of ${}^m\!{\mathscr {M}}_\tau$. Pick a point $(X_0,H_0)$ of M and choose a marking $\eta _0:H^2(X_0,\textbf{Z}){{\,\mathrm{{\mathop {\rightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}$. If $h_0:=\eta _0( H_0 )$, the $O(\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}})$-orbit of $h_0$ is the polarization type $\tau$, and the pair $(X_0,\eta _0)$ is in ${}^m{\mathfrak {M}}^{\,t_0,a}_{h_0^\perp }$, for some $t_0\in {\mathfrak {t}}$. As in [72, (7.4)], we consider the disjoint union

$$\begin{aligned} {}^m{\mathfrak {M}}^{\,a}_{{{\overline{h}}}_0 } := \coprod _{(h,t)\in O(\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}})\cdot (h_0,t_0)} {}^m{\mathfrak {M}}^{\,t,a}_{h^\perp }. \end{aligned}$$

It is acted on by $O(\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}})$ and, by [72, Lemma 8.3], there is an analytic bijection

$$\begin{aligned} M{{\,\mathrm{{\mathop {\longrightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}{}^m{\mathfrak {M}}^{\,a}_{{{\overline{h}}}_0 }/O(\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}). \end{aligned}$$

(41)

Proof of Theorem B.1

Let M be an irreducible component of ${}^m\!{\mathscr {M}}_\tau$. Pick a point $(X_0,H_0)$ of M and choose a marking $\eta _0:H^2(X_0,\textbf{Z}){{\,\mathrm{{\mathop {\rightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}$. As above, set $h_0:=\eta _0( H_0 )\in \Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}$ and let $t_0\in {\mathfrak {t}}$ be such that the pair $(X_0,\eta _0)$ is in ${}^m{\mathfrak {M}}^{\,t_0,a}_{h_0^\perp }$. Given another point (X, H) of M, by using the bijection (41), we can always find a marking $\eta :H^2(X,\textbf{Z}){{\,\mathrm{{\mathop {\rightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}{\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}}$ for which $(X, \eta )$ is in ${}^m{\mathfrak {M}}^{\,t_0,a}_{h_0^\perp }$ and $H =\eta ^{-1}(h_0)$.

By [72, Corollary 9.10], for all $(X,\eta )\in {}^m{\mathfrak {M}}^{\,t_0}$, the primitive embeddings

$$\begin{aligned} \theta _X\circ \eta ^{-1}:{\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}}\hookrightarrow {\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}} \end{aligned}$$

are all in the same $O({\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}})$-orbit. We fix one such embedding $\theta$ and a generator ${{\textbf{v}}}\in {\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}$ of $\theta (\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}})^\perp$. As in Proposition B.4, we consider the Heegner divisors ${\mathscr {D}}_{\tau ,K}$ in the quotient ${\mathscr {P}}_\tau := {\widehat{O}}({\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}} ,h_0)\backslash \Omega _{\tau }$, where K is a primitive, rank-2, signature-(1, 1), sublattice of ${\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}}$ such that $h_0\in K$ and there exist integers $0\le k \le m-1$ and $a\ge -1$, and $\kappa \in K$ with

$$\begin{aligned} \kappa ^2 = 2(m-1) ( 4 (m-1) a -k^2 ), \qquad \kappa \cdot h_0 =0, \qquad \frac{\theta (\kappa ) + k {{\textbf{v}}}}{2(m-1)} \in {\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}. \end{aligned}$$

(42)

There are finitely many such divisors. Indeed, since the signature of K is (1, 1), we have $\kappa ^2<0$. Moreover, we have $\kappa ^2\ge 2(m-1) ( -4 (m-1) -(m-1)^2 )$, hence $\kappa ^2$ may take only finitely many values. As explained in the proof of Lemma 3.28, this implies that there only finitely many Heegner divisors of the above form.

We claim that the image of the period map

$$\begin{aligned} \wp _\tau :M\longrightarrow {\mathscr {P}}_\tau \end{aligned}$$

coincides with the complement of the union of these Heegner divisors. We first show the image does not meet these divisors. Let $(X,H)\in M$ and, as explained above, choose a marking $\eta$ such that $(X,\eta )\in {}^m{\mathfrak {M}}^{\,t_0,a}_{h_0^\perp }$, where $h_0:=\eta (H)$. If $(X,H)\in {\mathscr {D}}_{\tau ,K}$, for K as above, the lattice $K\subset {{\,\textrm{NS}\,}}(X)$ satisfies condition (ii) of Proposition B.4, which is impossible.

Conversely, take a point $x\in {\mathscr {P}}_{\tau }$. The refined period map defined in [72, (7.3)] is surjective (this is a consequence of [45, Theorem 8.1]). Hence there exists $(X,\eta _X)\in {}^m{\mathfrak {M}}^{\,t_0,+}_{h_0^\perp }$ such that $H_X:=\eta _X^{-1}(h_0)$ is an algebraic class in the positive cone of X and $(X,H_X)$ has period point x. By [45, Theorem 3.11], X is projective. We can now apply Proposition B.4: if x is outside the union of the Heegner divisors described above, there exist a projective hyper-Kähler manifold Y of type $K3^{[m]}$, an ample primitive class $H_Y$, and a Hodge isometry $g:H^2(X,\textbf{Z}) {{\,\mathrm{{\mathop {\rightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}H^2(Y,\textbf{Z})$ such that $[\theta _X]=[\theta _Y\circ g]$ and $g(H_X)=H_Y$. By [72, Theorem 9.8], g is a parallel transport operator; by [72, Definition 1.1(1)], this means that if $\eta _Y:=\eta \circ g^{-1}$, the pair $(Y,\eta _Y)$ belongs to the same connected component ${}^m{\mathfrak {M}}^{\,t_0}$ of ${}^m{\mathfrak {M}}$. Moreover, $(Y,\eta _Y )$ is in ${}^m{\mathfrak {M}}^{\,t_0,+}_{h_0^\perp }$ and since $\eta _Y^{-1}(h_0)=H_Y$ is ample, it is even in ${}^m{\mathfrak {M}}^{\,t_0,a}_{h_0^\perp }$. By (41), it defines a point of M which still has period point x. This means that x is in the image of ${\mathscr {P}}_\tau$, which is what we wanted. $\square$

Remark B.6

Let us explain how to list the Heegner divisors referred to in the statement of Theorem B.1. Fix a representative $h_\tau \in \Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}$ of the polarization $\tau$. For each $O({\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}})$-equivalence class of primitive embeddings $\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}\hookrightarrow {\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}$,⁶¹ pick a representative $\theta$ and a generator ${{\textbf{v}}}$ of $\theta (\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}})^\perp$. Let T be the saturation in ${\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}$ of the sublattice generated by $\theta (h_\tau )$ and ${{\textbf{v}}}$. By [6, Theorem 2.1], the abstract isometry class of the pair $(T,\theta (h_\tau ))$ determines a component M of ${}^m\!{\mathscr {M}}_\tau$ (and all the components are obtained in this fashion). Now list, for all integers $0\le k \le m-1$ and $a\ge -1$, all ${\widehat{O}}({\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}},h_\tau )$-orbits of primitive, rank-2, signature-(1, 1), sublattices K of ${\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}}$ such that $h_\tau \in K$ and there exist $\kappa \in K$ satisfying (42).

The image $\wp _\tau (M)$ is then the complement in ${\mathscr {P}}_\tau$ of the union of the corresponding Heegner divisors ${\mathscr {D}}_{\tau ,K}$. The whole procedure is worked out in a particular case in Example B.7 below (there are more examples in Sect. 3.11 in the case $m=2$).

A priori, the image of $\wp _{\tau }$ may be different when restricted to different components M of ${}^m\!{\mathscr {M}}_\tau$ (see [105] for more results in this direction).

Example B.7

The moduli space ${}^4\!{\mathscr {M}}_2^{(2)}$ for hyper-Kähler manifolds of type ${{\,\mathrm{{K3}}\,}}^{[4]}$ with a polarization of square 2 and divisibility 2 is irreducible (Theorem 3.5). Let us show that the image of the period map ${^4}\wp _{2}^{(2)}$ is the complement of the three irreducible Heegner divisors ${^4}{\mathscr {D}}_{2,2}^{(2)}$, ${^4}{\mathscr {D}}_{2,6}^{(2)}$, and ${^4}{\mathscr {D}}_{2,8}^{(2)}$.

We begin with the more general case where m is divisible by 4 and $p:=m-1$ is an odd prime number. We follow the recipe given in Remark B.6. The irreducibility of ${}^m\!{\mathscr {M}}_2^{(2)}$ (Theorem 3.5) means that we can fix the embedding $\theta :{\Lambda _{{{\,\mathrm{{K3}}\,}}^{[4]}}}\hookrightarrow {\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}$ and the class $h=h_\tau \in {\Lambda _{{{\,\mathrm{{K3}}\,}}^{[4]}}}$ as we like. Let us fix bases $(u_i,v_i)$, for $i\in \{1,\dots ,4\}$, for each of the four copies of U in ${\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}}$. We choose the embedding

$$\begin{aligned} \Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}} = U^{\oplus 3}\oplus E_8(-1)^{\oplus 2}\oplus I_1(-2p) \hookrightarrow U^{\oplus 4}\oplus E_8(-1)^{\oplus 2}={\widetilde{\Lambda }}_{{{\,\mathrm{{K3}}\,}}} \end{aligned}$$

given by mapping the generator $\ell$ of $I_1(-2p)$ to $u_1-pv_1$. We also set ${{\textbf{v}}}:=u_1+pv_1$ (so that ${{\textbf{v}}}^\perp =\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}$) and $h:= 2(u_2+\frac{m}{4}v_2) + \ell$.

As explained in Remark B.6 (and using Proposition B.5), the image of the period map is the complement of the Heegner divisors of the form ${\mathscr {D}}_{\tau ,K}$, where K is a primitive, rank-2, sublattice of ${\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}}$ such that $h\in K$ and there exist integers $0\le k \le p$ and $a\ge -1$, and $\kappa \in K$ with

$$\begin{aligned} \kappa ^2 = 2p( 4pa -k^2 )<0, \qquad \kappa \cdot h =0, \qquad 2p\mid \mathop {\textrm{div}}\nolimits _{{\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}}}(\kappa ). \end{aligned}$$

Since we are interested in the lattices $K^\bot$, this is equivalent, by strange duality, to looking at the lattices $K^\bot$ in $h^\bot ={\Lambda _{{{\,\mathrm{{K3}}\,}}^{[2]}}}$, where K now contains the primitive class ${{\textbf{v}}}$, of square 2p. This is a computation that was done during the proof of Proposition 3.29(2)(c): in the notation of that proof, if we write $\kappa =b\kappa _\textrm{prim}$, with $\kappa _\textrm{prim}$ primitive, and ${{\,\textrm{disc}\,}}(K^\bot )=:-2e$,

• either $(\mathop {\textrm{div}}\nolimits _{{\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}}}(\kappa _\textrm{prim}),\kappa _\textrm{prim}^2)=(1,-2e/p)$ and $p\mid e$;

• or $(\mathop {\textrm{div}}\nolimits _{{\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}}}(\kappa _\textrm{prim}),\kappa _\textrm{prim}^2)=(p,-2pe)$ and $p\not \mid e$.

In the first case, we have $2p\mid \mathop {\textrm{div}}\nolimits _{{\Lambda _{{{\,\mathrm{{K3}}\,}}^{[m]}}}}(\kappa ) =2b$ and, writing $b=2pb'$ and $e=pe'$, we obtain

$$\begin{aligned} 2p( 4pa -k^2 )=\kappa ^2 = 4p^2b^{\prime 2}\kappa _\textrm{prim}^2= 4p^2b^{\prime 2}\kappa _\textrm{prim}^2=-8peb^{\prime 2}=-8p^2e'b^{\prime 2}. \end{aligned}$$

This implies $2p\mid k$, and since $0\le k \le p$, we get $k=0$, $a=-1$, and $e'=b'=1$. This corresponds to the irreducible divisor ${^m}{\mathscr {D}}_{2,2p}^{(2)}$.

In the second case, we write similarly $b=2b'$ and we obtain

$$\begin{aligned} 2p( 4pa -k^2 )=\kappa ^2 = 4b^{\prime 2}\kappa _\textrm{prim}^2= -8peb^{\prime 2}. \end{aligned}$$

Writing $k=2k'$, we get $k^{\prime 2} -pa= eb^{\prime 2}$, with $k'>0$.

Assume now $p=3$. The only possible pairs (k, a) are then $(0,-1)$, (2, 0), and $(2,-1)$. When $(k,a)=(0,-1)$, the discussion above shows that we obtain the divisor ${^4}{\mathscr {D}}_{2,6}^{(2)}$.

When $(k,a)=(2,0)$, we obtain $e=1$, hence the divisor ${^4}{\mathscr {D}}_{2,2}^{(2)}$. When $(k,a)=(2,-1)$, we obtain $e\in \{1,4\}$, hence the extra divisor ${^4}{\mathscr {D}}_{2,8}^{(2)}$.

Example B.8

(a) The image of the period map ${}^8\!{\mathscr {M}}_2^{(2)}\rightarrow {}^8\!{\mathscr {P}}_2^{(2)}$ is the complement of ${^8}{\mathscr {D}}_{2,2}^{(2)}\cup {^8}{\mathscr {D}}_{2,4}^{(2)}\cup {^8}{\mathscr {D}}_{2,8}^{(2)}\cup {^8}{\mathscr {D}}_{2,14}^{(2)}\cup {^8}{\mathscr {D}}_{2,16}^{(2)}\cup {^8}{\mathscr {D}}_{2,18}^{(2)}\cup {^8}{\mathscr {D}}_{2,22}^{(2)}\cup {^8}{\mathscr {D}}_{2,32}^{(2)}$.

(b) The image of the period map ${}^{12}\!{\mathscr {M}}_2^{(2)}\rightarrow {}^{12}\!{\mathscr {P}}_2^{(2)}$ is the complement of ${^{12}}{\mathscr {D}}_{2,2}^{(2)}\cup {^{12}}{\mathscr {D}}_{2,6}^{(2)}\cup {^{12}}{\mathscr {D}}_{2,8}^{(2)}\cup {^{12}}{\mathscr {D}}_{2,10}^{(2)}\cup {^{12}}{\mathscr {D}}_{2,18}^{(2)}\cup {^{12}}{\mathscr {D}}_{2,22}^{(2)}\cup {^{12}}{\mathscr {D}}_{2,24}^{(2)}\cup {^{12}}{\mathscr {D}}_{2,28}^{(2)}\cup {^{12}}{\mathscr {D}}_{2,30}^{(2)}\cup {^{12}}{\mathscr {D}}_{2,32}^{(2)}\cup {^{12}}{\mathscr {D}}_{2,40}^{(2)}\cup {^{12}}{\mathscr {D}}_{2,50}^{(2)}\cup {^{12}}{\mathscr {D}}_{2,54}^{(2)}\cup {^{12}}{\mathscr {D}}_{2,72}^{(2)}$.

(c) Let p be a prime number such that $p\equiv -1\pmod 4$. The complement of the image of the period map ${}^{p+1}\!{\mathscr {M}}_2^{(2)}\rightarrow {}^{p+1}\!{\mathscr {P}}_2^{(2)}$ contains ${^{p+1}}{\mathscr {D}}_{2,2(l^2-pa)}^{(2)}$ for all $0\le l\le \frac{p-1}{2}$ and $-1\le a< l^2/p$, but none of the divisors ${^{p+1}}{\mathscr {D}}_{2,2e}^{(2)}$ for $e> \frac{(p+1)^2}{4}$.

Remark B.9

The ample cone of a projective hyper-Kähler fourfold of type $K3^{[2]}$ was already described in Theorem 3.16. This description involved only classes ${{\textbf{s}}}\in {\mathscr {S}}_X$ such that ${{\textbf{s}}}^2=-2$ and ${{\textbf{s}}}\cdot {{\textbf{v}}}_X\in \{0,1\}$. We still find these classes in higher dimensions [77, Corollary 2.9]:

• classes ${{\textbf{s}}}$ with ${{\textbf{s}}}^2=-2$ and ${{\textbf{s}}}\cdot {{\textbf{v}}}_X=0$; in the notation of Proposition B.4, we have $a=-1$, $k=0$, and $\theta _X(\kappa _X)=2(m-1){{\textbf{s}}}$, and the lattice $\textbf{Z}h\oplus \textbf{Z}\frac{1}{2(m-1)}\kappa _X$ has intersection matrix $\begin{pmatrix} h^2 &{}\quad 0 \\ 0 &{}\quad -2\end{pmatrix}$ but may or may not be primitive;

• classes ${{\textbf{s}}}$ with ${{\textbf{s}}}^2=-2$ and ${{\textbf{s}}}\cdot {{\textbf{v}}}_X=1$; in the notation of Proposition B.4, we have $a=-1$, $k=1$, and $\theta _X(\kappa _X)=2(m-1){{\textbf{s}}}-{{\textbf{v}}}_X$; the lattice $\textbf{Z}h\oplus \textbf{Z}\kappa _X$ has intersection matrix $\begin{pmatrix} h^2 &{} 0 \\ 0 &{} -2(m-1)(4m-3)\end{pmatrix}$ and may or may not be primitive.

When $m\in \{3,4\}$, complete lists of possible pairs $(\kappa _{\text {prim}}^2,\mathop {\textrm{div}}\nolimits (\kappa _{\text {prim}}))$ are given in [77, Sects. 2.2 and 2.3]:

• when $m=3$, we have

  $$\begin{aligned} (\kappa _{\text {prim}}^2,\mathop {\textrm{div}}\nolimits (\kappa _{\text {prim}}))\in \{(-2,1),(-4,2),(-4,4), (-12,2), (-36,4)\}; \end{aligned}$$

• when $m=4$, we have

  $$\begin{aligned} (\kappa _{\text {prim}}^2,\mathop {\textrm{div}}\nolimits (\kappa _{\text {prim}}))\in \{(-2,1),(-6,2),(-6,3),(-6,6),(-14,2),(-24,3),(-78,6)\}. \end{aligned}$$

Depending on m and on the polarization, not all pairs in these lists occur: in Example B.7 (where $m=4$), only the pairs $(-6,3)$, $(-2,1)$, and $(-24,3)$ do occur.

Whenever $m-1$ is a power of a prime number, the pair $(\kappa _{\text {prim}}^2,\mathop {\textrm{div}}\nolimits (\kappa _{\text {prim}}))=(-2m-6,2)$ is also in the list [77, Corollary 2.9]: in our notation, it corresponds to ${{\textbf{s}}}^2=2a=-2$ and ${{\textbf{s}}}\cdot {{\textbf{v}}}_X =k=m-1$.

When the divisibility $\gamma$ of the polarization h is 1 or 2, we list some of the Heegner divisors that are avoided by the period map.

Proposition B.10

Consider the period map

$$\begin{aligned} {}^m\wp ^{(\gamma )}_{2n}:{}^m\!\!{\mathscr {M}}_{2n}^{(\gamma )}\longrightarrow O(L_{{{\,\mathrm{{K3}}\,}}^{[m]}},h_\tau )\backslash \Omega _{\tau } \end{aligned}$$

When $\gamma =1$, the image of $\,{}^m\wp ^{(1)}_{2n}$ does not meet

• $\nu$ of the components of the hypersurface ${}^m{\mathscr {D}}_{2n,2n(m-1)}^{(1)}$, where $\nu \in \{0,1,2\}$ is the number of the following congruences that hold: $n+m\equiv 2\pmod 4$, $m\equiv 2\pmod 4$, $n\equiv 1\pmod 4$;

• one of the components of the hypersurface ${}^m{\mathscr {D}}_{2n,8n(m-1)}^{(1)}$;

• when $m-1$ is a prime power, one component of the hypersurface ${}^m{\mathscr {D}}_{2n,2(m-1)(m+3)n}^{(1)}$.

When $\gamma =2$ (so that $n+m\equiv 1\pmod 4$), the image of $\, {}^m\wp ^{(2)}_{2n}$ does not meet

• one of the components of the hypersurface ${}^m{\mathscr {D}}_{2n,2n(m-1)}^{(2)}$;

• when $m-1$ is a power of 2, one component of the hypersurface ${}^m{\mathscr {D}}_{2n,(m-1)(m+3)n/2}^{(2)}$.

Proof

No algebraic class with square $-2$ can be orthogonal to $h_\tau$ (see Remark B.9). As in the proof of Theorem 3.32, there is such a class $\kappa$ if and only if the period point belongs to the hypersurface ${}^m{\mathscr {D}}^{(\gamma )}_{2n,K}$, where K is the rank-2 lattice with intersection matrix $\bigl ({\begin{matrix}2n&{}0\\ 0&{}-2\end{matrix}}\bigr )$. This hypersurface is a component of ${}^m{\mathscr {D}}^{(\gamma )}_{2n,d}$, where $d:=|{{\,\textrm{disc}\,}}(K^\bot )|$ can be computed using the formula

$$\begin{aligned} d=\left| \frac{\kappa ^2{{\,\textrm{disc}\,}}(h_\tau ^\bot )}{s^2}\right| \end{aligned}$$

from [38, Lemma 7.5] (see (19)), where $s:=\mathop {\textrm{div}}\nolimits _{h_\tau ^\bot }(\kappa )\in \{1,2\}$.

Assume $\gamma =1$, so that $D(h_\tau ^\bot )\simeq \textbf{Z}/(2m-2)\textbf{Z}\times \textbf{Z}/2n\textbf{Z}$ (see (6)). As in the proof of Proposition 3.29, we write $\kappa =a(u-nv)+b\ell +cw$, so that $1=-\frac{1}{2}\kappa ^2= na^2+(m-1)b^2+(\frac{1}{2} w^2)c^2$, and $s =\gcd (2na,2(m-1)b,c)$ is 1 if c is odd, and 2 otherwise.

If $s=1$, we have $d=8n(m-1)$ and $\kappa _*=0$, and, by Eichler’s criterion, this defines an irreducible component of ${}^2{\mathscr {D}}_{2n,8n(m-1)}^{(1)}$ (to prove that it is nonempty, just take $a=b=0$, $c=1$, and $w^2=-2$).

If $s=2$, we have $d=2n(m-1)$ and $\kappa _*$ has order 2. More precisely,

$$\begin{aligned} \kappa _*={\left\{ \begin{array}{ll} (0,n)&{}\quad \text {if }a\text { is odd and }b\text { is even;}\\ (m-1,0)&{}\quad \text {if }a\text { is even and }b\text { is odd;}\\ (m-1,n)&{}\quad \text {if }a\text { and }b\text { are odd.} \end{array}\right. } \end{aligned}$$

Conversely, to obtain such a class $\kappa$ with square $-2$, one needs to solve the equation $na^2+(m-1)b^2+rc^2=1$, with c even and r any integer. It is equivalent to solve $na^2+(m-1)b^2\equiv 1\pmod 4$ and one checks that this gives the first item of the proposition.

Assume $\gamma =2$, so that ${{\,\textrm{disc}\,}}(h_\tau ^\bot )=n(m-1)$ (see (7)) and $d=\frac{2}{s^2}n(m-1)\in \{2n(m-1),\frac{n(m-1)}{2}\}$. As in the proof of Proposition 3.29, we may take $h_\tau ^\bot =\textbf{Z}w_1\oplus \textbf{Z}w_2\oplus M$, with $w_1:= (m-1)v+\ell$ and $w_2:=-u+\frac{n+m-1}{4}v$ and write $\kappa =aw_1+bw_2+cw$. The equality $\kappa ^2=-2$ now reads $1=a(a+b)(m-1)+b^2 \bigl (\frac{n+m-1}{4}\bigr )-c^2\bigl (\frac{1}{2} w^2\bigr )$, and again, $s=1$ if c is odd, $s=2$ if c is even.

Again, the case $s=1$ gives rise to a single component of ${}^2{\mathscr {D}}_{2n,2n(m-1)}^{(1)}$, since it corresponds to $\kappa _*=0$ (take $a=b=0$ and $c=1$).⁶²

Assume now that $m-1$ is a prime power. In that case, no algebraic class with square $-2m-6$ and divisibility 2 can be orthogonal to $h_\tau$ (see Remark B.9). Assume that there is such a class $\kappa$.

When $\gamma =1$, keeping the same notation as above, we need to solve $na^2+(m-1)b^2+(\frac{1}{2} w^2)c^2=m+3$, with a and c even. Just taking $c=2$, $a=0$, and $b=1$ works, with $d=\frac{8(m+3)n(m-1) }{\gcd (2na,2(m-1)b,c)^2}=2(m-1)(m+3)n$. In the notation of Example B.7, we may take ${{\textbf{v}}}=u_1+(m-1)v_1$, $\ell =u_1-(m-1)v_1$, $h= u_2+nv_2$, $\kappa =(m-1)(2(u_3-v_3)+\ell )$, and ${{\textbf{s}}}=u_3-v_3+u_1$.

When $\gamma =2$, we need to solve $a(a+b)(m-1)+b^2 \bigl (\frac{n+m-1}{4}\bigr )-c^2\bigl (\frac{1}{2} w^2\bigr )=m+3$, with $a(m-1)$ and c even, and $d=\frac{2(m+3)n(m-1) }{\gcd (2na,2(m-1)b,c)^2}\mid (m-1)(m+3)n/2$, so that both n and $m-1$ are even. In that case, take $c=2$, $b=0$, and $a=1$. In the notation of Example B.7, we may take ${{\textbf{v}}}=u_1+(m-1)v_1$, $\ell =u_1-(m-1)v_1$, $h= 2(u_2+\frac{n+m-1}{4}v_2)+\ell$, $\kappa =(m-1)((m-1)v_2+2(u_3-v_3)+\ell )$, and ${{\textbf{s}}}=\frac{m-1}{2}v_2+u_3-v_3+u_1$. $\square$

1.1 The Period Map for Cubic Fourfolds

Let ${\mathscr {M}}_{\text {cub}}$ be the moduli space of smooth cubic hypersurfaces in $\textbf{P}^5$. With any cubic fourfold W, we can associate its period, given by the weight-2 Hodge structure on $H^4_{\text {prim}}(W,\textbf{Z})$. The Torelli Theorem for cubic fourfolds [110] says that the period map is an open embedding. By [16, Proposition 6], there is a Hodge isometry $H^4_{\text {prim}}(W,\textbf{Z})(-1) \simeq H^2_{\text {prim}}(F(W),\textbf{Z})$. Hence, we can equivalently consider the period map for (smooth) cubic fourfolds as the period map of their (smooth) Fano varieties of lines, which are hyper-Kähler fourfolds of type ${{\,\mathrm{{K3}}\,}}^{[2]}$ with a polarization of square 6 and divisibility 2. The latter form a dense open subset in the moduli space ${}^2\!{\mathscr {M}}_{6}^{(2)}$.

By [32, Theorem 8.1], the image of the corresponding period map ${}^2\!{\mathscr {M}}_{6}^{(2)} \rightarrow {}^2{\mathscr {P}}_{6}^{(2)}$ is exactly the complement of the Heegner divisor ${}^2{\mathscr {D}}_{6,6}^{(2)}$. The description of the image of the period map for cubic fourfolds, which is smaller, is part of a celebrated result of Laza and Looijenga [62, 68].

Theorem B.11

(Laza, Looijenga) The image of the period map for cubic fourfolds is the complement of the divisor ${}^2{\mathscr {D}}_{6,2}^{(2)}\cup {}^2{\mathscr {D}}_{6,6}^{(2)}$.

It had been known for some time (see [40]) that the image of the period map for cubic fourfolds is contained in the complement of ${}^2{\mathscr {D}}_{6,2}^{(2)}\cup {}^2{\mathscr {D}}_{6,6}^{(2)}$.

Let W be a smooth cubic fourfold containing no planes (this is equivalent to saying that its period point is not on the Heegner divisor ${}^2{\mathscr {D}}_{6,8}^{(2)}$). We explained in Sect. 3.6.3 the construction (taken from [64]) of a hyper-Kähler eightfold $X(W)\in {}^4\!{\mathscr {M}}_2^{(2)}$. It follows from [65, Proposition 1.3] that the varieties F(W) and X(W) are “strange duals” in the sense of Remark 3.24.⁶³

It follows from Theorem B.11 that the image of the set of X(W) by the period map ${}^4\!{\mathscr {M}}_{2}^{(2)} \rightarrow {}^4{\mathscr {P}}_{2}^{(2)}$ contains the complement of the Heegner divisors ${}^2{\mathscr {D}}_{6,2}^{(2)}$, ${}^2{\mathscr {D}}_{6,6}^{(2)}$, and ${}^2{\mathscr {D}}_{6,8}^{(2)}$. Since the image of this period map is exactly the complement of these Heegner divisors (Example B.7), we obtain that any element of ${}^4\!{\mathscr {M}}_{2}^{(2)}$ is of the form X(W). We provide below a different proof (due to Bayer and Mongardi) of this statement and of a weaker form of Theorem B.11.

Proposition B.12

(Bayer, Mongardi) (a) Any element of ${}^4\!{\mathscr {M}}_{2}^{(2)}$ is of the type X(W), for some smooth cubic fourfold $W\subset \textbf{P}^5$ containing no planes.

(b) The image of the period map for cubic fourfolds contains the complement of ${}^2{\mathscr {D}}_{6,2}^{(2)}\cup {}^2{\mathscr {D}}_{6,6}^{(2)}\cup {}^2{\mathscr {D}}_{6,8}^{(2)}$.

Sketch of proof of Proposition B.12

Any eightfold X in the moduli space $^{4}\!{\mathscr {M}}_{2}^{(2)}$ has, by Proposition 4.3, an anti-symplectic regular involution. Upon varying the eightfold in the moduli space, the fixed loci form a smooth family. In particular, since, when $X=X(W)$, the fixed locus contains a copy of the cubic fourfold W, and any smooth deformation of a cubic fourfold is a cubic fourfold as well (this is due to Fujita; see [53, Theorem 3.2.5]), there exists for any $X\in {^{4}\!{\mathscr {M}}_{2}^{(2)}}$ a smooth cubic fourfold $W_X\subset X$ fixed by the involution, and the periods of $W_X$ and X are compatible because they are strange duals, as explained above.

By Example B.7, the image of the period map for $^{4}\!{\mathscr {M}}_{2}^{(2)}$ is exactly the complement of the union of the Heegner divisors ${^4}{\mathscr {D}}_{2,2}^{(2)}$, ${^4}{\mathscr {D}}_{2,6}^{(2)}$, and ${^4}{\mathscr {D}}_{2,8}^{(2)}$, and these divisors correspond by strange duality to the (irreducible) Heegner divisors ${}^2{\mathscr {D}}_{6,2}^{(2)}$, ${}^2{\mathscr {D}}_{6,6}^{(2)}$, and ${}^2{\mathscr {D}}_{6,8}^{(2)}$ in the period domain of cubic fourfolds. $\square$