# On symplectic birational self-maps of projective hyperkähler manifolds of
K3[n]-type

Yajnaseni Dutta, Dominique Mattei Universität Bonn, Endenicher Allee 60, 53115
Bonn, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Yulieth
Prieto–Montañez The Abdus Salam International Centre for Theoretical Physics,
Str. Costiera, 11, 34151 Trieste TS, Italy<EMAIL_ADDRESS>
###### Abstract.

We prove that projective hyperkähler manifolds of K3[n]–type admitting a
symplectic birational self-map of finite order with a non–trivial action on
the discriminant group of $H^{2}(X,\mathbb{Z})$ are isomorphic to moduli
spaces of stable (twisted) coherent sheaves on a K3 surface. Motivated by
this, we analyse the reflections on the movable cone of moduli spaces of
sheaves with non-trivial action on the discriminant group and determine when
they come from a birational involution.

###### Key words and phrases:

Hilbert scheme of points, Birational maps, Lattices, Moduli of sheaves,
Bridgeland stability for K3 surfaces.

###### 2020 Mathematics Subject Classification:

primary 14J50, 14E05; secondary 14D20, 14C05.

###### Contents

  1. 1 Introduction
  2. 2 Cohomological action of symplectic birational maps
  3. 3 Picard rank $2$ case
  4. 4 Involutions for higher Picard ranks
  5. 5 Some Examples

##  1\. Introduction

A hyperkähler manifold (HK manifold for short) is a simply connected compact
Kähler manifold $X$ admitting an everywhere non-degenerate holomorphic
$2$-form $\sigma$ that generates $H^{0}(X,\Omega_{X}^{2})$. We say that a HK
manifold is of K3[n]-type if it is a smooth deformation of the Hilbert scheme
of $n\geq 2$ points on a K3 surface. The goal of this paper is to study
(symplectic) birational self-maps of finite order on such HK manifolds.
Birational self-maps that are not biregular are hard to come across (for
instance, on K3 surfaces there are none). On the Hilbert scheme of points on
K3 surfaces of Picard rank one, denoted K3[n], these have been classified by
[2].

A lot of geometry of HK manifolds is dictated by its second cohomology
$H^{2}(X,\mathbb{Z})$, which when $X$ is of K$3^{[n]}$-type, together with the
Beauville–Bogomolov–Fujiki form $q_{X}$ is an indefinite lattice of signature
$(3,20)$. In this set-up, Mongardi in [26, Thm. 26] shows that any (biregular)
automorphism acts as $\operatorname{Id}$ on the (lattice-theoretic)
discriminant group $A_{X}\coloneqq
H^{2}(X,\mathbb{Z})^{\vee}/H^{2}(X,\mathbb{Z})$.

We focus on finite order symplectic birational maps from $X$ to itself which
act as $-\operatorname{Id}$ on $A_{X}$. Our first main result is the
following, which is proved in §2.

######  Theorem A.

Let $X$ be a projective HK manifold of K3[n]-type admitting a finite order
symplectic birational map $g$ with $g^{*}_{|_{A_{X}}}=-\operatorname{Id}$.
Then $X$ is isomorphic to a moduli space of (twisted) sheaves on a K3 surface.

In particular, $X$ has Picard rank at least $2$. In other words, a general
projective HK manifold of K3[n]-type cannot admit finite order birational
self-maps that induce non-trivial action on $A_{X}$.

Typical (and, up to our knowledge, the only geometrically explicit) examples
of such symplectic birational maps are described by Markman in [25, sections
11.1, 11.2] (see §2.1 for one of the examples).

To elaborate on the action on the discriminant group, denote by
$\operatorname{Bir}(X)$ the group of birational self-maps of $X$, by
$\mathrm{O}(H^{2}(X,\mathbb{Z}),q_{X})$ the group of isometries of
$H^{2}(X,\mathbb{Z})$, and by $\mathrm{O}(A_{X},\bar{q}_{X})$ the group of
isometries of $A_{X}$. For $X$ a HK manifold, birational self-maps naturally
induce an automorphism of $H^{2}(X,\mathbb{C})$. Huybrechts [16, Lem. 2.6]
showed that the quadratic form $q_{X}$ behaves well under birational maps. In
other words, we have the following group homomorphism:

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

When $X$ if of K$3^{[n]}$-type it is in fact an injection ([3], [15]). We
denote by $\operatorname{Mon^{2}_{Bir}}(X)$ its image.

Monodromy operators (i.e. parallel transport isometry of
$H^{2}(X,\mathbb{Z})$), denoted $\operatorname{Mon^{2}}(X)$, are of great
importance due to the birational Torelli theorem ([32], [33], see [25, Thm.
$1.3$]).

The subgroup $\operatorname{Mon^{2}_{Hdg}}(X)<\operatorname{Mon^{2}}(X)$
preserving the Hodge structures splits as the semidirect product
$\operatorname{Mon^{2}_{Hdg}}(X)=W_{\text{Exc}}\rtimes\operatorname{Mon^{2}_{Bir}}(X)$,
where $W_{\text{Exc}}$ is the Weyl subgroup generated by reflections of the
form

$R_{E}:\alpha\mapsto\alpha-\dfrac{2(\alpha,[E])}{([E],[E])}[E]$

for all prime exceptional divisors (i.e. reduced and irreducible effective
divisor of negative Beauville-Bogomolov degree) $E\in\operatorname{Pic}(X)$.
While $R_{E}$ is a priori only defined over $\mathbb{Q}$, Markman proved that
it is in fact an integral monodromy operator [25].

A natural question is to identify monodromy operators within isometries of
$H^{2}(X,\mathbb{Z})$. When $X$ is of K$3^{[n]}$-type, by [23, Lem. 4.2] the
subgroup $\operatorname{Mon^{2}}(X)$ identifies with the preimage of
$\\{\operatorname{Id},-\operatorname{Id}\\}$ via the natural surjective [28,
Prop. 1.14.2] map

$\mathrm{O}(H^{2}(X,\mathbb{Z}),q_{X})\to\mathrm{O}(A_{X},\bar{q}_{X}).$

In the next part of this paper, we study the existence of finite order
symplectic birational self-maps on moduli spaces of stable pure sheaves
$M:=M(S,v)$ (and their generalizations) on K3 surfaces with non-trivial action
on the discriminant group, for $v$ a primitive Mukai vector. When the Picard
rank is $2$, it turns out that all such maps are automatically involutions.
More generally, we prove:

######  Theorem B.

Let $X$ be a HK manifold of K$3^{[n]}$-type. Assume that $X$ has Picard rank
$2$. Then for any finite order birational map $g\in\operatorname{Bir}(X)$, the
cohomological action $g^{*}\in\mathrm{O}(H^{2}(M,\mathbb{Z}))$ is an
involution.

It is known that whenever $\operatorname{Bir}(X)$ is finite, then
$\operatorname{Bir}(X)\simeq(\mathbb{Z}/2\mathbb{Z})^{\oplus m}$ with $m=1$ or
$2$ [8, Thm. 4.9]. Moreover, for any K3 surface $S$ of Picard rank $1$, the
group $\operatorname{Bir}(S^{[n]})$ is finite [30, Cor. 5.2].

This prompts us to study symplectic birational involutions on moduli spaces
$M=M(S,v)$ with $v$ a primitive Mukai vector.

######  Theorem C.

Let $S$ be a K3 surface and $v=(r,cD,s)$, $D\in\operatorname{NS}(S)$, be a
primitive Mukai vector, $r\neq 0$. Consider $M=M(S,v)$. For
$e=(r,cD,(cD)^{2}/r-s)$ the reflection $R_{e}\in O(H^{2}(M,\mathbb{Z}))$ is
induced by a birational involution acting non-trivially on $A_{M}$ if and only
if $v$ satisfies (4.13), $r\neq 1$ or 2 and $v$ is not one of the following
series of Mukai vectors

  1. (1)

$(r,kr,\frac{k^{2}H^{2}}{2r}-m)$ for some $k\in\mathbb{Z}$ and $m=1$ or $2$.

  2. (2)

$D^{2}\equiv 0\pmod{4}$, $v=(2a,baD,\dfrac{D^{2}}{4}m^{2}a-1)$ for some
integers $a\geq 2$, $m$ odd.

  3. (3)

$D^{2}\equiv 2\pmod{4}$, $v=(2a,baD,\dfrac{D^{2}m^{\prime 2}a-2}{4})$ for some
integers $a\geq 2$ odd, $m^{\prime}$ odd.

When $\operatorname{Pic}(S)=\mathbb{Z}\cdot D$, the condition (4.13) is simply
$r\mid 2c$ and $\gcd(r,s)=1$ or $2$. Condition (4.13) ensures that $R_{e}$ is
a monodromy operator, see Lemma 4.13 for more details.

The key input is Markman’s detailed analysis of the isometries of the
Néron–Severi lattice $\operatorname{NS}(M)\subset H^{2}(M,\mathbb{Z})$ in [24,
25]. The further conditions on $v$ determines whether $R_{e}$ lies in
$W_{\text{Exc}}$ or in $\operatorname{Mon^{2}_{Bir}}(M)$. To deduce these
conditions we use Bridgeland stability for K3 surfaces developed in [5, 6].

In fact we note in Corollary 4.18 that if $v$ falls into the last three cases,
$M\simeq M(S^{\prime},v^{\prime})$ for a derived equivalent K3 surface
$S^{\prime}$ and a Mukai vector
$v^{\prime}=(r^{\prime},c^{\prime}D^{\prime},s^{\prime})$ such that
$r^{\prime}=1$ or 2, and $R_{e}$ induces reflection at
$e^{\prime}=(r^{\prime},c^{\prime}D^{\prime},\frac{c^{\prime 2}D^{\prime
2}}{r}-s)$. So in a sense, rank 1 and 2 are the only problematic cases.

A complete classification of Mukai vectors that allows for such birational
involutions will entail studying other reflections in $\operatorname{Mov}(M)$.
To this end, we show in §4.3 that whenever a reflection $R$ is a monodromy
operator which acts non-trivially on $A_{M}$ (which, in particular, is the
case whenever such $R$ lies in $\operatorname{Mon^{2}_{Bir}}(M)$), then after
changing $S$ and $v$ to $S^{\prime}$ and $v^{\prime}$ under a Fourier–Mukai
transform, we can assume that $M=M(S^{\prime},v^{\prime})$ and $R$ is as in C
with respect to $v^{\prime}$.

###  1.1. Notations

A lot of notations have already been introduced. We need a few more:
Throughout this text, the pairing of a lattice $\Lambda$ is denoted by $(-,-)$
defined over the integers. We often conveniently denote $(w,w)$ by simply
$w^{2}$. The saturation of a sublattice $\Lambda_{1}\subset\Lambda$ is denoted
by $\overline{\Lambda_{1}}$. Given an element $w\in\Lambda$, the divisibility,
denoted $\operatorname{div}(w)$, is defined to be the integer $n$ such that
the image of the map $(\\_,w)\colon\Lambda\to\mathbb{Z}$ is given by the ideal
$n\mathbb{Z}$.

The K3 surfaces and the HK manifolds we consider are projective. For a HK
manifold $X$ of K3[n]-type, the lattice $(H^{2}(X,\mathbb{Z}),q_{X})$ is of
discriminant $-2n+2$ and signature $(3,20)$. Moreover, we have an isomorphism
$H^{2}(X,\mathbb{Z})\simeq\Lambda_{\operatorname{K3}^{[n]}}:=U^{\oplus
3}\oplus E_{8}(-1)^{\oplus 2}\oplus\langle-2(n-1)\rangle,$ where $U$ is the
hyperbolic plane and $E_{8}$ is the unique even, unimodular, positive definite
lattice of rank 8. The Mukai Lattice, denoted by $\widetilde{\Lambda}$, is the
abstract lattice $U^{\oplus 4}\oplus E_{8}(-1)^{\oplus 2}$.

Given a K3 surface $S$, one consider the weight 2 Hodge structure
$\widetilde{H}(S,\mathbb{Z})\simeq H^{0}(S,\mathbb{Z})(-1)\oplus
H^{2}(S,\mathbb{Z})\oplus H^{4}(S,\mathbb{Z})(1)$. As abstract lattices we
have $\widetilde{H}(S,\mathbb{Z})\simeq\widetilde{\Lambda}$. Given a primitive
Mukai vector $v\in\widetilde{H}(S,\mathbb{Z})$ and $H\in\operatorname{Pic}(S)$
an ample line bundle, $M_{H}(S,v)$ denotes the Mumford-Gieseker moduli space
of stable pure sheaves on $S$ with Mukai vector $v$ (we drop $S$ from the
notation when the context is clear). For $H$ generic this is a HK manifold of
K3[n]-type when $v^{2}\geq-2$, thanks to the work of many authors, notably
Mukai, Göttsche, Huybrechts, O’Grady and ultimately Yoshioka [34, Thm. 8.1].

Given a Hodge structure of weight 2 on a lattice $L_{\mathbb{Z}}$ (e.g.
$\widetilde{H}(S,\mathbb{Z})$ or $H^{2}(X,\mathbb{Z})$), we define the
algebraic part of $L$ as $L^{1,1}_{\mathbb{Z}}\coloneqq
L^{1,1}_{\mathbb{C}}\cap L_{\mathbb{Z}}$. Note that the algebraic part of the
weight 2 Hodge structure $\widetilde{H}(S,\mathbb{C})$ is given by

$\widetilde{H}^{1,1}(S,\mathbb{Z})\simeq H^{0}(S,\mathbb{Z})\oplus
H^{1,1}(S,\mathbb{Z})\oplus H^{4}(S,\mathbb{Z}).$

This is called the extended Néron–Severi lattice.

Finally, by $\operatorname{NS}(X)$ we denote the Néron–Severi lattice and by
$\operatorname{T}(X)$ we denote the transcendental lattice. Recall that inside
$H^{2}(X,\mathbb{Z})$, we have
$\operatorname{T}(X)\simeq\operatorname{NS}(X)^{\perp}$.

#### Acknowledgements

This article is indebted to the work of our predecessors in this area,
especially to the results of Arend Bayer, Emanuelle Macrì and Eyal Markman. We
are deeply grateful to the suggestions of Giovanni Mongardi which initiated
this project. We are thankful for numerous helpful conversations with Nick
Addington, Pietro Beri, Marcello Bernardara, Luca Giovenzana, Klaus Hulek,
Daniel Huybrechts, Giacomo Mezzedimi and Ángel Ríos. We thank them for their
inputs.

The first named author gratefully acknowledges funding through Hausdorff
Center of Mathematics, Bonn under Germany’s Excellence Strategy (DFG) -
EXC-2047/1 - 390685813 during the preparation of this project. The second
named author is funded by ERC Synergy Grant HyperK, agreement ID 854361. The
third named author was visiting the institutes of mathematics in Bonn,
Hannover and Toulouse, while working on this paper, and would like to thank
these institutes for their excellent hospitality and stimulating atmosphere.
She was supported by the Riemann Fellowship and LYSM project.

##  2\. Cohomological action of symplectic birational maps

The goal of this section is to prove A. The proof, that we give at the end of
the section, is a combination of several results. First, we need a
characterization of the birational models of $X$ via the lattice
$H^{2}(X,\mathbb{Z})$. In [24, §9], Markman describes $\widetilde{\Lambda}$ as
an overlattice of $H^{2}(X,\mathbb{Z})$ with the following properties:

###### Theorem 2.1 (see [1, Thm. 3]).

Let $X$ be a HK of K$3^{[n]}$–type, $n\geq 2$.

  1. (1)

The orthogonal $H^{2}(X,\mathbb{Z})^{\perp}\subset\widetilde{\Lambda}$ is
generated by a primitive vector $v$ of square $2n-2$.

  2. (2)

If $X$ is a moduli space of sheaves on a K3 surface $S$ with Mukai vector
$v\in\widetilde{H}(S,\mathbb{Z})$, then the extension
$H^{2}(X,\mathbb{Z})\subset\widetilde{\Lambda}$ is given by the Mukai isometry
(3).

  3. (3)

A HK manifold $Y$ is birational to $X$ if and only if there is a Hodge
isometry $\widetilde{\Lambda}\longrightarrow\widetilde{\Lambda}$ sending
$H^{2}(X,\mathbb{Z})$ isomorphically to $H^{2}(Y,\mathbb{Z})$.

The Hodge structure on $H^{2}(X,\mathbb{Z})$ can be extended to a Hodge
structure on $\widetilde{\Lambda}$ by setting $v\in\widetilde{\Lambda}^{1,1}$.
Moreover, the action of a map $g\in\operatorname{Bir}(X)$ extends to a Hodge
isometry on $\widetilde{\Lambda}$.

The following result is well-known. For the sake of completeness we include it
here.

######  Lemma 2.2.

If $g^{*}_{|_{A_{X}}}=\varepsilon\operatorname{Id}$ ($\varepsilon=1$ or $-1$),
there exists a unique extension of $g^{*}$ on $\widetilde{\Lambda}$ satisfying
$g^{*}v=\varepsilon v$.

###### Proof.

Define an extension $g^{*}$ on $H^{2}(X,\mathbb{Z})\oplus\langle v\rangle$ by
setting $v\mapsto\varepsilon v$, and extend it over $\mathbb{Q}$. Since
$H^{2}(X,\mathbb{Z})_{\mathbb{Q}}\oplus\langle
v\rangle_{\mathbb{Q}}=\widetilde{\Lambda}_{\mathbb{Q}}$, it suffices to prove
that the extension
$g^{*}_{\mathbb{Q}}\in\mathrm{O}(\widetilde{\Lambda}_{\mathbb{Q}})$ preserves
$\widetilde{\Lambda}$ . Any element
$x\in\widetilde{\Lambda}=\widetilde{\Lambda}^{\vee}\subset
H^{2}(X,\mathbb{Z})^{\vee}\oplus\langle v\rangle^{\vee}$ decomposes as
$x=x_{1}+x_{2}$, with $x_{1}\in H^{2}(X,\mathbb{Z})^{\vee}$ and
$x_{2}\in\langle v\rangle^{\vee}$. Since
$A_{X}\simeq\mathbb{Z}/(2n-2)\mathbb{Z}$, it is easy to see that
$g^{*}_{\mathbb{Q}}(x_{1})=\varepsilon x_{1}+(2n-2)y$ with $y\in
H^{2}(X,\mathbb{Z})$, therefore $g_{\mathbb{Q}}^{*}(x)=\varepsilon
x_{1}+(2n-2)y+\varepsilon x_{2}=\varepsilon x+(2n-2)y\in\widetilde{\Lambda}$.
∎

Let $\widetilde{\Lambda}^{1,1}$ be the algebraic part of $\widetilde{\Lambda}$
(see §1.1). The next lemma demonstrates that the property of being birational
to a moduli space of (twisted) sheaves on a K3 may be understood lattice-
theoretically from the Hodge structure of $\widetilde{\Lambda}$.

The non-twisted case was proved by Addington in [1, Prop. 4] and later
generalized for twisted K3 surfaces by Huybrechts in [20, Lem. 2.6].

######  Theorem 2.3.

Let $X$ be a HK manifold of K$3^{[n]}$-type, $n\geq 2$. Then the following are
equivalent:

  1. (1)

$\widetilde{\Lambda}^{1,1}$ contains a copy of $U$ (resp. $U(k)$);

  2. (2)

$X$ is birational to a moduli space of stable sheaves (resp. twisted sheaves)
on $S$.

Now assume that $g$ is as in A. Denote by $H^{2}(X,\mathbb{Z})^{g^{*}}$ (resp.
$\widetilde{\Lambda}^{g^{*}}$) the invariant sublattice of
$H^{2}(X,\mathbb{Z})$ (resp. $\widetilde{\Lambda}$). Similarly let
$S_{g^{*}}(X)=(H^{2}(X,\mathbb{Z})^{g^{*}})^{\perp}\subset
H^{2}(X,\mathbb{Z})$ (resp.
$S_{g^{*}}(\widetilde{\Lambda})=(\widetilde{\Lambda}^{g^{*}})^{\perp}\subset\widetilde{\Lambda}$)
be the co-invariant sublattice of $H^{2}(X,\mathbb{Z})$ (resp.
$\widetilde{\Lambda}$ ). The following is well-known.

###### Lemma 2.4.

$S_{g^{*}}(X)$ is a negative definite sublattice of $\operatorname{NS}(X)$.

###### Proof.

Pick $x\in S_{g^{*}}(X)$. Let $\sigma\in H^{2,0}(X)$ denote a generator. Since
$H^{2}(X,\mathbb{Z})^{g^{*}}$ is non-degenerate,
$x^{\prime}=\sum_{i=0}^{\operatorname{ord}(g)-1}(g^{i})^{*}x$ is
$g^{*}$-invariant and co-invariant, therefore $x^{\prime}=0$. Since
$(g^{i}(x),\sigma)=(x,\sigma)$ for all $i$, we obtain that
$0=(x^{\prime},\sigma)=\operatorname{ord}(g)(x,\sigma)$, therefore
$(x,\sigma)=0$. and hence $S_{g^{*}}(X)\subset\operatorname{NS}(X)$.

Finally, note that $H^{2}(X,\mathbb{Z})^{g^{*}}$ contains 3 positive vectors.
Indeed, $\sigma,\bar{\sigma}\in H^{2}(X,\mathbb{Z})^{g^{*}}$. Furthermore, let
$\alpha\in H^{1,1}(X)$ be a Kähler class of $X$. The class
$\alpha^{\prime}=\sum_{i=0}^{\operatorname{ord}(g)-1}(g^{i})^{*}\alpha$ is in
$H^{2}(X,\mathbb{Z})^{g^{*}}$ and it is different from $\sigma$ and
$\bar{\sigma}$. Hence,
$\text{sign}(H^{2}(X,\mathbb{Z})^{g^{*}})=(3,\operatorname{rank}H^{2}(X,\mathbb{Z})^{g^{*}}-3)$,
and so

$\text{sign}(S_{g^{*}}(X))=(0,23-\operatorname{rank}H^{2}(X,\mathbb{Z})^{g^{*}}).$

∎

######  Remark 2.5.

For a finite order group $G\subset\mathrm{O}(L)$ of isometries of a lattice
$L$, the quotient group $L/(S_{G}(L)\oplus L^{G})$ is $|G|$-torsion. Indeed,
for any element $x\in L$, we can write $|G|\cdot x=\sum_{g\in
G}g(x)+\sum_{g\in G}(x-g(x))$. If $L$ is unimodular, we know moreover that
$L/(S_{G}(L)\oplus L^{G})\simeq A_{S_{G}(L)}\simeq A_{L^{G}}$. As a
consequence, when the birational map $g\in\operatorname{Bir}(X)$ has order
$2$, we obtain that

$A_{S_{g^{*}}(\widetilde{\Lambda})}\simeq(\mathbb{Z}/2\mathbb{Z})^{\oplus\ell}$

for some integer $\ell$. A lattice with such a discriminant group is
$2$-elementary, and $2$-elementary lattices were classified by Nikulin in [29,
Thm. 4.3.2].

From 2.4 we deduce that it is enough for our purpose to prove that there
exists some twisted hyperbolic lattice $U(k)$ (for some $k\in\mathbb{Z}$)
lying in $S_{g^{*}}(\Lambda)$. To achieve this, we need the following well-
known lemma.

######  Lemma 2.6.

Let $L$ be an even lattice and $e\in L$ a non-trivial isotropic class with
$\text{div}(e)=m$. Then $U(m^{2})\subset L$.

###### Proof.

Pick $f^{\prime}\in L$ such that $(e,f^{\prime})=m$, and set $f^{2}=2k$ for
some $k\in\mathbb{Z}$. Pick $f=-ke+mf^{\prime}$. Then $f^{2}=0$ and
$(e,f)=m^{2}$, therefore $\langle e,f\rangle=U(m^{2})$. ∎

######  Remark 2.7.

As a consequence of a result of Hasse and Minkowski, any indefinite lattice of
rank at least $5$ contains an isotropic class, see [31, Cor. IV.3.2].

Let $N_{24}$ be the Leech lattice, i.e. the unique unimodular negative
definite lattice of rank $24$ without $(-2)$-classes. The final ingredient we
need is the following result that allows $g^{2}$ to be seen as an element of
the so-called Conway group $Co_{0}\coloneqq\mathrm{O}(N_{24})$. The advantage
of doing so, is that one can use the classification of $S_{G}(N_{24})$ for any
finite group action $G\subset Co_{0}$ as in [13]. The original idea of using
$N_{24}$ in this way is due to Gaberdiel, Hohenegger and Volpato [10, B2]. We
use the following more conveniant form, that appears in [9, Prop. 2.4], see
[19, §2.2].

######  Lemma 2.8.

Let $L$ be a lattice and let $G\subset\mathrm{O}(L)$ be a subgroup of
isometries which acts trivially on the discriminant group $A_{L}$. Assume that
$S_{G}(L)$ is negative definite and does not contain $(-2)$-classes. If there
exists a primitive embedding $S_{G}(L)\subset\Lambda_{1,25}$ into the unique
unimodular lattice of signature $(1,25)$, then $G$ is isomorphic to a subgroup
of the Conway group $Co_{0}$ such that

$S_{G}(L)=S_{G}(N_{24}).$

In order to use this, we need to prove the next property.

######  Lemma 2.9.

Let $f\in\operatorname{Bir}(X)$ be a symplectic birational self-map of finite
order. Then there are no $(-2)$-classes in $S_{f^{*}}(X)$.

###### Proof.

Let $\delta\in\operatorname{NS}(X)$ be a $(-2)$-class in $S_{f^{*}}(X)$. By
[24, Thm. 9.17], $\delta$ or $2\delta$ is the class of an effective divisor
$D$. Since $f$ is an isomorphism in codimension one,
$D^{\prime}=\sum_{i=0}^{\operatorname{ord}(f)-1}f(D)$ is also an effective
divisor. Its cohomology class
$\sum_{i=0}^{\operatorname{ord}(f)-1}f^{*}\delta$ is both invariant and co-
invariant, therefore $D^{\prime}$ is homologous to $0$. Since linear and
numerical equivalences coincide on HK manifolds, we obtain that $D^{\prime}$
is linearly trivial, which is impossible. ∎

######  Remark 2.10.

In fact, the result is a consequence of a more general phenomenon. Namely, for
a HK manifold $X$, the group $\operatorname{Mon^{2}_{Bir}}(X)$ preserves the
(interior of the) movable cone, which is cut out by a family of walls induced
by a subset $\mathcal{S}\subset\operatorname{NS}(X)_{\mathbb{R}}$. If a
subgroup $G\subset\mathrm{O}(H^{2}(Y,\mathbb{Z}))$ acts symplectically, with
$S_{G}(Y)$ negative definite, and preserves one of the chambers, then
$S_{G}(H^{2}(Y,\mathbb{Z}))\cap\mathcal{S}=\emptyset$ [11, Lem. 2.14]. The
classes in $\mathcal{S}$ are exactly the classes of the so-called stably prime
exceptional divisors [24, Thm. 6.17]. When $X$ is of K$3^{[n]}$-type, by [24,
Thm. 9.17] square $-2$ divisor classes (up to a square) lie in $\mathcal{S}$,
in particular they cannot lie in the co-invariant lattice.

We can now proceed to the proof of A.

###### Proof of Theorem A.

The crux of the proof lies in showing that $X$ is birational to a moduli of
(twisted) sheaves on a K3 surface. The isomorphism then follows from [6,
Section 7] (see also §4.3).

Thanks to Theorem 2.3 and Lemma 2.6, it is enough to prove that
$S_{g^{*}}(\widetilde{\Lambda})$ admits an isotropic class. If $g$ is of order
2, then $S_{g^{*}}(\widetilde{\Lambda})$ is $2$-elementary (see Remark 2.5),
hence it contains an isotropic class by [29, Thm. 4.3.3]. Otherwise, we use
the fact that whenever an indefinite lattice has rank at least $5$, it
contains an isotropic class (Remark 2.7). By Lemma 2.2, the sublattice
$S_{g^{*}}(\widetilde{\Lambda})$ is indefinite. Hence it is enough to prove
that $\operatorname{rank}S_{g^{*}}(\widetilde{\Lambda})\geq 5$, or
equivalentely $\operatorname{rank}S_{g^{*}}(X)\geq 4$. To this end, set
$f=g^{2}$ and $G=\langle f^{*}\rangle$. Note that $S_{f^{*}}(X)$ is negative-
definite by 2.4. Moreover, $S_{f^{*}}(X)$ embeds in $\Lambda_{1,25}$: indeed,
$S_{f^{*}}(X)$ embeds in the Mukai Lattice $\widetilde{\Lambda}$. Moreover,
from Remark 2.5 we have

$\displaystyle\ell(A_{S_{f^{*}}(X)})=\ell(A_{S_{f^{*}}(\widetilde{\Lambda})})$
$\displaystyle=$ $\displaystyle\ell(A_{\widetilde{\Lambda}^{f^{*}}})$
$\displaystyle\leq$ $\displaystyle
24-\operatorname{rank}S_{f^{*}}(\widetilde{\Lambda}).$

In particular,
$\ell(A_{S_{f^{*}}(X)})<26-\operatorname{rank}S_{f^{*}}(\widetilde{\Lambda})$,
and therefore [28, Cor. 1.12.3] applies. Finally, note that $f$ acts as
$\operatorname{Id}$ on the discriminant group $A_{X}$ and by Lemma 2.9, the
lattice $S_{f^{*}}(X)$ does not contain any ($-2$)-class. Therefore we can
apply Lemma 2.8 to lift the action of $f^{*}$ to the Leech lattice $N_{24}$
and identify $S_{f^{*}}(X)=S_{f^{*}}(N_{24})$. This enables us to use the
classification results of [13]. The group $G$ embeds in its pointwise
stabilizer $\widetilde{G}=\mathrm{O}_{o}(S_{G}(N_{24}))$ with
$N_{24}^{G}=N_{24}^{\widetilde{G}}$. Such fixed lattice are classified in [13,
Table 1], and for $G$ non-trivial the order of such fixed lattice is at most
$16$. In particular, $S_{f^{*}}(X)\simeq S_{G}(N_{24})$ has order at least
$24-16=8$. The inclusion $S_{f^{*}}(X)\subset S_{g^{*}}(X)$ concludes the
proof. ∎

###  2.1. Markman’s example $M_{H}(r,0,-s)$

In this section we recall an explicit geometric example given by Markman [25,
sections 2.2, 11.1] of symplectic birational self-map of order 2 on moduli
spaces of sheaves on K3 surfaces with a non-trivial action on the discriminant
group. In the latter part of the paper, where we study certain reflections
coming from the so-called “vertical wall” (see Definition 4.12) of the
stability manifold of the K3 surface, we revisit this example as it serves as
one such reflection (see §5.1).

Let $S$ be a projective K3 surface of Picard rank 1 and $H$ be an ample line
bundle on $S$. Suppose that $r,s$ are two integers satisfying $s\geq r\geq 1$
and $\gcd(r,s)=1$. Set $M\coloneqq M_{H}(r,0,-s)$ the moduli space of stable
sheaves on $S$ with Mukai vector $v=(r,0,-s)$. The Mukai isometry (3)
identifies $H^{2}(M,\mathbb{Z})$ with
$H^{2}(S,\mathbb{Z})\oplus\mathbb{Z}\cdot e$, where $e=\theta(r,0,s)$.

  * •

$\mathbf{r=1}$: In this case, $M_{H}(1,0,-s)=S^{[1+s]}$ and $e=[E]/2$ where
$E\subset S^{[1+s]}$ is the diagonal, i.e. $E$ is the exceptional divisor of
the Hilbert–Chow morphism $\varepsilon:S^{[1+s]}\longrightarrow S^{(1+s)}$.

  * •

$\mathbf{r=2}$: In this case, $e=[E]$ where $E\subset M_{H}(2,0,-s)$ is the
locus of stable sheaves which are not locally free. By [25, Lem. 10.16], $E$
is the exceptional divisor of Li’s morphism [21] from $M_{H}(2,0,-s)$ onto the
Uhlenbeck–Yau compactification of the moduli space of $H$-slope stable vector
bundles.

  * •

$\mathbf{r\geq 3}$: In this case, $e$ is not a $\mathbb{Q}$-effective class.
Let $Z\subset M_{H}(r,0,-s)$ be the locus of stable sheaves which are not
locally free or not $H$-slope stable. Denote by $U=M\setminus Z$ the locus of
locally free $H$-slope stable sheaves and $\iota:U\longrightarrow U$ the map
that sends $\mathcal{F}$ to its dual sheaf $\mathcal{F}^{\vee}$. By [25, Lem.
9.5], $Z$ is a closed subset of codimension $\geq 2$ in $M$ and so
$\iota:M\longrightarrow M$ is a birational involution. By [25, Prop. 11.1],
the induced map $\iota^{*}$ in cohomology corresponds to the reflection map
$R_{e}$.

##  3\. Picard rank $2$ case

Assume $X$ is a $K3^{[n]}$-type HK manifold with Picard rank $2$. We know by
Oguiso’s result (c.f. [8, Thm. 4.9]) that, while $\operatorname{Aut}(X)$ and
$\operatorname{Bir}(X)$ can be infinite, whenever they are finite they are of
the form $(\mathbb{Z}/2\mathbb{Z})^{r}$ with $r=1$ or $2$. In this section we
prove B that states that any finite order birational self-map of $X$ is in
fact of order $2$.

###### Proof of B.

Since $X$ has Picard rank $2$, its real Néron-Severi group
$\operatorname{NS}_{\mathbb{R}}(X)$ is a $2$-plane. By a result of Markman
[24, Thm. 6.25], there exist a polyhedral rational convex cone
$\Delta\subset\operatorname{Mov}(X)$ such that

$\overline{\operatorname{Mov}(X)}=\bigcup_{s\in\operatorname{Bir}(X)}s^{*}(\Delta).$

Since the map $g\in\operatorname{Bir}(X)$ has finite order, we can define

$\Delta_{g}^{0}:=\bigcup_{n\in\mathbb{Z}}(g^{n})^{*}\Delta$

which is a finite union of convex polyhedral rational cones in
$\overline{\operatorname{Mov}(X)}$. Define $\Delta_{g}$ as the convex hull of
$\Delta_{g}^{0}$. Denote by $l_{1},l_{2}$ the extremal rays of $\Delta_{g}$
(which lie already in $\Delta_{g}^{0}$). Then $g^{*}(\Delta_{g})$ lies in the
cone generated by $g^{*}(l_{1})$ and $g^{*}(l_{2})$, which lies in
$\Delta_{g}$. By bijectivity of $g$, we get $g^{*}(\Delta_{g})=\Delta_{g}$.

Since the extremal rays of $\Delta_{g}$ are rational, we can choose integral
generators $x_{1}$ of $l_{1}$ and $x_{2}$ of $l_{2}$. Then $g^{*}$ must either
preserve or flip $l_{1}$ and $l_{2}$. In particular,
$(g^{2})^{*}(x_{i})=\alpha_{i}x_{i}$ for some $\alpha_{i}>0$, $i=1,2$. Since
$g$ is an integral isometry, we get $\alpha_{1},\alpha_{2}\in\mathbb{Z}$ and
$\alpha_{1}\alpha_{2}=1$. We get $\alpha_{1}=\alpha_{2}=1$ and hence
$(g^{2})^{*}=\operatorname{Id}|_{\operatorname{NS}(X)}$.

The only integral Hodge involutions of $\operatorname{T}(X)$ are
$\pm\operatorname{Id}$, therefore
$(g^{2})^{*}=\operatorname{Id}|_{\operatorname{T}(X)}$, so in particular
$g^{*}$ is an involution on $H^{2}(X,\mathbb{Z})$. Since
$\operatorname{Bir}(X)\to\mathrm{O}(H^{2}(X,\mathbb{Z}))$ is injective, we
obtain that $g$ is an involution. ∎

######  Remark 3.1.

In fact, $g^{*}$ is a reflection in the line spanned by $x_{1}+x_{2}$. Indeed,
a non-trivial involution of a rank $2$ lattice $L$ is always a reflection in
the line spanned by $x+\iota(x)$, for any $x\neq 0\in L$.

######  Remark 3.2.

When $H^{2}=2n$ with $n\equiv-1\pmod{4}$ and the divisibility of $H$ is 2,
Debarre in [8, Prop. 4.19] shows that the non-trivial elements of finite order
in the orthogonal group $\mathrm{O}^{+}(\operatorname{Pic}(X))$ preserving the
positive cone are of order 2. Moreover, in some cases $\operatorname{Bir}(X)$
is isomorphic to the infinite dihedral group, in particular such order $2$
elements exist despite $\operatorname{Bir}(X)$ being infinite.

##  4\. Involutions for higher Picard ranks

The goal of this section is to show the existence of symplectic finite order
birational self-maps on certain HK manifold $X$ of K$3^{[n]}$-type. According
to the Torelli theorem, a way to proceed is to find monodromy operators of
$H^{2}(X,\mathbb{Z})$. In §3 we proved that when $\operatorname{Pic}(X)$ has
rank $2$, any such finite order operator is a reflection (see Remark 3.1).
Therefore, one candidate in higher Picard rank are reflections
$R_{e}\in\mathrm{O}(H^{2}(X,\mathbb{Z}))$ in the hyperplane orthogonal to some
class $e\in\operatorname{NS}(X)$. We will use the following result of Markman.

###### Proposition 4.1 [25, Prop. 1.5].

Let $e\in H^{2}(X,\mathbb{Z})$ be a primitive class with $e^{2}<0$. Then the
reflection $R_{e}$ belongs to $\operatorname{Mon^{2}}(X)$ if and only if $e$
satisfies one of the following:

  1. (1)

$e^{2}=-2$, or

  2. (2)

$e^{2}=2-2n$ and $n-1$ divides $(e,-)\in H^{2}(X,\mathbb{Z})^{\vee}$.

Moreover, $R_{e}$ acts on $A_{X}$ as $\operatorname{Id}$ in (1) and
$-\operatorname{Id}$ in (2).

######  Corollary 4.2.

If a symplectic birational involution $g\in\operatorname{Bir}(X)$ acts as a
reflection $g^{*}=R_{e}$ on $H^{2}(X,\mathbb{Z})$, then necessarily
$e^{2}=2-2n$. In particular, $g^{*}|_{A_{X}}=-\operatorname{Id}$.

###### Proof.

Note that $e\in S_{R_{e}}(X)$ since for any $w\in H^{2}(X,\mathbb{Z})^{R_{e}}$
we have $(e,w)=(-e,w)$. Moreover $e^{2}<0$ by 2.4. Then by Lemma 2.9
$e^{2}\neq-2$. ∎

Furthermore, we will focus on moduli $M$ of sheaves on K3 surfaces and use
certain walls in the stability manifold of the K3 surface to find these
reflection hyperplanes in $\operatorname{Mov}(M)$.

###  4.1. Reminders on stability conditions

Let $S$ be a K3 surface and $\alpha\in\operatorname{Br}(S)$ be a Brauer class
111We refer the readers to [18, Section 10.2.2, 16.4] for details. and fix
$B\in H^{2}(S,\mathbb{Q})$ a $B$-field lift of $\alpha$. We will not give the
general definition of stability conditions (we redirect the interested reader
to [27] for the general theory, and we refer to [7] for the case of K3
surfaces), but only recall how to construct the ones we will use in the
present paper and some of their properties.

Pick a pair of $\mathbb{R}$-divisors
$\omega,\beta\in\operatorname{NS}(S)_{\mathbb{R}}$, with
$\omega\in\operatorname{Amp}(S)$, the ample cone inside
$\operatorname{NS}(S)_{\mathbb{R}}$. For $E\in\operatorname{D}^{b}(S,\alpha)$,
define

$\displaystyle Z_{\omega,\beta}(E)$ $\displaystyle=$
$\displaystyle(\exp(i\omega+\beta+B),v(E))=\Re Z(E)+i\Im Z(E).$ (1)

In particular, writing $v(E)=(r,\theta,s)$, we have $\Im
Z(E)=\omega\cdot(\theta-r(\beta+B))$. Define a pair of additive subcategories
(called a torsion pair)
$\mathcal{T}^{\omega,\beta},\mathcal{F}^{\omega,\beta}\subset\operatorname{D}^{b}(S)$
as follows: the non-trivial objects of $\mathcal{T}^{\omega,\beta}$ (resp.
$\mathcal{F}^{\omega,\beta}$) are the twisted sheaves (resp. torsion free
twisted sheaves) $F\in\operatorname{Coh}(S,\alpha)$ such that every non-
trivial torsion free quotient $F\twoheadrightarrow G$ (resp. non-zero subsheaf
$E\hookrightarrow F$) satisfy $\Im Z_{\omega,\beta}(G)>0$ (resp. $\Im
Z_{\omega,\beta}(E)\leq 0$). Then define the abelian subcategory of
$\operatorname{D}^{b}(S,\alpha)$:

$\displaystyle\mathcal{A}_{\omega,\beta}:=\\{F\in\operatorname{D}^{b}(S,\alpha)\
|\
\mathcal{H}^{-1}(F)\in\mathcal{F}^{\omega,\beta},\mathcal{H}^{0}(F)\in\mathcal{T}^{\omega,\beta},\mathcal{H}^{i}(F)=0\text{
for }i\neq-1,0\\}$ (2)

The follolwing result is first due to Bridgeland [7] in the untwisted case,
then generalized to the twisted case by Huybrechts, Macrì, Stellari [14, Prop.
3.6].

######  Theorem 4.3.

The pair $\sigma_{\omega,\beta}=(\mathcal{A}_{\omega,\beta},Z_{\omega,\beta})$
is a stability condition on $\operatorname{D}^{b}(S)$ provided that
$Z_{\omega,\beta}(E)\notin\mathbb{R}_{\leq 0}$ for all spherical twisted sheaf
$E\in\operatorname{Coh}(X,\alpha)$. This condition is satisfied whenever
$\omega^{2}>2$. Moreover, the stability condition $\sigma_{\omega,\beta}$
depends continuously on
$(\omega,\beta)\in\operatorname{Amp}(S)\times\operatorname{NS}(S)_{\mathbb{R}}$.

We denote by $\operatorname{Stab}^{\dagger}(S)$ the component of the space of
stability condition $\operatorname{Stab}(S)$ containing all stability
conditions of the form $\sigma_{\omega,\beta}$. We drop the $\omega,\beta$
when the context is clear. Define the generalized rank
$\operatorname{rank}_{Z}=\Im Z:\mathcal{A}\to\mathbb{R}_{\geq 0}$ and
generalized degree $\deg_{Z}=-\Re Z:\mathcal{A}\to\mathbb{R}$. These maps
satisfy the usual properties of rank and degree of sheaves, such as
$\operatorname{rank}(E)=0\Rightarrow\deg(E)>0$ and additivity with respect to
exact sequences. Therefore one can define the generalized slope as

$\mu_{Z}(E):=\deg_{Z}(E)/\operatorname{rank}_{Z}(E),$

where $\mu_{Z}(E)=\infty$ when $\operatorname{rank}_{Z}(E)=0$. Objects
$E\in\operatorname{D}^{b}(S,\alpha)$ such that $E[k]\in\mathcal{A}$ for some
$k\in\mathbb{Z}$ and $E[k]$ is slope-(semi)stable with respect to $\mu_{Z}$
are called $Z$-(semi)stable (or $\sigma$-(semi)stable).

####  4.1.1. Walls and moduli spaces

Fix a Mukai vector $v\in\widetilde{H}^{*}(S,\alpha,\mathbb{Z})$.

###### Theorem 4.4 [4, Prop. 3.3].

There exists a locally finite set of walls (real codimension one submanifolds
with boundary) in $\operatorname{Stab}(S)$, depending on $v$, such that the
sets of $\sigma$-stable and $\sigma$-semistable objects of class $v$ do not
change when $v$ vary in a chamber (i.e. in a component of the complement of
the walls).

When $v$ is primitive, then $\sigma$ lies on a wall if and only if there
exists a strictly $\sigma$-semistable object of class $v$. A stability
condition is called generic with respect to $v$ if it does not lie on a wall.

In [6], the authors prove that for a generic
$\sigma\in\operatorname{Stab}^{\dagger}(S)$ there exists a coarse moduli space
$M_{\sigma}(v)$ of $\sigma$-semistable objects with Mukai vector $v$. It is a
normal projective irreducible variety with $\mathbb{Q}$-factorial
singularities. When $v$ is primitive, then $M_{\sigma}(v)=M_{\sigma}^{st}(v)$,
i.e. it consists only of $\sigma$-stable objects and hence is a smooth
projective HK manifold of K$3^{[n]}$-type.

###### Theorem 4.5 [5, Thm. 2.15].

Let $v=mv_{0}\in\widetilde{H}^{*}(S,\alpha,\mathbb{Z})$ be a vector with
$v_{0}$ primitive and $m>0$, and let
$\sigma\in\operatorname{Stab}^{\emph{\textdagger}}(S,\alpha)$ be a generic
stability condition with respect to $v$.

  1. (1)

The coarse moduli space $M_{\sigma}(v)$ is non-empty if and only if
$v_{0}^{2}\geq-2$,

  2. (2)

Either $\dim M_{\sigma}(v)=v^{2}+2$ and $M_{\sigma}^{st}\neq\emptyset$, or
$m>1$ and $v_{0}\leq 0$.

######  Remark 4.6.

Given a polarization $H\in\operatorname{Pic}(S)$ which is generic with respect
to $v$, there is always a Gieseker chamber: the moduli space $M_{\sigma}(v)$
for any $\sigma$ in this chamber is isomorphic to the moduli space of
$H$-Gieseker stable sheaves, see [7, Prop. 14.2].

The Hodge structure on the second cohomology of $M_{\sigma}(v)$ is closely
related to the one of $S$ by the so-called Mukai Hodge isometry

$\displaystyle\theta:H^{2}(M_{\sigma}(v),\mathbb{Z})\to\begin{cases}v^{\perp}\subset
H^{2}(M,\mathbb{Z})&\text{ if }v^{2}\neq 0\\\ v^{\perp}/v\subset
H^{2}(M,\mathbb{Z})&\text{ if }v^{2}=0.\end{cases}$ (3)

The latter have been shown by Yoshioka [34, Sections 7 & 8] for moduli of
sheaves, and generalized by Bayer and Macrì [6, Thm. 7.10] to Bridgeland
stability conditions.

####  4.1.2. Wall-crossing

Bayer and Macrì proved in [5] that the MMP for a moduli space of sheaves
$M_{H}(v)$ can be performed by wall-crossing in the space of stability
conditions; namely, they proved the following (see Thm. 1.1, 1.2 in loc.
cit.).

###### Theorem 4.7 [5, Thm. 1.1, 1.2].

Let $v$ be a primitive Mukai vector.

  1. (1)

Given $\sigma,\tau\in\operatorname{Stab}^{\emph{\textdagger}}(S)$ generic, the
two moduli spaces $M_{\sigma}(v)$ and $M_{\tau}(v)$ of Bridgeland-stable
objects are birational to each other.

  2. (2)

Fix a (generic) base point
$\sigma\in\operatorname{Stab}^{\emph{\textdagger}}(S)$. Every smooth
$K$-trivial birational model of $M_{\sigma}(v)$ appears as a moduli space
$M_{\tau}(v)$ for some $\tau\in\operatorname{Stab}^{\emph{\textdagger}}(S)$.

To be more precise, the authors construct a map

$\displaystyle
l:\operatorname{Stab}^{\dagger}(S)\to\operatorname{NS}(M_{\sigma}(v)),$ (4)

such that for any chamber $\mathcal{C}\subset\operatorname{Stab}^{\dagger}(S)$
and $\tau\in\mathcal{C}$ we have
$l(\mathcal{C})=\operatorname{Amp}(M_{\tau}(v))$. Given a wall $\mathcal{W}$
for $v$ and $\sigma_{0}$ a generic stability condition on the wall, let
$\sigma_{+},\sigma_{-}$ be two generic stability conditions nearby
$\mathcal{W}$ in opposite chambers.

###### Theorem 4.8 [6, Thm. 1.4].

There exists birational contractions

$\pi^{\pm}:M_{\sigma_{\pm}}(v)\to\overline{M}_{\pm}$

where $\overline{M}_{\pm}$ are normal irreducible projective varieties. The
curves contracted by $\pi^{\pm}$ are precisely the curves of objects that are
$S$-equivalent to each other with respect to $\sigma_{0}$.

The type of birational transformation are described as follows.

######  Definition 4.9.

We call a wall $\mathcal{W}$:

  1. (1)

a fake wall if there are no curves in $M_{\sigma_{\pm}}(v)$ of objects that
are $S$-equivalent to each other with respect to $\sigma_{0}$,

  2. (2)

a totally semistable wall if $M_{\sigma_{0}}^{st}(v)=\emptyset$,

  3. (3)

a flopping wall if we can identify $\overline{M}_{+}=\overline{M}_{-}$ and the
induced map $M_{\sigma_{+}}\dashrightarrow M_{\sigma_{-}}$ induces a flopping
contraction,

  4. (4)

a divisorial wall if the morphisms
$\pi^{\pm}:M_{\sigma_{\pm}}(v)\to\overline{M}_{\pm}$ are both divisorial
contractions.

The type of wall can be studied purely lattice-theoretically. Given a wall
$\mathcal{W}$, one can associate the hyperbolic rank $2$ primitive sublattice
$\mathcal{H}_{\mathcal{W}}\subset\widetilde{H}(S,\alpha,\mathbb{Z})$ of rank
$2$ given by

$\mathcal{H}_{\mathcal{W}}=\\{w\in\widetilde{H}(S,\alpha,\mathbb{Z})\ |\
\Im\dfrac{Z(w)}{Z(v)}=0\textit{ for all
}\sigma=(\mathcal{A},Z)\in\mathcal{W}\\}.$

Conversely, given a primitive rank $2$ hyperbolic sublattice containing $v$,
one can define a potential wall $\mathcal{W}$ associated to $\mathcal{H}$ as a
connected component of the real codimension one submanifold of stability
conditions $\sigma=(\mathcal{A},Z)$ which satisfy that $Z(\mathcal{H})$ is
contained in a line.

###### Theorem 4.10 [5, Thm. 5.7].

Let $\mathcal{H}\subset\widetilde{H}(S,\alpha,\mathbb{Z})$ be a primitive
hyperbolic rank $2$ lattice containing $v$ and
$\mathcal{W}\subset\operatorname{Stab}^{\emph{\textdagger}}(S)$ a potential
wall associated to it.

  1. (1)

The set $\mathcal{W}$ is a divisorial wall if there exists a class
$s\in\mathcal{H}$ such that $(s,v)=0$ and $s^{2}=-2$, or an isotropic class
$w\in\mathcal{H}$ such that $(w,v)=1$ or $2$.

  2. (2)

Otherwise, if $v$ can be written as $v=a_{1}+a_{2}$ with
$a_{i}\in\mathcal{H}$, $a_{i}^{2}\geq 0$ and $(a_{i},v)>0$, $i=1,2$, or if
there exists a spherical class $s\in\mathcal{H}$ with $0<(s,v)\leq v^{2}/2$,
the $\mathcal{W}$ is a flopping wall.

  3. (3)

In all other cases, $\mathcal{W}$ is either a fake wall or it is not a wall.

######  Remark 4.11.

Following [5, Lem. 10.1], the type of a wall is directly related to the
position of its image by (4) in $\overline{\operatorname{Mov}}(M)$: the image
$l(\mathcal{W})$ lies on the boundary of $\operatorname{Mov}(M)$ if and only
if it is divisorial. On the other hand, $\mathcal{W}$ is either a flopping or
fake wall if its image lies in the interior of $\operatorname{Mov}(M)$ (it is
fake if and only if its image lies in the interior of a chamber corresponding
to a birational model of $M$).

###  4.2. Reflection in the vertical wall

Set $M\coloneqq M_{\sigma}(S,v)$ for some K3 surface $S$ with arbitrary Picard
rank and $\sigma\in\operatorname{Stab}^{\dagger}(S)$. Denote

$v=(r,\theta,s)$

with $r,s\in\mathbb{Z}$ and $\theta\in\operatorname{NS}(S)$. For convenience
we denote $\theta=cD$ with $c\in\mathbb{Z}$ and $D$ primitive. We will freely
use the Mukai isometry 3 to identify $H^{2}(M,\mathbb{Z})$ with
$v^{\perp}\subset\widetilde{H}(S,\mathbb{Z})$.

If $\Phi:\operatorname{D}^{b}(S)\xrightarrow{\sim}\operatorname{D}^{b}(S)$ is
an autoequivalence, the induced isometry
$\varphi\in\mathrm{O}(\widetilde{H}(S,\mathbb{Z}))$ on the extended Mukai
lattice gives an isomorphism

$M_{\sigma}(v)\xrightarrow{\sim}M_{\Phi(\sigma)}(\varphi(v)).$

An additional useful observation is that by 4.7 (and assuming
$\Phi(\sigma)\in\operatorname{Stab}^{\dagger}(S)$), for any
$\tau\in\operatorname{Stab}^{\dagger}(S)$ we have

${M\simeq
M_{\Phi(\sigma)}(\varphi(v))}$${M_{\tau}(\varphi(v)).}$$\scriptstyle{\sim}$

In particular, it holds for $\tau$ in the Gieseker chamber, whenever it
exists.

We will use freely the following three types of autoequivalences:

  * •

The shift $[-1]$, which acts as $-\operatorname{Id}$ on
$\widetilde{H}(S,\mathbb{Z})$.

  * •

The tensor product $-\otimes L$ for $L\in\operatorname{Pic}(S)$, which acts as
cup product with $\exp(L)=(1,L,L^{2}/2)$ on $\widetilde{H}(S,\mathbb{Z})$.

  * •

The spherical twist $T_{\mathcal{O}_{X}}$, which acts as the reflection
$R_{(1,0,1)}$ in the hyperplane orthogonal to the spherical class
$(1,0,1)\in\widetilde{H}(S,\mathbb{Z})$.

From [12, Lem. 7.2, Prop. 7.6], these autoequivalences preserve the
distinguished component
$\operatorname{Stab}^{\dagger}(S)\subset\operatorname{Stab}(S)$. In
particular, applying some of these equivalences we assume $r>0$ from now on.

Consider the geometric stability conditions of the form
$\sigma=\sigma_{\omega,\beta}$ with
$\omega\in\operatorname{Amp}(S),\beta\in\operatorname{NS}(S)_{\mathbb{R}}$
defined by (1).

######  Definition 4.12.

We call the potential wall
$\mathcal{W}\subset\operatorname{Amp}(S)\times\operatorname{NS}(S)_{\mathbb{R}}$
(seen as a subspace of $\operatorname{Stab^{\dagger}}(S)$) defined by the
equation $\Im Z_{\omega,\beta}(v)=0$ the vertical wall of $S$ associated to
$v$.

In other words, $\mathcal{W}$ is spanned by the set of $(\omega,\beta)$
satisfying the equation $\omega\cdot\theta=r\omega\cdot\beta$. Consider the
corresponding hyperbolic rank $2$ lattice

$\mathcal{H}_{\mathcal{W}}:=\\{w=(\lambda,(\lambda/r)\theta,\mu),\
\lambda,\mu\in\mathbb{Z},\ r\mid c\lambda\\},$

i.e. the saturation of the sublattice spanned by $v$ and $(0,0,1)$, and note
that $Z_{\omega,\beta}(w)\in\mathbb{R}$ for all $(\omega,\beta)\in\mathcal{W}$
and $w\in\mathcal{H}_{\mathcal{W}}$.

Using [6, Lem. $9.2$], it is easy to see that the map $l$ as in (4) sends any
stability condition $\sigma_{\omega,\beta}\in\mathcal{W}$ (up to a positive
constant) to the vector
$a_{\omega,\beta}=(0,\omega,\beta\cdot\omega)=(0,\omega,\omega\cdot\theta/r)$.
These vectors form a cone $C_{\mathcal{W}}$ of codimension one (i.e. a wall)
in
$\overline{\operatorname{Mov}}(M)\subset\operatorname{NS}(M)_{\mathbb{R}}\simeq
v^{\perp}_{\mathbb{R}}\subset\widetilde{H}^{1,1}(S)_{\mathbb{R}}$. We consider
the reflection in this wall. To do so, consider the vector
$e:=(r,\theta,(\theta^{2}/r)-s)$. It is orthogonal both to $v$ and
$a_{\omega,\beta}$ for all $(\omega,\beta)\in\mathcal{W}$. In particular, $e$
is a generator of the line orthogonal to the wall
$C_{\mathcal{W}}\subset\operatorname{Mov}(M)$. Finally, note that
$e^{2}=-v^{2}=2-2n$. We now use Markman’s criterion in 4.1 to check when the
reflection $R_{e}$ is a monodromy operator.

######  Lemma 4.13.

The reflection $R_{e}$ belongs to $\operatorname{Mon^{2}}(M)$ with
$R_{e}\mid_{A_{M}}=-\operatorname{Id}$ if and only if

$r\mid 2c\dfrac{D\cdot L}{\operatorname{div}(L)}\ \forall
L\in\operatorname{NS}(S)\text{ and }\gcd(r,s)=1\text{ or }2,$

where
$\operatorname{div}(L)\colon=\gcd_{L^{\prime}\in\operatorname{NS}(S)}(L\cdot
L^{\prime})$ is the divisibility of $L$ in $\operatorname{NS}(S)$.

###### Proof.

By 4.1, all we need to check is that $e$ is integral and primitive, and
$n-1\mid\text{div}_{H^{2}(M,\mathbb{Z})}(e)$. Integrality of $e$ is equivalent
to $r\mid c^{2}D^{2}$ and primitivity is equivalent to
$\gcd(r,c,c^{2}D^{2}/r-s)=1$. For what follows, let $c^{2}D^{2}=kr$ for some
$k\in\mathbb{Z}$.

To see what conditions the divisibility criterion imposes on $r$ and $c$, pick
some $w=(a,\delta,f)\in v^{\perp}\subset\widetilde{H}(S,\mathbb{Z})$ for
$a,f\in\mathbb{Z}$ and $\delta\in H^{2}(S,\mathbb{Z})$. Recall that for any
unimodular lattice $\Lambda$ and primitive sublattice
$\Lambda_{1}\subset\Lambda$, the composition
$\Lambda/(\Lambda_{1}\oplus\Lambda_{1}^{\perp})\hookrightarrow
A_{\Lambda_{1}}\oplus A_{\Lambda_{1}^{\perp}}\twoheadrightarrow
A_{\Lambda_{1}}$ is an isomorphism. Applying this on the unimodular lattice
$H^{2}(S,\mathbb{Z})$ and letting $\Lambda_{1}=\operatorname{NS}(S)$, we
deduce that
$H^{2}(S,\mathbb{Z})^{\vee}\twoheadrightarrow\operatorname{NS}(S)^{\vee}$ is
surjective and hence $\delta$ can be written as
$b\frac{L}{\operatorname{div}(L)}+T$ for some $b\in\mathbb{Z}$,
$L\in\operatorname{NS}(S)$ and $T\in\operatorname{T}(S)^{\vee}$. Therefore, we
can assume that $w$ has the shape

$w=(a,b\dfrac{L}{\operatorname{div}(L)}+T,f).$

The condition $w\in v^{\perp}$ becomes

$bc\dfrac{L\cdot D}{\operatorname{div}(L)}=as+rf$ (4)

and a direct computation gives $(w,e)=\dfrac{2a}{r}(1-n)$. Therefore, the
condition $(n-1)|(w,e)$ is equivalent to $r|2a$. If $s=0$, then we must have
$r=1$ or $2$. Indeed, otherwise let $w=(1,cD,k)$ and note that
$w\in(r,cD,0)^{\perp}$ and it satisfies $(n-1)\nmid(w,e)$. If $c=0$,
primitivity of $v$ implies $\gcd(r,s)=1$, and $as=-rf$, so $r|a$. For both of
these situations, the lemma is proved.

Now assume $c,s\neq 0$. If there exist some $L\in\operatorname{NS}(S)$ such
that $r\nmid 2c\dfrac{D\cdot L}{\operatorname{div}(L)}$, then pick
$w=\left(c\dfrac{L\cdot
D}{\operatorname{div}(L)}-r,s\dfrac{L}{\operatorname{div}(L)}+T,s\right)$,
thus $(w,v)=0$ and $(w,e)=(n-1)\big{(}2-\dfrac{2cL\cdot
D}{\operatorname{div}(L)r}\big{)}$. Therefore, for any
$L\in\operatorname{NS}(S)$ we can write $2c(L\cdot
D)/\operatorname{div}(L)=m_{L}r$ for some integer $m_{L}$. Denote
$g=\gcd(r,s)$.

First pick $w=(r/g,2D/\operatorname{div}(D)+T,m_{D}-s/g)$. Then $(w,v)=0$ and
$(w,e)=(n-1)(-2/g)$, so the divisibility criterion implies that $g=1,2$. From
(4) we have

$\displaystyle 2as=r\left(\dfrac{2bcD\cdot
L}{\operatorname{div}(L)r}-2f\right)=r(m_{L}b-2f).$ (5)

Now if $g=1$, then $r\mid 2a$. If $g=2$, dividing both sides of equation (5)
yields $as=(r/2)(m_{L}b-2f)$, with $\gcd(r/2,s)=1$. Therefore $r/2\mid a$, so
$r\mid 2a$.

Finally note that by Remark 4.14 below, a posteriori the condition (4.13)
automatically implies $r\mid c^{2}D^{2}$ and $\gcd(r,c,c^{2}D^{2}/r-s)=1$,
i.e. $e$ is integral and primitive. ∎

######  Remark 4.14.

The condition (4.13) implies $r\mid cD^{2}$. We split the argument in two
cases. First if $\operatorname{div}(D)$ is even, then condition (4.13) implies
that $rm_{D}\dfrac{\operatorname{div}(D)}{2}=cD^{2}$ for the integer $m_{D}$
as defined in the proof. So in particular $r\mid cD^{2}$. If
$\operatorname{div}(D)$ is odd, then there is some $L\in\operatorname{NS}(S)$
such that $D\cdot L=\operatorname{div}(D)$. For this choice of $L$, the
condition (4.13) implies that
$rm_{L}\operatorname{div}(L)=2c\operatorname{div}(D)$. Since
$\operatorname{div}(D)\mid D^{2}/2$, we can write
$D^{2}/2=\operatorname{div}(D)m$ for some integer $m$. Then we can rewrite the
equation as $rm_{L}\operatorname{div}(L)m=cD^{2}$, from which the assertion
follows again.

Since $R_{e}\in\operatorname{Mon^{2}}(M)$, by [24, Thm. 1.6], it decomposes
uniquely (uniqueness is a consequence of the injectivity of
$\operatorname{Bir}(M)\to\mathrm{O}(H^{2}(M,\mathbb{Z}))$) as $R_{e}=\psi\circ
g^{*}$, with $g\in\operatorname{Bir}(M)$ a symplectic birational self-map and
$\psi\in W_{\text{Exc}}$.

We have already seen that the image $C_{\mathcal{W}}$ of $\mathcal{W}$ in
$\operatorname{NS}(M)$ lie in $\overline{\operatorname{Mov}}(M)$. By Remark
4.11, we deduce that if the image $C_{\mathcal{W}}$ lies on the boundary of
$\operatorname{Mov}(M)$, then $g=\operatorname{Id}$ and $\mathcal{W}$ is a
divisorial wall. Otherwise $C_{\mathcal{W}}$ lies in the interior of
$\operatorname{Mov}(M)$, which implies $\psi=\operatorname{Id}$ and
$\mathcal{W}$ is either a flopping or a fake wall. But since $R_{e}$ acts non-
trivially on $A_{M}$, the wall cannot be fake. Indeed, otherwise
$C_{\mathcal{W}}$ lies inside of a chamber corresponding to a birational model
$f:M^{\prime}\dashrightarrow M$ and hence $g^{*}$ preserves the nef cone of
$M^{\prime}$. The Torelli theorem then implies that the map $f^{-1}\circ
g\circ f$ extends to an isomorphism on $M^{\prime}$, which renders
$g^{*}|_{A_{M}}=\operatorname{Id}$ contradicting our assumption.

One can also directly check this using the numerical criteria 4.10 as follows.

######  Proposition 4.15.

The potential wall $\mathcal{W}$ is either a divisorial or a flopping wall.

###### Proof.

One can write $v=w+t$ with $w=(v+(0,0,1))$, $t=(0,0,-1)$. Note that
$w^{2}=v^{2}-2r$, $t^{2}=0$, $(w,v)=v^{2}-r$ and $(v,t)=r$. By Lemma 4.13, we
know that $r|v^{2}$. If $v^{2}>r$, then $v^{2}\geq 2r$ so the decomposition
$v=w+t$ implies that $\mathcal{W}$ is either a divisorial wall or a flopping
wall.

Assume $v^{2}=r$. Denote $2cD^{2}/\operatorname{div}(D)=m_{D}r$. Then
$c=(\operatorname{div}(D)m_{D}r)/(2D^{2})$ and
$v^{2}=c^{2}D^{2}-2rs=r\left(\frac{\lambda}{2}cm-2s\right)$. Therefore
$v^{2}=r$ if and only if $\frac{\operatorname{div}(D)}{2}cm_{D}=2s+1$. Such a
$v$ have the shape

$v=\left(2a,\dfrac{\operatorname{div}(D)am_{D}}{D^{2}}D,\dfrac{m_{D}^{2}a(\operatorname{div}(D)^{2}/D^{2})-2}{4}\right)$

with $a\in\mathbb{Z}$ odd. Denoting $b=\operatorname{div}(D)m/D^{2}$, $v$ has
exactly the shape of the last case of Lemma 4.16. ∎

Using 4.10 we can also deduce under condition (4.13), when $R_{e}$ does not
come from a birational involution. In other words we give necessary and
sufficient numerical criterion for when $\mathcal{W}$ is a divisorial wall.

######  Proposition 4.16.

Under the conditions (4.13), the vertical wall as in Definition 4.12 is a
divisorial wall if and only if one of the following cases occur.

  * •

$r=1,r=2$ or $v=(r,krD,\dfrac{D^{2}k^{2}r-m}{2})$ for some $k\in\mathbb{Z}$
and $m=1$ or 2.

  * •

$r>2$, $r\nmid c$, and one of the two possibilities occurs.

    1. (1)

$D^{2}\equiv 0\pmod{4}$, $v=(2a,maD,\dfrac{D^{2}m^{2}a}{4}-1)$ for some
integers $a\geq 2$, $m$ odd.

    2. (2)

$D^{2}\equiv 2\pmod{4}$, $v=(2a,m^{\prime}aD,\dfrac{D^{2}m^{\prime 2}a-2}{4})$
for some integers $a\geq 3$ odd, $m^{\prime}$ odd.

Note that the Mukai vector $w=(\lambda,(\lambda
c/r)D,\mu)\in\mathcal{H}_{\mathcal{W}}$ satisfies the following formulas,
which we use in the proof:

$\displaystyle w^{2}=\lambda^{2}\dfrac{D^{2}c^{2}}{r^{2}}-2\lambda\mu$ (6)
$\displaystyle(w,v)=\lambda\dfrac{D^{2}c^{2}}{r}-r\mu-s\lambda.$ (7)

###### Proof of Proposition 4.16.

By 4.10 it is enough to give a complete classification of Mukai vectors $v$,
for which there exists an isotropic $w\in\mathcal{H}_{\mathcal{W}}$ satisfying
$(w,v)=1$ or 2 or a spherical $w\in\mathcal{H}_{\mathcal{W}}$ satisfying
$(w,v)=0$.

Step 1: Assume $r=1$ or $2$.

Then $w=(0,0,-1)$ satisfies $w^{2}=0$ and $(w,v)=1$ or $2$, so the vertical
wall is divisorial.

Step 2: Assume $c=kr$ for some $k\in\mathbb{Z}$.

Let $w=(\lambda,\lambda\dfrac{c}{r}D,\mu)$ as before. If $\lambda=0$, then
$w^{2}=0$ and $(w,v)=\mu r=1$ or $2$ if and only if $r=1,2$. Hence we assume
$\lambda\neq 0$.

Equations (6) and (7) become

$\displaystyle w^{2}$ $\displaystyle=$
$\displaystyle\lambda^{2}D^{2}k^{2}-2\lambda\mu$ $\displaystyle(w,v)$
$\displaystyle=$ $\displaystyle\lambda D^{2}k^{2}r-r\mu-s\lambda$

First assume $w^{2}=0$. Then $\lambda D^{2}k^{2}=2\mu$. Hence

$\displaystyle(w,v)$ $\displaystyle=$ $\displaystyle\lambda
D^{2}k^{2}r-r\mu-s\lambda$ $\displaystyle=$ $\displaystyle\lambda
D^{2}k^{2}r-r\lambda k^{2}(D^{2}/2)-s\lambda$ $\displaystyle=$
$\displaystyle\lambda((D^{2}/2)k^{2}r-s)$

Now $(w,v)=1$ implies $\lambda=\pm 1$ and $(w,v)=2$ implies $\lambda=\pm 1,\pm
2$. It is direct to see that $\lambda<0$ leads to $v^{2}<0$, which is absurd
(recall we assumed $v^{2}\geq 2$). Hence we obtain $s=(D^{2}/2)k^{2}r-m$ with
$m=1$ or $2$.

Now assume $w^{2}=-2$, i.e. $\lambda^{2}D^{2}k^{2}=2\lambda\mu-2$, so we
obtain again $\lambda=\pm 1$ or $\pm 2$. In fact, $\lambda=\pm 2$ is
impossible because we would obtain $4D^{2}k^{2}=\pm 2(2\mu-1)$. Hence
$\lambda=\pm 1$ and

$(w,v)=0\iff\pm D^{2}k^{2}r=r\mu\pm s,$

so $r|s$ which implies that $r=1$ since $v=(r,krD,s)$ is primitive.

Step 3: General case.

From condition (4.13) and Remark 4.14, we can assume that $cD^{2}=kr$ and
$c\lambda=mr$ for some $k,m$. Note that as in Step 2, when $\lambda=0$ the
wall is divisorial only when $r=1$ or $2$, and $\lambda=\pm 1$ gives $r\mid
c$, hence these cases are already treated in Step $1$ and $2$. Therefore we
assume $|\lambda|\geq 2$. Equalities (6) and (7) give:

$\displaystyle w^{2}$ $\displaystyle=$ $\displaystyle\lambda mk-2\lambda\mu$
$\displaystyle(w,v)$ $\displaystyle=$ $\displaystyle rmk-r\mu-s\lambda$

First assume $w^{2}=-2$. Then $\lambda=\pm 2$. The wall is divisorial in this
case, if and only if $(w,v)=0$. Hence the second equality implies $r\mid 2s$,
which gives $r=1$ or $2$ and we are back to Step $1$.

Now assume $w^{2}=0$, that is $mk=2\mu$ and hence $(w,v)=r\mu-s\lambda$. Since
$2\mu r=mrk=\lambda ck$, we obtain

$\displaystyle 2(w,v)=\lambda(ck-2s).$

Assume $(w,v)=1$. Then $\lambda=\pm 2$, so $r\mid 2c$ and $ck-2s=\pm 1$, in
particular $c,k$ are odd. If $r$ is even then $ck=\pm\mu r$ is even, which
leads to a contradiction, therefore $r$ is odd and hence $r\mid c$, so we are
done by Step $2$.

The last case is $(w,v)=2$, which gives $\lambda=\pm 2$ or $\pm 4$. Note that
$\lambda<0$ is impossible, since it would give $ck-2s<0$, but
$v^{2}=r(kc-2s)\geq 0$.

First, let $\lambda=2$. We obtain $ck-2s=2$ and $r\mid 2c$. We can assume
$r=2a$ for some $a>0$ (otherwise $r\mid c$ and we are back to Step $2$),
therefore $c=ma$ and $k=mD^{2}/2$. We obtain

$s=\dfrac{ck}{2}-1=\dfrac{D^{2}}{4}m^{2}a-1.$

Note that $\mu=mck/r=m^{2}aD^{2}/4a=m^{2}D^{2}/4$, in particular $D^{2}\equiv
0\pmod{4}$. For any such choice of $a$ and $m$, the integral vector
$w=(2,mD,m^{2}D^{2}/4)$ gives $(w,v)=2$.

The situation with $\lambda=4$ is similar: we get $ck-2s=1$, in particular
$c,k$ are odd. As before we can assume $r=2a$ for some $a$, but moreover if
$a$ is even we get $2\mu=ck$ which is impossible. Therefore $a$ is odd,
$m=2m^{\prime}$ is even (with $m^{\prime}$ odd), $c=m^{\prime}a$,
$k=m^{\prime}D^{2}/2$ and

$s=\dfrac{m^{\prime 2}a(D^{2}/2)-1}{2},$

which in particular implies that $D^{2}\equiv 2\pmod{4}$. For any such choice
of $a$ and $m^{\prime}$, the integral vector
$w=(4,2m^{\prime}D,\dfrac{D^{2}}{2}m^{\prime 2})$ gives $(w,v)=2$.

∎

We conclude that whenever $v$ does not satisfies the condition of the previous
proposition, $\mathcal{W}$ is a flopping wall and
$R_{e}\in\operatorname{Mon^{2}_{Bir}}(M)$.

######  Corollary 4.17.

Let $M=M_{\sigma}(S,v)$ be a moduli space of dimension $2n$ for some K3
surface $S$ and $\sigma\in\operatorname{Stab}^{\dagger}(S)$. Assume that $v$
satisfies conditions (4.13) and is not equal to any of the vectors in
Proposition 4.16. Then there exists a symplectic birational involution
$\iota\in\operatorname{Bir}(M)$ with
$\iota^{*}=R_{e}\in\mathrm{O}(H^{2}(M,\mathbb{Z})$ with $e^{2}=2-2n$ and
$\iota^{*}|_{A_{M}}=-\operatorname{Id}$.

Given a Mukai vector $v$ as above, it would be interesting to know an explicit
geometric description of the involution $\iota$. We discuss in §5.1 one such
explicit case.

###  4.3. Reflection along any wall

The goal of this subsection is to show that reflection along the image under
$l$ of any wall $\mathcal{W}$ in $\operatorname{Stab}(M)$ that induce a
monodromy operator with non-trivial action on the discriminant group $A_{M}$
may be seen as reflection along the image of the vertical wall on an
isomorphic model of $M$, as considered in the previous subsection. We need the
following key construction.

####  4.3.1. Construction

Consider a wall $\mathcal{W}$ as above. We will see in Corollary 4.19 that the
associated hyperbolic lattive $\mathcal{H}_{\mathcal{W}}$ contains an
isotropic class $w$, so that $\mathcal{H}_{\mathcal{W}}=\overline{\langle
v,w\rangle}$. For such an isotropic class, we let $S^{\prime}\coloneqq
M_{H}(S,w)$ and $v^{\prime}=\Phi(v)$ with
$\Phi:\operatorname{D}^{b}(S)\to\operatorname{D}^{b}(S^{\prime},\alpha)$ the
induced Fourier–Mukai transform and $\alpha$ is some Brauer class on
$S^{\prime}$. Define
$\mathcal{W}^{\prime}\subset\operatorname{Stab}(S^{\prime})$ to be the induced
wall, i.e. defined by the hyperbolic plane
$\mathcal{H}_{\mathcal{W}^{\prime}}=\overline{\langle\Phi(w),v^{\prime}\rangle}$.
Let $v^{\prime}=(r^{\prime},\theta^{\prime},s^{\prime})$. Pick
$\sigma_{0}\in\mathcal{W}$ and set $\tau_{0}\coloneqq\phi(\sigma_{0})$. Note
that $\tau_{0}$ has the shape
$\tau_{0}=\sigma_{\omega^{\prime},\beta^{\prime}}$ for some
$\omega^{\prime},\beta^{\prime}\in\operatorname{NS}(S^{\prime})_{\mathbb{R}}$
(see [12, Prop. 7.5] for $\alpha$ trivial, or [17, Prop. 5.2]). Since
$\Phi(w)=(0,0,1)$ we get $\Im Z_{\omega^{\prime},\beta^{\prime}}(\Phi(w))=0$.
Therefore $\Im Z_{\omega^{\prime},\beta^{\prime}}(v^{\prime})=0$, i.e.
$\mathcal{W}^{\prime}$ is the vertical wall of $S^{\prime}$.

Moreover, $-r^{\prime}=(v^{\prime},(0,0,1))=(v,w)$. Composing the equivalence
$\Phi$ with the shift $[1]$ we get for any
$\sigma\in\operatorname{Stab^{\dagger}}(S)$ an isomorphism

$M=M_{\sigma}(S,v)\xrightarrow{\sim}M^{\prime}=M_{\tau}(S^{\prime},-v^{\prime},\alpha)$

for $\tau\coloneqq\Phi(\sigma)[1]$ and the image of $\mathcal{W}$ is the
vertical wall of $S^{\prime}$ with respect to $-v^{\prime}$. Note that if
$-r^{\prime}=1$ then necessarily $\alpha$ is trivial.

An easy upshot of this construction is that it simplifies the Mukai vectors in
Proposition 4.16 significantly when we allow ourselves to choose a possibly
different K3 surface $S$.

######  Corollary 4.18.

Up to replacing $S$ by a twisted Fourier–Mukai partner $(S^{\prime},\alpha)$,
under the condition (4.13) the only cases for which the vertical wall is
divisorial are the cases $r=1,2$.

###### Proof.

Consider the cases in Proposition 4.16. First, when $c=kr$ and
$s=(D^{2}/2)k^{2}r-m$ for $m=1$ or 2, replacing $v$ with its image
$v^{\prime}$ under the isometry $(-)\cdot\exp(-kD)$ and $R_{(1,0,1)}$ gives
$v^{\prime}=(m,0,-r)$. It is straightforward to check that these isometries
send the vertical wall associated to $v$ to the one associated to
$v^{\prime}$. For each of the other cases, there exists an isotropic class
$w\in\mathcal{H}_{\mathcal{W}}$ with $(w,v)=2$, so the construction in §4.3.1
above applies. Recall that $-r^{\prime}=(v,w)=2$ and hence the rank of the new
Mukai vector is 2. ∎

For any wall $\mathcal{W}$, denote $\mathcal{W}\coloneqq
l(\mathcal{W})\subset\operatorname{Mov}(M)$ by abuse of notation. Another
implication of the construction above is the following.

######  Corollary 4.19.

Assume that the reflection $R_{e}$ in the wall $\mathcal{W}$ orthogonal to the
vector $e\in\operatorname{NS}(M)$ lies in $\operatorname{Mon^{2}_{Hdg}}(M)$
and acts non-trivially on $A_{M}$. Then, up to replacing $S$ by a twisted
Fourier–Mukai partner $(S^{\prime},\alpha)$, $R_{e}$ is the reflection in the
vertical wall. In particular, for a birational involution
$g\in\operatorname{Bir}(M)$ with
$g^{*}=R_{e}\in\mathrm{O}(H^{2}(M,\mathbb{Z}))$, up to an autoequivalence
$g^{*}$ is the reflection in the vertical wall.

###### Proof.

As we have seen in Proposition 4.15, an element
$R_{e}\in\operatorname{Mon^{2}_{Hdg}}(M)$ lies either in
$\operatorname{Mon^{2}_{Bir}}(M)$ or $W_{\text{Exc}}$. In the former case,
$e^{2}<0$ by 2.4. In the latter case, $e^{2}<0$ by [24, Thm. 6.18]. By 4.1, we
have $e^{2}=2-2n$, in particular $v+e\in\mathcal{H}_{\mathcal{W}}$ is an
isotropic class. Therefore, the construction applies and we can assume that
$\mathcal{W}$ is the vertical wall.

∎

######  Remark 4.20.

The contraction map $\pi:M\to\overline{M}$ of 4.8 induced by the vertical wall
is the map to the Uhlenbeck compactification constructed by Li [22], see [5,
Section 8]. When $r=1$, an isometry given by applying a combination of
$\exp{(-kD)}$ for some $k\in\mathbb{Z}$ and the reflection $R_{(1,0,1)}$
reduces $v$ to the case when $v=(1,0,1-n)$, i.e. $M$ is birational to a
Hilbert scheme of points. In this case, the map $\pi$ is nothing but the
Hilbert–Chow morphism, see [6, Example $10.1$].

Recall that in A we have shown that the existence of finite order birational
self-map with non-trivial action on the discriminant group is somewhat
special. Therefore it is natural to ask the following.

######  Question 4.21.

If $M$ is a HK manifold of K$3^{[n]}$-type which admits a finite order
symplectic birational self-map with non-trivial action on $A_{M}$, does $M$
also admits a symplectic involution which action on $H^{2}(M,\mathbb{Z})$ is
the reflection in the vertical wall?

##  5\. Some Examples

###  5.1. Markman’s example redux

In this section, we revisit in the light of stability conditions the
reflection maps discussed in §2.1. We show that the reflection map $R_{e}$
associated to some particular $e$ can be obtained as a reflection on the
vertical (flopping) wall in the halfplane of stability conditions.

Let $S$ be a K3 surface with $\operatorname{Pic}(S)=\mathbb{Z}\cdot H$, $H$
ample. Pick $v=(r,0,-s)$ with $s\geq r\geq 1$ and $\gcd(r,s)=1$. Set
$M:=M_{H}(v)$. The image of the vertical wall in
$\operatorname{Stab}^{\dagger}(S)$ in $H^{2}(M,\mathbb{Z})=v^{\perp}$ is
generated by $e=(r,0,s)$. We see that $v$ satisfies the conditions (4.13), in
particular the the reflection $R_{e}$ lies in $\operatorname{Mon^{2}}(M)$. We
consider the same trichotomy as in Section 2.1.

  * •

$\mathbf{r=1}$: In this case, $M\simeq S^{[1+s]}$. The wall is easily seen to
be divisorial (of type Hilbert-Chow). Indeed, the vector $w=(1,0,0)$ is an
isotropic vector satisfying $(w,v)=1$. In particular, by Remark 4.20 the
contraction morphism induced on the wall is the Hilbert-Chow morphism
$S^{[1+s]}\to S^{(1+s)}$.

  * •

$\mathbf{r=2}$: In this case, the vector $w=(0,0,-1)$ gives $(w,v)=2$. Hence
by 4.10 the vertical wall is divisorial (of type Li–Gieseker–Uhlenbeck), and
by Remark 4.20 the contraction morphism induced on the wall is the morphism
onto the Uhlenbeck–Yau compactification of the moduli space of $H$-slope
stable vector bundles.

  * •

$\mathbf{r>2}$: Since $s>r$, according to Proposition 4.16, the vertical wall
is not divisorial. Though, the class $w=(1,0,1)$ is spherical and satisfies
$0<(w,v)\leq(v,v)/2$, hence $\mathcal{W}$ is a flopping wall. The birational
map $M\dashrightarrow M,\mathcal{F}\mapsto\mathcal{F}^{\perp}$ (for
$\mathcal{F}$ $H$-slope stable locally free) induces the reflection $R_{e}$ in
cohomology.

###  5.2. Hilbert schemes

Let $S$ be a K3 surface with $\operatorname{Pic}(S)=\mathbb{Z}H$, $H^{2}=2d$,
$d>1$, and let $M=S^{[n]}$. In this set-up, the authors in [2] have given a
precise description of $\operatorname{Bir}(M)$ for all values of $n,d$. In
some cases, $M$ admits a symplectic birational involution $\sigma$ acting non-
trivially on $A_{M}$. In these cases, $\sigma^{*}$ acts as a reflection on
$H^{2}(M,\mathbb{Z})$, and hence by Corollary 4.19, up to replacing $S$ by a
Fourier-Mukai partner, this reflection is in fact the reflection in the
vertical wall. In what follows, we can construct in some cases an explicit
isometry to relate birationally the involution $\sigma$ on $M$ with Markman’s
example described in §2.1 and in §5.1.

Pick any $r>3$, set $n=r^{2}d-r+1$, so that $M=S^{[n]}=M_{H}(v)$ with
$v=(1,0,r(1-rd))$. Interestingly, in this case the vertical wall is
divisorial. Nonetheless by [2, Thm. 1.1] $M$ admits a symplectic birational
involution $\sigma$ and the line $\ell\subset\operatorname{NS}(M)$ generated
by $2rdH_{M}-2d\delta$ is fixed by $\sigma$ where $H_{M}=(0,H,0)$ (a
polarization of $M$) and $\delta=(-1,0,r(1-rd))$ (half the class of the big
diagonal) generate $\operatorname{NS}(M)$. Consider the isometry of
$\widetilde{H}(S,\mathbb{Z})$ given by the composition

$\varphi=\exp(-H)\circ-R_{(1,0,1)}\circ\exp(-rH).$

By direct computations, one can check that $\varphi(v)=(r,0,1-rd)$, so we
obtain a birational map $\widetilde{\varphi}:M\dashrightarrow
M^{\prime}:=M_{H}(r,0,1-rd)$. Note that, the image of the fixed line $\ell$ in
$\operatorname{NS}(M^{\prime})$ is generated by $(0,H,0)$, so the action of
the composition

$\psi:=\widetilde{\varphi}^{-1}\circ\sigma\circ\widetilde{\varphi}$

acts on $\operatorname{NS}(M)$ just as in the Markman’s example described §2.1
and 5.1. By injectivity of
$\operatorname{Bir}(S)\to\mathrm{O}(H^{2}(M,\mathbb{Z}))$, the birational map
$\psi$ is precisely given by $M^{\prime}\dashrightarrow M^{\prime},\
\mathcal{F}\mapsto\mathcal{F}^{\vee}$.

######  Remark 5.1.

We should point out that this example is somewhat special in view of [24, Rmk.
9.21]; two moduli spaces $M_{H}(r_{i},0,-s_{i})$, $i=1,2$ with
$r_{1}s_{1}=r_{2}s_{2}$ are not birational in general.

As a matter of course, the example leads to the following

######  Question 5.2.

Is $S^{[n]}$ birational to $M_{H}(r,0,-s)$ with $s>r>2$ whenever it admits a
symplectic birational involution?

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