# Rational Hodge isometries of hyper-Kähler varieties of $K3^{[n]}$-type are
algebraic

Eyal Markman  Department of Mathematics and Statistics, University of
Massachusetts, Amherst, MA 01003, USA<EMAIL_ADDRESS>
###### Abstract.

Let $X$ and $Y$ be compact hyper-Kähler manifolds deformation equivalence to
the Hilbert scheme of length $n$ subschemes of a $K3$ surface. A class in
$H^{p,p}(X\times Y,{\mathbb{Q}})$ is an analytic correspondence, if it belongs
to the subring generated by Chern classes of coherent analytic sheaves. Let
$f:H^{2}(X,{\mathbb{Q}})\rightarrow H^{2}(Y,{\mathbb{Q}})$ be a rational Hodge
isometry with respect to the Beauville-Bogomolov-Fujiki pairings. We prove
that $f$ is induced by an analytic correspondence. We furthermore lift $f$ to
an analytic correspondence $\tilde{f}:H^{*}(X,{\mathbb{Q}})[2n]\rightarrow
H^{*}(Y,{\mathbb{Q}})[2n]$, which is a Hodge isometry with respect to the
Mukai pairings and which preserves the gradings up to sign. When $X$ and $Y$
are projective the correspondences $f$ and $\tilde{f}$ are algebraic.

###### Contents

  1. 1 Introduction
    1. 1.1 Algebraicity of rational Hodge isometries
    2. 1.2 Fourier-Mukai kernels of positive rank and the Lefschetz Standard Conjecture
    3. 1.3 The Pontryagin product
    4. 1.4 Outline of the proof of Theorem 1.4
    5. 1.5 Why does the generalized Kummer deformation type require a modification?
  2. 2 Generators for the rational isometry group
  3. 3 The LLV Lie algebra
    1. 3.1 The subring generated by $H^{2}(X,{\mathbb{Q}})$
    2. 3.2 The functor $\widetilde{H}$
  4. 4 The degree reversing Hodge isometry of a Fourier-Mukai kernel of non-zero rank
  5. 5 Hyperholomorphic vector bundles deforming a Fourier-Mukai kernel
    1. 5.1 Twistor lines
    2. 5.2 Moduli spaces of rational Hodge isometries
    3. 5.3 Diagonal twistor lines
    4. 5.4 Diagonal twistor paths
    5. 5.5 Deforming vector bundles along diagonal twistor paths
    6. 5.6 Deforming the Fourier-Mukai kernel
  6. 6 The BKR equivalence
  7. 7 A universal bundle over $M^{[n]}\times S^{[n]}$ from a universal bundle over $M\times S$
    1. 7.1 A universal bundle over $M^{[n]}\times S^{[n]}$
    2. 7.2 The functor $\widetilde{\Theta}_{n}:{\mathcal{G}}^{[1]}\rightarrow{\mathcal{G}}^{[n]}$
    3. 7.3 The functor $\Theta_{n}:{\mathcal{G}}^{[1]}\rightarrow{\mathcal{G}}^{[n]}$
  8. 8 Deforming the universal vector bundle over $M^{[n]}\times S^{[n]}$
  9. 9 Algebraicity of a rational Hodge isometry
  10. 10 The Pontryagin product on the cohomology of IHSMs of $K3^{[n]}$-type
    1. 10.1 The Pontryagin product for an IHSM with vanishing odd cohomology
    2. 10.2 The Pontryagin product for an IHSM of $K3^{[n]}$-type
  11. 11 Acknowledgements

##  1\. Introduction

###  1.1. Algebraicity of rational Hodge isometries

An irreducible holomorphic symplectic manifold (IHSM) is a simply connected
compact Kähler manifold $X$ such that $H^{0}(X,\Omega^{2}_{X})$ is spanned by
a nowhere degenerate holomorphic $2$-form. Such manifolds admit hyper-Kähler
structures and are examples of compact hyper-Kähler manifolds. When two-
dimensional, such a manifold is a $K3$ surface. If $X$ is a Kähler manifold
which is deformation equivalent to the Hilbert scheme $S^{[n]}$ of length $n$
subschemes of a $K3$ surface $S$, then $X$ is an irreducible holomorphic
symplectic manifold [Be1]. The latter are said to be of $K3^{[n]}$-type. The
second integral cohomology $H^{2}(X,{\mathbb{Z}})$ of an irreducible
holomorphic symplectic manifold $X$ is endowed with a symmetric integral
primitive non-degenerate bilinear pairing of signature $(3,b_{2}(X)-3)$ known
as the Beauville-Bogomolov-Fujiky (BBF) pairing, where $b_{2}(X)$ is the
second Betti number [Be1]. A homomorphism $f:H^{2}(X,{\mathbb{Z}})\rightarrow
H^{2}(Y,{\mathbb{Z}})$ between the second cohomologies of two irreducible
holomorphic symplectic manifolds is said to be an integral Hodge isometry, if
it is an isometry with respect to the BBF-pairings as well as an isomorphism
of Hodge structures. If $f:H^{2}(X,{\mathbb{Q}})\rightarrow
H^{2}(Y,{\mathbb{Q}})$ has these properties it is said to be a rational Hodge
isometry.

Let $X$ be an irreducible holomorphic symplectic manifold. Given a class
$\alpha$ in the even cohomology $H^{ev}(X,{\mathbb{Q}})$ of $X$, denote by
$\alpha_{i}$ its graded summand in $H^{2i}(X,{\mathbb{Q}})$. Let
$\alpha^{\vee}\in H^{ev}(X,{\mathbb{Q}})$ be the class satisfying
$(\alpha^{\vee})_{i}=(-1)^{i}\alpha_{i}$. The Mukai pairing on the even
cohomology $H^{ev}(X,{\mathbb{Q}})$ is defined by

$(\alpha,\beta):=\int_{X}\alpha^{\vee}\cup\beta.$

The Mukai vector of an object $F$ in the bounded derived category $D^{b}(X)$
of coherent sheaves on $X$ is $v(F):=ch(F)\sqrt{td_{X}}\in
H^{ev}(X,{\mathbb{Q}}).$ Grothendieck-Riemann-Roch yields

$(v(E),v(F)):=\chi(E^{\vee}\otimes F)$

for objects $E$ and $F$ in $D^{b}(X)$, where
$E^{\vee}:=R{\mathcal{H}}om(E,{\mathcal{O}}_{X})$ is the derived dual object.
When $X$ is of $K3^{[n]}$-type the odd cohomology vanishes and
$H^{ev}(X,{\mathbb{Z}})=H^{*}(X,{\mathbb{Z}})$ [Ma1].

Let $X$ be a $2n$-dimensional IHSM. Denote by $H^{*}(X,{\mathbb{Q}})[2n]$ the
shifted cohomology of $X$, where
$H^{k}(X,{\mathbb{Q}})[2n]=H^{k+2n}(X,{\mathbb{Q}})$. We say that a
homomorphism $f:H^{*}(X,{\mathbb{Q}})[2n]\rightarrow
H^{*}(Y,{\mathbb{Q}})[2n]$ is degree reversing, if it maps
$H^{k}(X,{\mathbb{Q}})[2n]$ to $H^{-k}(Y,{\mathbb{Q}})[2n]$, for all $k$. The
homomorphism is degree preserving up to sign, if it is either degree
preserving or degree reversing.

Let $X$ and $Y$ be IHSMs of $K3^{[n]}$-type and
$f:H^{2}(X,{\mathbb{Q}})\rightarrow H^{2}(Y,{\mathbb{Q}})$ a Hodge isometry.

######  Theorem 1.1.

There exists an analytic correspondence
$\tilde{f}:H^{*}(X,{\mathbb{Q}})\rightarrow H^{*}(Y,{\mathbb{Q}})$, which is
an isometry with respect to the Mukai pairings and satisfies one of the
following:

  1. (1)

$\tilde{f}$ is degree preserving and it restricts to $H^{2}(X,{\mathbb{Q}})$
as $f$,

  2. (2)

$\tilde{f}$ is degree reversing and the composition

$H^{*}(X,{\mathbb{Q}})\stackrel{{\scriptstyle
c_{2}(X)^{n-1}\cup}}{{\longrightarrow}}H^{*}(X,{\mathbb{Q}})\stackrel{{\scriptstyle\tilde{f}}}{{\longrightarrow}}H^{*}(X,{\mathbb{Q}})$

of $\tilde{f}$ with cup product with $c_{2}(X)^{n-1}$ restricts to
$H^{2}(X,{\mathbb{Q}})$ as a non-zero rational multiple of $f$.

In particular, $f$ is algebraic whenever $X$ and $Y$ are projective.

The Theorem is proved in section 9. When $n=1$ and $X$ and $Y$ are $K3$
surfaces Theorem 1.1 was proved by Mukai for projective surfaces with Picard
number $\geq 11$ [Mu1], by Nikulin for projective surfaces with Picard number
$\geq 5$ [N], announced by Mukai in [Mu2], and proved by Buskin in complete
generality [Bu]. Another proof for projective $K3$ surfaces $X$ and $Y$,
without further restrictions on their Picard numbers, is due to Huybrechts
[Hu5].

A more direct relationship between the Hodge isometries $f$ and $\tilde{f}$ in
Theorem 1.1 is described in terms of Taelman’s LLV lattice in Theorem 1.4
below.

######  Definition 1.2.

Let $X_{1}$ and $X_{2}$ be deformation equivalent compact Kähler manifolds. An
isomorphism $f:H^{*}(X_{1},{\mathbb{Z}})\rightarrow H^{*}(X_{2},{\mathbb{Z}})$
is said to be a parallel-transport operator, if it arrises as follows. There
exist some smooth and proper family of compact Kähler manifolds
$\pi:{\mathcal{X}}\rightarrow B$ over an analytic space $B$, points $b_{1}$,
$b_{2}$ in $B$, isomorphisms $g_{i}:X_{i}\rightarrow{\mathcal{X}}_{b_{i}}$
with the fibers ${\mathcal{X}}_{b_{i}}$ of $\pi$ over $b_{i}$, $i=1,2$, and a
continuous path $\gamma$ from $b_{1}$ to $b_{2}$, such that parallel-transport
along the path $\gamma$ in the local system $R\pi_{*}{\mathbb{Z}}$ induces
$g_{2,*}\circ f\circ
g_{1}^{*}:H^{*}({\mathcal{X}}_{b_{1}},{\mathbb{Z}})\rightarrow
H^{*}({\mathcal{X}}_{b_{2}},{\mathbb{Z}})$. When $X_{1}=X_{2}$ the parallel-
transport operator $f$ is also called a monodromy operator. The monodromy
group ${\rm Mon}(X)$ of a compact Kähler manifold $X$ is the subgroup of
$GL(H^{*}(X,{\mathbb{Z}}))$ consisting of all monodromy operators.

Recall that a groupoid is a category all of whose morphisms are isomorphisms.
Let

(1.1) ${\mathcal{G}}$

be the groupoid, whose objects are pairs $(X,\epsilon)$ consisting of an
irreducible holomorphic symplectic manifold $X$ and an orientation111The
morphisms of ${\mathcal{G}}$ do not depend on the orientation. The orientation
is not needed if $n:=\dim(X)/2$ is odd. If $n$ is even, the orientation is
used in [Ta] in the definition of the functor $\widetilde{H}$ introduced below
in Equation (1.4). $\epsilon$ of $H^{2}(X,{\mathbb{Q}})$. If $\dim(X)$ is
divisible by $4$, assume222The assumption holds for all known irreducible
holomorphic symplectic manifolds. that $b_{2}(X)$ is odd. Morphisms in ${\rm
Hom}_{\mathcal{G}}((X,\epsilon),(Y,\epsilon^{\prime}))$ are compositions of
three types of rational isometries from $H^{*}(X,{\mathbb{Q}})$ to
$H^{*}(Y,{\mathbb{Q}})$ with respect to the Mukai pairing:

  1. (1)

A parallel-transport operator $f:H^{*}(X,{\mathbb{Q}})\rightarrow
H^{*}(Y,{\mathbb{Q}})$.

  2. (2)

The isometry $H^{*}(X,{\mathbb{Q}})\rightarrow H^{*}(Y,{\mathbb{Q}})$ induced
by the correspondence $ch({\mathcal{P}})\sqrt{td_{X\times Y}}$, where
${\mathcal{P}}\in D^{b}(X\times Y)$ is the Fourier-Mukai kernel of an
equivalence $D^{b}(X)\rightarrow D^{b}(Y)$ of derived categories between two
deformation equivalent IHSMs $X$ and $Y$.

  3. (3)

An isometry of $H^{*}(X,{\mathbb{Q}})$ induced by cup product with
$\exp(\lambda):=\sum_{k=0}^{\dim(X)}\lambda^{k}/k!$ for some $\lambda\in
H^{2}(X,{\mathbb{Q}})$.

The set of morphisms ${\rm
Hom}_{\mathcal{G}}((X,\epsilon),(Y,\epsilon^{\prime}))$ is independent of the
orientations and will be denoted also by ${\rm Hom}_{\mathcal{G}}(X,Y)$. Note
that the latter is empty, by definition, if $X$ and $Y$ are not deformation
equivalent.

######  Definition 1.3.

Given an object $F$ of positive rank $r$ in the derived category of coherent
sheaves, set $\kappa(F):=ch(F)\exp\left(-c_{1}(F)/r\right)$, where
$\exp(x)=\sum_{j=0}^{\infty}x^{j}/j!$. Then $\kappa(F)^{r}=ch(F^{\otimes
r}\otimes\det(F)^{-1})$, where the tensor power is taken in the derived
category. If $F$ is an object of positive rank $r$ in the derived category of
twisted coherent sheaves with respect to some Brauer class, then $F^{\otimes
r}\otimes\det(F)^{-1}$ is an untwisted object and we define $\kappa(F)$ as the
$r$-th root of the Chern character of the latter with degree $0$ summand equal
to $r$. If $F$ has negative rank, set $\kappa(F):=-\kappa(F[1])$.

Let

(1.2) ${\mathcal{G}}_{an}$

be the subgroupoid of ${\mathcal{G}}$ with the same objects. Morphisms of
${\mathcal{G}}_{an}$ are compositions of two types of rational isometries: (1)
Parallel-transport operators $f:H^{*}(X,{\mathbb{Q}})\rightarrow
H^{*}(Y,{\mathbb{Q}})$, which are Hodge isometries. (2) Hodge isometries

(1.3) $[\kappa({\mathcal{P}})\sqrt{td_{X\times
Y}}]_{*}:H^{*}(X,{\mathbb{Q}})\rightarrow H^{*}(Y,{\mathbb{Q}})$

induced by classes $\kappa({\mathcal{P}})\sqrt{td_{X\times Y}}$, where
${\mathcal{P}}$ is a locally free twisted sheaf over $X\times Y$, such that
the Azumaya ${\mathcal{O}}_{X\times Y}$-algebra
${\mathcal{E}}nd({\mathcal{P}})$ is a deformation (in the sense of Definition
5.18) of the endomorphism algebra of a locally free Fourier-Mukai kernel of an
equivalence of derived categories.333 Note that
$[\kappa({\mathcal{P}})\sqrt{td_{X\times Y}}]_{*}$ is indeed a morphism of
${\mathcal{G}}$, so that ${\mathcal{G}}_{an}$ is indeed a subgroupoid of
${\mathcal{G}}$. This is seen as follows. If $(X\times
Y,{\mathcal{E}}nd({\mathcal{P}}))$ is a deformation of $(X_{0}\times
Y_{0},{\mathcal{E}}nd({\mathcal{P}}_{0}))$ and ${\mathcal{P}}_{0}$ is a
locally free untwisted sheaf, which is the Fourier-Mukai kernel of an
equivalence of derived categories, then
$[\kappa({\mathcal{P}})\sqrt{td_{X\times
Y}}]_{*}=f_{2}\circ[\kappa({\mathcal{P}}_{0})\sqrt{td_{X_{0}\times
Y_{0}}}]_{*}\circ f_{1}$, where $f_{1}$ and $f_{2}$ are parallel-transport
operators, and so morphisms in ${\mathcal{G}}$. Furthermore, writing ${\rm
rank}({\mathcal{P}}_{0})=r$ and
$c_{1}({\mathcal{P}}_{0})=\pi_{X_{0}}^{*}(\lambda_{1})+\pi_{Y_{0}}^{*}(\lambda_{2})$
we see that $[\kappa({\mathcal{P}}_{0})\sqrt{td_{X_{0}\times
Y_{0}}}]_{*}(\bullet)=\exp(-\lambda_{2}/r)\cup[ch({\mathcal{P}}_{0})\sqrt{td_{X_{0}\times
Y_{0}}}]_{*}(\exp(-\lambda_{1}/r)\cup(\bullet))$ is the composition of three
morphisms in ${\mathcal{G}}$.  Morphisms in ${\mathcal{G}}_{an}$ are induced
by analytic correspondences. Indeed, this follows from the Torelli theorem for
parallel-transport Hodge isometries (Theorem 5.2 and Lemma 5.3) and by
definition for morphisms of type (2) above.

The rational LLV lattice of an irreducible holomorphic symplectic manifold $X$
is the vector space

$\widetilde{H}(X,{\mathbb{Q}}):={\mathbb{Q}}\alpha\oplus
H^{2}(X,{\mathbb{Q}})\oplus{\mathbb{Q}}\beta$

endowed with the non-degenerate symmetric bilinear pairing, which restricts to
$H^{2}(X,{\mathbb{Q}})$ as the BBF-pairing, such that $\alpha$ and $\beta$ are
isotropic and orthogonal to $H^{2}(X,{\mathbb{Q}})$ and satisfy
$(\alpha,\beta)=-1$. An orientation for $H^{2}(X,{\mathbb{Q}})$ determines an
orientation of $\widetilde{H}(X,{\mathbb{Q}})$, and so the LLV lattice of
objects of ${\mathcal{G}}$ has a chosen orientation. We assign
$\widetilde{H}(X,{\mathbb{Q}})$ the grading where $\alpha$ has degree $-2$,
$\beta$ has degree $2$, and $H^{2}(X,{\mathbb{Q}})$ has degree $0$.

Let $\widetilde{{\mathcal{G}}}$ be the groupoid, whose objects are irreducible
holomorphic symplectic manifolds and such that morphisms are isometries of
their rational LLV lattices. Results of Taelman’s yield a functor

(1.4) $\widetilde{H}:{\mathcal{G}}\rightarrow\widetilde{{\mathcal{G}}}$

(see Definition 3.3).

We show that the morphism (1.3) is degree reversing (Lemma 4.1). It follows
that every morphism $\phi$ in ${\rm
Hom}_{{\mathcal{G}}_{an}}((X,\epsilon),(Y,\epsilon^{\prime}))$ preserves the
grading up to sign. Consequently, $\widetilde{H}(\phi)$ preserves the grading
of $\widetilde{H}(X,{\mathbb{Q}})$ up to sign and maps $H^{2}(X,{\mathbb{Q}})$
to $H^{2}(Y,{\mathbb{Q}})$ and restricts to $H^{2}(X,{\mathbb{Q}})$ as a Hodge
isometry $\widetilde{H}(\phi)_{0}$ (Lemma 3.4).

Let ${\mathcal{H}dg}$ be the groupoid, whose objects are irreducible
holomorphic symplectic manifolds, and whose morphisms in ${\rm
Hom}_{{\mathcal{H}dg}}(X,Y)$ are rational Hodge isometries
$f:H^{2}(X,{\mathbb{Q}})\rightarrow H^{2}(Y,{\mathbb{Q}})$ preserving the
orientations of the positive cones (Definition 5.5). We get the functor

(1.5) $\widetilde{H}_{0}:{\mathcal{G}}_{an}\rightarrow{\mathcal{H}dg}$

forgetting the orientation of objects and mapping a morphism $\phi$ to
$\widetilde{H}(\phi)_{0}$ or $-\widetilde{H}(\phi)_{0}$ (whichever preserves
the orientations of the positive cones). Let ${\mathcal{G}}_{an}^{[n]}$ be the
full subgroupoid of ${\mathcal{G}}_{an}$, whose objects are of
$K3^{[n]}$-type. Define the subgroupoid ${\mathcal{H}dg}^{[n]}$ of
${\mathcal{H}dg}$ analogously. The following theorem is proved in section 9.

######  Theorem 1.4.

The functor
$\widetilde{H}_{0}:{\mathcal{G}}_{an}^{[n]}\rightarrow{\mathcal{H}dg}^{[n]}$
is full.444This means that $\widetilde{H}_{0}:{\rm
Hom}_{{\mathcal{G}}_{an}^{[n]}}(X_{1},X_{2})\rightarrow{\rm
Hom}_{{\mathcal{H}dg}}(X_{1},X_{2})$ is surjective, for any two objects
$X_{1}$ and $X_{2}$ of ${\mathcal{G}}_{[an]}^{[n]}$. In particular, the
analytic correspondence $\tilde{f}$ of Theorem 1.1 can be chosen to be a
morphism in ${\rm
Hom}_{{\mathcal{G}}^{[n]}_{an}}((X,\epsilon),(Y,\epsilon^{\prime}))$
satisfying $\widetilde{H}_{0}(\tilde{f})=f$, possibly after replacing $f$ by
$-f$, so that it would preserve the orientations of the positive cones.

Let $f:H^{2}(X,{\mathbb{Q}})\rightarrow H^{2}(Y,{\mathbb{Q}})$ be a rational
isometry between two IHSMs of $K3^{[n]}$-type $X$ and $Y$. We say that $f$ is
of $r$-cyclic type, for some positive integer $r$, if there exists a primitive
class $u\in H^{2}(X,{\mathbb{Z}})$, satisfying $(u,u)=2r$ and $(u,\bullet)\in
H^{2}(X,{\mathbb{Z}})^{*}$ is primitive as well, and there exists a parallel-
transport operator $g:H^{2}(X,{\mathbb{Z}})\rightarrow H^{2}(Y,{\mathbb{Z}})$,
such that $f=-g\rho_{u}$, where $\rho_{u}\in O(H^{2}(X,{\mathbb{Q}}))$ is the
reflection in $u$ given by $\rho_{u}(x)=x-\frac{2(u,x)}{(u,u)}u.$ Such $f$ is
necessarily compatible with the orientations of the positive cones. The proof
of Theorem 1.4 relies on the following.

######  Theorem 1.5.

(Corollary 8.5). Let $f:H^{2}(X,{\mathbb{Q}})\rightarrow
H^{2}(Y,{\mathbb{Q}})$ be a rational Hodge isometry of $r$-cyclic type between
two IHSMs $X$ and $Y$ of $K3^{[n]}$-type. There exists over $X\times Y$ a
(possibly twisted) locally free coherent sheaf $E$ of rank $n!r^{n}$, such
that $\phi:=[\kappa(E)\sqrt{td_{X\times Y}}]_{*}$ is a degree reversing
morphism in ${\rm Hom}_{{\mathcal{G}}^{[n]}_{an}}(X,Y)$ and
$\widetilde{H}_{0}(\phi)=f$.

###  1.2. Fourier-Mukai kernels of positive rank and the Lefschetz Standard
Conjecture

Theorem 1.5 leads to an alternative short proof of the Lefschetz standard
conjecture for projective IHSMs of $K3^{[n]}$-type, originally proved in [CM].
Let $X$ be a projective IHSM of $K3^{[n]}$-type. Let $\lambda\in
H^{1,1}(X,{\mathbb{Q}})$ be an ample class. Let $h\in{\rm
End}(H^{*}(X,{\mathbb{Q}}))$ multiply $H^{k}(X,{\mathbb{Q}})[2n]$ by $k$.
Denote by $e_{\lambda}\in{\rm End}(H^{*}(X,{\mathbb{Q}}))$ the Lefschetz
operator of cup product with $\lambda$ and let $e^{\vee}_{\lambda}\in{\rm
End}(H^{*}(X,{\mathbb{Q}}))$ be dual Lefschetz operator, i.e., the unique
element satisfying

$[e_{\lambda},e_{\lambda}^{\vee}]=h,\
[h,e_{\lambda}^{\vee}]=-2e_{\lambda}^{\vee}.$

Choose an IHSM $Y$, such that there exists a rational Hodge isometry
$f:H^{2}(X,{\mathbb{Q}})\rightarrow H^{2}(Y,{\mathbb{Q}})$ of $r$-cyclic type,
for some some integer $r>0$. The existence of such $Y$ follows, for every
$r>0$, by Theorem 1.5 and the surjectivity of the period map555 Choose an
isometry $\eta_{X}:H^{2}(X,{\mathbb{Z}})\rightarrow\Lambda$ with a fixed
lattice $\Lambda$. The period map sends $(X,\eta)$ to
$\eta(H^{2,0}(X))\in{\mathbb{P}}(\Lambda\otimes_{\mathbb{Z}}{\mathbb{C}})$.
Choose $Y$ to be an IHSM of $K3^{[n]}$-type with an isometry
$\eta_{Y}:H^{2}(Y,{\mathbb{Z}})\rightarrow\Lambda$, such that $(X,\eta_{X})$
and $(Y,\eta_{Y})$ belong to the same connected component of the moduli space
of marked IHSM and such that
$\eta_{Y}(H^{2,0}(Y))=\eta_{X}(\rho_{u}(H^{2,0}(X)))$, for a class $u\in
H^{2}(X,{\mathbb{Z}})$ as in Theorem 1.5. Then
$f:=\eta_{Y}^{-1}\eta_{X}(-\rho_{u}):H^{2}(X,{\mathbb{Q}})\rightarrow
H^{2}(Y,{\mathbb{Q}})$ is a rational Hodge isometry of $r$-cyclic type
preserving the orientations of the positive cones.  [Hu1]. The morphism
$\phi\in{\rm Hom}_{{\mathcal{G}}^{[n]}_{an}}(X,Y)$ in Theorem 1.5 is degree
reversing. We get that $e_{\lambda}^{\vee}$ is algebraic, by the following
elementary lemma.

Let $X$ and $Y$ be IHSMs and let $\phi\in{\rm
Hom}_{{\mathcal{G}}_{an}}((X,\epsilon),(Y,\epsilon^{\prime}))$ be a degree
reversing morphism. Then $\tilde{\phi}:=\widetilde{H}(\phi)$ is degree
reversing as well (Lemma 3.4). Let $t\in{\mathbb{Q}}$ be such that
$\tilde{\phi}(\beta)=t\alpha$. Let $\lambda\in H^{2}(X,{\mathbb{Q}})$ be such
that $(\lambda,\lambda)\neq 0$.

######  Lemma 1.6.

The dual Lefschetz operator is given by
$e_{\lambda}^{\vee}=\frac{2}{t(\lambda,\lambda)}\phi^{-1}e_{\tilde{\phi}(\lambda)}\phi$
and is algebraic if $\lambda$ is, as $e_{\tilde{\phi}(\lambda)}$ and the
morphisms $\phi$ and $\phi^{-1}$ are.

The short proof of the lemma is given at the end of Section 4.

######  Example 1.7.

If $X$ is an IHSM of $K3^{[n]}$-type, $n=1,2$, then ${\rm
Aut}_{{\mathcal{G}}^{[n]}_{an}}(X)$ contains a degree reversing involution.
When $X$ is a $K3$ surface such an involution is induced by the ideal sheaf of
the diagonal in $X\times X$, as the ideal sheaf is the kernel of a derived
auto-equivalence. When $X$ is of $K3^{[2]}$-type, a rank $2$ twisted reflexive
sheaf $E$ was constructed on $X\times X$ in [Ma5]. When $X=S^{[2]}$, for a
$K3$ surface $S$, Addington showed that $E$ is the Fourier-Mukai kernel of an
auto-equivalence of $D^{b}(S^{[2]})$ acting on cohomology as an involution
[A]. It follows that $[\kappa(E)\sqrt{td_{X\times X}}]_{*}$ is a degree
reversing involution in ${\rm Aut}_{{\mathcal{G}}^{[2]}_{an}}(X)$, for every
$X$ of $K3^{[2]}$-type. Does a degree reversing automorphism exist in ${\rm
Aut}_{{\mathcal{G}}^{[n]}_{an}}(X)$ for every $X$ of $K3^{[n]}$-type for
$n\geq 3$? What about other deformation types?

###  1.3. The Pontryagin product

In Section 10 we define a Pontryagin product $\star$ on the cohomology
$H^{*}(X,{\mathbb{Q}})$ of every irreducible holomorphic symplectic manifold
$X$ with vanishing odd cohomology. The unit with respect to $\star$ is
$c_{X}[pt]/n!$, where $\dim(X)=2n$, $[pt]\in H^{top}(X,{\mathbb{Q}})$ is the
class Poincaré dual to the class of a point, and $c_{X}$ is the Fujiki
constant.666 The Fujiki constant $c_{X}$ is the positive rational number such
that the equality
$\int_{X}\lambda^{2n}=\frac{(2n)!}{2^{n}n!}c_{X}(\lambda,\lambda)^{n}$ holds
for all $\lambda\in H^{2}(X,{\mathbb{Q}})$. The Fujiki constant is calculated
for all known irreducible holomorphic symplectic manifolds in [Ra]. If $X$ of
$K3^{[n]}$-type, then $c_{X}=1$.  The product $\star$ maps
$H^{k}(X,{\mathbb{Q}})\otimes H^{l}(X,{\mathbb{Q}})$ to
$H^{k+l-4n}(X,{\mathbb{Q}})$, so that $\star$ is compatible with the grading
for which $H^{k}(X,{\mathbb{Q}})$ has degree $4n-k$. Parallel-transport
operators induce isomorphisms of the Pontryagin rings (Lemma 10.2).

The Pontryagin product $\star$ on the cohomology $H^{*}(S,{\mathbb{Z}})$ of a
$K3$ surface $S$ is simple to describe. The unit with respect to $\star$ is
the class $[pt]\in H^{4}(X,{\mathbb{Z}})$ Poincaré dual to a point. The
product $\star$ maps $H^{k}(S,{\mathbb{Z}})\otimes H^{l}(S,{\mathbb{Z}})$ to
$H^{k+l-4}(S,{\mathbb{Z}})$ and satisfies
$\lambda_{1}\star\lambda_{2}=(\lambda_{1},\lambda_{2})\in
H^{0}(S,{\mathbb{Z}})$, for classes $\lambda_{1},\lambda_{2}\in
H^{2}(S,{\mathbb{Z}})$. Let
$\rho_{u}:\widetilde{H}(S,{\mathbb{Z}})\rightarrow\widetilde{H}(S,{\mathbb{Z}}),$
be the reflection $\rho_{u}(r,c,s)=(-s,c,-r)$ in the Mukai vector $u:=(1,0,1)$
of the structure sheaf ${\mathcal{O}}_{S}$. Then $-\rho_{u}$ conjugates the
usual product on $H^{*}(S,{\mathbb{Z}})$ to the Pontryagin product. Note that
$-\rho_{u}$ corresponds to the action on cohomology of the autoequivalence
$\Phi_{{\mathcal{I}}_{\Delta}}$ of $D^{b}(S)$, whose Fourier-Mukai kernel is
the ideal sheaf ${\mathcal{I}}_{\Delta}$ of the diagonal $\Delta$ in $S\times
S$.

Morphisms in ${\mathcal{G}}_{[an]}$, which are associated to equivalences of
derived categories with a Fourier-Mukai kernel of positive rank, are degree
reversing. This corresponds to a normalization which annihilates the first
Chern class of the Fourier-Mukai kernel. We introduce a further normalization,
which assures that the compositions of any two such degree reversing morphisms
would map the usual identity $1\in H^{0}(X,{\mathbb{Q}})$ to itself. We
speculate below that thus normalized, the degree reversing morphisms conjugate
the cup product to the Pontryagin product. We introduce this further
normalization in the next paragraph.

Let ${\mathbb{Q}}^{\times}$ be the multiplicative group of rational numbers.
Let

$\chi:{\mathcal{G}}_{an}\rightarrow{\mathbb{Q}}^{\times}$

be the groupoid homomorphism sending every object to ${\mathbb{Q}}^{\times}$
and sending a morphism $\phi$ to $t$, if $\phi$ is degree preserving and
$\widetilde{H}(\phi)(\alpha)=t\alpha$, or if $\phi$ is degree reversing and
$\widetilde{H}(\phi)(\alpha)=\frac{1}{t}\beta.$ Given a non-zero rational
number $t$, denote by $\mu_{t}:H^{*}(X,{\mathbb{Q}})[2n]\rightarrow
H^{*}(X,{\mathbb{Q}})[2n]$ the graded linear transformation multiplying
$H^{2k}(X,{\mathbb{Q}})[\dim(X)]$ by $t^{k}$. Let

(1.6) ${\mathcal{R}}^{[n]}_{an}$

be the subgroupoid of ${\mathcal{G}}^{[n]}_{an}$ with the same objects, but
whose morphisms $\phi$ have the property that $\mu_{\chi(\phi)}\circ\phi$
conjugates the usual product on $H^{*}(X,{\mathbb{Q}})$ to itself, if $\phi$
is degree-preserving, and to $\star$ if $\phi$ is degree reversing.

######  Conjecture 1.8.

The subgroupoid ${\mathcal{R}}^{[n]}_{an}$ is in fact the whole of
${\mathcal{G}}^{[n]}_{an}$. Furthermore, if an object of positive rank $F$ in
$D^{b}(X\times Y)$ is the Fourier-Mukai kernel of an equivalence
$\Phi_{F}:D^{b}(X)\rightarrow D^{b}(Y)$ of derived categories of projective
irreducible holomorphic symplectic manifolds $X$ and $Y$ of $K3^{[n]}$-type,
then the rank of $F$ is $n!t^{n}$, for some rational number $t$, and setting
$\phi:=\mu_{t}\circ[\kappa(F)\sqrt{td_{X\times
Y}}]_{*}:H^{*}(X,{\mathbb{Q}})\rightarrow H^{*}(Y,{\mathbb{Q}})$ we have

(1.7) $\phi(\gamma_{1}\cup\gamma_{2})=\phi(\gamma_{1})\star\phi(\gamma_{2}),$

for all $\gamma_{1},\gamma_{2}\in H^{*}(X,{\mathbb{Q}})$.

The above stated equality ${\rm rank}(F)=n!t^{n}$, for some rational number
$t$, is known [Bec, Theorem 4.14]. As evidence for the conjecture we show that
the examples of morphisms of ${\mathcal{G}}^{[n]}_{an}$ used in the proof of
Theorem 1.4 all satisfy Conjecture 1.8. Consequenly, Theorem 1.4 holds even
after replacing ${\mathcal{G}}^{[n]}_{an}$ by ${\mathcal{R}}^{[n]}_{an}$ and
we get the following result proven in Section 10.

######  Proposition 1.9.

(Proposition 10.7). The restriction of the functor
$\widetilde{H}_{0}:{\mathcal{G}}^{[n]}_{an}\rightarrow{\mathcal{H}dg}^{[n]}$
to the subgroupoid ${\mathcal{R}}^{[n]}_{an}$ remains full.

Let $X$ and $Y$ be projective IHSMs of $K3^{[n]}$-type. The proof of
Proposition 10.7 implies that the degree reversing correspondence
$\kappa(E)\sqrt{td_{X\times Y}}$ in Theorem 1.5 satisfies Conjecture 1.8. Set
$\phi:=\mu_{t}\circ[\kappa(E)\sqrt{td_{X\times Y}}]_{*}$, as in the
conjecture. Let $[\Delta]\in H^{*}(X^{3},{\mathbb{Q}})$ the class of the small
diagonal $\Delta$ in $X^{3}$. The following Corollary is proved in Section
10.2.

######  Corollary 1.10.

The Pontryagin product $\star:H^{*}(Y\times Y,{\mathbb{Q}})\rightarrow
H^{*}(Y,{\mathbb{Q}})$ is induced by the algebraic class $\phi^{3}([\Delta])$
in $H^{*}(Y^{3},{\mathbb{Q}})$. In particular, Pontryagin product
$\gamma\star:H^{*}(Y,{\mathbb{Q}})\rightarrow H^{*}(Y,{\mathbb{Q}})$ by an
algebraic class $\gamma$ is induced by an algebraic correspondence.

The algebraicity of the dual Lefschetz operator $e_{\lambda}^{\vee}$ in Lemma
1.6 is a special case of the above corollary, since $e_{\lambda}^{\vee}$ is a
scalar multiple of the operator of Pontryagin product with the algebraic class
$\phi^{-1}(\tilde{\phi}(\lambda))$.

###  1.4. Outline of the proof of Theorem 1.4

The reader familiar with Buskin’s proof of the case $n=1$ of the Theorem will
notice the similarity of our strategy to his. Buskin considers examples of two
dimensional moduli spaces $M$ of vector bundles over a $K3$ surface $S$ and
the universal bundle ${\mathcal{U}}$ over $M\times S$. $M$ is a $K3$ surface
and so the product $M\times S$ is a hyper-Kähler manifold. Buskin uses
Verbitsky’s results on hyperholomorphic vector bundles over hyper-Kähler
manifolds to deform the pair $(M\times S,{\mathcal{U}})$ to twisted vector
bundles $(X_{1}\times X_{2},{\mathcal{U}}^{\prime})$ over products of not
necessarily projective $K3$ surfaces $X_{1}$ and $X_{2}$. Buskin proves that
the groupoid ${\mathcal{H}dg}^{[1]}$ is generated by parallel-transport
operators, which are isomorphisms of Hodge structures, and by the restrictions
to $H^{2}(X_{1},{\mathbb{Q}})$ of the degree reversing isometries
$[\kappa({\mathcal{U}}^{\prime})\sqrt{td_{X_{1}\times
X_{2}}}]_{*}:\widetilde{H}(X_{1},{\mathbb{Q}})\rightarrow\widetilde{H}(X_{2},{\mathbb{Q}})$.

We first construct the functor
$\widetilde{H}_{0}:{\mathcal{G}}_{an}\rightarrow{\mathcal{H}dg}$ and establish
its properties in Sections 3 and 4. The construction depends heavily on the
recent work of Taelman [Ta], as well as on the work of the author and
independently of Beckmann describing the action of equivalences of derived
categories on the LLV lattice [Bec, Ma6]. In section 7 we lift the equivalence
of derived categories $\Phi_{\mathcal{U}}:D^{b}(M)\rightarrow D^{b}(S)$, with
the universal bundle ${\mathcal{U}}$ as a kernel, to an equivalence
$\Phi_{{\mathcal{U}}^{[n]}}:D^{b}(M^{[n]})\rightarrow D^{b}(S^{[n]})$ of
derived categories of Hilbert schemes. This is done by conjugating the
equivalence of equivariant derived categories on the cartesian products
$\Phi_{{\mathcal{U}}^{\boxtimes
n}}:D^{b}_{{\mathfrak{S}}_{n}}(M^{n})\rightarrow
D^{b}_{{\mathfrak{S}}_{n}}(S^{n})$, obtained from the cartesian power of
$\Phi_{\mathcal{U}}$, with the Briedgeland-King-Reid equivalence
$BKR:D^{b}_{{\mathfrak{S}}_{n}}(S^{n})\rightarrow D^{b}(S^{[n]})$ and its
analogue for $M$. The Briedgeland-King-Reid equivalence is reviewed in Section
6. The assignment $S\mapsto S^{[n]}$ and
$\Phi_{\mathcal{U}}\mapsto\Phi_{{\mathcal{U}}^{[n]}}$ extends to define a
functor

(1.8) $\Theta_{n}:{\mathcal{G}}_{an}^{[1]}\rightarrow{\mathcal{G}}_{an}^{[n]}$

constructed in (7.12). The Fourier-Mukai kernel ${\mathcal{U}}^{[n]}$ is again
a locally free sheaf over $M^{[n]}\times S^{[n]}$. Again we use Verbitsky’s
results on hyperholomorphic vector bundles on hyper-Kähler manifolds to deform
the pair $(M^{[n]}\times S^{[n]},{\mathcal{U}}^{[n]})$ to pairs $(X_{1}\times
X_{2},E)$ consisting of a twisted vector bundle $E$ over the product
$X_{1}\times X_{2}$ of not necessarily projective IHSMs of $K3^{[n]}$-type.
The general techniques are developed in Section 5 and applied to
${\mathcal{U}}^{[n]}$ in Section 8.

It remains to prove that we get a full functor already when we restrict the
functor
$\widetilde{H}_{0}:{\mathcal{G}}^{[n]}_{an}\rightarrow{\mathcal{H}dg}^{[n]}$
to the subgroupoid of ${\mathcal{G}}_{an}^{[n]}$ generated by parallel-
transport operators, which are Hodge isometries, and morphisms in the image of
$\Theta_{n}$. This is done as follows.

Step 1: (Generators for the group of rational isometries) Let $L$ be a lattice
isometric to the second cohomology of a $K3$ surface. Let $\Lambda$ be the
orthogonal direct sum $L\oplus{\mathbb{Z}}\delta$, where
$(\delta,\delta)=2-2n$. Then $\Lambda$ is isometric to the second cohomology
of an irreducible holomorphic symplectic manifold $X$ of $K3^{[n]}$-type
endowed with the BBF-pairing [Be1]. The monodromy group of $X$ acts on
$H^{2}(X,{\mathbb{Z}})$ via a normal subgroup of $O(H^{2}(X,{\mathbb{Z}}))$
and so corresponds to a well defined subgroup ${\rm Mon}(\Lambda)$ of
$O(\Lambda)$, by [Ma2, Theorem 1.6 and Lemma 4.10]. We first prove that the
group $O(\Lambda_{\mathbb{Q}})$ of rational isometries is generated by ${\rm
Mon}(\Lambda)$ and $O(L_{\mathbb{Q}})$ (Proposition 2.1). It follows from the
Cartan-Dieudonné Theorem that every rational isometry
$\psi:\Lambda_{\mathbb{Q}}\rightarrow\Lambda_{\mathbb{Q}}$, which preserves
the orientation of the positive cone, decomposes as the composition
$\psi=\psi_{k}\circ\cdots\circ\psi_{1}$, where $\psi_{i}$ belongs to the
double orbit ${\rm Mon}(\Lambda)(-\rho_{u}){\rm Mon}(\Lambda)$, where
$\rho_{u}$ is the reflection in a primitive integral class $u\in L$ with
$(u,u)>0$ (Corollary 2.5). The double orbit ${\rm Mon}(\Lambda)(-\rho_{u}){\rm
Mon}(\Lambda)$ depends only on $(u,u)$ and is otherwise independent of the
class $u$, since so does the double orbit $O^{+}(L)(-\rho_{u})O^{+}(L),$ by
[Bu, Prop. 3.3], and ${\rm Mon}(\Lambda)$ contains $O^{+}(L)$.

Step 2: (Reduction to the double orbit of a reflection) Let
$f:H^{2}(X,{\mathbb{Q}})\rightarrow H^{2}(Y,{\mathbb{Q}})$ be a rational Hodge
isometry preserving the orientations of the positive cones, where $X$ and $Y$
are of $K3^{[n]}$-type. Choose isometries
$\eta_{X}:H^{2}(X,{\mathbb{Z}})\rightarrow\Lambda$ and
$\eta_{Y}:H^{2}(Y,{\mathbb{Z}})\rightarrow\Lambda$, so that the marked pairs
$(X,\eta_{X})$ and $(Y,\eta_{Y})$ belong to the same connected component
${\mathfrak{M}}_{\Lambda}^{0}$ of the moduli space of marked IHSMs of
$K3^{[n]}$-type. Let $\Omega_{\Lambda}$ be the period domain of marked IHSMs
of $K3^{[n]}$-type and let
$P:{\mathfrak{M}}_{\Lambda}^{0}\rightarrow\Omega_{\Lambda}$ be the period map
(see (5.1) for the definitions of $\Omega_{\Lambda}$ and $P$). Set
$\psi:=\eta_{Y}\circ f\circ\eta_{X}^{-1}\in O(\Lambda_{\mathbb{Q}})$ and
choose a decomposition $\psi=\psi_{k}\circ\cdots\circ\psi_{1}$ as above. The
group $O^{+}(\Lambda_{\mathbb{Q}})$ acts on $\Omega_{\Lambda}$ and
$\psi(P(X,\eta_{X}))=P(Y,\eta_{Y})$. Set
$\ell_{i}:=(\psi_{i}\circ\cdots\circ\psi_{1})(P(X,\eta_{X}))$, $0\leq i\leq
k.$ The surjectivity of the period map implies the existence of marked pairs
$(X_{i},\eta_{i})\in{\mathfrak{M}}_{\Lambda}^{0}$, such that
$P(X_{i},\eta_{i})=\ell_{i}$, where we choose $(X_{0},\eta_{0}):=(X,\eta_{X})$
and $(X_{k},\eta_{k}):=(Y,\eta_{Y})$. Set
$f_{i}:=\eta_{i}^{-1}\psi_{i}\circ\eta_{i-1}:H^{2}(X_{i-1},{\mathbb{Q}})\rightarrow
H^{2}(X_{i},{\mathbb{Q}})$. Then $f=f_{k}\circ\cdots\circ f_{1}$. If $k=0$, so
that $\psi=\eta_{Y}\circ f\circ\eta_{X}^{-1}$ is the identity, then
$f=\eta_{Y}^{-1}\eta_{X}$ is a parallel-transport operator and a Hodge
isometry, and so $f$ lifts to a morphism $\phi\in{\rm
Hom}_{{\mathcal{G}}_{an}}(X,Y)$ with $\widetilde{H}_{0}(\phi)=f$, by
Verbitsky’s Torelli Theorem 5.2. It remains to prove that each $f_{i}$ belongs
to the image of $\widetilde{H}_{0}$.

Step 3: The equivalence of derived categories
$\Phi_{{\mathcal{U}}^{[n]}}:D^{b}(M^{[n]})\rightarrow D^{b}(S^{[n]})$
described above yields a morphism in ${\mathcal{G}}^{[n]}_{an}$ which is
mapped via $\widetilde{H}_{0}$ to a Hodge isometry
$H^{2}(M^{[n]},{\mathbb{Q}})\rightarrow H^{2}(S^{[n]},{\mathbb{Q}})$ which in
turn is simply the extension777 We note that
$\widetilde{H}([ch({\mathcal{U}}^{[n]})\sqrt{td_{M^{[n]}\times
S^{[n]}}}]_{*}):\widetilde{H}(M^{[n]},{\mathbb{Q}})\rightarrow\widetilde{H}(S^{[n]},{\mathbb{Q}})$
is not the naive extension of $[ch({\mathcal{U}})\sqrt{td_{M\times
S}}]_{*}:\widetilde{H}(M,{\mathbb{Q}})\rightarrow\widetilde{H}(S,{\mathbb{Q}})$,
by [Ta, Theorem 9.4] in the $K3^{[2]}$-case and by [Ma6, Theorem 12.2] and
[Bec, Theorem 7.4] in the $K3^{[n]}$-case, $n\geq 2$ (see Lemma 7.2).
Nevertheless, when we replace $ch({\mathcal{U}}^{[n]})$ by
$\kappa({\mathcal{U}}^{[n]})$ the resulting isometry
$\widetilde{H}([\kappa({\mathcal{U}}^{[n]})\sqrt{td_{M^{[n]}\times
S^{[n]}}}]_{*}):\widetilde{H}(M^{[n]},{\mathbb{Q}})\rightarrow\widetilde{H}(S^{[n]},{\mathbb{Q}})$
is the naive extension of $[\kappa({\mathcal{U}})\sqrt{td_{M\times
S}}]_{*}:\widetilde{H}(M,{\mathbb{Q}})\rightarrow\widetilde{H}(S,{\mathbb{Q}})$,
by Corollary 7.3.  of the isometery $H^{2}(M,{\mathbb{Q}})\rightarrow
H^{2}(S,{\mathbb{Q}})$ associated to $\Phi_{\mathcal{U}}$ (Corollary 7.3). The
resulting Hodge isometry $H^{2}(M^{[n]},{\mathbb{Q}})\rightarrow
H^{2}(S^{[n]},{\mathbb{Q}})$ is thus in the double orbit ${\rm
Mon}(\Lambda)(-\rho_{u}){\rm Mon}(\Lambda)$, where $\rho_{u}$ is the
reflection in a primitive integral class $u\in L$ with $(u,u)>0$. The proof of
Corollary 7.3 relies on [Ma6, Theorem 12.2] obtained independently by Beckman
[Bec, Theorem 7.4]. All such double orbits are obtained that way by varying
the Mukai vector of the two-dimensional moduli space $M$, by [Bu, Prop. 3.3].
We prove that every rational Hodge isometry
$f:H^{2}(X,{\mathbb{Q}})\rightarrow H^{2}(Y,{\mathbb{Q}})$, with $f$ in a
fixed double orbit of a reflection in $u\in L$ and preserving the orientation
of the positive cones, is of the form
$\widetilde{H}_{0}([\kappa(E)\sqrt{td_{X\times Y}}]_{*})$, for a pair
$(X\times Y,E)$, which is a deformation of $(M^{[n]}\times
S^{[n]},{\mathcal{U}}^{[n]})$ for a suitable choice of $M$ (Corollary 8.5).

###  1.5. Why does the generalized Kummer deformation type require a
modification?

Let $X$ be an abelian surface. The fiber $Y$ over $0\in X$ of the albanese
morphism $alb:X^{[n+1]}\rightarrow X$ is a $2n$-dimensional IHSM known as a
generalized Kummer variety [Be1]. Section 2 applies both to varieties of
$K3^{[n]}$-type and to those of generalized Kummer deformation type. Sections
3 to 5 are written for every IHSM. The lift of a derived equivalence of
surfaces to their Hilbert schemes in Section 6 applies also for abelian
surfaces. In particular, the functor
$\widetilde{H}_{0}:{\mathcal{G}}_{an}\rightarrow{\mathcal{H}dg}$ is well
defined for the respective subgroupoids corresponding to $2n$-dimensional
IHSMs of generalized Kummer deformation type, for all $n\geq 2$. The
difficulty is that for IHSMs of generalized Kummer deformation type we do not
have an analogue of the functor
$\Theta_{n}:{\mathcal{G}}^{[1]}\rightarrow{\mathcal{G}}^{[n]}$ (given in (1.8)
and constructed for $K3$ surfaces in Section 7).

The albanese morphism $alb:X^{[n+1]}\rightarrow X$ is invariant under the
action on $X^{[n+1]}$ of the subgroup $X[n+1]$ of $X$ generated by torsion
points of order $n+1$. Hence, $X[n+1]$ embeds as a subgroup ${\rm Aut}_{0}(Y)$
of the automorphism group of $Y$, as well as a subgroup ${\rm Mon}_{0}(Y)$ of
its monodromy group. Indeed, ${\rm Mon}_{0}(Y)$ is the monodromy group of the
isotrivial fibration $alb$. ${\rm Aut}_{0}(Y)$ is the subgroup of the
automorphism group of $Y$ acting trivially on $H^{k}(Y,{\mathbb{Z}})$, for
$k=2$ and $k=3$, by [BNS, Theorem 3 and Corollary 5] and [Ogu, Theorem 1.3].

There are two obstructions to the construction of the naive analogue of
$\Theta_{n}$. First, parallel-transport operators between abelian surfaces
$X_{1}$ and $X_{2}$ lift only to cosets of parallel-transport operators of
their $2n$-dimensional generalized Kummer varieties $Y_{i}$ by the subgroups
${\rm Mon}_{0}(Y_{i})$ [Ma8, Theorem 1.4]. Furthermore, it is not clear that
equivalences of derived categories of abelian surfaces induce equivalences of
the derived categories of their generalized Kummers (see the discussion in
Section 4.4 of Ploog’s thesis [P2]).

The group ${\rm Aut}_{0}(Y)$ varies with $Y$ in a local system over moduli, by
[HT, Theorem 2.1]. Denote by $H^{*}(Y,{\mathbb{Q}})^{{\rm Aut}_{0}(Y)}$ the
subring of $H^{*}(Y,{\mathbb{Q}})$ consisting of elements invariant under
${\rm Aut}_{0}(Y)$. The first obstruction mentioned above is resolved when we
redefine the morphisms in ${\mathcal{G}}$ so that ${\rm
Hom}_{\mathcal{G}}(Y_{1},Y_{2})$ consists of isometries from
$H^{*}(Y_{1},{\mathbb{Q}})^{{\rm Aut}_{0}(Y_{1})}$ to
$H^{*}(Y_{2},{\mathbb{Q}})^{{\rm Aut}_{0}(Y_{2})}$. We believe that the
functor $\Theta_{n}$ has an analogue associating to every equivalence $\Phi$,
of derived categories of abelian surfaces with a Fourier-Mukai kernel of non-
zero rank, a degree reversing Hodge isometry $\Theta_{n}(\Phi)$ of the
invariant subrings of their generalized Kummers. We plan to return to this
construction in the near future.

##  2\. Generators for the rational isometry group

Let $U$ be the rank $2$ lattice with basis $\\{e_{1},e_{2}\\}$ satisfying
$(e_{i},e_{i})=0$, $i=1,2$, and $(e_{1},e_{2})=-1$. Let $L$ be an even
unimodular lattice of signature $(3,s_{-})$, $s_{-}\geq 3$, which contains a
sublattice isometric to the orthogonal direct sum of three copies of $U$. Let
the lattice $\Lambda$ be the orthogonal direct sum
$L\oplus{\mathbb{Z}}\delta$, where $(\delta,\delta)=-2d$ for some positive
integer $d$. Given a non-degenerate lattice $M$, denote by $O^{+}(M)$ the
subgroup of isometries of $M$ which preserve the orientation of the positive
cone of $M\otimes_{\mathbb{Z}}{\mathbb{R}}$ [Ma4, Sec. 4]. Let $\Gamma\subset
O^{+}(\Lambda)$ be the subgroup which acts on the discriminant group
$\Lambda^{*}/\Lambda$ by multiplication by $\pm 1$. Set
$\Lambda_{\mathbb{Q}}:=\Lambda\otimes_{\mathbb{Z}}{\mathbb{Q}}$ and
$L_{\mathbb{Q}}:=L\otimes_{\mathbb{Z}}{\mathbb{Q}}$. We regard
$O(L_{\mathbb{Q}})$ as a subgroup of $O(\Lambda_{\mathbb{Q}})$ by extending
each isometry so that it leaves $\delta$ invariant.

######  Proposition 2.1.

$O(\Lambda_{\mathbb{Q}})$ is generated by $\Gamma$ and $O(L_{\mathbb{Q}})$.

The following Lemmas will be needed for the proof of the above Proposition.

######  Lemma 2.2.

Let $\alpha=\lambda+k\delta$, where $\lambda$ is a primitive class in $L$ and
$k\in{\mathbb{Z}}$. There exists an isometry $g\in\Gamma$, such that
$g(\alpha)$ belongs to $L$.

###### Proof.

Let $\widetilde{\Lambda}$ be the orthogonal direct sum of $L$ and $U$ and let
$\iota:\Lambda\rightarrow\widetilde{\Lambda}$ be the isometric embedding
restricting to the identity on $L$ and sending $\delta$ to the element of the
second direct summand $U$ with coordinate $(1,d)$ in the basis
$\\{e_{1},e_{2}\\}$. Let $v\in\widetilde{\Lambda}$ be the element with
coordinates $(1,-d)$ of the second direct summand $U$, so that $(v,v)=2d$ and
$v^{\perp}=\iota(\Lambda)$, where $v^{\perp}$ is the co-rank $1$ sublattice of
$\widetilde{\Lambda}$ orthogonal to $v$. The sublattice $\Sigma_{1}:={\rm
span}_{\mathbb{Z}}\\{\iota(\alpha),v\\}$ of $\widetilde{\Lambda}$ is
saturated. Indeed, if $a,b\in{\mathbb{Q}}$ and
$a\iota(\alpha)+bv=a\lambda+(ak+b)e_{1}+(adk-bd)e_{2}$ is an integral class of
$\widetilde{\Lambda}$, then $a$ must be an integer, since $\lambda$ is
primitive, hence $b$ must be an integer as well. Let $\beta\in L$ be a
primitive class satisfying $(\beta,\beta)=(\alpha,\alpha)$. The sublattice
$\Sigma_{2}:={\rm span}_{\mathbb{Z}}\\{\beta,v\\}$ is saturated in
$\widetilde{\Lambda}$ and is isometric to $\Sigma_{1}$. There exists an
isometry $g\in O(\widetilde{\Lambda})$ satisfying $g(v)=v$ and
$g(\iota(\alpha))=\beta$, by results of Nikulin [N, Ma2, Lemma 8.1]. Set
$\tilde{\gamma}:=g$, if $g$ belongs to $O^{+}(\widetilde{\Lambda}),$ and
$\tilde{\gamma}:=-g$ otherwise. Then
$\tilde{\gamma}\circ\iota=\iota\circ\gamma$, for a unique element
$\gamma\in\Gamma$, by [Ma2, Theorem 1.6 and Lemma 4.2], and $\gamma(\alpha)$
belongs to $L$. ∎

Let $G$ be the subgroup of $O(\Lambda_{\mathbb{Q}})$ generated by $\Gamma$ and
$O(L_{\mathbb{Q}})$.

######  Lemma 2.3.

Let $\lambda\in L_{\mathbb{Q}}$ and $t\in{\mathbb{Q}}$ and assume that
$(\lambda,\lambda)\neq 0$. There exists an isometry $g\in G$, such that
$g(\lambda+t\delta)$ belongs to $L_{\mathbb{Q}}$.

###### Proof.

Let $q\in{\mathbb{Q}}^{\times}$ be such that $q(\lambda+t\delta)$ is integral.
Choose an integral and primitive class $\lambda^{\prime}\in L$ satisfying
$(\lambda^{\prime},\lambda^{\prime})=q^{2}(\lambda,\lambda)$. There exists an
isometry $f\in O(L_{\mathbb{Q}})$, such that $\lambda^{\prime}=f(q\lambda)$,
by [G, Prop. 2.35]. Then $f(q(\lambda+t\delta))=\lambda^{\prime}+qt\delta$ is
integral and there exists $h\in\Gamma$, such that $h(f(q(\lambda+t\delta)))$
is a primitive class in $L$, by Lemma 2.2. The statement follows, since
$h\circ f$ belongs to $G$. ∎

###### Proof of Proposition 2.1.

Choose $u\in L$ satisfying $(u,u)=2d+2$. Then $(u+\delta,u+\delta)=2$. Let
$\rho_{u+\delta}$ be the reflection of $\Lambda$ in $(u+\delta)^{\perp}$

$\rho_{u+\delta}(x)=x-(u+\delta,x)(u+\delta).$

Then $\rho_{u+\delta}(\delta)=(2d)u+(1+2d)\delta$. Furthermore,
$-\rho_{u+\delta}$ belongs to $\Gamma$. There exists $g_{1}\in G$, such that
$g_{1}(-\rho_{u+\delta}(\delta))$ belongs to $L_{\mathbb{Q}}$, by Lemma 2.3.

Let $\phi$ be an isometry in $O(\Lambda_{\mathbb{Q}})$. Write
$\phi(\delta)=\lambda+t\delta$. If $(\lambda,\lambda)=0$ choose $u\in L$, such
that $(u,\lambda)=0$ and $(u,u)=2d+2$. Then
$\rho_{u+\delta}(\phi(\delta))=\lambda+(2d)u+(1+2d)\delta$ and the self-
intersection $(\lambda+(2d)u,\lambda+(2d)u)=4d^{2}(2d+2)$ is non-zero. Now,
$\phi$ belongs to $G$, if and only if $-\rho_{u+\delta}\phi$ belongs to $G$.
Hence, we may assume that $(\lambda,\lambda)\neq 0$. Then there exists
$g_{2}\in G$ such that $(g_{2}\circ\phi)(\delta)$ belongs to $L_{\mathbb{Q}}$,
by Lemma 2.3. There exists $h\in O(L_{\mathbb{Q}})$, such that
$h(g_{2}(\phi(\delta)))=g_{1}(-\rho_{u+\delta}(\delta))$, by [G, Prop. 2.35].
Hence, $(-\rho_{u+\delta}g_{1}^{-1}hg_{2}\phi)(\delta)=\delta$, and so
$-\rho_{u+\delta}g_{1}^{-1}hg_{2}\phi$ belongs to $O(L_{\mathbb{Q}})$. Hence,
$\phi$ belongs to $G$. ∎

######  Lemma 2.4.

$O(L_{\mathbb{Q}})$ is generated by $O(L)$ and reflections $\rho_{u}$ in
$u^{\perp}$, for $u\in L$ satisfying $(u,u)>0$.

###### Proof.

$O(L_{\mathbb{Q}})$ is generated by reflections $\rho_{u}$, with $(u,u)\neq
0$, by [G, Prop. 2.36]. The isometry group $O(L)$ acts transitively on the set
$\\{u\ :(u,u)=2d\\}$, $d\in{\mathbb{Z}}$, by [N, Th. 1.14.4]. Consider the
above basis $\\{e_{1},e_{2}\\}$ of $U$ and observe that
$\rho_{e_{1}-de_{2}}\rho_{e_{1}+de_{2}}$ is minus the identity of $U$. Hence,
for every element $u\in L$ with $(u,u)<0$, there exists an element $w\in L$
with $(w,w)=-(u,u)$, such that $\rho_{u}\rho_{w}$ belongs to $O(L)$. Thus, it
sufficed to consider reflections $\rho_{u}$ with $(u,u)>0$. ∎

######  Corollary 2.5.

Let $\phi$ be an isometry in $O^{+}(\Lambda_{\mathbb{Q}})$. There exist a
positive integer $k$, integral elements $u_{i}\in L$ satisfying
$(u_{i},u_{i})\geq 2$, $1\leq i\leq k$, and elements $\gamma_{i}\in\Gamma$,
$0\leq i\leq k$, such that

$\phi=(-1)^{k}\gamma_{k}\rho_{u_{k}}\gamma_{k-1}\rho_{u_{k-1}}\cdots\gamma_{1}\rho_{u_{1}}\gamma_{0},$

where $\rho_{u_{i}}(x)=x-\frac{2(u_{i},x)}{(u_{i},u_{i})}u_{i}$ is the
reflection of $\Lambda_{\mathbb{Q}}$ in $u_{i}^{\perp}$.

###### Proof.

Follows immediately from Proposition 2.1, Lemma 2.4, the fact that that
$O^{+}(L)$ is contained in $\Gamma$, and the fact that $-\rho_{u_{i}}$ belongs
to $O^{+}(\Lambda_{\mathbb{Q}})$ if $(u_{i},u_{i})>0$, since the signature of
$\Lambda$ is $(3,1+s_{-})$ and $3$ is odd. ∎

######  Remark 2.6.

Note that in Corollary 2.5 we can require $\gamma_{i}$ to belong to any
subgroup $\Gamma_{0}$ of $\Gamma$, such that $\Gamma_{0}$ and $O^{+}(L)$
generate $\Gamma$. When $L$ is the $K3$ lattice, the group $\Gamma$ is the
monodromy group of IHSMs of $K3^{[d+1]}$ deformation type [Ma2, Theorem 1.6].
If $L=U^{\oplus 3}$, the monodromy of generalized Kummer varieties of
dimension $2d-2$, for $d\geq 3$, is the following index $2$ subgroup
$\Gamma_{0}$ of $\Gamma$. $\Gamma$ acts of the discriminant froup
$\Lambda^{*}/\Lambda$ by a character $\xi:\Gamma\rightarrow\\{\pm 1\\}$, by
definition of $\Gamma$. Let $\det:\Gamma\rightarrow\\{\pm 1\\}$ be the
determinant character. Then $\Gamma_{0}$ is the kernel of the product
character $\det\cdot\xi:\Gamma\rightarrow\\{\pm 1\\}$, by [Ma8, Theorem 1.4].
The character $\det\cdot\xi$ has value $-1$ on a reflection $\rho_{\omega}\in
O^{+}(L)$ in a class $w\in L$ with $(w,w)=-2$, and so $\Gamma_{0}$ and
$O^{+}(L)$ generate $\Gamma$.

##  3\. The LLV Lie algebra

In Section 3.1 we recall the LLV Lie algebra ${\mathfrak{g}}_{X}$ and its
action on the cohomology of a $2n$-dimensional IHSM X. We review the
${\mathfrak{g}}_{X}$-equivariant isomorphism between the subring
$SH^{*}(X,{\mathbb{Q}})$ of $H^{*}(X,{\mathbb{Q}})$ generated by
$H^{2}(X,{\mathbb{Q}})$ and the $n$-th symmetric power of the LLV lattice
$\widetilde{H}(X,{\mathbb{Q}})$. In Section 3.2 we recall Taelman’s definition
of the functor
$\widetilde{H}:{\mathcal{G}}\rightarrow\widetilde{{\mathcal{G}}}$, metioned in
(1.4), sending equivalences of derived categories to isometries of LLV
lattices. Given a morphism $\phi\in{\rm Hom}_{\mathcal{G}}(X,Y)$, which is
degree preserving up to sign, we relate in Lemmas 3.6 and 3.7 the restrictions
of $\phi$ and of $\widetilde{H}(\phi)$ to $H^{2}(X,{\mathbb{Q}})$.

###  3.1. The subring generated by $H^{2}(X,{\mathbb{Q}})$

Let $X$ be a $2n$-dimensional irreducible holomorphic symplectic manifold.
Given a class $\lambda\in H^{2}(X,{\mathbb{Q}})$ denote by
$e_{\lambda}:H^{*}(X,{\mathbb{Q}})\rightarrow H^{*}(X,{\mathbb{Q}})$ the
endomorphism given by cup product with $\lambda$. The grading operator

(3.1) $h_{X}:H^{*}(X,{\mathbb{Q}})\rightarrow H^{*}(X,{\mathbb{Q}}).$

acts on $H^{k}(X,{\mathbb{Q}})[2n]=H^{2n+k}(X,{\mathbb{Q}})$, $-2n\leq k\leq
2n$, via multiplication by $k$. A triple $(e,h,f)$ of endomorphisms of
$H^{*}(X,{\mathbb{Q}})$ is called an ${\mathfrak{s}l}_{2}$-triple, if

$[e,f]=h,\ [h,e]=2e,\ [h,f]=-2f.$

Given $e$, if $f$ exists, then it is the unique endomorphism completing the
pair $(e,h)$ to an ${\mathfrak{s}l}_{2}$-triple. The Looijenga-Lunts-Verbitsky
(LLV) Lie algebra ${\mathfrak{g}}_{X}\subset{\rm End}(H^{*}(X,{\mathbb{Q}}))$
is the Lie algebra generated by all ${\mathfrak{s}l}_{2}$-triples
$(e_{\lambda},h,f_{\lambda})$, $\lambda\in H^{2}(X,{\mathbb{Q}})$.

Denote by $S_{[n]}\widetilde{H}(X,{\mathbb{Q}})$ the subspace of ${\rm
Sym}^{n}(\widetilde{H}(X,{\mathbb{Q}}))$ generated by $n$-th powers of
isotropic elements of $\widetilde{H}(X,{\mathbb{Q}})$. Verbitsky proved that
the subring $SH^{*}(X,{\mathbb{Q}})$ is an irreducible
${\mathfrak{g}}_{X}$-submodule of $H^{*}(X,{\mathbb{Q}})$, appearing with
multiplicity one, there exists a Lie algebra isomorphism

(3.2)
$\dot{\rho}:{\mathfrak{s}o}(\widetilde{H}(X,{\mathbb{Q}}))\rightarrow{\mathfrak{g}}_{X},$

and there exists a unique graded isomorphism of
${\mathfrak{s}o}(\widetilde{H}(X,{\mathbb{Q}}))$-modules

(3.3) $\Psi:SH^{*}(X,{\mathbb{Q}})[2n]\rightarrow
S_{[n]}\widetilde{H}(X,{\mathbb{Q}}),$

satisfying $\Psi(1)=\frac{\alpha^{n}}{n!}$, where the
${\mathfrak{s}o}(\widetilde{H}(X,{\mathbb{Q}}))$-module structure of
$SH^{*}(X,{\mathbb{Q}})[2n]$ is via $\dot{\rho}$ [Ta, Prop. 3.5]. The pairing
on $\widetilde{H}(X,{\mathbb{Q}})$ induces a pairing $b_{[n]}$ on
$S_{[n]}\widetilde{H}(X,{\mathbb{Q}})$, the Mukai pairing on
$H^{*}(X,{\mathbb{Q}})$ restricts to a pairing $b_{SH}$ on
$SH^{*}(X,{\mathbb{Q}})$, and $c_{X}b_{[n]}(\Psi(x),\Psi(y))=b_{SH}(x,y),$
where $c_{X}$ is the Fujiki constant [Ta, Prop. 3.8].

Following is the explicit description of the isomorphism $\Psi$ (see [Ta,
Prop. 3.5]). Given $\lambda\in H^{2}(X,{\mathbb{Q}})$, denote by

(3.4) $e_{\lambda}\in{\mathfrak{s}o}(\widetilde{H}(X,{\mathbb{Q}}))$

also the endomorphism of $\widetilde{H}(X,{\mathbb{Q}})$ sending $\alpha$ to
$\lambda$, annihilating $\beta$, and sending $\lambda^{\prime}\in
H^{2}(X,{\mathbb{Q}})$ to $(\lambda,\lambda^{\prime})\beta$. Extend
$e_{\lambda}$ as an endomorphism of ${\rm
Sym}^{n}\widetilde{H}(X,{\mathbb{Q}})$ via the product rule, so that
$e_{\lambda}(x_{1}\cdots x_{n})=\sum_{i=1}^{n}x_{1}\cdots
e_{\lambda}(x_{i})\cdots x_{n}.$ Then

$\Psi(\lambda_{1}\cdots\lambda_{k})=e_{\lambda_{1}}\cdots
e_{\lambda_{k}}(\alpha^{n}/n!).$

Let

(3.5)
$\tilde{h}_{X}:\widetilde{H}(X,{\mathbb{Q}})\rightarrow\widetilde{H}(X,{\mathbb{Q}}).$

be the grading operator multiplying $\alpha$ by $-2$, $\beta$ by $2$, and
$H^{2}(X,{\mathbb{Q}})$ by $0$.

###  3.2. The functor $\widetilde{H}$

Let $\phi$ be a morphism in ${\rm
Hom}_{\mathcal{G}}((X,\epsilon),(Y,\epsilon^{\prime}))$, where ${\mathcal{G}}$
is the groupoid (1.1). Then $\phi$ is in particular an isometry
$\phi:H^{*}(X,{\mathbb{Q}})\rightarrow H^{*}(Y,{\mathbb{Q}})$ with respect to
the Mukai pairing and we denote by

$Ad_{\phi}:{\rm End}(H^{*}(X,{\mathbb{Q}}))\rightarrow{\rm
End}(H^{*}(Y,{\mathbb{Q}}))$

the Lie algebra isomorphism given by $Ad_{\phi}(\xi)=\phi\xi\phi^{-1}$. Then
$Ad_{\phi}$ restricts to a Lie algebra isomorphism from ${\mathfrak{g}}_{X}$
onto ${\mathfrak{g}}_{Y}$, by [Ta, Theorem A]. Hence, $\phi$ restricts to an
isometry $\phi_{\mid}:SH^{*}(X,{\mathbb{Q}})\rightarrow
SH^{*}(Y,{\mathbb{Q}})$. Consequently, when $X$ and $Y$ are deformation
equivalent, so that $c_{X}=c_{Y}$, then
$\Psi_{Y}\circ\phi_{\mid}\circ\Psi_{X}^{-1}:S_{[n]}\widetilde{H}(X,{\mathbb{Q}})\rightarrow
S_{[n]}\widetilde{H}(Y,{\mathbb{Q}})$ is an isometry, which conjugates
$S_{[n]}{\mathfrak{s}o}(\widetilde{H}(X,{\mathbb{Q}}))$ to
$S_{[n]}{\mathfrak{s}o}(\widetilde{H}(Y,{\mathbb{Q}}))$, where
$S_{[n]}{\mathfrak{s}o}(\widetilde{H}(X,{\mathbb{Q}}))$ is the image of
${\mathfrak{s}o}(\widetilde{H}(X,{\mathbb{Q}}))$ in ${\rm
End}(S_{[n]}\widetilde{H}(X,{\mathbb{Q}}))$.

Given an isomorphism $f:V_{1}\rightarrow V_{2}$ of oriented vector spaces, we
set $\det(f)=1$, if $f$ is orientation preserving and $\det(f)=-1$ otherwise.

######  Proposition 3.1.

[Ta, Prop. 4.4, Theorem 4.10, and Theorem 4.11] Let $V_{1}$ and $V_{2}$ be
$d$-dimensional vector spaces over ${\mathbb{Q}}$ with non-degenerate bilinear
parings and $\phi:S_{[n]}V_{1}\rightarrow S_{[n]}V_{2}$ an isometry, such that
$\phi(S_{[n]}{\mathfrak{s}o}(V_{1}))\phi^{-1}=S_{[n]}{\mathfrak{s}o}(V_{2})$.
If $n$ is even assume that $d$ is odd and choose an orientation for each of
$V_{1}$ and $V_{2}$. Then there exists a unique isometry
$\widetilde{H}(\phi):V_{1}\rightarrow V_{2}$, such that the restriction
$S_{[n]}(\widetilde{H}(\phi))$ of its $n$-symmetric power to $S_{[n]}V_{1}$
satisfies $S_{[n]}(\widetilde{H}(\phi))=\phi$, if $n$ is odd, and
$\det(\widetilde{H}(\phi))S_{[n]}(\widetilde{H}(\phi))=\phi$, if $n$ is even.

######  Remark 3.2.

Fix a positive integer $n$. Consider the two groupoids ${\mathcal{V}}$ and
$S_{[n]}{\mathcal{V}}$, whose objects are vector spaces over ${\mathbb{Q}}$,
which are odd dimensional if $n$ is even, with non-degenerate symmetric
quadratic forms and with the additional choice of orientations, if $n$ is
even. The morphisms of ${\mathcal{V}}$ are isometries. The morphisms in ${\rm
Hom}_{S_{[n]}{\mathcal{V}}}(V_{1},V_{2})$ are isometries
$\phi:S_{[n]}V_{1}\rightarrow S_{[n]}V_{2}$ satisfying
$\phi(S_{[n]}{\mathfrak{s}o}(V_{1}))\phi^{-1}=S_{[n]}{\mathfrak{s}o}(V_{2})$.
We have the obvious functor $S_{[n]}:{\mathcal{V}}\rightarrow
S_{[n]}{\mathcal{V}}$, which is the identity on objects, and which sends an
isometry from $V_{1}$ to $V_{2}$ to the restriction of its $n$-th symmetric
power to the subrepresentation $S_{[n]}V_{1}$. Proposition 3.1 gives rise to a
functor $\widetilde{H}:S_{[n]}{\mathcal{V}}\rightarrow{\mathcal{V}}$. If $n$
is odd, the functors $S_{[n]}$ and $\widetilde{H}$ are inverses of each other.
If $n$ is even, then $S_{[n]}$ is not faithful, since
$S_{[n]}(\phi)=S_{[n]}(-\phi)$. Proposition 3.1 asserts that if $n$ is even,
then the functor $F:{\mathcal{V}}\rightarrow S_{[n]}{\mathcal{V}}$, which is
the identity on objects, and which sends $f\in{\rm
Hom}_{\mathcal{V}}((V_{1},\epsilon_{1}),(V_{2},\epsilon_{2}))$ to
$\det(f)S_{[n]}(f)$, is an equivalence and defines $\widetilde{H}$ as its
inverse. We have $S_{[n]}(-f)=-S_{[n]}(f)$, if $n$ is odd, and $F(-f)=-F(f)$,
if $n$ is even, and so $\widetilde{H}(-\phi)=-\widetilde{H}(\phi)$. Let
$f_{i}\in O(V_{i})$, $i=1,2$. We have

(3.6) $\widetilde{H}(S_{[n]}(f_{2})\circ\phi\circ
S_{[n]}(f_{1}))=\left\\{\begin{array}[]{ccc}f_{2}\circ\widetilde{H}(\phi)\circ
f_{1}&\mbox{if}&n\ \mbox{is odd},\\\
\det(f_{1})\det(f_{2})f_{2}\circ\widetilde{H}(\phi)\circ f_{1}&\mbox{if}&n\
\mbox{is even}.\end{array}\right.$

######  Definition 3.3.

The functor $\widetilde{H}:{\mathcal{G}}\rightarrow\widetilde{{\mathcal{G}}}$,
given in (1.4), sends a morphism $\phi$ in ${\rm
Hom}_{\mathcal{G}}((X,\epsilon_{X}),(Y,\epsilon_{Y}))$ to
$\widetilde{H}(\Psi_{Y}\circ\phi_{\mid}\circ\Psi_{X}^{-1}):\widetilde{H}(X,{\mathbb{Q}})\rightarrow\widetilde{H}(Y,{\mathbb{Q}})$,
where the latter functor $\widetilde{H}$ is from Remark 3.2 and $\phi_{\mid}$
is the restriction of $\phi$ to $SH^{*}(X,{\mathbb{Q}})$.

######  Lemma 3.4.

Let $X$ and $Y$ be $2n$-dimensional irreducible holomorphic symplectic
manifolds. Let $\phi$ be a morphism in ${\rm
Hom}_{{\mathcal{G}}}((X,\epsilon_{X}),(Y,\epsilon_{Y}))$. Set
$\tilde{\phi}:=\widetilde{H}(\phi):\widetilde{H}(X,{\mathbb{Q}})\rightarrow\widetilde{H}(Y,{\mathbb{Q}})$.
The equality $\tilde{\phi}\tilde{h}_{X}=(-1)^{k}\tilde{h}_{Y}\tilde{\phi}$,
for some $k\in\\{0,1\\}$, is equivalent to $\phi h_{X}=(-1)^{k}h_{Y}\phi$ for
the same $k$.

###### Proof.

The proof is modeled after that of [Ta, Theorem 5.3]. The isomorphism
$\phi:H^{*}(X,{\mathbb{Q}})\rightarrow H^{*}(Y,{\mathbb{Q}})$ conjugates the
LLV Lie algebra ${\mathfrak{g}}_{X}$ to ${\mathfrak{g}}_{Y}$, by [Ta, Theorem
A], and it conjugates $SH^{*}(X,{\mathbb{Q}})$ to $SH^{*}(Y,{\mathbb{Q}})$, by
[Ta, Theorem B]. Hence, $\phi h_{X}\phi^{-1}$ belongs to ${\mathfrak{g}}_{Y}$.
Theorems 4.10 and 4.11 in [Ta] yield the commutative diagram

(3.7)
$\textstyle{{\mathfrak{s}o}(\widetilde{H}(X,{\mathbb{Q}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ad(\tilde{\phi})}$$\textstyle{{\mathfrak{s}o}(\widetilde{H}(Y,{\mathbb{Q}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{s}l}(SH^{*}(X,{\mathbb{Q}}))}$$\textstyle{{\mathfrak{s}l}(SH^{*}(Y,{\mathbb{Q}}))}$$\textstyle{{\mathfrak{g}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{ad(\phi)}$$\textstyle{{\mathfrak{g}}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$

where the left bottom vertical arrow is the restriction to the
subrepresentation $SH^{*}(X,{\mathbb{Q}})$ of $H^{*}(X,{\mathbb{Q}})$ and the
right bottom vertical arrow is the analogue. The top left vertical arrow
corresponds to $f\mapsto\Psi_{X}^{-1}\circ(S_{[n]}f)\circ\Psi_{X}$ and the
right is the analogue. All four vertical arrows are injective and each pair in
the same column have the same image yielding the curved arrows which are
isomorphisms (inverses of $\dot{\rho}$ in (3.2)). The grading operator
$h_{X}\in{\mathfrak{g}}_{X}$ restricts to the same element of
${\mathfrak{s}l}(SH^{*}(X,{\mathbb{Q}}))$ as the image of the grading operator
$\tilde{h}_{X}\in{\mathfrak{s}o}(\widetilde{H}(X,{\mathbb{Q}}))$, since
$\Psi_{X}$ is a graded isomorphism. The same holds for $h_{Y}$ and
$\tilde{h}_{Y}$. Hence, the equality
$\tilde{\phi}\tilde{h}_{X}\tilde{\phi}^{-1}=(-1)^{k}\tilde{h}_{Y}$ is
equivalent to the equality of the restrictions of $\phi h_{X}\phi^{-1}$ and
$(-1)^{k}h_{Y}$ to elements of ${\mathfrak{s}l}(SH^{*}(Y,{\mathbb{Q}}))$, by
the commutativity of the diagram and the injectivity of the top right vertical
arrow. The latter is equivalent to the equality $\phi
h_{X}\phi^{-1}=(-1)^{k}h_{Y}$, since the restriction homomorphism
${\mathfrak{g}}_{Y}\rightarrow{\mathfrak{s}l}(SH^{*}(Y,{\mathbb{Q}}))$ is
injective. ∎

The relation between $h_{X}$ and $\tilde{h}_{X}$ explains the following result
of Verbitsky and Looijenga-Lunts.

######  Lemma 3.5.

[LL, Prop. 4.5(ii)] The commutator of $h$ in ${\mathfrak{g}}_{X}$ decomposes
as a direct sum $\bar{{\mathfrak{g}}}_{X}\oplus{\mathbb{Q}}h_{X}$, where
$\bar{{\mathfrak{g}}}_{X}$ is the derived Lie subalgebra of the commutator,
and $\bar{{\mathfrak{g}}}_{X}$ is isomorphic to
${\mathfrak{s}o}(H^{2}(X,{\mathbb{Q}})).$

The Hodge operator

(3.8) $h^{\prime}_{X}\in{\rm End}(H^{*}(X,{\mathbb{C}}))$

acts on $H^{p,q}(X)$ by multiplication by $q-p$. Verbitsky proved that it is
an element of the subalgebra $\bar{{\mathfrak{g}}}_{X,{\mathbb{C}}}$ of
${\mathfrak{g}}_{X,{\mathbb{C}}}$ ([Ve1, Theorem 8.1], see also [Ta, Cor.
2.12]).

######  Lemma 3.6.

Assume that $\phi\in{\rm
Hom}_{\mathcal{G}}((X,\epsilon_{X}),(Y,\epsilon_{Y}))$ is degree preserving
and let $\phi_{2}$ be the restriction of $\phi$ to $H^{2}(X,{\mathbb{Q}})$.
Let $t\in{\mathbb{Q}}^{\times}$ be such that
$\widetilde{H}(\phi)(\alpha)=t^{-1}\alpha$. Then the restriction
$\widetilde{H}(\phi)_{0}$ of $\widetilde{H}(\phi)$ to $H^{2}(X,{\mathbb{Q}})$
satisfies

$\widetilde{H}(\phi)_{0}=\left\\{\begin{array}[]{ccc}t^{n-1}\phi_{2},&\mbox{if}&n\
\mbox{is odd},\\\ \det(\widetilde{H}(\phi))t^{n-1}\phi_{2},&\mbox{if}&n\
\mbox{is even}.\end{array}\right.$

###### Proof.

Assume that $\phi\in{\rm
Hom}_{\mathcal{G}}((X,\epsilon_{X}),(Y,\epsilon_{Y}))$ is degree-preserving.
Then so is $\widetilde{H}(\phi)$, by Lemma 3.4. Let $\lambda$ be a class in
$H^{2}(X,{\mathbb{Q}})$. We have

$\displaystyle(S_{[n]}\widetilde{H}(\phi))(\Psi(\lambda))$ $\displaystyle=$
$\displaystyle(S_{[n]}\widetilde{H}(\phi))(e_{\lambda}(\alpha^{n}/n!))=(S_{[n]}\widetilde{H}(\phi))(\alpha^{n-1}\lambda/(n-1)!)$
$\displaystyle=$
$\displaystyle\frac{t^{1-n}}{(n-1)!}\alpha^{n-1}\widetilde{H}(\phi)(\lambda),$
$\displaystyle(S_{[n]}\widetilde{H}(\phi))(\Psi(\lambda))$ $\displaystyle=$
$\displaystyle
a\Psi(\phi(\lambda))=ae_{\phi(\lambda)}(\alpha^{n}/n!)=a\alpha^{n-1}\phi(\lambda)/(n-1)!,$

where $a=1$, if $n$ is odd, and $a=\det(\widetilde{H}(\phi))$, if $n$ is even.
The statement follows from the equality of the right hand sides in the two
equations above (Proposition 3.1). ∎

######  Lemma 3.7.

Let $\phi\in{\rm Hom}_{\mathcal{G}}((X,\epsilon_{X}),(Y,\epsilon_{Y}))$ be a
degree reversing morphism, where $X$ and $Y$ are $2n$-dimensional. Then the
value $\widetilde{H}_{0}(\phi):H^{2}(X,{\mathbb{Q}})\rightarrow
H^{2}(Y,{\mathbb{Q}})$ at $\phi$ of the functor $\widetilde{H}_{0}$ given in
(1.5) is a scalar multiple of the composition

(3.9) $H^{2}(X,{\mathbb{Q}})\stackrel{{\scriptstyle\cup
c_{2}(X)^{n-1}}}{{\longrightarrow}}H^{4n-2}(X,{\mathbb{Q}})\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}H^{2}(Y,{\mathbb{Q}}).$

###### Proof.

Note first that $H^{2}(X,{\mathbb{Q}})$ and $H^{4n-2}(X,{\mathbb{Q}})$ are
subspaces of $SH^{*}(X,{\mathbb{Q}})$. Let $\phi_{\mid}$ be the restriction of
$\phi$ to $SH^{*}(X,{\mathbb{Q}})$. Given an isometry $f\in
O(\widetilde{H}(X,{\mathbb{Q}}))$, let

$\eta_{f}:SH^{*}(X,{\mathbb{Q}})\rightarrow SH^{*}(X,{\mathbb{Q}})$

be given by $\eta_{f}:=\Psi_{X}^{-1}\circ S_{[n]}(f)\circ\Psi_{X}$. Then

$\widetilde{H}(\phi_{\mid}\circ\eta_{f})=\left\\{\begin{array}[]{ccc}\widetilde{H}(\phi)\circ
f,&\mbox{if}&n\ \mbox{is odd},\\\ \det(f)\widetilde{H}(\phi)\circ
f,&\mbox{if}&n\ \mbox{is even},\end{array}\right.$

by (3.6). The left hand side above is defined to be
$\widetilde{H}(\Psi_{X}\circ(\phi_{\mid}\circ\eta_{f})\circ\Psi_{X}^{-1})$
where $\widetilde{H}$ is the functor in Proposition 3.1.

Let $\tau\in O(\widetilde{H}(X,{\mathbb{Q}}))$ be the element which
interchanges $\alpha$ and $\beta$ and multiplies $H^{2}(X,{\mathbb{Q}})$ by
$-1$. Then $\tau\tilde{h}_{X}=-\tilde{h}_{X}\tau$ and so $\eta_{\tau}$ is
degree reversing. Hence, $\phi_{\mid}\circ\eta_{\tau}$ is degree preserving
and so the restriction $\widetilde{H}(\phi_{\mid}\circ\eta_{\tau})_{0}$ of
$\widetilde{H}(\phi_{\mid}\circ\eta_{\tau})$ to $H^{2}(X,{\mathbb{Q}})$ is a
scalar multiple of $(\phi_{\mid}\circ\eta_{\tau})_{2}$, by Lemma 3.6.
Furthermore,
$\widetilde{H}(\phi_{\mid}\circ\eta_{\tau})_{0}=\widetilde{H}(\phi)_{0}\circ\widetilde{H}(\eta_{\tau})_{0}$
and $\widetilde{H}(\eta_{\tau})_{0}=-id,$ so
$\widetilde{H}(\phi_{\mid}\circ\eta_{\tau})_{0}=-\widetilde{H}(\phi)_{0}=\pm\widetilde{H}_{0}(\phi).$
We conclude that $\widetilde{H}_{0}(\phi)$ is a scalar multiple of
$(\phi_{\mid}\circ\eta_{\tau})_{2}$.

It remains to prove that $\eta_{\tau}:H^{2}(X,{\mathbb{Q}})\rightarrow
H^{4n-2}(X,{\mathbb{Q}})$ is a scalar multiple of cup product with
$c_{2}(X)^{n-1}$. Let $G$ be the subgroup of
$SO(\widetilde{H}(X,{\mathbb{Q}}))$ leaving each of $\alpha$ and $\beta$
invariant. Then $G$ commutes with $\tau$ and its $\eta$-action on
$SH^{*}(X,{\mathbb{Q}})$ is via degree preserving isometries with respect to
which $\eta_{\tau}$ is $G$-equivariant. Now, $H^{2}(X,{\mathbb{Q}})$ and
$H^{4n-2}(X,{\mathbb{Q}})$ are dual irreducible $G$-representations and $\cup
c_{2}(X)^{n-1}:H^{2}(X,{\mathbb{Q}})\rightarrow H^{4n-2}(X,{\mathbb{Q}})$ is
$G$-equivariant as well and non-vanishing, by [LL, Theorem 4.7]. Hence, a
scalar multiple of the latter is equal to the restriction of $\eta_{\tau}$ to
$H^{2}(X,{\mathbb{Q}})$. ∎

Let $X$ and $Y$ be projective IHSMs and ${\mathcal{P}}\in D^{b}(X\times Y)$
the Fourier-Mukai kernel of an equivalence
$\Phi_{\mathcal{P}}:D^{b}(X)\rightarrow D^{b}(Y)$. Assume that the rank $r$ of
${\mathcal{P}}$ does not vanish. Let $x$ be a point of $X$, denote by
${\mathcal{P}}_{x}\in D^{b}(Y)$ the restriction of ${\mathcal{P}}$ to
$\\{x\\}\times Y$, and set $\lambda:=c_{1}({\mathcal{P}}_{x})\in
H^{2}(Y,{\mathbb{Z}})$. Note that the object ${\mathcal{P}}_{x}$ of $D^{b}(Y)$
is the image of the sky-scraper sheaf at $x$ via $\Phi_{\mathcal{P}}$. Set
$\widetilde{H}(\Phi_{\mathcal{P}}):=\widetilde{H}([ch({\mathcal{P}})\sqrt{td_{X\times
Y}}]_{*})$. The following was proved independently by Beckmann [Bec, Lemma
4.13(iv)] and the author [Ma6, Theorem 6.13(3)].

######  Proposition 3.8.

The isometry
$\widetilde{H}(\Phi_{\mathcal{P}}):\widetilde{H}(X,{\mathbb{Q}})\rightarrow\widetilde{H}(Y,{\mathbb{Q}})$
maps $\beta$ to the isotropic line spanned by $r\alpha+\lambda+s\beta$, where
$s=(\lambda,\lambda)/2r$ so that the line is isotropic.

##  4\. The degree reversing Hodge isometry of a Fourier-Mukai kernel of non-
zero rank

Consider an object $E$ of non-zero rank in the derived category of a product
$X\times Y$ of $2n$-dimensional deformation equivalent irreducible holomorphic
symplectic manifolds $X$ and $Y$. Assume that $E$ is the Fourier-Mukai kernel
of an equivalence of derived categories. This means that the functor
$\Phi_{E}:D^{b}(X)\rightarrow D^{b}(Y)$, given by
$\Phi_{E}(\bullet)=R\pi_{Y,*}(L\pi_{X}^{*}(\bullet)\otimes E)$, is an
equivalence of triangulated categories.

######  Lemma 4.1.

  1. (1)

The class $\kappa(E)\sqrt{td_{X\times Y}}$, with $\kappa(E)$ as in Definition
1.3, induces a degree reversing Hodge isometry

$\phi:H^{*}(X,{\mathbb{Q}})\rightarrow H^{*}(Y,{\mathbb{Q}}).$

  2. (2)

The Hodge isometry
$\widetilde{H}(\phi):\widetilde{H}(X,{\mathbb{Q}})\rightarrow\widetilde{H}(Y,{\mathbb{Q}})$,
associated to $\phi$ and any choice of orientations of $H^{2}(X,{\mathbb{Q}})$
and $H^{2}(Y,{\mathbb{Q}})$, restricts to a Hodge isometry

(4.1) $\widetilde{H}(\phi)_{0}:H^{2}(X,{\mathbb{Q}})\rightarrow
H^{2}(Y,{\mathbb{Q}}).$

  3. (3)

The Lie algebras isomorphism
$Ad_{\phi}:{\mathfrak{g}}_{X}\rightarrow{\mathfrak{g}}_{Y}$ is graded and
restricts to

$Ad_{\widetilde{H}_{0}(\phi)}:\bar{{\mathfrak{g}}}_{X}\rightarrow\bar{{\mathfrak{g}}}_{Y}.$

###### Proof.

Let $r$ be the rank of $E$ and let $\lambda_{X}\in H^{2}(X,{\mathbb{Z}})$ and
$\lambda_{Y}\in H^{2}(Y,{\mathbb{Z}})$ satisfy
$c_{1}(E)=\pi_{X}^{*}(\lambda_{X})+\pi_{Y}^{*}(\lambda_{Y})$. Given
$\lambda\in H^{2}(X,{\mathbb{Q}})$, let $e_{\lambda}$ be the endomorphism of
$\widetilde{H}(X,{\mathbb{Q}})$ given in (3.4). Set
$B_{\lambda}:=\exp(e_{\lambda})$.

The isometry $\widetilde{H}(\Phi_{E})$ maps ${\rm span}\\{\beta_{X}\\}$ to
${\rm
span}\\{r\alpha_{Y}+\lambda_{Y}+\frac{(\lambda_{Y},\lambda_{Y})}{2r}\beta_{Y}\\}$,
by Proposition 3.8. We have $B_{-\lambda_{X}/r}(\beta_{X})=\beta_{X}$ and

$B_{-\lambda_{Y}/r}\left(r\alpha_{Y}+\lambda_{Y}+\frac{(\lambda_{Y},\lambda_{Y})}{2r}\beta_{Y}\right)=r\alpha_{Y}.$

The equality
$\widetilde{H}(\phi)=B_{-\lambda_{Y}/r}\widetilde{H}(\Phi_{E})B_{-\lambda_{X}/r}$
implies that $\widetilde{H}(\phi)$ maps ${\rm span}\\{\beta_{X}\\}$ to ${\rm
span}\\{\alpha_{Y}\\}$. The adjoint $\Phi_{E}^{\dagger}$ has Fourier-Mukai
kernel $E^{*}[2n]$. The argument above applied to the adjoint shows that
$\widetilde{H}(\phi)^{-1}$ maps ${\rm span}\\{\beta_{Y}\\}$ to ${\rm
span}\\{\alpha_{X}\\}$. Hence, $\widetilde{H}(\phi)$ maps ${\rm
span}\\{\alpha_{X}\\}$ to ${\rm span}\\{\beta_{Y}\\}$. It follows that
$\widetilde{H}(\phi)$ maps the subspace $H^{2}(X,{\mathbb{Q}})$ orthogonal to
${\rm span}\\{\alpha_{X},\beta_{X}\\}$ to the subspace $H^{2}(Y,{\mathbb{Q}})$
orthogonal to ${\rm span}\\{\alpha_{Y},\beta_{Y}\\}$. This proves part (2) of
the Lemma. We denote by $\widetilde{H}(\phi)_{0}$ the restriction of
$\widetilde{H}(\phi)$ to $H^{2}(X,{\mathbb{Q}})$.

The isometry $\widetilde{H}(\phi)$ conjugates the grading operator
$\tilde{h}_{X}$ of $\widetilde{H}(X,{\mathbb{Q}})$, given in (3.5), to
$-\tilde{h}_{Y}$, where $\tilde{h}_{Y}$ is the grading operator of
$\widetilde{H}(Y,{\mathbb{Q}})$. Lemma 3.4 implies that $\phi$ is degree
reversing and part (1) is proven.

(3) The subalgebra $\bar{{\mathfrak{g}}}_{X}$ is the semi-simple part of the
degree $0$ subalgebra of the LLV algebra ${\mathfrak{g}}_{X}$, by [LL, Prop.
4.5(ii)], and $Ad_{\phi}:{\mathfrak{g}}_{X}\rightarrow{\mathfrak{g}}_{Y}$ is a
graded Lie algebra isomorphism, by part (1), and so $Ad_{\phi}$ maps
$\bar{{\mathfrak{g}}}_{X}$ to the semi-simple part $\bar{{\mathfrak{g}}}_{Y}$
of the degree $0$ subalgebra of ${\mathfrak{g}}_{Y}$. Furthermore, the
restriction of $Ad_{\phi}$ to $\bar{{\mathfrak{g}}}_{Y}$ is equal to
$Ad_{\widetilde{H}(\phi)_{0}}$, by part (2) and the commutativity of the outer
square in Diagram (3.7). Finally,
$Ad_{\widetilde{H}(\phi)_{0}}=Ad_{\widetilde{H}_{0}(\phi)}$, as
$\widetilde{H}_{0}(\phi)=\pm\widetilde{H}(\phi)_{0}$, by definition (1.5). ∎

Set $\gamma:=\kappa(E)\sqrt{td_{X\times Y}}$ and let $\gamma_{i}$ be the
graded summand in $H^{2i}(X\times Y,{\mathbb{Q}})$.

######  Corollary 4.2.

If the odd cohomology of $X$ vanishes, then $\gamma_{i}=0$, for odd $i$. In
that case $\kappa(E)=\kappa(E^{\vee})$.

###### Proof.

The class $\gamma_{i}$ induces the homomorphism
$H^{k-2n}(X,{\mathbb{Q}})[2n]\rightarrow H^{k+2i-6n}(Y,{\mathbb{Q}})[2n]$. As
$\phi$ is degree reversing, this homomorphism vanishes unless $k+i-4n=0$.
Hence, it vanishes unless the parities of $i$ and $k$ are the same, and it
vanishes if $k$ is odd, by assumption. ∎

###### Proof of Lemma 1.6.

Set $\tilde{\phi}:=\widetilde{H}(\phi)$. The morphism $\phi$ conjugates the
LLV Lie algebra of $X$ to that of $Y$, by [Ta, Theorem A]. The LLV Lie
algebras act faithfully on the rational LLV lattices. Hence, it suffices to
verify the equality
$e_{\lambda}^{\vee}=\frac{2}{t(\lambda,\lambda)}\tilde{\phi}^{-1}e_{\tilde{\phi}(\lambda)}\tilde{\phi}$
for the endomorphisms of $\widetilde{H}(X,{\mathbb{Q}})$. We have

$\displaystyle(\tilde{\phi}^{-1}e_{\tilde{\phi}(\lambda)}\tilde{\phi})(\alpha)$
$\displaystyle=$
$\displaystyle\tilde{\phi}^{-1}(e_{\tilde{\phi}(\lambda)}(t^{-1}\beta))=0,$
$\displaystyle(\tilde{\phi}^{-1}e_{\tilde{\phi}(\lambda)}\tilde{\phi})(\beta)$
$\displaystyle=$
$\displaystyle\tilde{\phi}^{-1}(e_{\tilde{\phi}(\lambda)}(t\alpha))=t\tilde{\phi}^{-1}(\tilde{\phi}(\lambda))=t\lambda,$
$\displaystyle(\tilde{\phi}^{-1}e_{\tilde{\phi}(\lambda)}\tilde{\phi})(\lambda^{\prime})$
$\displaystyle=$
$\displaystyle\tilde{\phi}^{-1}((\tilde{\phi}(\lambda),\tilde{\phi}(\lambda^{\prime}))\beta)=t(\lambda,\lambda^{\prime})\alpha.$

Set $\psi:=\tilde{\phi}^{-1}e_{\tilde{\phi}(\lambda)}\tilde{\phi}$. We get

$\displaystyle[e_{\lambda},\psi](\alpha)$ $\displaystyle=$
$\displaystyle-\psi(e_{\lambda}(\alpha))=-t(\lambda,\lambda)\alpha,$
$\displaystyle[e_{\lambda},\psi](\beta)$ $\displaystyle=$ $\displaystyle
e_{\lambda}(t\lambda)=t(\lambda,\lambda)\beta,$
$\displaystyle[e_{\lambda},\psi](\lambda^{\prime})$ $\displaystyle=$
$\displaystyle
t(\lambda,\lambda^{\prime})\lambda-t(\lambda,\lambda^{\prime})\lambda=0.$

Hence, $[e_{\lambda},\psi]=\frac{t(\lambda,\lambda)}{2}h.$ Clearly,
$[h,\psi]=-2\psi$, since $\psi$ has degree $-2$. So,
$e_{\lambda}^{\vee}=\frac{2}{t(\lambda,\lambda)}\psi$. ∎

##  5\. Hyperholomorphic vector bundles deforming a Fourier-Mukai kernel

Let $X$ and $Y$ be deformation equivalent IHSMs and $E$ a locally free sheaf
over $X\times Y$ which is the Fourier-Mukai kernel of an equivalence
$\Phi_{E}:D^{b}(X)\rightarrow D^{b}(Y)$ of derived categories. Let
$\phi:H^{*}(X,{\mathbb{Q}})\rightarrow H^{*}(Y,{\mathbb{Q}})$ be the morphism
in ${\rm Hom}_{{\mathcal{G}}_{an}}(X,Y)$ induced by
$\kappa(E)\sqrt{td_{X\times Y}}$. Let
$\widetilde{H}_{0}(\phi):H^{2}(X,{\mathbb{Q}})\rightarrow
H^{2}(Y,{\mathbb{Q}})$ be the value on $\phi$ of the functor
$\widetilde{H}_{0}$, given in (1.5). Let $\pi_{X}$ and $\pi_{Y}$ be the
projections from $X\times Y$. We prove in this section the following theorem.

######  Theorem 5.1.

(Proposition 5.21 and Lemma 5.14). Assume that there exists an open subcone of
the Kähler cone of $X$, such that $\widetilde{H}_{0}(\phi)$ maps each Kähler
class $\omega_{1}$ in this subcone to a Kähler class
$\omega_{2}:=\widetilde{H}_{0}(\phi)(\omega_{1})$ on $Y$ with respect to which
$E$ is $[\pi_{X}^{*}(\omega_{1})+\pi_{Y}^{*}(\omega_{2})]$-slope stable. Then
for every IHSM $X^{\prime}$ deformation equivalent to $X$ there exists an IHSM
$Y^{\prime}$ and a locally free (possibly twisted) coherent sheaf $E^{\prime}$
over $X^{\prime}\times Y^{\prime}$, such that the pair $(X^{\prime}\times
Y^{\prime},E^{\prime})$ is deformation equivalent to $(X\times Y,E)$.

Note that the assumptions of the theorem are symmetric with respect to $X$ and
$Y$, upon replacing $E$ by $E^{*}$ and $\phi$ by $\phi^{-1}$, and so the
conclusions are symmetric as well. Proposition 5.21 is a slightly stronger
version of the theorem, allowing the assumptions in the statement of the
theorem to hold instead for a deformation $(X_{0}\times Y_{0},E_{0})$ of the
Fourier-Mukai kernel $(X\times Y,E)$. Theorem 5.1 may be viewed as a global
analogue, in the special case of classical and gerby deformations of locally
free Fourier-Mukai kernels between IHSMs, of the infinitesimal deformability
result of Toda for generalized deformations of Fourier-Mukai kernels [To,
Theorem 1.1].

The proof of Theorem 5.1 uses Verbitsky’s results on hyperholomorphic vector
bundles over hyper-Kähler manifolds and their deformability along twistor
paths in moduli [Ve2]. In Section 5.1 we recall the notion of a twistor line
in the moduli space of marked IHSMs. In section 5.2 we define the moduli space
${\mathfrak{M}}_{\psi}$ of rational Hodge isometries. Given a lattice
$\Lambda$ and a rational isometry $\psi\in O(\Lambda_{\mathbb{Q}})$, the
moduli space ${\mathfrak{M}}_{\psi}$ parametrizes isomorphism classes of
quadruples $(X_{1},\eta_{1},X_{2},\eta_{2})$ of deformation equivalent marked
pairs $(X_{i},\eta_{i})$, where
$\eta_{i}:H^{2}(X_{i},{\mathbb{Z}})\rightarrow\Lambda$ is an isometry,
$i=1,2$, such that
$\eta_{2}^{-1}\psi\eta_{1}:H^{2}(X_{1},{\mathbb{Q}})\rightarrow
H^{2}(X_{2},{\mathbb{Q}})$ is a Hodge isometry which maps some Kähler class to
a Kähler class. In Section 5.3 we define diagonal twistor lines in
${\mathfrak{M}}_{\psi}$. In Section 5.4 we define the notion of generic
twistor paths and prove that every two points in the same connected component
of ${\mathfrak{M}}_{\psi}$ are connected by such a path. In Section 5.5 we
review Verbitsky’s result about hyperholomorphic vector bundles and their
deformability along twistor paths in ${\mathfrak{M}}_{\psi}.$ In Section 5.6
we prove Theorem 5.1.

###  5.1. Twistor lines

A marking of an irreducible holomorphic symplectic manifold $X$ is an isometry
$\eta:H^{2}(X,{\mathbb{Z}})\rightarrow\Lambda$ with a fixed lattice $\Lambda$.
Two marked pairs $(X_{1},\eta_{1})$ and $(X_{2},\eta_{2})$ are isomorphic, if
there exists an isomorphism $f:X_{1}\rightarrow X_{2}$, such that
$\eta_{2}=\eta_{1}\circ f^{*}$. The moduli space ${\mathfrak{M}}_{\Lambda}$ of
isomorphism classes of marked irreducible holomorphic symplectic manifolds is
a non-Hausdorff complex manifold of dimension ${\rm rank}(\Lambda)-2$ [Hu1].

Set $\Omega_{\Lambda}:=\\{\ell\in{\mathbb{P}}(\Lambda_{\mathbb{C}})\ :\
(\ell,\ell)=0\ \mbox{and}\ (\lambda,\bar{\lambda})>0\ \mbox{for all non-zero}\
\lambda\in\ell\\}$. The period map

(5.1) $P:{\mathfrak{M}}_{\Lambda}\rightarrow\Omega_{\Lambda},$

is given by $P(X,\eta)=\eta(H^{2,0}(X))$. The restriction of the period map to
every connected component of ${\mathfrak{M}}_{\Lambda}$ is known to be
surjective and generically injective, by Verbitsky’s Torelli Theorem [Hu2,
Ve3]. Furthermore, if ${\rm Pic}(X)$ is trivial, or cyclic generated by a
class of non-negative BBF-degree, then $(X,\eta)$ is the unique point in the
fiber of $P$ through the connected component of ${\mathfrak{M}}_{\Lambda}$
containing $(X,\eta)$.

The Torelli Theorem solves the case of Theorem 1.1 in which the Hodge isometry
$f$ is a parallel-transport operator.

######  Theorem 5.2.

Let $(X,\eta)$ and $(X^{\prime},\eta^{\prime})$ be two marked pairs in the
same connected component of ${\mathfrak{M}}_{\Lambda}$ satisfying
$P(X,\eta)=P(X^{\prime},\eta^{\prime})$. There exist an analytic cycle $Z$ in
$X\times X^{\prime}$, such that
$[Z]_{*}:H^{*}(X^{\prime},{\mathbb{Z}})\rightarrow H^{*}(X,{\mathbb{Z}})$ is a
parallel-transport operator, which restricts to
$H^{2}(X^{\prime}_{i},{\mathbb{Z}})$ as $\eta^{-1}\circ\eta^{\prime}$.

###### Proof.

The assumption that $(X,\eta)$ and $(X^{\prime},\eta^{\prime})$ belong to the
same connected component of ${\mathfrak{M}}_{\Lambda}$ and have the same
period implies that $(X,\eta)$ and $(X^{\prime},\eta^{\prime})$ are
inseparable points of ${\mathfrak{M}}_{\Lambda}$, by Verbitsky’s Torelli
Theorem [Hu2, Ve3]. Note also that $\eta^{\prime}\circ\eta^{-1}$ is a
parallel-transport operator, which is a Hodge isometry. The existence of the
analytic cycle $Z$ with the claimed properties thus follows from results of
Huybrechts in [Hu6] (see [Ma4, Theorem 3.2]). ∎

######  Lemma 5.3.

Let $X$ and $X^{\prime}$ be IHSMs and let
$f_{i}:H^{*}(X,{\mathbb{Z}})\rightarrow H^{*}(X^{\prime},{\mathbb{Z}})$ be a
parallel-transport operator, $i=1,2$. Assume that $f_{1}$ and $f_{2}$ restrict
to the same parallel-transport operator $f:H^{2}(X,{\mathbb{Z}})\rightarrow
H^{2}(X^{\prime},{\mathbb{Z}})$. Then there exist automorphisms $g\in{\rm
Aut}(X)$ and $g^{\prime}\in{\rm Aut}(X^{\prime})$, acting trivially on the
second cohomologies, such that $f_{2}=g^{\prime}_{*}\circ f_{1}\circ g_{*}$.

###### Proof.

There exist families $\pi_{i}:{\mathcal{X}}_{i}\rightarrow B_{i}$ over
connected analytic spaces $B_{i}$, points $b_{i}$, $b_{i}^{\prime}\in B_{i}$,
and continuous paths $\gamma_{i}$ from $b_{i}$ to $b_{i}^{\prime}$ in $B_{i}$,
such that $f_{i}$ is the parallel-transport operator along $\gamma_{i}$ in the
local system $R\pi_{i,*}{\mathbb{Z}}$, $i=1,2$. We may and do assume that
$B_{i}$ is simply connected, possibly after passing to the universal cover and
lifting the path $\gamma_{i}$. Choose an isometric trivialization
$\eta_{i}:R^{2}\pi_{i,*}{\mathbb{Z}}\rightarrow\underline{\Lambda}$, where
$\underline{\Lambda}$ is the trivial local system over $B_{i}$ with a fixed
lattice $\Lambda$, with
$\eta_{1,b_{1}}=\eta_{2,b_{2}}:H^{2}(X,{\mathbb{Z}})\rightarrow\Lambda$. Set
$\eta:=\eta_{1,b_{1}}$. Set $\eta^{\prime}:=\eta\circ f^{-1}$. We get that
$\eta_{i,b_{i}^{\prime}}=\eta^{\prime}$, for $i=1,2$.

Let ${\mathfrak{M}}_{\Lambda}^{0}$ be the connected component of the moduli
space of isomorphism classes of marked IHSMs containing the isomorphism class
$[(X,\eta)]$ of $(X,\eta)$. Let
$s_{i}:B_{i}\rightarrow{\mathfrak{M}}_{\Lambda}^{0}$ be the classifying
morphism associated to $\eta_{i}$. Set
$q:=[(X,\eta)]=s_{1}(X,\eta_{1,b_{1}})=s_{2}(X,\eta_{2,b_{2}})$ and
$q^{\prime}:=[(X,\eta^{\prime})]=s_{1}(X^{\prime},\eta_{1,b_{1}^{\prime}})=s_{2}(X^{\prime},\eta_{2,b_{2}^{\prime}})$.

There exists over ${\mathfrak{M}}_{\Lambda}^{0}$ a universal family
$\Pi:{\mathcal{Y}}\rightarrow{\mathfrak{M}}_{\Lambda}^{0}$ and isomorphisms
$h_{i}:{\mathcal{X}}_{i}\rightarrow s_{i}^{-1}({\mathcal{Y}})$ of the two
families $\pi_{i}$ and $s_{i}^{-1}(\Pi)$ over $B_{i}$, by [Ma7, Theorem 1.1]
and our assumption that $B_{i}$ is simply connected, for $i=1,2$. Furthermore,
the local system $R\Pi_{*}{\mathbb{Z}}$ over ${\mathfrak{M}}_{\Lambda}^{0}$ is
trivial, by [Ma7, Lemma 2.1]. Hence, the paths $s_{i}\circ\gamma_{i}$,
$i=1,2$, from $q$ to $q^{\prime}$ in ${\mathfrak{M}}_{\Lambda}^{0}$ induce the
same parallel-transport operator from the fiber $Y_{q}$ of ${\mathcal{Y}}$
over $q$ to the fiber $Y_{q^{\prime}}$ of ${\mathcal{Y}}$ over $q^{\prime}$.
Let $h_{i,b_{i}}:X\rightarrow Y_{q}$ and
$h_{i,b_{i}^{\prime}}:X^{\prime}\rightarrow Y_{q^{\prime}}$ be the isomorphism
associated to $h_{i}$, $i=1,2$. We conclude that $h_{i,b_{i}^{\prime}}\circ
f_{i}\circ h^{-1}_{i,b_{i}}$ is the parallel-transport along
$s_{i}\circ\gamma_{i}$, $i=1,2$, and so $h_{1,b_{1}^{\prime}}\circ f_{1}\circ
h_{1,b_{1}}^{-1}=h_{2,b_{2}^{\prime}}\circ f_{2}\circ h_{2,b_{2}}^{-1}.$ Then
$g:=h_{1,b_{1}}^{-1}\circ h_{2,b_{2}}\in{\rm Aut}(X)$ and
$g^{\prime}:=h_{2,b_{2}^{\prime}}^{-1}\circ h_{1,b_{1}^{\prime}}\in{\rm
Aut}(X^{\prime})$ are automorphisms satisfying $f_{2}=g^{\prime}_{*}\circ
f_{1}\circ g_{*}$. ∎

Given a positive definite $3$-dimensional subspace $W$ of
$\Lambda_{\mathbb{R}}$, set
$W_{\mathbb{C}}:=W\otimes_{\mathbb{R}}{\mathbb{C}}$. We get the projective
conic curve ${\mathbb{P}}(W_{\mathbb{C}})\cap\Omega_{\Lambda}$, which is
called a twistor line. Let $(X,\eta)$ be a marked pair in
${\mathfrak{M}}_{\Lambda}$ and $\omega\in H^{1,1}(X,{\mathbb{R}})$ a Kähler
class. The intersection of $W_{\mathbb{C}}:=H^{2,0}(X)\oplus
H^{0,2}(X)\oplus{\mathbb{C}}\omega$ with $H^{2}(X,{\mathbb{R}})$ is a positive
definite $3$-dimensional subspace $W$ and the twistor line
$Q_{(X,\eta,\omega)}:={\mathbb{P}}(\eta(W_{\mathbb{C}}))\cap\Omega_{\Lambda}$
passes through $P(X,\eta)$. We endow $W$ with the orientation associated to
the basis

(5.2) $\\{Re(\sigma),Im(\sigma),\omega\\},$

for a non-zero class $\sigma\in H^{2,0}(X)$. There exists a family

(5.3) $\pi:{\mathcal{X}}\rightarrow Q_{(X,\eta,\omega)}$

of irreducible holomorphic symplectic manifolds with fiber $X$ over the point
$P(X,\eta)$, known as the twistor family associated to $\omega$, with the
following properties [Hu1, paragraph 1.17 page 76]. The marking $\eta$ extends
to a trivialization of the local system $R^{2}\pi_{*}{\mathbb{Z}}$, as
$Q_{(X,\eta,\omega)}$ is simply connected. Denote by $\eta_{t}$ the resulting
marking of the fiber $X_{t}$ of $\pi$ over $t\in Q_{(X,\eta,\omega)}$. Then

$P(X_{t},\eta_{t})=t,$

for each $t\in Q_{(X,\eta,\omega)}$ and the line in
$H^{2}(X_{t},{\mathbb{R}})$, which is contained in $\eta_{t}^{-1}(\eta(W))$
and orthogonal to $H^{2,0}(X_{t})$, is spanned by a Kähler class $\omega_{t}$.
The twistor family depends only on the ray in the Kähler cone of $X$ through
$\omega$. Denote by

(5.4) $R_{t}\subset{\mathcal{K}}_{X_{t}}\cap\eta_{t}^{-1}(\eta(W))$

the ray through $\omega_{t}$ in the Kähler cone ${\mathcal{K}}_{X_{t}}$ of
$X_{t}$. The curve

$\widetilde{Q}_{(X,\eta,\omega)}:=\\{(X_{t},\eta_{t})\ :\ t\in
Q_{(X,\eta,\omega)}\\}$

in ${\mathfrak{M}}^{0}_{\Lambda}$ is mapped isomorphically onto
$Q_{(X,\eta,\omega)}$ via the period map. The curve
$\widetilde{Q}_{(X,\eta,\omega)}$ is called the twistor line through
$(X,\eta)$ associated to $\omega$. The twistor line
$\widetilde{Q}_{(X,\eta,\omega)}$ is said to be generic, if ${\rm Pic}(X_{t})$
is trivial, for some $t\in Q_{(X,\eta,\omega)}$. This is the case for $\omega$
in the complement of a countable union of hyperplanes in the Kähler cone.

###  5.2. Moduli spaces of rational Hodge isometries

Let $X$ and $Y$ be the irreducible holomorphic symplectic manifolds in Lemma
4.1. Choose markings $\eta_{X}:H^{2}(X,{\mathbb{Z}})\rightarrow\Lambda$ and
$\eta_{Y}:H^{2}(Y,{\mathbb{Z}})\rightarrow\Lambda$ such that $(X,\eta_{X})$
and $(Y,\eta_{Y})$ belong to the same connected component
${\mathfrak{M}}_{\Lambda}^{0}$ of ${\mathfrak{M}}_{\Lambda}$.

######  Convention 5.4.

Choose an orientation $\epsilon_{\Lambda}$ of
$\Lambda\otimes_{\mathbb{Z}}{\mathbb{Q}}$. Given two pairs $(X_{1},\eta_{1})$
and $(X_{2},\eta_{2})$ in ${\mathfrak{M}}_{\Lambda}$, we endow
$H^{2}(X_{i},{\mathbb{Q}})$ with the orientation $\epsilon_{i}$, such that
$\eta_{i}$ is orientation preserving. The homomorphism

$\widetilde{H}:{\rm Hom}_{\mathcal{G}}(X_{1},X_{2})\rightarrow{\rm
Hom}(\widetilde{H}(X_{1},{\mathbb{Q}}),\widetilde{H}(X_{2},{\mathbb{Q}})),$

induced by the functor (1.4), is then always evaluated in terms of the
orientations $\epsilon_{1}$ and $\epsilon_{2}$. The resulting homomorphism
depends on the markings $\eta_{1}$ and $\eta_{2}$, but is independent of the
choice of $\epsilon_{\Lambda}$.

The unit sphere in any positive definite $3$-dimensional subspace of
$H^{2}(X,{\mathbb{R}})$ is a deformation retract of the positive cone
${\mathcal{C}}^{\prime}_{X}:=\\{x\in H^{2}(X,{\mathbb{R}})\ ;\ (x,x)>0\\}$
(see [Ma4, Sec 4]). Hence, the second cohomology of the positive cone is a
one-dimensional representation of $O(H^{2}(X,{\mathbb{Q}}))$ corresponding to
a character $\nu:O(H^{2}(X,{\mathbb{Q}}))\rightarrow\mu_{2}:=\\{\pm 1\\}$. Set

$O^{+}(H^{2}(X,{\mathbb{Q}})):=\ker(\nu).$

The orientation given by the basis (5.2) is independent of the choices of a
holomorphic $2$-form and of a Kähler class and determines a positive generator
of the cyclic group $H^{2}({\mathcal{C}}^{\prime}_{X},{\mathbb{Z}})$.

######  Definition 5.5.

The orientation of the positive cone is the positive generator of
$H^{2}({\mathcal{C}}^{\prime}_{X},{\mathbb{Z}})$. An isometry
$g:H^{2}(X_{1},{\mathbb{Q}})\rightarrow H^{2}(X_{2},{\mathbb{Q}})$ is
orientation preserving, if
$g_{*}:H^{2}({\mathcal{C}}^{\prime}_{X_{1}},{\mathbb{Z}})\rightarrow
H^{2}({\mathcal{C}}^{\prime}_{X_{2}},{\mathbb{Z}})$ maps the positive
generator to the positive generator.

Set

$\nu(g)=\left\\{\begin{array}[]{ccc}1&\mbox{if}&g\ \mbox{is orientation
preserving}\\\ -1&\mbox{if}&g\ \mbox{is orientation
reveresing}.\end{array}\right.$

Note that if $g$ is a Hodge isometry, which maps some Kähler class to a Kähler
class, then $\nu(g)=1$, as it maps a basis of the form (5.2) to such a basis.

######  Remark 5.6.

Keep the notation of Convention 5.4. Let
$g:H^{*}(X_{1},{\mathbb{Z}})\rightarrow H^{*}(X_{2},{\mathbb{Z}})$ be a
parallel-transport operator and denote by $\bar{g}$ its restriction to
$H^{2}(X_{1},{\mathbb{Z}})$. One checks that if $\bar{g}$ is orientation
preserving, so that $\det(\eta_{2}\bar{g}\eta_{1}^{-1})=1$, then
$\widetilde{H}(g)$ maps $\alpha$ to $\alpha$, $\beta$ to $\beta$, and
restricts to $H^{2}(X_{1},{\mathbb{Q}})$ as $\bar{g}$, so that
$\widetilde{H}_{0}(g)=\bar{g}$ (see888The statement is true for any parallel-
transport operator, when $n:=\dim(X_{i})/2$ is odd. When $n$ is even we
explain in [Ma6, Remark 5.4] Taelman’s sign convention in terms of a functor
$\chi$ from ${\mathcal{G}}$ to $\mu_{2}:=\\{\pm 1\\}$, such that
$\chi(\phi)\widetilde{H}(\phi)$ is orientation preserving, for every
$\phi\in{\rm Hom}_{\mathcal{G}}(X_{1},X_{2})$, and $\chi(g)=1$, for every
parallel-transport operator. So when $n$ is even, the restrictions of
$\widetilde{H}(g)$ to $H^{2}(X_{1},{\mathbb{Z}})$ is equal to that of $g$, if
$g$ is orientation preserving, and is equal to that of $-g$, if $g$ is
orientation reversing.  [Ma6, Remark 5.4]). In particular, this is the case
when $(X_{1},\eta_{1})$ and $(X_{2},\eta_{2})$ belong to the same connected
component of ${\mathfrak{M}}_{\Lambda}$ and $\bar{g}=\eta_{2}^{-1}\eta_{1}$.

Let $E$ be the Fourier-Mukai kernel of non-zero rank of an equivalence
$\Phi_{E}:D^{b}(X)\rightarrow D^{b}(Y)$ of derived categories of two
deformation equivalent IHSMs $X$ and $Y$. Set
$\phi:=[\kappa(E)\sqrt{td_{X\times Y}}]_{*}:H^{*}(X,{\mathbb{Q}})\rightarrow
H^{*}(Y,{\mathbb{Q}})$ and let $\widetilde{H}(\phi)_{0}$ be the restriction of
$\widetilde{H}(\phi)$ to $H^{2}(X,{\mathbb{Q}})$. Set

(5.5)
$\widetilde{H}_{0}(\phi):=\nu(\widetilde{H}(\phi)_{0})\widetilde{H}(\phi)_{0},$

We denote $\widetilde{H}_{0}(\phi)$ also by $\psi_{E}$ to emphasis it
dependence on $E$.

$\psi_{E}:=\widetilde{H}_{0}([\kappa(E)\sqrt{td_{X\times Y}}]_{*}).$

Let $\psi:\Lambda_{\mathbb{Q}}\rightarrow\Lambda_{\mathbb{Q}}$, be the
isometry given by

(5.6) $\psi=\eta_{Y}\widetilde{H}_{0}(\phi)\eta_{X}^{-1}.$

Then $\psi$ belongs to $O^{+}(\Lambda_{\mathbb{Q}})$ (see [Ma4, Sec 4]).

######  Definition 5.7.

An Azumaya ${\mathcal{O}}_{X}$-algebra of rank $r$ over a complex manifold $X$
is a sheaf ${\mathcal{A}}$ of locally free coherent
${\mathcal{O}}_{X}$-modules, with a global section $1_{\mathcal{A}}$, and an
${\mathcal{O}}_{X}$-linear associative multiplication
$m:{\mathcal{A}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{A}}\rightarrow{\mathcal{A}}$
with identity $1_{\mathcal{A}}$, admitting an open covering $\\{U_{\alpha}\\}$
of $X$ and an isomorphism
$\eta_{\alpha}:{\mathcal{A}}_{\mid_{U_{\alpha}}}\rightarrow{\mathcal{E}}nd(F_{\alpha})$
of unital associative algebras for some locally free
${\mathcal{O}}_{U_{\alpha}}$-module $F_{\alpha}$ of rank $r$ over each
$U_{\alpha}$. We will also refer to such ${\mathcal{A}}$ as an Azumaya algebra
over $X$.

######  Remark 5.8.

The endomorphism sheaf ${\mathcal{E}}nd(F)$ of a locally free twisted sheaf
$F$ over $X$ is an Azumaya algebra over $X$. Conversely, an Azumaya algebra
corresponds to a natural equivalence class of twisted sheaves over $X$, modulo
tensorization by line-bundles [C]. The class $\kappa(F)$ depends only on this
equivalence class and so $\kappa({\mathcal{A}})$ is well defined [Ma5, Sec.
2.3]

######  Assumption 5.9.

  1. (1)

Assume that the Fourier-Mukai kernel $E$ in Lemma 4.1 is represented by a
locally free sheaf, which we denote by $E$ as well.

  2. (2)

Assume that there exist999If the isometry $\psi_{E}$, given in Equation (5.5),
maps some Kähler class $\omega_{1}$ on $X$ to a Kähler class $\omega_{2}$ on
$Y$, then one can take the constant families with $B$ a point. This is the
case in the $K3$ surface examples used in [Bu].  proper and smooth families of
Kähler manifolds $p:{\mathcal{X}}\rightarrow B$ and
$q:{\mathcal{Y}}\rightarrow B$, over the same simply connected analytic base
$B$, a possibly twisted vector bundle ${\mathcal{E}}$ over
${\mathcal{X}}\times_{B}{\mathcal{Y}}$, a point $b\in B$, isomorphisms
$f:X\rightarrow X_{b}$ and $g:Y\rightarrow Y_{b}$, and an isomorphism of
Azumaya algebras ${\mathcal{E}}nd(E)\cong(f\times
g)^{*}{\mathcal{E}}nd({\mathcal{E}})$, where $X_{b}$ and $Y_{b}$ are the
fibers of ${\mathcal{X}}$ and ${\mathcal{Y}}$ over $b$, such that the value of
the parallel-transport $\psi_{E_{0}}$ of $\psi_{E}$ in the local system
$(R^{2}p_{*}{\mathbb{Q}})^{*}\otimes R^{2}q_{*}{\mathbb{Q}}$ at some point
$0\in B$ maps some Kähler class $\omega_{1}$ on the fiber $X_{0}$ of $p$ to a
Kähler class $\omega_{2}$ over the fiber $Y_{0}$ of $q$.

  3. (3)

$E_{0}$ is
$[\pi_{X_{0}}^{*}(\omega)+\pi_{Y_{0}}^{*}(\psi_{E_{0}}(\omega))]$-slope-
stable, for classes $\omega$ in a non-empty subcone of the intersection
${\mathcal{K}}_{X_{0}}\cap\psi_{E_{0}}^{-1}({\mathcal{K}}_{Y_{0}})$.

######  Remark 5.10.

Note that the class $\kappa(E_{0})\sqrt{td_{X_{0}\times Y_{0}}}$ induces the
degree reversing Hodge isometry (with respect to the Mukai pairings)

(5.7) $\phi_{0}:H^{*}(X_{0},{\mathbb{Q}})\rightarrow
H^{*}(Y_{0},{\mathbb{Q}}),$

which is the parallel-transport of $\phi$ in Lemma 4.1, hence a morphism in
${\rm Hom}_{\mathcal{G}}(X_{0},Y_{0})$, $\widetilde{H}(\phi_{0})$ restricts to
a Hodge isometry
$\widetilde{H}(\phi_{0})_{0}:H^{2}(X_{0},{\mathbb{Q}})\rightarrow
H^{2}(Y_{0},{\mathbb{Q}})$, which is the parallel-transport of
$\widetilde{H}(\phi)_{0}$ in the Lemma, and the parallel-transport
$\psi_{E_{0}}:H^{2}(X_{0},{\mathbb{Q}})\rightarrow H^{2}(Y_{0},{\mathbb{Q}})$
of $\psi_{E}$ in Assumption 5.9(2) is equal to $\widetilde{H}_{0}(\phi_{0})$.

Let ${\mathfrak{M}}_{\psi}$ be the moduli space of isomorphism classes of
quadruples $(X_{1},\eta_{1},X_{2},\eta_{2})$, where
$\eta_{i}:H^{2}(X_{i},{\mathbb{Z}})\rightarrow\Lambda$ is a marking, such that
$\eta_{2}^{-1}\psi\eta_{1}:H^{2}(X_{1},{\mathbb{Q}})\rightarrow
H^{2}(X_{2},{\mathbb{Q}})$ is a Hodge isometry which maps some Kähler class to
a Kähler class, where $\psi$ is given in (5.6). Two quadruples
$(X_{1},\eta_{1},X_{2},\eta_{2})$ and
$(X^{\prime}_{1},\eta^{\prime}_{1},X^{\prime}_{2},\eta^{\prime}_{2})$ are
isomorphic, if the marked pair $(X_{i},\eta_{i})$ is isomorphic to
$(X^{\prime}_{i},\eta^{\prime}_{i})$, for $i=1,2$. The moduli space
${\mathfrak{M}}_{\psi}$ has a natural structure of a non-Hausdorff complex
manifold. The proof of the latter statement is identical to that of [Bu, Prop.
4.9]. The isometry $\psi$ induces an automorphism of $\Omega_{\Lambda}$ which
we denote by $\psi$ as well.

######  Lemma 5.11.

If $(X_{1},\eta_{1},X_{2},\eta_{2})$ belongs to ${\mathfrak{M}}_{\psi}$, then
$P(X_{2},\eta_{2})=\psi(P(X_{1},\eta_{1})).$

###### Proof.

The composition
$\eta_{2}^{-1}\psi\eta_{1}:H^{2}(X_{1},{\mathbb{Q}})\rightarrow
H^{2}(X_{2},{\mathbb{Q}})$ is a Hodge isometry. Hence,
$P(X_{2},\eta_{2}):=\eta_{2}(H^{2,0}(X_{2}))=[\eta_{2}\circ(\eta_{2}^{-1}\psi\eta_{1})](H^{2,0}(X_{1}))=\psi(P(X_{1},\eta_{1})).$
∎

Let $\eta_{X_{0}}$ be the marking of $X_{0}$ resulting in the extension of
$\eta_{X}$ to a trivialization of $R^{2}p_{*}{\mathbb{Z}}$ for the family
$p:{\mathcal{X}}\rightarrow B$ in Assumption 5.9. Define $\eta_{Y_{0}}$
similarly. Note that $\psi(\eta_{X_{0}}(\omega_{1}))=\eta_{Y_{0}}(\omega_{2})$
for some Kähler classes $\omega_{1}$ and $\omega_{2}$, by Assumption 5.9, and
so $(X_{0},\eta_{X_{0}},Y_{0},\eta_{Y_{0}})$ belongs to
${\mathfrak{M}}_{\psi}$. Denote by

(5.8) ${\mathfrak{M}}_{\psi}^{0}$

the connected component of ${\mathfrak{M}}_{\psi}$ containing
$(X_{0},\eta_{X_{0}},Y_{0},\eta_{Y_{0}})$.

Given a quadruple $q:=(X_{1},\eta_{1},X_{2},\eta_{2})$ in
${\mathfrak{M}}_{\psi}$ set

(5.9)
$\psi_{q}:=\eta_{2}^{-1}\psi\eta_{1}:H^{2}(X_{1},{\mathbb{Q}})\rightarrow
H^{2}(X_{2},{\mathbb{Q}}).$

Let $\omega_{1}$ be a Kähler class on $X_{1}$, such that
$\omega_{2}:=\psi_{q}(\omega_{1})$ is a Kähler class on $X_{2}$. Note that
$\psi:\Omega_{\Lambda}\rightarrow\Omega_{\Lambda}$ maps the twistor line
$Q_{(X_{1},\eta_{1},\omega_{1})}$ to the twistor line
$Q_{(X_{2},\eta_{2},\omega_{2})}$. Let $(X_{i,t},\eta_{i,t})$, $t\in
Q_{(X_{i},\eta_{i},\omega_{i})}$, be the point of
$\widetilde{Q}_{(X_{i},\eta_{i},\omega_{i})}$ over $t$, for $i=1,2$.

######  Lemma 5.12.

The rays $R_{t}$ and $R_{\psi(t)}$ in the Kähler cones of $X_{1,t}$ and
$X_{2,\psi(t)}$, given in (5.4), satisfy the equality
$\psi(\eta_{1,t}(R_{t}))=\eta_{2,\psi(t)}(R_{\psi(t)})$, for each $t\in
Q_{(X_{1},\eta_{1},\omega_{1})}$. Consequently, the quadrupe
$(X_{1,t},\eta_{1,t},X_{2,\psi(t)},\eta_{2,\psi(t)})$ belongs to
${\mathfrak{M}}_{\psi}$, for all $t\in Q_{(X_{1},\eta_{1},\omega_{1})}$.

###### Proof.

Let ${\mathcal{C}}_{X_{i}}:=\\{\lambda\in H^{1,1}(X_{i},{\mathbb{R}})\ :\
(\lambda,\lambda)>0\\}$ be the positive cone, $i=1,2$. Each
${\mathcal{C}}_{X_{i}}$ has two connected components and we denote by
${\mathcal{C}}_{X_{i}}^{+}$ the one which contains the Kähler cone. The
isometry $\psi_{q}$ maps ${\mathcal{C}}_{X_{1}}$ isomorphically onto
${\mathcal{C}}_{X_{2}}$ and $\psi_{q}({\mathcal{K}}_{X_{1}})\cap K_{X_{2}}$ is
non-empty, hence
$\psi_{q}({\mathcal{C}}_{X_{1}}^{+})={\mathcal{C}}_{X_{2}}^{+}$. Equivalently,
$\psi(\eta_{1,t_{0}}({\mathcal{C}}^{+}_{X_{t_{0}}}))=\eta_{2,\psi(t_{0})}({\mathcal{C}}^{+}_{X_{\psi(t_{0})}})$,
where $t_{0}=P(X_{1},\eta_{1})$. Hence,
$\psi(\eta_{1,t}({\mathcal{C}}^{+}_{X_{t}}))=\eta_{2,\psi(t)}({\mathcal{C}}^{+}_{X_{\psi(t)}})$,
for all $t\in Q_{(X_{1},\eta_{1},\omega_{1})}$, by continuity and
connectedness of $Q_{(X_{1},\eta_{1},\omega_{1})}$. We know that $\psi$ maps
the line spanned by $\eta_{1,t}(R_{t})$ to the line spanned by
$\eta_{2,\psi(t)}(R_{\psi(t)})$, since $\eta_{2,t}^{-1}\psi\eta_{1,t}$ is a
Hodge isometry which maps the positive definite $3$-dimensional subspace
$W_{1}$ associated to $(X_{1},\eta_{1},\omega_{1})$ to the positive definite
$3$-dimensional subspace $W_{2}$ associated to $(X_{2},\eta_{2},\omega_{2})$
and $\eta_{1,t}(R_{t})$ spans the line in $W_{1}$ orthogonal to
$\eta_{1,t}(H^{2,0}(X_{t}))$ and $\eta_{2,t}(R_{\psi(t)})$ spans the line in
$W_{2}$ orthogonal to $\eta_{2,t}(H^{2,0}(X_{\psi(t)}))$. Now, $R_{t}$ is the
intersection of the line spanned by it with ${\mathcal{C}}^{+}_{X_{t}}$ and
the similar statement holds for $R_{\psi(t)}$. Hence,
$\psi(\eta_{1,t}(R_{t}))=\eta_{2,\psi(t)}(R_{\psi(t)})$. ∎

Lemma 5.12 enables us to define the twistor line in
${\mathfrak{M}}_{\psi}^{0}$ through $(X_{1},\eta_{1},X_{2},\eta_{2})$
determined by
$\omega_{1}\in{\mathcal{K}}_{X_{1}}\cap\psi_{q}^{-1}({\mathcal{K}}_{X_{2}})$
to be the curve

(5.10)
$\widetilde{Q}_{(X_{1},\eta_{1},X_{2},\eta_{2}),\omega_{1}}:=\\{(X_{1,t},\eta_{1,t},X_{2,t},\eta_{2,\psi(t)})\
:\ t\in Q_{(X_{1},\eta_{1},\omega_{1})}\\}.$

###  5.3. Diagonal twistor lines

We explain next that the twistor line (5.10) in the above definition is
associated to a hyper-Kähler structure on the differential manifold underlying
the product $X_{1}\times X_{2}$. This fact will be used in the proof of
Proposition 5.19 below. A Kähler class on an irreducible holomorphic
symplectic manifold $X$ determines a Ricci-flat hermitian metric $g$ on the
manifold $M$ underlying $X$, by the Calabi-Yau theorem [Be1]. The metric $g$
admits three complex structures $I$, $J$, and $K$, satisfying the quaternionic
relation $IJ=K$, such that $X$ is the complex manifold $(M,I)$. For each point
$t$ in the unit sphere $S^{2}:=\\{(a,b,c)\ :\ a^{2}+b^{2}+c^{2}=1\\}$, we get
the unit purely imaginary quaternion $I_{t}:=aI+bJ+cK$, which is a complex
structure on $M$. The $2$-form $\omega_{I_{t}}:=g(I_{t}(\bullet),(\bullet))$
is a Kähler form. The original class $\omega$ is the cohomology class of the
$2$-form $\omega_{I}:=g(I(\bullet),(\bullet))$. The classes of
$\\{\omega_{I},\omega_{J},\omega_{K}\\}$ form a basis for the positive
definite $3$-dimensional subspace $W\subset H^{2}(X,{\mathbb{R}})$ whose
complexification is $H^{2,0}(X)\oplus H^{0,2}(X)\oplus{\mathbb{C}}\omega$. The
latter basis is orthogonal with respect to the BBF-pairing, and the BBF-
degrees of the three classes are the same. Conversely, a choice of an
orthogonal basis of $W$ with the same orientation consisting of $\omega$ and
two classes of the same BBF-degree as $\omega$ determines complex structures
$J$ and $K$ and an action of the unit purely imaginary quaternions on the
tangent bundle of $M$ via parallel complex structures. The above $2$-sphere of
complex structures is thus naturally identified with the unit sphere in $W$.

The standard complex structure on the unit sphere combines with the above
construction to define a complex structure on $S^{2}\times M$, such that the
first projection $\pi:S^{2}\times M\rightarrow S^{2}$ is holomorphic and the
fiber over $t$ is $(M,I_{t})$, giving rise to the twistor family (5.3) (see
[HKLR]). The base of the twistor family is thus naturally identified with the
unit sphere in the positive definite $3$-dimensional subspace $W$ of
$H^{2}(X,{\mathbb{R}})$ via the isomorphism sending a class $\omega^{\prime}$
in the unit sphere in $W$ to the unique101010 The intersection of the conic
$Q$ in ${\mathbb{P}}(W_{\mathbb{C}})$ of isotropic lines in $W_{\mathbb{C}}$
with the line ${\mathbb{P}}((\omega^{\prime})^{\perp}_{\mathbb{C}}\cap
W_{\mathbb{C}})$ orthogonal to $\omega^{\prime}$ consists of two complex
conjugate points inducing on $W$ distinct orientations. isotropic line in
${\mathbb{P}}(W_{\mathbb{C}})$, which is orthogonal to $\omega^{\prime}$ and
induces on $W$ the same orientation induced by $H^{2,0}(X)$ in (5.2).

Let $M_{i}$ be the differential manifold underlying $X_{i}$, $i=1,2$. Let
$g_{1}$ be the Ricci-flat hermitian metric on $M_{1}$ associated with the
Kähler class $\omega_{1}$ in (5.10) and let $g_{2}$ be the metric on $M_{2}$
associated with the Kähler class $\omega_{2}:=\psi_{q}(\omega_{1})$. Let
$W_{i}$ be the positive definite $3$-dimensional subspace of
$H^{2}(X_{i},{\mathbb{R}})$, whose complexification is $H^{2,0}(X_{i})\oplus
H^{0,2}(X_{i})\oplus{\mathbb{C}}w_{i}$. We have the equality
$W_{2}=\psi_{q}(W_{1})$, and $\psi_{q}$ maps the unit sphere in $W_{1}$ to
that in $W_{2}$, as $\psi_{q}$ is a Hodge isometry. Hence, $\psi$ restricts to
an isometry from $\eta_{1}(W_{1})$ onto $\eta_{2}(W_{2})$, which maps the
twistor line $Q_{(X_{1},\eta_{1},\omega_{1})}$ isomorphically onto
$Q_{(X_{2},\eta_{2},\omega_{2})}$. Denote by

$Q_{(X_{1},\eta_{1},X_{2},\eta_{2}),\omega_{1}}\subset
Q_{(X_{1},\eta_{1},\omega_{1})}\times Q_{(X_{2},\eta_{2},\omega_{2})}$

the graph of the restriction of $\psi$. Let
$\pi_{i}:{\mathcal{X}}_{i}\rightarrow Q_{(X_{i},\eta_{i},\omega_{i})}$ be the
twistor family through $(X_{i},\eta_{i})$ associated to $\omega_{i}$, $i=1,2$.
The twistor line (5.10) is the lift of
$Q_{(X_{1},\eta_{1},X_{2},\eta_{2}),\omega_{1}}$ to a curve in
${\mathfrak{M}}^{0}_{\psi}$ obtained via the extensions of $\eta_{i}$ to a
trivialization of $R\pi_{i,*}^{2}{\mathbb{Z}}$, $i=1,2$. Choose an orthogonal
basis of $W_{1}$, extending $\omega_{1}$ and consisting of classes of the same
BBF-degree, and let $I_{1}$, $J_{1}$, and $K_{1}$ be the resulting basis of
the purely imaginary quaternions acting via complex structures on the tangent
bundle of $M_{1}$. Define the complex structures $I_{2}$, $J_{2}$, $K_{2}$ on
$M_{2}$ associated to the orthogonal basis of $W_{2}$, which is the image via
$\psi_{q}$ of the chosen basis of $W_{1}$. Given a point $t$ in the unit
sphere in $W_{i}$ denote by $I_{i,t}$ the corresponding complex structure on
$M_{i}$ such that $g_{i}(I_{i,t}(\bullet),\bullet)$ is the Kähler form with
class $t$, $i=1,2$. We get the $2$-sphere of complex structures
$I_{t}:=(I_{1,t},I_{2,\psi(t)})$ on $M_{1}\times M_{2}$ compatible with the
Ricci-flat hermitian metric $(g_{1},g_{2})$ on $M_{1}\times M_{2}$. Then the
fiber product

${\mathcal{X}}_{1}\times_{Q_{(X_{1},\eta_{1},X_{2},\eta_{2}),\omega_{1}}}{\mathcal{X}}_{2}\rightarrow
Q_{(X_{1},\eta_{1},X_{2},\eta_{2}),\omega_{1}}$

of the two twistor families is the twistor family associated to the above
sphere of complex structures on $M_{1}\times M_{2}$.

###  5.4. Diagonal twistor paths

Let

$\Pi_{i}:{\mathfrak{M}}_{\psi}\rightarrow{\mathfrak{M}}_{\Lambda}$

be the morphism sending the isomorphism class of
$(X_{1},\eta_{1},X_{2},\eta_{2})$ to that of $(X_{i},\eta_{i})$. Note that
$\Pi_{i}$ maps $\widetilde{Q}_{(X_{1},\eta_{1},X_{2},\eta_{2}),\omega_{1}}$
isomorphically onto $\widetilde{Q}_{(X_{i},\eta_{i},\omega_{i})}$, $i=1,2$.

######  Definition 5.13.

(1) A twistor path in ${\mathfrak{M}}_{\Lambda}^{0}$ from a marked pair
$(X,\eta)$ to a marked pair $(X^{\prime},\eta^{\prime})$ consists of the data
of a sequence of marked pairs $(X_{i},\eta_{i})$, $0\leq i\leq k+1$, with
$(X_{0},\eta_{0})=(X,\eta)$ and
$(X_{k+1},\eta_{k+1})=(X^{\prime},\eta^{\prime})$, together with a twistor
line $\widetilde{Q}_{(X_{i},\eta_{i},\omega_{i})}$ through $(X_{i},\eta_{i})$
passing through $(X_{i+1},\eta_{i+1})$ as well, for $0\leq i\leq k$. The
twistor path is said to be generic if ${\rm Pic}(X_{i})$ is trivial, for
$1\leq i\leq k$.
(2) A twistor path in ${\mathfrak{M}}_{\psi}^{0}$ from a quadruple
$(X_{1},\eta_{1},X_{2},\eta_{2})$ to a quadruple $(Y_{1},e_{1},Y_{2},e_{2})$
consists of the data of a sequence $(X_{1,i},\eta_{1,i},X_{2,i},\eta_{2,i})$,
$0\leq i\leq k+1$, with
$(X_{1,0},\eta_{1,0},X_{2,0},\eta_{2,0})=(X_{1},\eta_{1},X_{2},\eta_{2})$ and
$(X_{1,k+1},\eta_{1,k+1},X_{2,k+1},\eta_{2,k+1})=(Y_{1},e_{1},Y_{2},e_{2})$,
together with a twistor line
$\widetilde{Q}_{(X_{1,i},\eta_{1,i},X_{2,i},\eta_{2,i}),\omega_{i})}$ through
$(X_{1,i},\eta_{1,i},X_{2,i},\eta_{2,i})$ passing through
$(X_{1,i+1},\eta_{1,i+1},X_{2,i+1},\eta_{2,i+1})$ as well, for $0\leq i\leq
k$. The twistor path is said to be generic if ${\rm Pic}(X_{1,i})$ is trivial,
for $1\leq i\leq k$.

######  Lemma 5.14.

The morphism
$\Pi_{i}:{\mathfrak{M}}_{\psi}^{0}\rightarrow{\mathfrak{M}}_{\Lambda}^{0}$ is
surjective, for $i=1,2$.

###### Proof.

Let $q:=(X_{1},\eta_{1},X_{2},\eta_{2})$ be a quadruple in
${\mathfrak{M}}_{\Lambda}^{0}$ and let $(Y^{\prime}_{1},\eta^{\prime}_{1})$ be
a marked pair in ${\mathfrak{M}}_{\Lambda}^{0}$. Choose a generic Kähler class
$\omega_{1}$ in
${\mathcal{K}}_{X_{1}}\cap\psi_{q}^{-1}({\mathcal{K}}_{X_{2}})$ so that the
twistor line $\widetilde{Q}_{(X_{1},\eta_{1},\omega_{1})}$ is generic and let
$(Y_{1},e_{1})$ be a marked pair in
$\widetilde{Q}_{(X_{1},\eta_{1},\omega_{1})}$ with a trivial ${\rm
Pic}(Y_{1})$. A pair $(Y_{2},e_{2})$ in the fiber of the period map through
${\mathfrak{M}}_{\Lambda}^{0}$ over $\psi(P(Y_{1},e_{1}))$ must be the unique
marked pair in this fiber, since ${\rm Pic}(Y_{2})$ is trivial as well. The
equalities ${\mathcal{K}}_{Y_{i}}={\mathcal{C}}_{Y_{i}}^{+}$, $i=1,2$, imply
that ${\mathcal{K}}_{Y_{1}}=(e_{1}^{-1}\psi e_{2})({\mathcal{K}}_{Y_{2}})$ and
so the quadruple $(Y_{1},e_{1},Y_{2},e_{2})$ belongs to
${\mathfrak{M}}_{\psi}^{0}$. Furthermore, every twistor path
$Q_{(Y_{1},e_{1},\omega^{\prime}_{1})}$ through $(Y_{1},e_{1})$ lifts to a
twistor path $Q_{(Y_{1},e_{1},Y_{2},e_{2}),\omega^{\prime}_{1}}$ through
$(Y_{1},e_{1},Y_{2},e_{2})$, by Lemma 5.12. There exists a generic twistor
path from $(Y_{1},e_{1})$ to $(Y^{\prime}_{1},\eta^{\prime}_{1})$, by [Ve4,
Theorems 3.2 and 5.2.e]. Repeating the above argument we get that this twistor
path lifts to a twistor path from $(Y_{1},e_{1},Y_{2},e_{2})$ to
$(Y^{\prime}_{1},\eta^{\prime}_{1},Y^{\prime}_{2},\eta^{\prime}_{2})$, for
some $(Y^{\prime}_{2},\eta^{\prime}_{2})$ in ${\mathfrak{M}}^{0}_{\Lambda}$,
by the genericity assumption. Hence, $(Y^{\prime}_{1},\eta^{\prime}_{1})$
belongs to the image of $\Pi_{1}$ and $\Pi_{1}$ is surjective. The isomorphism
${\mathfrak{M}}_{\psi}\rightarrow{\mathfrak{M}}_{\psi^{-1}}$ sending
$(X_{1},\eta_{1},X_{2},\eta_{2})$ to $(X_{2},\eta_{2},X_{1},\eta_{1})$
interchanges the roles of $\Pi_{1}$ and $\Pi_{2}$. It follows that $\Pi_{2}$
is surjective as well. ∎

######  Lemma 5.15.

Every two points in ${\mathfrak{M}}^{0}_{\psi}$ are connected by a generic
twistor path in ${\mathfrak{M}}^{0}_{\psi}$.

###### Proof.

Let $q:=(X_{1},\eta_{1},X_{2},\eta_{2})$ be a quadruple in
${\mathfrak{M}}^{0}_{\psi}$, such that ${\rm Pic}(X_{1})$ is trivial. Then
${\rm Pic}(X_{2})$ is trivial as well, since
$\psi_{q}:H^{2}(X_{1},{\mathbb{Q}})\rightarrow H^{2}(X_{2},{\mathbb{Q}})$ is a
Hodge isometry. Hence, $(X_{i},\eta_{i})$ is the unique pair in its connected
component in ${\mathfrak{M}}_{\Lambda}$ over its period $P(X_{i},\eta_{i})$,
by Verbitski’s Torelli theorem [Hu2, Ve3]. Consequently, $q$ is the unique
point in ${\mathfrak{M}}^{0}_{\psi}$ mapping via $P\circ\Pi_{i}$ to
$P(X_{i},\eta_{i})$, for each of $i=1,2$. It follows that every two such
quadruples are connected by a generic twistor path in
${\mathfrak{M}}^{0}_{\psi}$, by the analogous theorem for marked pairs and the
liftability of such paths to ${\mathfrak{M}}^{0}_{\psi}$ demonstrated in the
proof of Lemma 5.14. Now, through each point in ${\mathfrak{M}}^{0}_{\psi}$
passes a generic twistor line. Hence, every two points in
${\mathfrak{M}}^{0}_{\psi}$ are connected by a generic twistor path. ∎

###  5.5. Deforming vector bundles along diagonal twistor paths

A universal family $f:{\mathcal{X}}\rightarrow{\mathfrak{M}}_{\Lambda}^{0}$
exists, by [Ma7]. We get the universal families
$\Pi_{i}^{*}(f):\Pi_{i}^{*}{\mathcal{X}}\rightarrow{\mathfrak{M}}_{\psi}^{0}$,
$i=1,2$, as well as the universal fiber product

(5.11)
$\tilde{f}:\Pi_{1}^{*}{\mathcal{X}}\times_{{\mathfrak{M}}_{\psi}^{0}}\Pi_{2}^{*}{\mathcal{X}}\rightarrow{\mathfrak{M}}_{\psi}^{0}.$

The local system $Rf_{*}{\mathbb{Z}}$ over ${\mathfrak{M}}_{\Lambda}^{0}$ is
trivial, by [Ma7, Lemma 2.1]. Hence, so are the local system
$R\tilde{f}_{*}{\mathbb{Z}}$ and $R\tilde{f}_{*}{\mathbb{Q}}$.

######  Definition 5.16.

Let $M$ be a $d$-dimensional compact Kähler manifold. Set
$H^{ev}(M,{\mathbb{Q}}):=\oplus_{i=0}^{2d}H^{2i}(M,{\mathbb{Q}})$. Given a
class $\gamma\in H^{ev}(M,{\mathbb{Q}})$ denote by $\gamma_{i}$ the graded
summand in $H^{2i}(M,{\mathbb{Q}})$. We say that $\gamma$ is a Hodge class, if
$\gamma_{i}$ belongs to $H^{i,i}(M)$, for all $0\leq i\leq 2d$.

######  Definition 5.17.

Let $q:=(X_{1},\eta_{1},X_{2},\eta_{2})$ be a quadruple in
${\mathfrak{M}}^{0}_{\psi}$ and let $\gamma$ be a Hodge class in
$H^{ev}(X_{1}\times X_{2},{\mathbb{Q}})$. We say that $\gamma$ remains a Hodge
class under every deformation in ${\mathfrak{M}}^{0}_{\psi}$, if for every
quadruple
$q^{\prime}:=(X^{\prime}_{1},\eta^{\prime}_{1},X^{\prime}_{2},\eta^{\prime}_{2})$
in ${\mathfrak{M}}^{0}_{\psi}$ the parallel-transport $\gamma^{\prime}\in
H^{ev}(X^{\prime}_{1}\times X^{\prime}_{2},{\mathbb{Q}})$ of $\gamma$ in the
local system $R\tilde{f}_{*}{\mathbb{Q}}$ along some, hence any, continuous
path from $q$ to $q^{\prime}$ in ${\mathfrak{M}}^{0}_{\psi}$ is a Hodge class.

######  Definition 5.18.

Let $M_{1}$ and $M_{2}$ be compact Kähler manifolds. Let ${\mathcal{A}}_{i}$
be an Azumaya algebra over $M_{i}$. We say that ${\mathcal{A}}_{2}$ is
deformation equivalent to ${\mathcal{A}}_{1}$, if there exists a proper family
$\pi:{\mathcal{M}}\rightarrow B$ of compact Kähler manifolds over a connected
analytic base $B$, points $b_{i}\in B$, isomorphisms
$f_{i}:M_{i}\rightarrow{\mathcal{M}}_{b_{i}}$ with the fiber of $\pi$ over
$b_{i}$, and an Azumaya algebra ${\mathcal{A}}$ over ${\mathcal{M}}$, such
that $f_{i}^{*}{\mathcal{A}}$ is isomorphic to ${\mathcal{A}}_{i}$, for
$i=1,2$.

######  Proposition 5.19.

Let $q:=(X_{1},\eta_{1},X_{2},\eta_{2})$ be a quadruple in
${\mathfrak{M}}^{0}_{\psi}$ and let $F$ be a locally free sheaf over
$X_{1}\times X_{2}$ with the following properties.

  1. (1)

$F$ is
$\left(\pi_{X_{1}}^{*}(\omega)+\pi_{X_{2}}^{*}(\psi_{q}(\omega))\right)$-slope-
stable, for every Kähler class $\omega$ in some non-empty open subcone
${\mathcal{C}}_{F}$ of
${\mathcal{K}}_{X_{1}}\cap\psi_{q}^{-1}({\mathcal{K}}_{X_{2}})$.

  2. (2)

The class $c_{2}({\mathcal{E}}nd(F))$ remains of Hodge type $(2,2)$ along
every deformation in ${\mathfrak{M}}^{0}_{\psi}$.

For every point
$(X^{\prime}_{1},\eta^{\prime}_{1},X^{\prime}_{2},\eta^{\prime}_{2})$ in
${\mathfrak{M}}^{0}_{\psi}$ there exists a possibly twisted locally free sheaf
F’ over $X^{\prime}_{1}\times X^{\prime}_{2}$, such that the Azumaya algebra
${\mathcal{E}}nd(F^{\prime})$ is deformation equivalent to
${\mathcal{E}}nd(F)$.

###### Proof.

The slope stability property (1) of $F$ implies that the first Chern class of
every direct summand of ${\mathcal{E}}nd(F)$ vanishes, by [Ma5, Lemma 7.2].
Property (2), that $c_{2}({\mathcal{E}}nd(F))$ remains of Hodge type, implies
that the Azumaya algebra ${\mathcal{E}}nd(F)$ deforms along the twistor family
over a generic twistor line
$\widetilde{Q}_{(X_{1},\eta_{1},X_{2},\eta_{2}),\omega_{1}}$ in
${\mathfrak{M}}^{0}_{\psi}$ associated to a Kähler class $\omega_{1}$ in
${\mathcal{C}}_{F}$, by [Ve2, Theorem 3.19], which is generalized to Azumaya
algebras in [Ma5, Cor. 6.12]. Here we use the fact that the twistor line in
${\mathfrak{M}}^{0}_{\psi}$ is associated to a hyper-Kähler structure, as
explained in Section 5.3, as well as the fact that the first Chern class of
every direct summand of ${\mathcal{E}}nd(F)$ vanishes. Choose a quadruple
$q_{0}:=(Y_{1},e_{1},Y_{2},e_{2})$ in
$\widetilde{Q}_{(X_{1},\eta_{1},X_{2},\eta_{2}),\omega_{1}}$ with a trivial
Picard group ${\rm Pic}(Y_{1})$ and let ${\mathcal{A}}$ be the Azumaya algebra
over $Y_{1}\times Y_{2}$ resulting by deforming ${\mathcal{E}}nd(F)$ along
$\widetilde{Q}_{(X_{1},\eta_{1},X_{2},\eta_{2}),\omega_{1}}$. The Azumaya
algebra ${\mathcal{A}}$ further deforms along every generic twistor path in
${\mathfrak{M}}^{0}_{\psi}$ from $q_{0}$ to any point in
${\mathfrak{M}}^{0}_{\psi}$, by [Ma5, Prop. 6.17]. The existence of an Azumaya
algebra ${\mathcal{A}}^{\prime}$ over $X^{\prime}_{1}\times X^{\prime}_{2}$
deformation equivalent to ${\mathcal{E}}nd(F)$ follows from that fact that
every point $q^{\prime}$ in ${\mathfrak{M}}_{\psi}^{0}$ is connected to the
point $q_{0}$ via a generic twistor path, by Lemma 5.15. The existence of a
locally free sheaf $F^{\prime}$, such that the Azumaya algebras
${\mathcal{E}}nd(F^{\prime})$ and ${\mathcal{A}}^{\prime}$ are isomorphic,
follows from the well known correspondence between Azumaya algebras and
equivalence classes of twisted sheaves modulo tensorization by line bundles
[C]. ∎

###  5.6. Deforming the Fourier-Mukai kernel

######  Definition 5.20.

A quadruple $(X_{1},\eta_{1},X_{2},\eta_{2})$ in ${\mathfrak{M}}_{\psi}$ is
said to support a compatible vector bundle if there exists a (possibly
twisted) locally free coherent sheaf $F$ over $X_{1}\times X_{2}$ with the
following properties.

  1. (1)

The class $\kappa(F)\sqrt{td_{X_{1}\times X_{2}}}$ induces a morphism

$\phi_{F}:H^{*}(X_{1},{\mathbb{Q}})\rightarrow H^{*}(X_{2},{\mathbb{Q}})$

in ${\rm Hom}_{{\mathcal{G}}_{an}}(X_{1},X_{2})$, where ${\mathcal{G}}_{an}$
is the groupoid given in (1.2).

  2. (2)

The equality $\psi=\eta_{2}\widetilde{H}_{0}(\phi_{F})\eta_{1}^{-1}$ holds,
where $\widetilde{H}_{0}$ is given in (5.5).

######  Proposition 5.21.

Every quadruple $(X_{1},\eta_{1},X_{2},\eta_{2})$ in the connected component
${\mathfrak{M}}_{\psi}^{0}$ given in (5.8) supports a compatible vector
bundle.

###### Proof.

Step 1: Let $q\in{\mathfrak{M}}_{\psi}^{0}$ be the isomorphism class of
$(X_{1},\eta_{1},X_{2},\eta_{2})$. Consider the local system
$R\tilde{f}_{*}{\mathbb{Q}}$ over ${\mathfrak{M}}^{0}_{\psi}$ associated to
the universal fiber product (5.11). Let $\gamma_{q}\in H^{*}(X_{1}\times
X_{2},{\mathbb{Q}})$ be a flat deformation of the class

$\gamma:=\kappa(E_{0})\sqrt{td_{X_{0}\times Y_{0}}}$

via a path in ${\mathfrak{M}}_{\psi}^{0}$ from
$(X_{0},\eta_{X_{0}},Y_{0},\eta_{Y_{0}})$ to $q$. If
$\tilde{\eta}_{1}:H^{*}(X_{1},{\mathbb{Q}})\rightarrow
H^{*}(X_{0},{\mathbb{Q}})$ denotes the parallel-transport operator in the
local system $\Pi_{1}^{*}Rf_{*}{\mathbb{Q}}$ and
$\tilde{\eta}_{2}:H^{*}(X_{2},{\mathbb{Q}})\rightarrow
H^{*}(Y_{0},{\mathbb{Q}})$ denotes the parallel-transport operator in the
local system $\Pi_{2}^{*}Rf_{*}{\mathbb{Q}}$, both with respect to the chosen
path (backward), then
$\gamma=(\tilde{\eta}_{1}\otimes\tilde{\eta}_{2})(\gamma_{q})$. The class
$\gamma_{q}$ is independent of the path chosen, since the local system
$R\tilde{f}_{*}{\mathbb{Q}}$ over ${\mathfrak{M}}_{\Lambda}^{0}$ is trivial,
by [Ma7, Lemma 2.1]. We prove in this step that $\gamma_{q}$ is a Hodge class.
In particular, $c_{2}({\mathcal{E}}nd(E_{0}))$ remains of Hodge type over
$X_{1}\times X_{2}$, by the equality $c_{2}({\mathcal{E}}nd(E_{0}))=-2\,{\rm
rank}(E_{0})\kappa_{2}(E_{0})$. Once it is proven that $\gamma_{q}$ is a Hodge
class, for all $q\in{\mathfrak{M}}^{0}_{\psi}$, Proposition 5.19 implies the
existence of a locally free sheaf $F$ over $X_{1}\times X_{2}$ such that
${\mathcal{E}}nd(F)$ is deformation equivalent to ${\mathcal{E}}nd(E_{0})$.

The isometry111111The isomorphism $\phi_{0}$ is an isometry with respect to
the Mukai pairings. When the odd cohomology of $X$ vanishes $\phi_{0}$ is also
an isometry with respect to the Poincaré pairings, by Corollary 4.2, but we do
not assume the vanishing of the odd cohomology here.
$\phi_{0}:H^{*}(X_{0},{\mathbb{Q}})\rightarrow H^{*}(Y_{0},{\mathbb{Q}})$,
induced by $\gamma$, is invariant under the action of
$GL(H^{*}(X_{0},{\mathbb{Q}}))$ on ${\rm
Hom}(H^{*}(X_{0},{\mathbb{Q}}),H^{*}(Y_{0},{\mathbb{Q}}))$, where an element
$g\in GL(H^{*}(X_{0},{\mathbb{Q}}))$ acts by
$h\mapsto(\phi_{0}g\phi_{0}^{-1})hg^{-1}$. The isometry $\phi_{0}$ is related
to the class $\gamma:=\kappa(E_{0})\sqrt{td_{X_{0}\times Y_{0}}}$ via

$\phi_{0}\in{\rm Hom}(H^{*}(X_{0}),H^{*}(Y_{0}))\cong H^{*}(X_{0})^{*}\otimes
H^{*}(Y_{0})\cong H^{*}(X_{0})\otimes H^{*}(Y_{0})\cong H^{*}(X_{0}\times
Y_{0})\ni\gamma,$

where the middle isomorphism is induced by the Poincaré duality isomorphism
$PD:H^{*}(X_{0},{\mathbb{Q}})\rightarrow H^{*}(X_{0},{\mathbb{Q}})^{*}$.
Assume that $g$ is an isometry with respect to the Poincaré pairing and so
$(g^{-1})^{*}=PD\circ g\circ PD^{-1}$. In this case $\gamma$ is invariant with
respect to the action by $g\otimes(\phi_{0}g\phi_{0}^{-1})$. The Lie algebra
$\bar{{\mathfrak{g}}}_{X_{0},{\mathbb{R}}}$ acts on
$H^{*}(X_{0},{\mathbb{R}})$ by derivations [LL, Prop. 4.5(ii)], and so it
annihilates the cup product and is thus contained in the Lie algebra of the
isometry group with respect to the Poincaré pairing. We get that the class
$\gamma$ is annihilated by $(\xi\otimes 1)+(1\otimes Ad_{\phi_{0}}(\xi))$, for
every element $\xi\in\bar{{\mathfrak{g}}}_{X_{0}}$. We conclude that $\gamma$
is annihilated by the Lie subalgebra

$\bar{{\mathfrak{g}}}_{\phi_{0}}\ :=\\{(\xi\otimes 1)+(1\otimes
Ad_{\phi_{0}}(\xi))\ :\ \xi\in\bar{{\mathfrak{g}}}_{X_{0}}\\}\ \subset\
\bar{{\mathfrak{g}}}_{X_{0}}\times{\mathfrak{g}}_{Y_{0}},$

where the inclusion follows from the equality
$Ad_{\phi_{0}}({\mathfrak{g}}_{X_{0}})={\mathfrak{g}}_{Y_{0}}$ established in
[Ta, Theorem A]. Note that $\bar{{\mathfrak{g}}}_{\phi_{0}}$ is in fact the
subalgebra

$\bar{{\mathfrak{g}}}_{\psi_{E_{0}}}\ :=\ \\{(\xi\otimes 1)+(1\otimes
Ad_{\psi_{E_{0}}}(\xi))\ :\ \xi\in\bar{{\mathfrak{g}}}_{X_{0}}\\}\ \subset\
\bar{{\mathfrak{g}}}_{X_{0}}\times\bar{{\mathfrak{g}}}_{Y_{0}},$

by Lemma 4.1(3) and Remark 5.10.

We have the commutative diagram

$\textstyle{H^{2}(X_{0},{\mathbb{Q}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{1}^{-1}\eta_{X_{0}}}$$\scriptstyle{\eta_{Y_{0}}^{-1}\psi\eta_{X_{0}}=\psi_{E_{0}}}$$\textstyle{H^{2}(X_{1},{\mathbb{Q}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{q}:=\eta_{2}^{-1}\psi\eta_{1}}$$\textstyle{H^{2}(Y_{0},{\mathbb{Q}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{2}^{-1}\eta_{Y_{0}}}$$\textstyle{H^{2}(X_{2},{\mathbb{Q}}),}$

whose horizontal arrows are parallel-transport operators. The Lie algebra
${\mathfrak{g}}_{X_{0}}$ deforms flatly to ${\mathfrak{g}}_{X_{1}}$, by the
topological nature of both, and so the semi-simple part of its degree $0$
summand $\bar{{\mathfrak{g}}}_{X_{0}}$ deforms flatly to
$\bar{{\mathfrak{g}}}_{X_{1}}$. The Lie subalgebra
${\mathfrak{g}}_{\psi_{E_{0}}}$ deforms flatly to

$\bar{{\mathfrak{g}}}_{\psi_{q}}\ :=\ \\{(\xi\otimes 1)+(1\otimes
Ad_{\psi_{q}}(\xi))\ :\ \xi\in\bar{{\mathfrak{g}}}_{X_{1}}\\}\ \subset\
\bar{{\mathfrak{g}}}_{X_{1}}\times\bar{{\mathfrak{g}}}_{X_{2}},$

by the commutativity of the above diagram. Indeed,

$\displaystyle(Ad_{\eta_{1}^{-1}\eta_{X_{0}}}\otimes
Ad_{\eta_{2}^{-1}\eta_{Y_{0}}})\left((\xi\otimes 1)+(1\otimes
Ad_{\psi_{E_{0}}}(\xi))\right)$ $\displaystyle=$
$\displaystyle(Ad_{\eta_{1}^{-1}\eta_{X_{0}}}(\xi)\otimes 1)+(1\otimes
Ad_{\eta_{2}^{-1}\eta_{Y_{0}}\psi_{E_{0}}}(\xi))$ $\displaystyle=$
$\displaystyle(\xi^{\prime}\otimes 1)+(1\otimes Ad_{\psi_{q}}(\xi^{\prime})),$

where
$\xi^{\prime}:=Ad_{\eta_{1}^{-1}\eta_{X_{0}}}(\xi)\in\bar{{\mathfrak{g}}}_{X_{1}}$.
Let

$\gamma_{q}\in H^{*}(X_{1}\times X_{2},{\mathbb{Q}})$

be the flat deformation of the class $\gamma$ via a path in
${\mathfrak{M}}_{\psi}^{0}$. It follows that $\gamma_{q}$ is annihilated by
$\bar{{\mathfrak{g}}}_{\psi_{q}}$, as we have seen that $\gamma$ is
annihilated by $\bar{{\mathfrak{g}}}_{\psi_{E_{0}}}$.

Let $h^{\prime}_{X_{i}}$ be the Hodge operator of $X_{i}$, $i=1,2$, given in
(3.8). Then $h^{\prime}_{X_{2}}=Ad_{\psi_{q}}(h^{\prime}_{X_{1}})$, since
$\psi_{q}$ is an isomorphism of Hodge structures. Hence, the Hodge operator of
$X_{1}\times X_{2}$

$(h^{\prime}_{X_{1}}\otimes 1)+(1\otimes
h^{\prime}_{X_{2}})=(h^{\prime}_{X_{1}}\otimes 1)+(1\otimes
Ad_{\psi_{q}}(h^{\prime}_{X_{1}}))$

belongs to $\bar{{\mathfrak{g}}}_{\psi_{q}}$ and thus annihilates
$\gamma_{q}$. We conclude that $\gamma_{q}$ is a Hodge class.

Step 2: Consider first the quadruple
$q_{0}:=(X_{0},\eta_{X_{0}},Y_{0},\eta_{Y_{0}})$, used in the definition of
the connected component ${\mathfrak{M}}_{\psi}^{0}$ in (5.8), and set
$F:=E_{0}$. Property (1) in Definition 5.20 follows in this case from
Assumption 5.9 and property (2) follows from the definition of $\psi$ in
Equation (5.6).

Let $\phi_{\gamma_{q}}:H^{*}(X_{1},{\mathbb{Q}})\rightarrow
H^{*}(X_{2},{\mathbb{Q}})$ be the homomorphism induced by $\gamma_{q}$. The
equality $\phi_{\gamma_{q}}=\tilde{\eta}_{2}^{-1}\phi_{0}\tilde{\eta}_{1}$
exhibits $\phi_{\gamma_{q}}$ as a composition of morphisms in the groupoid
${\mathcal{G}}$, hence in ${\rm Hom}_{\mathcal{G}}(X_{1},X_{2})$. Choose a
twistor path $C$ in ${\mathfrak{M}}^{0}_{\psi}$ from $q_{0}$ to $q$ along
which the Azumaya algebra ${\mathcal{E}}nd(E_{0})$ deforms via an Azumaya
algebra ${\mathcal{A}}$ over the twistor family
$\Pi_{1}^{*}{\mathcal{X}}\times_{C}\Pi_{2}^{*}{\mathcal{X}}\rightarrow C$.
Such a path $C$ exists, by Assumption 5.9 and Proposition 5.19. There exists a
locally free twisted sheaf ${\mathcal{F}}$ over
$\Pi_{1}^{*}{\mathcal{X}}\times_{C}\Pi_{2}^{*}{\mathcal{X}}$, such that
${\mathcal{E}}nd({\mathcal{F}})$ is isomorphic to ${\mathcal{A}}$.
${\mathcal{F}}$ restricts to a locally free twisted sheaf $F_{0}$ over
$X_{0}\times Y_{0}$ and a locally free sheaf $F$ over $X_{1}\times X_{2}$,
such that the Azumaya algebras ${\mathcal{E}}nd(F_{0})$ is isomorphic to
${\mathcal{E}}nd(E_{0})$ and $\kappa(F)$ is a parallel-transport of
$\kappa(F_{0})$ along a (real) path in $C$. Hence, $\gamma_{q}$ is equal to
$\kappa(F)\sqrt{td_{X_{1}\times X_{2}}}$ and property (1) in Definition 5.20
follows. Property (2) in Definition 5.20 is equivalent to the equality
$\psi_{q}=\widetilde{H}_{0}([\kappa(F)\sqrt{td_{X_{1}\times X_{2}}}]_{*})$.
The latter equality follows at $q$ via Remark 5.6 from the fact that it holds
for $q_{0}$. Explicitly,

$\displaystyle\psi_{q}$ $\displaystyle=$
$\displaystyle(\eta_{2}^{-1}\eta_{Y_{0}})\psi_{E_{0}}(\eta_{X_{0}}^{-1}\eta_{1})=(\eta_{2}^{-1}\eta_{Y_{0}})\widetilde{H}_{0}([\kappa(E_{0})\sqrt{td_{X_{0}\times
Y_{0}}}]_{*})(\eta_{X_{0}}^{-1}\eta_{1})$
$\displaystyle\stackrel{{\scriptstyle{\rm Rem.}\ \ref{remark-convention-
behaves-well-with-respect-to-parallel-transport-operators}}}{{=}}$
$\displaystyle\widetilde{H}_{0}\left((\tilde{\eta}_{2}^{-1})\circ[\kappa(E_{0})\sqrt{td_{X_{0}\times
Y_{0}}}]_{*}\circ\tilde{\eta}_{1}\right)=\widetilde{H}_{0}([\kappa(F)\sqrt{td_{X_{1}\times
X_{2}}}]_{*}).$

∎

##  6\. The BKR equivalence

Let $S$ be a projective surface and denote by $S^{[n]}$ the Hilbert scheme of
length $n$ subscheme of $S$. The symmetric group ${\mathfrak{S}}_{n}$ on $n$
letters acts on the cartesian product $S^{n}$ permuting the factors. Let
$Hilb_{{\mathfrak{S}}_{n}}(S^{n})$ be the Hilbert scheme of length $n!$
subschemes $Z$ of $S^{n}$, which are ${\mathfrak{S}}_{n}$-invariant, and such
that $H^{0}(Z,{\mathcal{O}}_{Z})$ is the regular representation of
${\mathfrak{S}}_{n}$. Then $S^{[n]}$ is isomorphic to the closure in
$Hilb_{{\mathfrak{S}}_{n}}(S^{n})$ of the locus of reduced subschemes which
are ${\mathfrak{S}}_{n}$-orbits of $n$-tuples consisting of $n$ distinct
points of $S$, by [Hai]. The restriction

$\Gamma_{S}\subset S^{[n]}\times S^{n},$

to $S^{[n]}$ of the universal subscheme of
$Hilb_{{\mathfrak{S}}_{n}}(S^{n})\times S^{n}$ is called121212 The isospectral
Hilbert scheme $\Gamma_{S}$ is defined in [Hai] as the reduced subscheme
associated to the support of the fiber product of $S^{[n]}$ and $S^{n}$ over
the symmetric product $S^{(n)}$. Haiman proves that $\Gamma_{S}$ is flat over
$S^{[n]}$, and so it is indeed the universal subscheme.  the isospectral
Hilbert scheme. The projections $S^{[n]}\stackrel{{\scriptstyle
q}}{{\leftarrow}}\Gamma_{S}\stackrel{{\scriptstyle b}}{{\rightarrow}}S^{n}$
have the following properties. The morphism $q$ is flat and
${\mathfrak{S}}_{n}$-invariant of degree $n!$, by the proof of [Hai, Prop.
3.7.4]. The morphism $b$ restricts to an isomorphism over the complement of
the union of the diagonals in $S^{n}$. The isospectral Hilbert scheme
$\Gamma_{S}$ is Gorenstein, by [Hai, Theorem 3.1], and its dualizing sheaf
$\omega_{\Gamma_{S}}$ is isomorphic to the line bundle
$q^{*}(\omega_{S^{[n]}}(\delta))$, by [Hai, Theorem 3.1, Prop. 3.4.3, comment
after Lemma 3.4.2]. Above $\delta$ is the divisor class, such that
${\mathcal{O}}_{S^{[n]}}(-\delta)\cong\left(q_{*}({\mathcal{O}}_{\Gamma_{S}})\right)^{\chi}$,
where $\chi$ is the sign character of ${\mathfrak{S}}_{n}$. Note that
$2\delta$ is the class of the effective divisor in $S^{[n]}$ of non-reduced
subschemes.

Let $D^{b}_{{\mathfrak{S}}_{n}}(S^{n})$ be the bounded derived category of
${\mathfrak{S}}_{n}$-equivariant coherent sheaves on $S^{n}$ and
$D^{b}_{{\mathfrak{S}}_{n}}(S^{[n]}\times S^{n})$ its analogue with respect to
the permutation action of ${\mathfrak{S}}_{n}$ on the factor $S^{n}$ and the
trivial action on the factor $S^{[n]}$. Let $\rho$ be the natural
${\mathfrak{S}}_{n}$-linearization of ${\mathcal{O}}_{\Gamma_{S}}$. The object
$({\mathcal{O}}_{\Gamma_{S}},\rho)$ of
$D^{b}_{{\mathfrak{S}}_{n}}(S^{[n]}\times S^{n})$ is the Fourier-Mukai kernel
of an equivalence of derived categories

$BKR:D^{b}_{{\mathfrak{S}}_{n}}(S^{n})\rightarrow D^{b}(S^{[n]}),$

given by the composition $Rq_{*}^{{\mathfrak{S}}_{n}}Lb^{*}$ of the functors
$Lb^{*}:D^{b}_{{\mathfrak{S}}_{n}}(S^{n})\rightarrow
D^{b}_{{\mathfrak{S}}_{n}}(\Gamma_{S})$ and
$Rq_{*}^{{\mathfrak{S}}_{n}}:D^{b}_{{\mathfrak{S}}_{n}}(\Gamma_{S})\rightarrow
D^{b}(S^{[n]})$, by [BKR].

##  7\. A universal bundle over $M^{[n]}\times S^{[n]}$ from a universal
bundle over $M\times S$

In Section 7.1 we recall the construction of a locally free Fourier-Mukai
kernel of an equivalence of the derived categories of the Hilbert schemes
$S^{[n]}$ and $M^{[n]}$, where $M$ is a $2$-dimensional smooth and projective
moduli space of vector bundles on a $K3$ surface $S$. In Section 7.2 we
construct a functor
$\widetilde{\Theta}_{n}:{\mathcal{G}}^{[1]}\rightarrow{\mathcal{G}}^{[n]}$,
which sends a morphism associated to an equivalence $\Phi$ of derived
categories of $K3$ surfaces to a morphism associated to the equivalence of the
derived categories of their Hilbert schemes, which is the BKR-conjugate of the
cartesian power $\Phi^{n}$. In Section 7.3 we normalize
$\widetilde{\Theta}_{n}$ to obtain the functor
$\Theta_{n}:{\mathcal{G}}^{[1]}\rightarrow{\mathcal{G}}^{[n]}$, mentioned in
the introduction in (1.8), which maps ${\mathcal{G}}^{[1]}_{an}$ to
${\mathcal{G}}^{[n]}_{an}$. The functor $\Theta_{n}$ sends a parallel-
transport operator between two $K3$ surfaces $S_{1}$ and $S_{2}$ to the
associated parallel-transport operator between their Hilbert schemes
$S_{1}^{[n]}$ and $S_{2}^{[n]}$. If $\phi\in{\rm
Hom}_{{\mathcal{G}}_{an}^{[1]}}(S_{1},S_{2})$ is induced by the class
$\kappa({\mathcal{U}})\sqrt{td_{S_{1}\times S_{2}}}$ of the Fourier-Mukai
kernel ${\mathcal{U}}$ of an equivalence
$\Phi_{\mathcal{U}}:D^{b}(S_{1})\rightarrow D^{b}(S_{2})$, then
$\Theta_{n}(\phi)$ is induced by the analogous class associated to the BKR-
conjugate of the $n$-th cartesian power of $\Phi_{\mathcal{U}}$. In Corollary
7.4 we show that
$\widetilde{H}(\Theta_{n}(\phi)):\widetilde{H}(S_{1}^{[n]},{\mathbb{Q}})\rightarrow\widetilde{H}(S_{2}^{[n]},{\mathbb{Q}})$
is the naive extension of
$\widetilde{H}(\phi):\widetilde{H}(S_{1},{\mathbb{Q}})\rightarrow\widetilde{H}(S_{2},{\mathbb{Q}})$.

###  7.1. A universal bundle over $M^{[n]}\times S^{[n]}$

Assume that $S$ is a $K3$ surface,
$v=(r,\lambda,s)\in\widetilde{H}(S,{\mathbb{Z}})$ is a primitive and isotropic
Mukai vector of rank $r\geq 2$, and $H$ is a $v$-generic polarization. Then
every $H$-slope-semistable sheaf on $S$ is $H$-slope-stable and locally free.
Let $M$ be the moduli space of $H$-slope-stable sheaves on $S$ with Mukai
vector $v$. Then $M$ is a $K3$ surface, by [Mu1]. Assume that a universal
sheaf ${\mathcal{U}}$ exists over $M\times S$. Then ${\mathcal{U}}$ is the
Fourier-Mukai kernel of an equivalence

$\Phi_{\mathcal{U}}:D^{b}(M)\rightarrow D^{b}(S)$

(see [Hu3, Prop. 10.25]). We get the equivalence