Calibrations in hyperkähler geometry

Gueo Grantcharov, Misha Verbitsky111Misha Verbitsky is partially supported by
RFBR grant 10-01-93113-NCNIL-a, RFBR grant 09-01-00242-a, Simons-IUM
fellowship, Science Foundation of the SU-HSE award No. 10-09-0015 and AG
Laboratory HSE, RF government grant, ag. 11.G34.31.0023.

Abstract

We describe a family of calibrations arising naturally on a hyperkähler
manifold $M$. These calibrations calibrate the holomorphic Lagrangian,
holomorphic isotropic and holomorphic coisotropic subvarieties. When $M$ is an
HKT (hyperkähler with torsion) manifold with holonomy $SL(n,{\mathbb{H}})$, we
construct another family of calibrations $\Phi_{i}$, which calibrates
holomorphic Lagrangian and holomorphic coisotropic subvarieties. The
calibrations $\Phi_{i}$ are (generally speaking) not parallel with respect to
any torsion-free connection on $M$.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Calibrations in Riemannian geometry
    2. 2.2 Hyperkähler manifolds and calibrations
    3. 2.3 Calibrations in HKT-geometry
  3. 3 $SL(n,{\mathbb{H}})$-manifolds
    1. 3.1 An introduction to $SL(n,{\mathbb{H}})$-geometry
    2. 3.2 Balanced HKT-manifolds
  4. 4 Differential forms on hypercomplex manifolds
    1. 4.1 The quaternionic Dolbeault complex
    2. 4.2 Positive $(2,0)$-forms on hypercomplex manifolds
    3. 4.3 The map ${\cal V}_{p,q}:\;\Lambda^{p+q,0}_{I}(M){\>\longrightarrow\>}\Lambda^{n+p,n+q}_{I}(M)$ on $SL(n,{\mathbb{H}})$-manifolds
    4. 4.4 Algebra generated by $\omega_{I}$, $\omega_{J}$, $\omega_{K}$
  5. 5 Calibrations on hyperkähler manifolds
    1. 5.1 Hodge decomposition and $U(1)$-action
    2. 5.2 An $SU(2)$-invariant calibration
    3. 5.3 A holomorphic Lagrangian calibration
    4. 5.4 Isotropic and coisotropic calibrations
    5. 5.5 Holomorphic Lagrangian calibrations of degree two
    6. 5.6 Examples
  6. 6 Calibrations on $SL(n,{\mathbb{H}})$-manifolds

##  1 Introduction

The theory of calibrations was developed by R. Harvey and B. Lawson in [HL],
and proved to be very useful in describing the geometric structures associated
with special holonomies. Since then calibrations have become a central notion
in many geometric developments in string physics and M-theory. Up to dimension
8, the calibrations are thoroughly studied and pretty much understood ([DHM]),
but in the higher dimensions, the classification problem seems to be immense.
Even in more special situations, such as in hyperkähler geometry, the problem
of classification of natural111For hyperkähler geometry, “natural” would mean
“$Sp(n)$-invariant”. calibrations is unsolved.

On a Kähler manifold, the normalized power of the Kähler form
$\frac{\omega^{p}}{p!}$ is a calibration. A subvariety is complex analytic if
and only if it is calibrated. This is actually very easy to see, because a
subspace $V\subset TM$ is a face of $\frac{\omega^{p}}{p!}$ if and only if $V$
is complex linear (this follows from the so-called “Wirtinger inequalities”,
see e.g. [HL]).

In this paper we study a family of calibrations which appear naturally in
quaternionic geometry, and describe the corresponding calibrated subvarieties.
These calibrations are in many ways analogous to the powers of the Kähler
form. We define several new calibrations, for hyperkähler, hypercomplex and
HKT-geometry. From the calibration-theoretic point of view, the last of these
is most interesting, because it is (generally speaking) not preserved by any
torsionless connection on $M$. Some of these forms were considered previously
in [V6, AV2, V7].

In hyperkähler geometry, the role of a Kähler form is played by a 4-form
$\Theta:=\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2}$. In Section 5.2 we show
that the normalized powers $\Theta^{p}$ are calibrations. It is easy to see
that $V\subset TM$ is a face of $\Theta$ if and only if $V$ is a quaternionic
subspace (5.2).

The corresponding calibrated subvarieties are those which are complex analytic
with respect to $I$, $J$ and $K$. Such subvarieties are called trianalytic. In
[V1, V2], the theory of trianalytic subvarieties was developed to some extent.
It was shown that the trianalytic subvarieties admit a canonical
desingularizaton, which is hyperkähler. Also it was shown that any complex
analytic subvariety of $(M,I)$ is trianalytic, if the complex structure $I$ is
generic in its twistor family.

Any homogeneous polynomial $P(x,y,z)$ of degree $p$ gives a closed $2p$-form
$P(\omega_{I},\omega_{J},\omega_{K})$ on $M$, and (when the holonomy of $M$ is
maximal) all parallel differential forms on $M$ are obtained this way. When
$P(x,y,z)=\frac{x^{p}}{p!}$, it is a Kähler calibration, when
$P(x,y,z)=c_{p}(x^{2}+y^{2}+z^{2})^{p}$, where
$c_{p}=\sum_{k=0}^{p}\frac{(p!)^{2}}{(k!)^{2}}(2k)!4^{p-k}$, it is the
trianalytic calibration defined above( Theorem 5.3). It would be interesting
to classify all calibrations obtained this way.

The calibrations $\Psi_{k}$ and $\Phi_{n+k}$ we study in this paper are also
polynomials on $\omega_{I},\omega_{J},\omega_{K}$. These calibrations are
called holomorphic Lagrangian, holomorphic isotropic and holomorphic
coisotropic calibrations. The form $\Psi_{k}$ is obtained as a
$(k,k)$-component of
$\operatorname{Re}(\omega_{I}-\sqrt{-1}\>\omega_{K})^{k}$, normalized in
appropriate way, where $\omega_{I}-\sqrt{-1}\>\omega_{K}$ is a holomorphic
symplectic form on $(M,J)$, and the $(k,k)$-part is taken with respect to the
complex structure $I$. In [V6, AV2] it was proven that this form is closed and
weakly positive.

We show in Section 5.4 that a subvariety $Z\subset M$ is calibrated by
$\Psi_{k}$ if and only if $Z$ is holomorphic Lagrangian in $(M,I)$ (for
$k=\frac{1}{2}\dim_{\mathbb{C}}M$) and isotropic (for
$k<\frac{1}{2}\dim_{\mathbb{C}}M$) (5.3, 5.4). Note that holomorphic
Lagrangian calibrations have been found previously in [BrH] in dimension
eight.

In [F] a different holomorphic Lagrangian calibration in any dimension was
constructed as part of an investigation relating the faces of some
calibrations to intersecting supersymmetric branes in M-theory. In String
Theory the holomorphic Lagrangian submanifolds were related to 3-dimensional
topological field theory with target hyperkähler manifold [KRS]. In Section
5.6 we provide some examples of holomorphic Lagrangian subvarieties of
hypercomplex manifolds which are not hyperkähler.

The proof of this result relies on a particular partial order defined on the
set of precalibrations. We say that $\eta\preceq\eta_{1}$ if all faces of
$\eta$ are also faces of $\eta_{1}$. For instance, the calibrations
$c_{p}\Theta^{p}$,
$c_{p}=\sum_{k=0}^{p}\frac{(p!)^{2}}{(k!)^{2}}(2k)!4^{p-k}$, and
$\frac{\omega_{I}^{k}}{k!}$ defined above can be compared:

$c_{p}\Theta^{p}\preceq\frac{\omega_{I}^{2p}}{(2p)!}$

because the faces of $c_{p}\Theta^{p}$ are quaternionic subspaces in $TM$, and
the faces of $\frac{\omega_{I}^{2p}}{(2p)!}$ are complex subspaces (5.2).

Let $\rho$ be a precalibration on a complex manifold (2.1), and $\rho^{p,p}$
be its $(p,p)$-part. We show that a plane $V\subset TM$ is a face of
$\rho^{p,p}$ if and only for $\zeta(V)$ is a face of $\rho$ for all $\zeta\in
U(1)$, for the standard $U(1)$-action on $TM$ (5.1).

Applying this result to the special Lagrangian calibration on $(M,J)$ defined
in [HL] (see also [McL]), we obtain the form $\Psi_{n}$,
$n=\dim_{\mathbb{H}}M$, which calibrates complex analytic Lagrangian
subvarieties on $(M,I)$ (these subvarieties are known to be special Lagrangian
on $(M,J)$; see e.g. [Hit]). This argument is not hard to generalize to
arbitrary dimension.

In most cases listed in [HL] and elsewhere, a calibration form is parallel
with respect to the Levi-Civita connection. An interesting side effect of our
construction of holomorphic Lagrangian calibrations is an appearance of a
family of calibrations which are not parallel, under any torsionless
connection (6). These calibrations are associated with the so-called HKT
structures in hypercomplex geometry. In physics the HKT manifolds appear as
target manifolds with $N=(4,0)$ supersymmetric $\sigma$-models with Wess-
Zumino term [HP].

We construct calibrations on a special class of hypercomplex manifolds with
holonomy of its Obata connection in $SL(n,{\mathbb{H}})$, the commutator
subgroup of $GL(n,{\mathbb{H}})$. Such manifolds are called
$SL(n,{\mathbb{H}})$-manifolds. For more examples and an introduction to
$SL(n,{\mathbb{H}})$-geometry, see Section 3. For any
$SL(n,{\mathbb{H}})$-manifold $M$, and an induced complex structure $I$, there
is a holomorphic volume form $\Phi\in\Lambda^{2n,0}(M,J)$, which is parallel
with respect to the Obata connection ([V5], [BDV]). The space $V$ of parallel
holomorphic volume forms is 1-dimensional. A choice of an auxiliary induced
complex structure such that $I\circ J=-J\circ I$ endows $V$ with a real
structure and a positive direction (Subsection 4.2). We choose $\Phi$ to be
real and positive. Denote by $\Pi^{n,n}_{I}$ the projection to
$(n,n)$-component with respect to the complex structure $I$, such that $I\circ
J=-J\circ I$.

In Section 6 we show that $\operatorname{Re}(\Pi^{n,n}_{I}\Phi)$ is a
calibration for any quaternionic Hermitian metric $g$ for which $|\Phi|=2^{n}$
(6). This calibration calibrates complex subvarieties of $Z\subset(M,I)$ which
are Lagrangian with respect to the $(2,0)$-form
$\Omega=\omega_{J}+\sqrt{-1}\>\omega_{K}$, defined as in (2.2).

This calibration is defined for any quaternionic Hermitian metric, subject to
the condition $|\Phi|=1$ (and there are always many). When $(M,I,J,K,\Phi,g)$
is an HKT manifold with $\operatorname{Hol}(M)\subset SL(n,{\mathbb{H}})$,
more calibrations can be defined.

We choose $\Phi$ to be positive, real $(2n,0)$-form on $(M,J)$, and let
$\Phi_{n}:=\operatorname{Re}\Pi^{n,n}_{I}(\Phi)$. In [V7] it was shown that
the form $\Phi_{n+k}:=\frac{1}{2^{k}k!}\Phi_{n}\wedge\omega_{I}^{k}$ is always
closed and positive (4.3). In 6, we prove that this form is a calibration, for
a metric
$g^{\prime}:=g\cdot\left|\frac{\Phi_{n+k}}{2^{n}}\right|^{(2n+2k)^{-1}}$,
conformally equivalent to $g$. When $g$ is also balanced, $|\Phi|=const$, the
conformal weight $\left|\frac{\Phi_{n+k}}{2^{n}}\right|^{(2n+2k)^{-1}}$ is
constant (6), and $g^{\prime}$ is also HKT, but otherwise $g^{\prime}$ is not
an HKT metric. In either case, the calibration $\Phi_{n+k}$ is (generally
speaking) not parallel with respect to any connection on $M$ (6).

We show that $\Phi_{n+k}$ calibrates complex subvarieties of $(M,I)$ which are
coisotropic with respect to the (2,0)-form
$\Omega=\omega_{J}+\sqrt{-1}\>\omega_{K}$ (6). The situation with isotropic
subvarieties is completely different. Using the examples from Section 5.6, we
notice in Remark 6.5 that complex isotropic submanifolds in this case do not
have to be calibrated by any form, since they could be homologous to zero.

##  2 Preliminaries

###  2.1 Calibrations in Riemannian geometry

We provide here the basic definitions of the theory of calibrations which we
use in the paper. The standard reference for this material is [HL] and the
reader may also consult [J2] for recent progress and developments related to
manifolds with restricted holonomy.

Definition 2.1: Let $W\subset V$ be a $p$-dimensional subspace in a Euclidean
space, and $\operatorname{Vol}(W)$ denote the Riemannian volume form of
$W\subset V$, defined up to a sign. For any $p$-form $\eta\in\Lambda^{p}V$,
let comass $\operatorname{\sf comass}(\eta)$ be the maximum of
$\frac{\eta(v_{1},v_{2},...,v_{p})}{|v_{1}||v_{2}|...|v_{p}|}$, for all
$p$-tuples $(v_{1},...,v_{p})$ of vectors in $V$ and face be the set of planes
$W\subset V$ where $\frac{\eta}{\operatorname{Vol}(W)}=\operatorname{\sf
comass}(\eta)$.

Definition 2.2:  A precalibration on a Riemannian manifold is a differential
form with comass $\leqslant 1$ everywhere.

Definition 2.3: A calibration is a precalibration which is closed.

Definition 2.4: Let $\eta$ be a $k$-dimensional precalibration on a Riemannian
manifold, and $Z\subset M$ a $k$-dimensional subvariety (we usually assume
that the Hausdorff dimension of the set of singular points of $Z$ is
$\leqslant k-2$, because in this case a compactly supported differential form
can be integrated over $Z$). We say that $Z$ is calibrated by $\eta$ if at any
smooth point $z\in Z$, the space $T_{z}Z$ is a face of the precalibration
$\eta$.

Remark 2.5: Clearly, for any precalibration $\eta$,

$\operatorname{Vol}(Z)\geqslant\int_{Z}\eta,$ (2.1)

where $\operatorname{Vol}(Z)$ denotes the Riemannian volume of a compact $Z$,
and the equality happens iff $Z$ is calibrated by $\eta$. If, in addition,
$\eta$ is closed, $\int_{Z}\eta$ is a cohomological invariant, and the
inequality (2.1) implies that $Z$ minimizes the Riemannian volume in its
homology class.

###  2.2 Hyperkähler manifolds and calibrations

The following definitions are standard.

Definition 2.6: A manifold $M$ is called hypercomplex if $M$ is equipped with
a triple of complex structures $I,J,K$, satisfying the quaternionic relations
$I\circ J=-J\circ I=K$. If, in addition, $M$ is equipped with a Riemannian
metric $g$ which is Kähler with respect to $I,J,K$, $(M,I,J,K,g)$ is called
hyperkähler. This is equivalent to $\nabla I=\nabla J=\nabla K=0$, where
$\nabla$ is the Levi-Civita connection of $g$; see [Bes].

Remark 2.7:  It has been known since 1955 that any hypercomplex manifold
admits a torsion-free connection preserving $I,J$ and $K$, which is
necessarily unique. This connection is called the Obata connection, after M.
Obata, who discovered it in [Ob]. Any almost complex structure which is
preserved by a torsion-free connection is necessarily integrable (this is an
easy consequence of Newlander-Nirenberg theorem). Therefore, for any
$a,b,c\in{\mathbb{R}}$, with $a^{2}+b^{2}+c^{2}=1$, the almost complex
structure $aI+bJ+cK$ is in fact integrable. We denote by $(M,L)$ the manifold
$M$ considered as a complex manifold with the complex structure induced by
$L=aI+bJ+cK$.

Definition 2.8: Such complex structures are called induced by quaternions, and
the corresponding family, parametrized by $S^{2}$ – the twistor family, or the
hypercomplex family. This family is holomorphic, and its total space (fibered
over ${\mathbb{C}}P^{1}$) is called the twistor space of $M$. It is a complex
analytic space, non-Kähler even in simplest cases (for $M$ a torus or a K3
surface).

Hyperkähler geometry has a long history and is already well established. For
more details and background definitions, please see [Bes, J2]. In algebraic
geometry, the word hyperkähler is essentialy synonymous with “holomorphic
symplectic”. The reason is that any hyperkähler manifold is equipped with a
complex-valued form $\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$.111We always
write $\omega_{I},\omega_{J},\omega_{K}$ for the corresponding Kähler forms.
This form has Hodge type (2,0) on $(M,I)$ and is closed, hence holomorphically
symplectic.

The converse follows from the Yau’s proof of Calabi’s conjecture: a
holomorphically symplectic, Kähler manifold admits a unique hyperkähler metric
in a given Kähler class ([Bes]). For survey of recent advances in hyperkähler
geometry see [H1, H2].

Some of the main objects of this paper are holomorphic Lagrangian, isotropic
and coisotropic subvarieties of $(M,I)$, where $(M,I,J,K,g)$ is hyperkähler.

Definition 2.9: A complex analytic subvariety $Z$ of a holomorphically
symplectic manifold $(M,\Omega)$ is called holomorphic Lagrangian if
$\Omega{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}=0$, and
$\dim_{\mathbb{C}}Z=\frac{1}{2}\dim_{\mathbb{C}}M$, and isotropic if
$\Omega{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}=0$, and
$\dim_{\mathbb{C}}Z<\frac{1}{2}\dim_{\mathbb{C}}M$. It is called coisotropic
if $\Omega$ has rank
$\frac{1}{2}\dim_{\mathbb{C}}M-\operatorname{codim}_{\mathbb{C}}Z$ on $TZ$ in
all smooth points of $Z$, which is the minimal possible rank for a
$2n-p$-dimensional subspace in a $2n$-dimensional symplectic space.

###  2.3 Calibrations in HKT-geometry

Let $(M,I,J,K)$ be a hypercomplex manifold. Then the tangent bundle $TM$ is
equipped with a natural quaternionic action. In particular, the group $SU(2)$
of unitary quaternions acts on $TM$, in a canonical way. A Riemannian metric
on $M$ is called quaternionic Hermitian if it is $SU(2)$-invariant. A
hyperkähler metric is obviously quaternionic Hermitian, but the converse is
manifestly false, as we shall explain presently.

With every quaternionic Hermitian metric $g$ we associate 2-forms
$\omega_{I}:=g(I\cdot,\cdot),\omega_{J}:=g(J\cdot,\cdot)$ and
$\omega_{K}:=g(K\cdot,\cdot)$ which are clearly antisymmetric, because $g$ is
$SU(2)$-invariant. It is easy to check that

$\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$ (2.2)

is a (2,0)-form on $(M,I)$. This form is closed if and only if $(M,I,J,K,g)$
is hyperkähler ([Bes]).

For a weaker form of this condition, consider the $(1,0)$-part of the de Rham
differential,

$\partial:\;\Lambda^{p,q}(M,I){\>\longrightarrow\>}\Lambda^{p+1,q}(M).$

A quaternionic Hermitian hypercomplex manifold is called HKT (short for
“hyperkähler with torsion”) if $\partial\Omega=0$.

The theory of HKT-manifolds is a rapidly developing subfield of quaternionic
geometry. Originally this notion appeared in physics ([HP]), but
mathematicians found it very useful. For an early survey of HKT-geometry,
please see [GP].

Another ingredient of an HKT calibration theory is the notion of Obata
connection (2.2). Since this connection preserves the quaternionic structure,
its holonomy $\operatorname{Hol}(M)$ lies in $GL(n,{\mathbb{H}})$. The
holonomy of the Obata connection is one of the most important invariants of a
hypercomplex manifold. Many properties of $M$ can be related directly to its
holonomy group. In particular, the group $\operatorname{Hol}(M)$ is compact if
and only if $(M,I,J,K)$ admits a hyperkähler metric.

There seems to be no holonomy characterization of HKT structures. In fact the
holonomy of Obata connection is rarely known explicitly, except on hyperkähler
manifolds, where it is equal to the Levi-Civita connection. However the
knowledge of holonomy is still quite useful for the study of HKT geometry. For
many examples of compact hypercomplex manifolds, the group
$\operatorname{Hol}(M)\subset GL(n,{\mathbb{H}})$ is strictly smaller than
$GL(n,{\mathbb{H}})$. Only recently it was found that the group $SU(3)$ with
the left-invariant hypercomplex structure has $GL(n,{\mathbb{H}})$ as its
holonomy group ([Sol]).

An important subgroup inside $GL(n,{\mathbb{H}})$ is its commutator
$SL(n,{\mathbb{H}})$. This group can be defined as a group of quaternionic
matrices $A\subset\operatorname{End}({\mathbb{H}}^{n})$ preserving a non-zero
complex-valued form
$\Phi\in\Lambda^{2n,0}_{\mathbb{C}}({\mathbb{H}}^{n}_{I})$, where
${\mathbb{H}}^{n}_{I}$ is ${\mathbb{H}}^{n}$ considered as a $2n$-dimensional
complex space, with the complex structure $I$ induced by quaternions. The
coefficient $\lambda:=\frac{A(\Phi)}{\Phi}$ is called the Moore determinant of
the matrix $A$ ([A], [AV1]); it is always a positive real number, with
$\lambda^{4}$ equal to the determinant of $A$, considered as an element of
$GL(4n,{\mathbb{R}})$. The group $SL(n,{\mathbb{H}})$ is a group of
quaternionic matrices with Moore determinant 1.

##  3 $SL(n,{\mathbb{H}})$-manifolds

###  3.1 An introduction to $SL(n,{\mathbb{H}})$-geometry

As Obata has shown ([Ob]), a hypercomplex manifold $(M,I,J,K)$ admits a
necessarily unique torsion-free connection, preserving $I,J,K$. The converse
is also true: if a manifold $M$ equipped with an action of ${\mathbb{H}}$
admits a torsion-free connection preserving the quaternionic action, it is
hypercomplex. This implies that a hypercomplex structure on a manifold can be
defined as a torsion-free connection with holonomy in $GL(n,{\mathbb{H}})$.
This connection is called the Obata connection on a hypercomplex manifold.

Connections with restricted holonomy are one of the central notions in
Riemannian geometry, due to Berger’s classification of irreducible holonomy of
Riemannian manifolds. However, a similar classification exists for general
torsion-free connections ([MS]). In the Merkulov-Schwachhöfer list, only three
subroups of $GL(n,{\mathbb{H}})$ occur. In addition to the compact group
$Sp(n)$ (which defines hyperkähler geometry), also $GL(n,{\mathbb{H}})$ and
its commutator $SL(n,{\mathbb{H}})$ appear, corresponding to hypercomplex
manifolds and hypercomplex manifolds with trivial determinant bundle,
respectively. Both of these geometries are interesting, rich in structure and
examples, and deserve detailed study.

It is easy to see that $(M,I)$ has holomorphically trivial canonical bundle,
for any $SL(n,{\mathbb{H}})$-manifold $(M,I,J,K)$ ([V5]). For a hypercomplex
manifold with trivial canonical bundle admitting an HKT metric, a version of
Hodge theory was constructed ([V3]). Using this result, it was shown that a
compact hypercomplex manifold with trivial canonical bundle has holonomy in
$SL(n,{\mathbb{H}})$, if it admits an HKT-structure ([V5]).

In [BDV], it was shown that holonomy of all hypercomplex nilmanifolds lies in
$SL(n,{\mathbb{H}})$. Many working examples of hypercomplex manifolds are in
fact nilmanifolds, and by this result they all belong to the class of
$SL(n,{\mathbb{H}})$-manifolds.

The $SL(n,{\mathbb{H}})$-manifolds were studied in [AV2] and [V6], because on
such manifolds the quaternionic Dolbeault complex is identified with a part of
de Rham complex (4.3). Under this identification, ${\mathbb{H}}$-positive
forms become positive in the usual sense, and $\partial$,
$\partial_{J}$-closed or exact forms become
$\partial,\overline{\partial}$-closed or exact (see Section 3.1). This linear-
algebraic identification is especially useful in the study of the quaternionic
Monge-Ampère equation ([AV2]).

###  3.2 Balanced HKT-manifolds

The following lemma is contained in [BDV] (Theorem 3.2; see also [V7], Lemma
4.3). Recall that the map $\eta{\>\longrightarrow\>}J(\overline{\eta})$
defines a real structure on $\Lambda^{2p,0}(M,I)$. A $(p,0)$-form $\eta$ is
called ${\mathbb{H}}$-real if $J(\eta)=\overline{\eta}$.

Lemma 3.1:  Let $(M,I,J,K)$ be a hypercomplex manifold, and $\eta$ a top
degree $(2n,0)$-form, which is ${\mathbb{H}}$-real and holomorphic. Then
$\eta$ is Obata-parallel.

Definition 3.2: Let $(M,I,g)$ be a complex Hermitian manifold,
$\dim_{\mathbb{C}}M=n$, and $\omega\in\Lambda^{1,1}(M)$ its Hermitian form.
One says that $M$ is balanced if $d(\omega^{n-1})=0$.

Remark 3.3: It is easy to see that $d(\omega^{m})=0$ for $1\leqslant
m\leqslant n-2$ implies that $\omega$ is Kähler; the balancedness makes sense
as the only non-trivial condition of form $d(\omega^{m})=0$ which is not
equivalent to the Kähler property.

Theorem 3.4:  Let $(M,I,J,K,\Omega)$ be an HKT-manifold as in Section 2.3,
$\dim_{\mathbb{H}}M=n$. If $\overline{\partial}$ is the standard Dolbeault
operator on $(M,I)$, then the following conditions are equivalent.

(i)

$\overline{\partial}(\Omega^{n})=0$

(ii)

$\nabla(\Omega^{n})=0$, where $\nabla$ is the Obata connection

(iii)

The manifold $(M,I)$ with the induced quaternionic Hermitian metric is
balanced as a Hermitian manifold:

$d(\omega_{I}^{2n-1})=0.$

Proof: [V7], Theorem 4.8.

Remark 3.5: A balanced HKT-manifold has holonomy in $SL(n,{\mathbb{H}})$. This
statement follows immediately from the implication (iii) $\Rightarrow$ (ii) of
3.2. However the balanced HKT condition is a little stronger. It is shown in
[IP] that an HKT manifold has (restricted) holonomy of the Obata connection in
$SL(n,{\mathbb{H}})$ if and only if it is (locally) conformally balanced.

Remark 3.6: The condition $\nabla(\Omega^{n})=0$ is independent from the
choice of a basis $I,J,K$, $IJ=-JI=K$ of ${\mathbb{H}}$. Indeed, suppose that
$g\in SU(n)$, and $(I_{1},J_{1},K_{1})=(g(I),g(J),g(K))$ is a new basis in
${\mathbb{H}}$. The corresponding HKT-form
$\Omega_{1}=\omega_{J_{1}}+\sqrt{-1}\>\omega_{K_{1}}$ can be expressed as
$\Omega_{1}=g(\Omega)$, hence

$\nabla(\Omega_{1}^{n})=\nabla(g(\Omega_{1}^{n}))=g(\nabla(\Omega^{n}))=0.$

Therefore, 3.2 leads to the following corollary.

Corollary 3.7: Let $(M,I,J,K,\Omega)$ be an HKT-manifold, such that the
corresponding complex Hermitian manifold $(M,I)$ is balanced. Then $(M,I_{1})$
is balanced for any complex structrure $I_{1}$ induced by the quaternions.
Moreover, $(M,I,J,K,\Omega)$ is an $SL({\mathbb{H}},n)$-manifold.

##  4 Differential forms on hypercomplex manifolds

In this section, we give an introduction to the linear algebraic structures on
the de Rham algebra of a hypercomplex manifold. We follow [V6] and [V7].

###  4.1 The quaternionic Dolbeault complex

It is well-known that any irreducible representation of $SU(2)$ over
${\mathbb{C}}$ can be obtained as a symmetric power $S^{i}(V_{1})$, where
$V_{1}$ is a fundamental 2-dimensional representation. We say that a
representation $W$ has weight $i$ if it is isomorphic to $S^{i}(V_{1})$. A
representation is said to be pure of weight $i$ if all its irreducible
components have weight $i$.

Remark 4.1:  The Clebsch-Gordan formula (see [Hu]) claims that the weight is
multiplicative, in the following sense: if $i\leqslant j$, then

$V_{i}\otimes V_{j}=\bigoplus_{k=0}^{i}V_{i+j-2k},$

where $V_{i}=S^{i}(V_{1})$ denotes the irreducible representation of weight
$i$.

Let $M$ be a hypercomplex manifold, $\dim_{\mathbb{H}}M=n$. There is a natural
multiplicative action of $SU(2)\subset{\mathbb{H}}^{*}$ on $\Lambda^{*}(M)$,
associated with the hypercomplex structure.

Let $V^{i}\subset\Lambda^{i}(M)$ be a maximal $SU(2)$-invariant subspace of
weight $<i$. The space $V^{i}$ is well defined, because it is a sum of all
irreducible representations $W\subset\Lambda^{i}(M)$ of weight $<i$. Since the
weight is multiplicative (4.1), $V^{*}=\bigoplus_{i}V^{i}$ is an ideal in
$\Lambda^{*}(M)$.

It is easy to see that the de Rham differential $d$ increases the weight by 1
at most. Therefore, $dV^{i}\subset V^{i+1}$, and $V^{*}\subset\Lambda^{*}(M)$
is a differential ideal in the de Rham DG-algebra $(\Lambda^{*}(M),d)$.

Definition 4.2:  Denote by $(\Lambda^{*}_{+}(M),d_{+})$ the quotient algebra
$\Lambda^{*}(M)/V^{*}$. It is called the quaternionic Dolbeault algebra of
$M$, or the quaternionic Dolbeault complex (qD-algebra or qD-complex for
short).

Remark 4.3: The complex $(\Lambda^{*}_{+}(M),d_{+})$ was constructed earlier
by Capria and Salamon ([CS]) in a different (and more general) situation, and
much studied since then.

The Hodge bigrading is compatible with the weight decomposition of
$\Lambda^{*}(M)$, and gives a Hodge decomposition of $\Lambda^{*}_{+}(M)$
([V3]):

$\Lambda^{i}_{+}(M)=\bigoplus_{p+q=i}\Lambda^{p,q}_{+,I}(M).$

The spaces $\Lambda^{p,q}_{+,I}(M)$ are the weight spaces for a particular
choice of a Cartan subalgebra in $\mathfrak{su}(2)$. The
$\mathfrak{su}(2)$-action induces an isomorphism of the weight spaces within
an irreducible representation. This gives the following result.

Proposition 4.4:  Let $(M,I,J,K)$ be a hypercomplex manifold and

$\Lambda^{i}_{+}(M)=\bigoplus_{p+q=i}\Lambda^{p,q}_{+,I}(M)$

the Hodge decomposition of qD-complex defined above. Then there is a natural
isomorphism

$\Lambda^{p,q}_{+,I}(M)\cong\Lambda^{p+q,0}(M,I).$ (4.1)

Proof: See [V3].

This isomorphism is compatible with a natural algebraic structure on

$\bigoplus_{p+q=i}\Lambda^{p+q,0}(M,I),$

and with the Dolbeault differentials, in the following way.

Let $(M,I,J,K)$ be a hypercomplex manifold. We extend

$J:\;\Lambda^{1}(M){\>\longrightarrow\>}\Lambda^{1}(M)$

to $\Lambda^{*}(M)$ by multiplicativity. Recall that

$J(\Lambda^{p,q}(M,I))=\Lambda^{q,p}(M,I),$

because $I$ and $J$ anticommute on $\Lambda^{1}(M)$. Denote by

$\partial_{J}:\;\Lambda^{p,q}(M,I){\>\longrightarrow\>}\Lambda^{p+1,q}(M,I)$

the operator $J\circ\overline{\partial}\circ J$, where
$\overline{\partial}:\;\Lambda^{p,q}(M,I){\>\longrightarrow\>}\Lambda^{p,q+1}(M,I)$
is the standard Dolbeault operator on $(M,I)$, that is, the $(0,1)$-part of
the de Rham differential. Since $\overline{\partial}^{2}=0$, we have
$\partial_{J}^{2}=0$. In [V3] it was shown that $\partial$ and $\partial_{J}$
anticommute:

$\\{\partial_{J},\partial\\}=0.$ (4.2)

Consider the quaternionic Dolbeault complex $(\Lambda^{*}_{+}(M),d_{+})$
constructed in 4.1. Using the Hodge bigrading, we can decompose this complex,
obtaining a bicomplex

$\Lambda^{*,*}_{+,I}(M)\ext@arrow
0099\arrowfill@\relbar\relbar\longrightarrow{}{d^{1,0}_{+,I},d^{0,1}_{+,I}}\Lambda^{*,*}_{+,I}(M)$

where $d^{1,0}_{+,I}$, $d^{0,1}_{+,I}$ are the Hodge components of the
quaternionic Dolbeault differential $d_{+}$, taken with respect to $I$.

Theorem 4.5:  Under the multiplicative isomorphism

$\Lambda^{p,q}_{+,I}(M)\cong\Lambda^{p+q,0}(M,I)$

constructed in 4.1, $d^{1,0}_{+}$ corresponds to $\partial$ and $d^{0,1}_{+}$
to $\partial_{J}$:

$\textstyle{\Lambda^{0}_{+}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{0,1}_{+}}$$\scriptstyle{d^{1,0}_{+}}$$\textstyle{\Lambda^{0,0}_{I}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{\partial_{J}}$$\textstyle{\Lambda^{1,0}_{+}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{0,1}_{+}}$$\scriptstyle{d^{1,0}_{+}}$$\textstyle{\Lambda^{0,1}_{+}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{0,1}_{+}}$$\scriptstyle{d^{1,0}_{+}}$$\cong$$\textstyle{\Lambda^{1,0}_{I}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{\partial_{J}}$$\textstyle{\Lambda^{1,0}_{I}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{\partial_{J}}$$\textstyle{\Lambda^{2,0}_{+}(M)}$$\textstyle{\Lambda^{1,1}_{+}(M)}$$\textstyle{\Lambda^{0,2}_{+}(M)}$
$\textstyle{\Lambda^{2,0}_{I}(M)}$$\textstyle{\Lambda^{2,0}_{I}(M)}$$\textstyle{\Lambda^{2,0}_{I}(M)}$
(4.3)

Moreover, under this isomorphism, the form
$\omega_{I}\in\Lambda^{1,1}_{+,I}(M)$ corresponds to
$\Omega\in\Lambda^{2,0}_{I}(M)$.

Proof: See [V3] or [V4].

###  4.2 Positive $(2,0)$-forms on hypercomplex manifolds

The notion of positive $(2p,0)$-forms on hypercomplex manifolds (sometimes
called q-positive, or ${\mathbb{H}}$-positive) was developed in [AV1] (see
also [AV2] and [V6]).

Let $\eta\in\Lambda^{p,q}_{I}(M)$ be a differential form. Since $I$ and $J$
anticommute, $J(\eta)$ lies in $\Lambda^{q,p}_{I}(M)$. Clearly,
$J^{2}{\left|{}_{{\phantom{|}\\!\\!}_{\Lambda^{p,q}_{I}(M)}}\right.}=(-1)^{p+q}$.
For $p+q$ even,
$J{\left|{}_{{\phantom{|}\\!\\!}_{\Lambda^{p,q}_{I}(M)}}\right.}$ is an
anticomplex involution, that is, a real structure on $\Lambda^{p,q}_{I}(M)$. A
form $\eta\in\Lambda^{2p,0}_{I}(M)$ is called real if
$J(\overline{\eta})=\eta$.

For a real $(2,0)$-form $\eta$,

$\eta\left(x,J(\overline{x}))\right)=\overline{\eta}\left(J(x),J^{2}(\overline{x})\right)=\overline{\eta}\left(\overline{x},J(x)\right),$

for any $x\in T^{1,0}_{I}(M)$. From the definition of a real form, we obtain
that the scalar $\eta\left(x,J(\overline{x})\right)$ is always real.

Definition 4.6: A real $(2,0)$-form $\eta$ on a hypercomplex manifold is
called positive if $\eta\left(x,J(\overline{x})\right)\geqslant 0$ for any
$x\in T^{1,0}_{I}(M)$, and strictly positive if this inequality is strict, for
all $x\neq 0$.

An HKT-form $\Omega\in\Lambda^{2,0}_{I}(M)$ of any HKT-structure is strictly
positive. Moreover, HKT-structures on a hypercomplex manifold are in one-to-
one correspondence with $\partial$-closed, strictly positive $(2,0)$-forms.

The analogy between Kähler forms and HKT-forms can be pushed further; it turns
out that any HKT-form $\Omega\in\Lambda^{2,0}_{I}(M)$ has a local potential
$\varphi\in C^{\infty}(M)$, in such a way that
$\partial\partial_{J}\varphi=\Omega$ ([AV1]). Here $\partial\partial_{J}$ is a
composition of $\partial$ and $\partial_{J}$ defined on quaternionic Dolbeault
complex as above (these operators anticommute).

###  4.3 The map ${\cal
V}_{p,q}:\;\Lambda^{p+q,0}_{I}(M){\>\longrightarrow\>}\Lambda^{n+p,n+q}_{I}(M)$
on $SL(n,{\mathbb{H}})$-manifolds

Let $(M,I,J,K)$ be an $SL(n,{\mathbb{H}})$-manifold, $\dim_{\mathbb{R}}M=4n$,
and

${\cal
R}_{p,q}:\;\Lambda^{p+q,0}_{I}(M){\>\longrightarrow\>}\Lambda^{p,q}_{I,+}(M)$

the isomorphism induced by $\mathfrak{su}(2)$-action as in 4.1. Consider the
projection

$\Lambda^{p,q}_{I}(M){\>\longrightarrow\>}\Lambda^{p,q}_{I,+}(M),$ (4.4)

and let $R:\;\Lambda^{p,q}_{I}(M){\>\longrightarrow\>}\Lambda^{p+q,0}_{I}(M)$
denote the composition of (4.4) and ${\cal R}_{p,q}^{-1}$.

Let $\Phi_{I}$ be a nowhere degenerate holomorphic section of
$\Lambda^{2n,0}_{I}(M)$. Assume that $\Phi_{I}$ is real, that is,
$J(\Phi_{I})=\overline{\Phi}_{I}$, and positive. Existence of such a form is
equivalent to $\operatorname{Hol}(M)\subset SL(n,{\mathbb{H}})$ (3.2). It is
often convenient to define $SL(n,{\mathbb{H}})$-structure by fixing the
quaternionic action and the holomorphic form $\Phi_{I}$.

Define the map

${\cal
V}_{p,q}:\;\Lambda^{p+q,0}_{I}(M){\>\longrightarrow\>}\Lambda^{n+p,n+q}_{I}(M)$

by the relation

${\cal V}_{p,q}(\eta)\wedge\alpha=\eta\wedge
R(\alpha)\wedge\overline{\Phi}_{I},$ (4.5)

for any test form $\alpha\in\Lambda^{n-p,n-q}_{I}(M)$.

The map ${\cal V}_{p,p}$ is especially remarkable, because it maps closed,
positive $(2p,0)$-forms to closed, positive $(n+p,n+p)$-forms, as the
following proposition implies.

Proposition 4.7:  Let $(M,I,J,K,\Phi_{I})$ be an
$SL(n,{\mathbb{H}})$-manifold, and

${\cal
V}_{p,q}:\;\Lambda^{p+q,0}_{I}(M){\>\longrightarrow\>}\Lambda^{n+p,n+q}_{I}(M)$

the map defined above. Then

(i)

${\cal V}_{p,q}(\eta)={\cal R}_{p,q}(\eta)\wedge{\cal V}_{0,0}(1)$.

(ii)

The map ${\cal V}_{p,q}$ is injective, for all $p$, $q$.

(iii)

$(\sqrt{-1}\>)^{(n-p)^{2}}{\cal V}_{p,p}(\eta)$ is real if and only
$\eta\in\Lambda^{2p,0}_{I}(M)$ is real, and positive if and only if $\eta$ is
positive.

(iv)

${\cal V}_{p,q}(\partial\eta)=\partial{\cal V}_{p-1,q}(\eta)$, and ${\cal
V}_{p,q}(\partial_{J}\eta)=\overline{\partial}{\cal V}_{p,q-1}(\eta)$.

(v)

${\cal V}_{0,0}(1)=\lambda{\cal R}_{n,n}(\Phi_{I})$, where $\lambda$ is a
positive rational number, depending only on the dimension $n$.

Proof: See [V6], Proposition 4.2, or [AV2], Theorem 3.6.

Remark 4.8:  For the purposes of the present paper, we are interested in 4.3
for the case $\eta=\Omega^{k}$, where $\Omega$ is an HKT-form. In this case,
${\cal R}_{p,p}(\Omega^{k})$ is a projection of $\omega_{I}^{k}$ to the
component of maximal weight (see 4.4 below). Now, ${\cal
V}_{p,q}(\Omega^{k})={\cal R}_{p,q}(\Omega^{k})\wedge{\cal V}_{0,0}(1)$, as
follows from 4.3 (i). However, ${\cal V}_{0,0}(1)$ has weight $2n$, by 4.3
(v), and $\omega_{I}^{k}$ has weight $\leqslant 2k$, hence their product is of
weight $\geqslant 2n-2k$. Since this product is $(2n-2k)$-form, it is pure of
weight $(2n-2k)$, and components of $\omega_{I}^{k}$ of weight $<2k$ do not
contribute to the product $\omega^{k}_{I}\wedge{\cal V}_{0,0}(1)$. We obtain
that the closed, positive form ${\cal V}_{k,k}(\Omega^{k})$ is proportional to
$\omega^{k}_{I}\wedge{\cal V}_{0,0}(1)$, with positive coefficient.

###  4.4 Algebra generated by $\omega_{I}$, $\omega_{J}$, $\omega_{K}$

Let $(M,I,J,K,g)$ be a quaternionic Hermitian manifold. Consider the algebra
$A^{*}=\oplus A^{2i}$ generated by $\omega_{I}$, $\omega_{J}$, and
$\omega_{K}$. In [V1], this algebra was computed explicitly. It was shown
that, up to the middle degree, $A^{*}$ is a symmetric algebra with generators
$\omega_{I}$, $\omega_{J}$, $\omega_{K}$. The algebra $A^{*}$ has Hodge
bigrading $A^{k}=\bigoplus\limits_{p+q=k}A^{p,q}$. From the Clebsch-Gordan
formula, we obtain that $A^{2i}_{+}:=\Lambda^{2i}_{+}(M)\cap A^{2i}$, for
$i\leqslant n$, is an orthogonal complement to $Q(A^{2i-4})$, where
$Q(\eta)=\eta\wedge(\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2})$. Moreover,
$A^{2i}_{+}$ is irreducible as a representation of $SU(2)$. Therefore, the
space $A^{p,p}_{+}=\ker
Q^{*}{\left|{}_{{\phantom{|}\\!\\!}_{A^{p,p}}}\right.}$ is 1-dimensional. This
argument also implies that the form ${\cal V}_{0,0}(1)$ is proportional to
$\Phi_{J}|^{n,n}_{I}$, where $\Phi_{J}$ is a holomorphic volume form on
$(M,J)$, obtained as a top power of the appropriate holomorphic symplectic
form, and $\Phi_{J}|^{n,n}_{I}$ its $(n,n)$-part, taken with respect to $I$.

Proposition 4.9:  Let $(M,I,J,K,\Phi_{I})$ be an
$SL(n,{\mathbb{H}})$-manifold, equipped with an HKT-structure $\Omega$. Assume
that $\Omega^{n}=\Phi_{I}$. Let

$\Pi_{+}:\;\Lambda^{n+k,n+k}_{I}(M){\>\longrightarrow\>}\Lambda^{n+k,n+k}_{I,+}(M)$

be the projection to the component of maximal weight with respect to the
$SU(2)$-action. Then $\Xi_{k}:=\Pi_{+}(\omega_{I}^{n+k,n+k})$ is a closed,
weakly positive $(n+k,n+k)$-form, which is proportional to
$\omega^{k}_{I}\wedge\Phi_{J}|^{n,n}_{I}$ and to $\omega^{k}_{I}\wedge{\cal
V}_{0,0}(1)$.

Proof: The form $\omega^{k}_{I}\wedge\Phi_{J}|^{n,n}_{I}$ is proportional to
$\omega^{k}_{I}\wedge{\cal V}_{0,0}(1)$ as indicated above. Consider the
algebra $A^{*}=\oplus A^{2i}$ generated by $\omega_{I}$, $\omega_{J}$, and
$\omega_{K}$. The map $R^{p,q}$ is induced by the $SU(2)$-action, hence it
maps $A^{*,*}$ to itself. Since ${\cal V}_{p,q}(\eta)={\cal
R}_{p,q}(\eta)\wedge{\cal V}_{0,0}(1)$, and ${\cal V}_{0,0}(1)$ is
proportional to ${\cal R}_{n,n}(\Phi_{I})\in A^{*}$, we obtain

${\cal V}_{p,q}(A^{p+q,0})\subset A^{n+p,n+q}.$

Since ${\cal V}_{p,p}(\Omega^{p})\subset A^{n+p,n+p}_{+}$, the 1-dimensional
space $A^{n+p,n+p}_{+}$ is generated by ${\cal V}_{p,p}(\Omega^{p})$. This
form is closed and positive by 4.3. Therefore, the projection of
$\omega_{I}^{n+p}$ to $A^{n+p,n+p}_{+}$ is closed and positive (see 4.3).

##  5 Calibrations on hyperkähler manifolds

###  5.1 Hodge decomposition and $U(1)$-action

Let $I$ be a complex structure on a vector space $V$ and
$\rho:\;U(1){\>\longrightarrow\>}\operatorname{End}(V)$ a real
$U(1)$-representation given by $\rho(t)(X)=(\cos t+\sin tI)X$. This is
extended by multiplicativity to a representation in the tensor powers of $V$
with $\rho(t)(\alpha)(X)=\alpha(\rho(t)X)$ for a 1-form $\alpha$. In the usual
fashion, we define the weight decomposition associated with this
$U(1)$-action: the tensor $z$ has weight $p$ if $\rho(t)z=(\cos
pt)z+\sqrt{-1}\>(\sin pt)z$. We need also the definition of average over
$U(1)$ of $Y$:

$\operatorname{\sf Av}_{\rho}Y=\frac{1}{2\pi}\int_{0}^{2\pi}\rho(t)Ydt.$

Note that $\rho(t)Y=Y$ for every $t$ implies $IY=Y$ for any tensor $Y$ and
that $I\operatorname{\sf Av}_{\rho}Y=\operatorname{\sf Av}_{\rho}Y$.

Lemma 5.1:  Let $\rho$ be a U(1)-action on $W$, and $W=\bigoplus W^{i}$ the
corresponding weight decomposition. Then the projection to $W^{0}$ along the
sum of other $W^{i}$, $i\neq 0$, coincides with taking the average over
$U(1)$.

Proof: For each $\eta\in W^{i}$, $i\neq 0$, one has $\int_{U(1)}\rho(t)\eta
dt=0$, because $\int_{0}^{2\pi}\cos(t)dt=0$.

Theorem 5.2:  Let $\eta$ be a $2p$-form on a complex vector space $W$, with
$\operatorname{\sf comass}(\eta)\leqslant 1$, and
$\eta^{p,p}=\operatorname{\sf Av}_{\rho}\eta$ be the $(p,p)$-part of $\eta$.
Then $\operatorname{\sf comass}(\eta^{p,p})\leqslant 1$. Moreover, a
$2p$-dimensional plane $V$ is a face of $\eta^{p,p}$ if and only if
$\rho(t)(V)$ is a face of $\eta$ for all $t\in{\mathbb{R}}$.

Proof: For any decomposable $2p$-vector $\xi$, its image $\rho(t)(\xi)$ is
again decomposable for any $t$ and $|\rho(t)(\xi)|=|\xi|$. Then

$\eta^{(p,p)}(\xi)=(\operatorname{\sf
Av}_{\rho}(\eta))(\xi)=\frac{1}{2\pi}\int_{0}^{2\pi}\eta(\rho(t)(\xi))\leqslant
1$

since $\eta(\rho(t)\xi)\leqslant 1$ for every $t$. The equality holds iff
$\eta(\rho(t)\xi)=1$ for every $t$.

###  5.2 An $SU(2)$-invariant calibration

The most obvious example of a calibration on a hyperkähler manifold is
provided by the following theorem (see [Ber] for similar statement about a
quaternionic Wirtinger’s inequality).

Theorem 5.3:  Let $(M,I,J,K,g)$ be a hyperkähler manifold,
$\omega_{I},\omega_{J},\omega_{K}$ the corresponding symplectic forms, and
$\Theta_{p}:=\frac{(\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2})^{p}}{c_{p}}$
the standard $SU(2)$-invariant $4p$-form normalized by
$c_{p}=\sum_{k=1}^{p}\frac{(p!)^{2}}{(k!)^{2}}(2k)!4^{p-k}$. Then $\Theta_{p}$
is a calibration, and its faces are $p$-dimensional quaternionic subspaces of
$TM$. Moreover, the form
$\Xi_{p}:=\frac{(\omega_{J}^{2}+\omega_{K}^{2})^{p}}{(p!)^{2}4^{p}}$ is also a
calibration, with the same faces.

Proof: Consider the form $\widetilde{\Xi}_{p}:=\frac{\omega_{J}^{2p}}{(2p)!}$.
By 5.1, ${(\tilde{\Xi}_{p})}^{2p,2p}_{I}=\Xi_{p}$, where $(\cdot)^{2p,2p}_{I}$
is an operation of taking $(2p,2p)$-part under the complex structure $I$.
Indeed, $\omega_{J}^{2p}=\frac{(\Omega+\overline{\Omega})^{2p}}{4^{p}}$, where
$\Omega$ is the standard $(2,0)$ form on $(M,I)$. Then the $(2p,2p)$-part of
$\omega_{J}^{2p}$ is equal to

$\frac{(2p)!}{(p!)^{2}}\frac{\Omega^{p}\wedge\overline{\Omega}^{p}}{4^{p}}=\frac{(2p)!(\omega_{J}^{2}+\omega_{K}^{2})^{p}}{(p!)^{2}4^{p}}.$

By 5.1, a subspace $V\subset TM$ is a face of $\Xi_{p}$ if and only if
$\rho_{I}(t)(V)$ is a face of ${\widetilde{\Xi}_{p}}$ for all $t$, with
$\rho_{I}(t)$ the $U(1)$-action associated with $I$. The form
${\tilde{\Xi}_{p}}$ is a standard Kähler calibration associated with $J$; it
follows from [HL] that $V\subset TM$ is a face of ${\widetilde{\Xi}_{p}}$ if
and only if it is $J$-linear, that is, ${\mathbb{C}}$-linear with respect to
the action of ${\mathbb{C}}$ induced by $J$. Since $\rho(t)(V)$ is $J$-linear
for all $t$, it remains $J$-linear if we act on $V$ by a group $G$ generated
by $\rho_{I}$ and $\rho_{J}$, with $\rho_{J}$ a $U(1)$-action associated with
$J$. Clearly, $G\cong SU(2)$ is the group of unitary quaternions acting on
$\Lambda^{*}M$. Therefore, $V$ is a face of $\Xi_{p}$ if and only if $V$ is
$g(J)$-linear, for all $g\in SU(2)$. This is equivalent to $V$ being a
quaternionic subspace. Taking the average of $\Xi_{p}$ with respect to $SU(2)$
will not change its faces, because they are already $SU(2)$-invariant.
Therefore, $\operatorname{\sf Av}_{SU(2)}(\Xi_{p})$ is a calibration with its
faces quaternionic subspaces. Moreover it is $Sp(n)Sp(1)$-invariant $4p$-form,
so it is proportional to $(\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2})^{p}$.
Then, using 5.4 below, we obtain that that $\operatorname{\sf
Av}_{SU(2)}(\Xi_{p})=\Theta_{p}$ by evaluating both forms on a fixed
quaternionic subspace.

Remark 5.4: Subvarieties calibrated by $\Theta_{p}$ are called trianalytic
subvarieties. They were studied, at some length, in [V1] and [V2].

###  5.3 A holomorphic Lagrangian calibration

Proposition 5.5:  Let $(V^{4p},I,J,K,g)$ be a quaternionic Hermitian vector
space with fundamental forms $\omega_{I},\omega_{J},\omega_{K}$, and
$\Psi\in\Lambda^{2p}(V)$ a $2p$-form which is the real part of
$\frac{1}{p!}(\omega_{I}-\sqrt{-1}\>\omega_{K})^{p}$ (it is a $(2p,0)$-form
with respect to $J$). Denote by $\Psi^{p,p}_{I}$ the $(p,p)$-part of $\Psi$
with respect to $I$. Then $\Psi^{p,p}_{I}$ has comass 1. Moreover, a
$2p$-dimensional subspace $W\subset V$ is calibrated by $\Psi^{p,p}_{I}$ if
and only if $W$ is complex $I$-linear and calibrated by $\Psi$.

Proof: The real part of $\frac{1}{p!}(\omega_{I}-\sqrt{-1}\>\omega_{K})^{p}$
calibrates special Lagrangian subspaces taken with respect to the symplectic
form $\omega_{J}$ (see [HL]). Therefore, any face of
$\frac{1}{p!}(\omega_{I}-\sqrt{-1}\>\omega_{K})^{p}$ is
$\omega_{J}$-Lagrangian. By 5.1, a $2p$-dimensional plane $W$ is a face of
$\Psi^{p,p}_{I}$ if and only if $\rho(t)(W)$ is a face of $\Psi$ for all
$t\in{\mathbb{R}}$. It follows by taking $t=0$ that $W$ is
$\omega_{J}$-Lagrangian and by taking $t=\pi/2$ that $I(W)$ is
$\omega_{J}$-Lagrangian too. But $I(W)$ is $\omega_{J}$-Lagrangian iff $W$ is
$\omega_{K}$-Lagrangian. By [Hit] (see also 5.3 below) $W$ determines an
$I$-complex subspace.

Remark 5.6:  Let $V$ be a quaternionic Hermitian space,
$\dim_{\mathbb{H}}V=p$, and $\xi\in\Lambda^{2p}V$ a decomposable $2p$-vector
which is associated with a $2p$-dimensional subspace $W\subset V$. Clearly,
$W$ is Lagrangian with respect to $\omega_{J}$ if and only if
$L_{\omega_{J}}\xi=0$ and $\Lambda_{\omega_{J}}\xi=0$, where $L_{\omega_{J}}$,
$\Lambda_{\omega_{J}}$ are the corresponding Hodge operators,
$L_{\omega_{J}}(\eta):=\eta\wedge\omega_{J}$, and
$\Lambda_{\omega_{J}}=*L_{\omega_{J}}*$ its Hermitian adjoint. If $W$ is
Lagrangian with respect to $J$ and $K$, one has

$[L_{\omega_{J}},\Lambda_{\omega_{K}}]\xi=0.$ (5.1)

However, the commutator $[L_{\omega_{J}},\Lambda_{\omega_{K}}]$ acts on forms
of type $(p,q)$ with respect to $I$ as a multiplication by $(p-q)\sqrt{-1}\>$
(see [V0]). Then (5.1) implies that $\xi$ is of type $(p,p)$ with respect to
$I$.

Claim 5.7:  Let $V$ be an $n$-dimensional quaternionic Hermitian space, and
${\cal V}^{0,0}:\;{\mathbb{R}}{\>\longrightarrow\>}\Lambda^{n,n}_{I}(V)$ be a
map defined in Subsection 4.3 (in Subsection 4.3 it was defined for
$SL(n,{\mathbb{H}})$-manifolds, but the definition can be repeated for
quaternionic spaces word by word). Then ${\cal V}^{0,0}(1)=\Psi^{n,n}_{I}$,
where $\Psi^{n,n}_{I}$ is a form defined as in 5.3.

Proof: From 4.3 (v), we know that ${\cal V}^{0,0}(1)$ and $\Psi^{n,n}_{I}$ are
proportional and we only have to calculate the coefficient of proportionality.
For this we use ${\cal
V}^{0,0}(1)\wedge\alpha=R(\alpha)\wedge\overline{\Phi}_{I}$ for a particular
choice of $\alpha$ as

$\alpha=\xi_{1}\wedge...\wedge\xi_{n}\wedge\overline{\xi_{n+1}}\wedge...\overline{\xi_{2n}},$

where $\xi_{i}$ are orthogonal and of unit norm. Then

$R(\alpha)=\xi_{1}\wedge...\wedge
J\overline{\xi_{n+1}}\wedge...J\overline{\xi_{2n}}.$

From here if ${\cal V}^{0,0}(1)=\lambda\Psi^{n,n}_{I}$, then $\lambda=1$.

Comparing 4.3 and 5.3, we find that the form $\Psi^{n,n}_{I}$ is positive.

###  5.4 Isotropic and coisotropic calibrations

A similar argument can be applied to other powers of $\Omega_{J}$.

Proposition 5.8:  Consider an $n$-dimensional quaternionic Hermitian space
$V$, and let $\Omega_{J}:=\omega_{I}-\sqrt{-1}\>\omega_{K}$ be the usual
$(2,0)$-form on the complex space $(V,J)$. When $p\leqslant n$ denote by
$\Psi_{p}:=\frac{1}{p!}\operatorname{Re}(\Omega_{J}^{p})$, and let
$\Psi^{p,p}_{I}$ be its $(p,p)$-part taken with respect to $I$. Then
$\Psi^{p,p}_{I}$ has comass 1, and its faces are complex isotropic subspaces
of $(V,I)$

Proof: Let $W\subset V$ be a real $2p$-dimensional subspace, and $W_{1}$ be
the smallest complex subspace of $(V,J)$ containing $W$. Adding more vectors
if necessary, we can always assume that $\dim_{\mathbb{C}}W_{1}=2p$. Denote by
$\xi$ the decomposable $4p$-vector associated with $W_{1}$, and $I(\xi)$ its
image under the action of a quaternion $I$. Then $\frac{1}{p!}\Omega_{J}^{p}$
is a $(2p,0)$-form on $W_{1}$, proportional to the unit holomorphic volume
form $\operatorname{Vol}^{2p,0}(W_{1})$ with a coefficient $\kappa$ which
satisfies

$|\kappa|=\frac{(\xi,I(\xi))}{|\xi|^{2}}$

where $(,)$ is the induced scalar product. By Cauchy-Schwarz inequality
$|\xi|\leqslant 1$, where the equality holds iff $I\xi=\xi$ or, equivalently,
$W_{1}$ is quaternionic. Since $\operatorname{Vol}^{2p,0}(W_{1})$ has comass
1,

$\operatorname{\sf comass}\left(\frac{1}{p!}\Omega_{J}^{p}\right)\leqslant 1$

with equality if and only if $W_{1}$ is quaternionic. In the latter case, $W$
is a face of $\frac{1}{p!}\Omega_{J}^{p}$ if and only if $W$ is complex
Lagrangian in $W_{1}$, as follows from 5.3.

We provide also an expression of $\Psi^{p,p}$ as a polynomial of
$\omega_{I},\omega_{J}$ and $\omega_{K}$ for even $p$.

Proposition 5.9:  Let $\Psi^{p,p}$ be the $(p,p)$ part with respect to $I$ of
$Re(\omega_{I}-\sqrt{-1}\>\omega_{K})^{p}$. Then

$\Psi^{p,p}=\sum_{k=0}^{q}\frac{(-1)^{k}}{4^{k}}\binom{p}{2k}\binom{2k}{k}\omega_{I}^{p-2k}\wedge(\omega_{K}^{2}+\omega_{J}^{2})^{k}$

where $q=\llcorner\frac{p}{2}\lrcorner$ is the greatest integer not exceeding
$\frac{p}{2}$.

Proof: First we notice that

$Re(\omega_{I}-\sqrt{-1}\>\omega_{K})^{p}=\sum_{k=0}^{\llcorner\frac{p}{2}\lrcorner}(-1)^{k}\binom{p}{2k}\omega_{I}^{p-2k}\wedge\omega_{K}^{2k}.$

Since $\omega_{I}^{p-2k}$ is of type $(p-2k,p-2k)$ with respect to $I$ we need
to determine the type of $\omega_{K}^{2k}$. To do this we use the fact that
$\omega_{K}=\frac{1}{2}\Omega+\frac{1}{2}\overline{\Omega}$ is the
decomposition of $\omega_{K}$ in $(2,0)+(0,2)$ parts with respect to $I$ where
$\Omega=\omega_{K}+\sqrt{-1}\>\omega_{J}$. Then

$\omega_{K}^{2k}=\frac{1}{4^{k}}\sum_{s=0}^{p-2k}\binom{2k}{s}\Omega^{s}\wedge\overline{\Omega}^{2k-s}$

and each term in the sum has degree $(2s,4k-s)$ with respect to $I$. So the
only term which will contribute to $\Psi^{p,p}$ above will be when $s=k$.
Obviously the term is
$\frac{1}{4^{k}}\binom{2k}{k}\Omega^{k}\wedge\overline{\Omega}^{k}$. Then the
proposition follows from the fact that
$\Omega\wedge\overline{\Omega}=\omega_{K}^{2}+\omega_{J}^{2}$.

Notice that one can take the imaginary part of $\Omega_{J}^{p}$ instead of the
real part. The resulting calibrated subspaces are again complex isotropic. To
identify the complex coisotropic subspaces, however, one has to be more
careful.

Proposition 5.10:  Consider an $n$-dimensional quaternionic Hermitian space
$V$, and let $\Omega_{J}:=\omega_{I}-\sqrt{-1}\>\omega_{K}$ be the usual
$(2,0)$-form on the complex space $(V,J)$. Let
$\Phi_{p}+\sqrt{-1}\Phi^{\prime}_{p}:=\frac{1}{2^{p}p!n!}(\Omega_{J})^{n}\wedge\omega_{I}^{p}$,
and $\Phi^{p,p}_{I}$ (resp. $\Phi^{\prime p,p}_{I}$) be the corresponding
$(n+p,n+p)$-parts taken with respect to $I$. Then $\Phi^{p,p}_{I}$ (resp.
$\Phi^{\prime p,p}_{I}$) have comass 1 and their faces are complex coisotropic
subspaces of $(V,I)$

Proof: First we notice that if a form $\alpha$ is calibration, then its Hodge
dual $*\alpha$ is again calibration and its faces are orthogonal complements
to the faces of $\alpha$. Then the form $*\Psi^{p,p}$ is a calibration with
faces $I$-complex $\Omega_{J}$-coisotropic subspaces. The same is true also if
we consider the imaginary part of $\Omega_{J}^{p}$ instead of $\Psi^{p}$. Then
it remains to check that the complex form in the proposition is Hodge dual to
$\Omega_{J}^{p}$ up to a real constant. To this end we first notice that
$*\Omega_{J}^{n-p}=c_{1}\overline{\Omega_{J}^{n}}\wedge\Omega_{J}^{p}$ for a
real positive constant $c_{1}$. Then $\Phi^{p,p}+\sqrt{-1}\Phi^{\prime p,p}$
and $\overline{\Omega_{J}^{n}}\wedge\Omega_{J}^{p}$ are both highest vectors
in an irreducible representation $A^{2n+2p}$ of $SU(2)$ (see Subsection 4.4),
hence they are proportional up to a complex constant. More explicitly we have:

$(\omega_{I}-\sqrt{-1}\>\omega_{K})^{n}\wedge(\omega_{I}-\sqrt{-1}\>\omega_{K})^{p}=\Omega_{J}^{n}\wedge(2\omega_{I}-\Omega_{J})^{p}$
$=(\Omega_{J})^{n}\wedge\sum_{s=0}^{p}\binom{p}{s}(-\Omega_{J})^{s}\wedge
2^{p-s}\omega_{I}^{p-s}$

Since $\Omega_{J}^{n+s}=0$ for $s>0$ all terms in the sum above vanish except
the first one. Then

$(\omega_{I}-\sqrt{-1}\>\omega_{K})^{n}\wedge(\omega_{I}+\sqrt{-1}\>\omega_{K})^{p}=(\omega_{I}-\sqrt{-1}\>\omega_{K})^{n}\wedge
2^{p}\omega_{I}^{p}$

From here and 5.4 $ii)$ the proposition follows.

To calculate the comass of the forms above we need the following well-known
preliminary Lemma:

Lemma 5.11:  If $(V^{2n},I,g)$ is an Hermitian vector space and $\omega$ is
the fundamental 2-form, then for any subset $X_{1},...,X_{2k}$ of a given
unitary basis $(e_{1},Ie_{1},...,e_{n},Ie_{n})$ we have:

i) $\omega^{k}(X_{1},....,X_{2k})=\pm k!$ if $span\\{X_{1},...,X_{2k}\\}$ is
complex and

ii) $\omega^{k}(X_{1},....,X_{2k})=0$ otherwise.

The proof of $i)$ is standard, while $ii)$ follows from the definition of
wedge product and the fact that $\omega(X_{i},X_{j})\neq 0$ only if
$IX_{i}=\pm X_{j}$.

Lemma 5.12:  Let $(V^{4n},I,J,K,g)$ be a real vector space with anti-commuting
complex structures $I,J,K$ compatible with the positive scalar product $g$.
Denote by $\omega_{I},\omega_{J},\omega_{k}$ the fundamental 2-forms
corresponding to $I,J$ and $K$ respectively and
$\Omega_{I}=\omega_{J}+\sqrt{-1}\omega_{K}$ be the standard $I$-complex
symplectic 2-form. Consider the form
$\Psi_{I}^{n}=Re(\omega_{I}+\sqrt{-1}\omega_{J})^{n}|_{I}^{(n,n)}$, where
$|_{I}^{(n,n)}$ denotes the $(n,n)$ component with respect to $I$. Then:

i)

$\Omega_{I}^{n}\wedge\overline{\Omega_{I}}^{n}=4^{n}(n!)^{2}\operatorname{Vol}$
for the volume form $\operatorname{Vol}$ on $V$.

ii)

$(\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2})^{n}=c_{n}\operatorname{Vol}$
where $c_{n}=\sum_{k=0}^{n}\frac{(n!)^{2}}{(k!)^{2}}(2k)!4^{n-k}$

iii)

$\omega_{I}^{k}\wedge\Psi^{n}=2^{k}k!n!\operatorname{Vol}_{E_{n+k}}$, where
$E_{n+k}$ is an $(n+k)$-dimensional $I$-complex and $\omega_{J}$-coisotropic
subspace.

Proof: Fix a quaternionic-Hermitian co-basis

$(e^{1},Ie^{1},Je^{1},Ke^{1},e^{2},Ie^{2},...,Ke^{n})$

of $V^{*}$ so that $\operatorname{Vol}=e^{1}\wedge...\wedge Ke^{n}$ and let
$e_{1},Ie_{1},...,Ke_{n}$ be the dual basis of $V$. From the fact that
$\Omega_{I}=\sum_{i}dz^{i}\wedge dw^{i}$ for coordinates
$dz_{i}=e^{i}+\sqrt{-1}Ie^{i}$ and $dw_{i}=Je^{i}+\sqrt{-1}Ke^{i}$, follows
that $\Omega_{I}^{n}=n!dz^{1}\wedge dw^{1}...dz^{n}\wedge dw^{n}$. Then to
obtain $i)$ we notice that $dz_{i}\wedge
d\overline{z_{i}}=-2\sqrt{-1}e^{i}\wedge Ie^{i}$ and $dw_{i}\wedge
d\overline{w_{i}}=-2\sqrt{-1}Je^{i}\wedge Ke^{i}$.

To prove $ii)$ we write

$(\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2})^{n}=(\omega_{I}^{2}+\Omega_{I}\wedge\overline{\Omega_{I}})^{n}=\sum_{k=0}^{n}\binom{n}{k}\omega_{I}^{2k}\wedge\Omega_{I}^{n-k}\wedge\overline{\Omega_{I}}^{n-k}.$

Then we consider the term
$\omega_{I}^{2k}\wedge\Omega_{I}^{n-k}\wedge\overline{\Omega_{I}}^{n-k}$. Let
$s_{i}=e^{i}\wedge Ie^{i}+Je^{i}\wedge Ke^{i}$ and $t_{j}=dz^{j}\wedge
dw^{j}$, so $\omega_{I}=\sum s_{i}$ and $\Omega_{I}=\sum t_{j}$. Then
$s_{i}^{3}=s_{i}t_{i}=t_{i}^{2}=0$ $s_{i},t_{j}$ commute and
$s_{i}^{2}=2\operatorname{Vol}_{i},t_{i}\overline{t_{i}}=4\operatorname{Vol}_{i}$,
where $\operatorname{Vol}_{i}=e^{i}\wedge Ie^{i}\wedge Je^{i}\wedge Ke^{i}$.
Fix $n-k$ indexes $(i_{k+1},i_{k+2},...,i_{n})$. Then notice that in the
product $\omega_{I}^{2k}\wedge
t_{i_{k+1}}t_{i_{k+2}}...t_{i_{n}}\wedge\overline{\Omega_{I}}^{n-k}$ the only
non-vanishing terms are of the form

$s_{i_{1}}^{2}s_{i_{2}}^{2}...s_{i_{k}}^{2}t_{i_{k+1}}t_{i_{k+2}}...t_{i_{n}}\overline{t_{i_{k+1}}t_{i_{k+2}}...t_{i_{n}}}$

for the complementary indexes $(i_{1},...,i_{k})$, such that
$(i_{1},...,i_{n})$ is a permutation of $(1,2...,n)$. Every such product is
equal to $2^{k}4^{n-k}\operatorname{Vol}$. Then we may select
$i_{1}=1,..,i_{k}=k,i_{k+1}=k+1,...,i_{n}=n$ and count the number of terms
corresponding to it; clearly, this number does not depend on the choice of the
permutation. The number is the product of the coefficients in front of
$s_{1}^{2}...s_{k}^{2}$ $t_{k+1}...t_{n}$ and
$\overline{t_{k+1}}...\overline{t_{n}}$ in the expansions of
$(s_{1}+...s_{k})^{2k}$ $(t_{k+1}+...+t_{n})^{n-k}$ and
$(\overline{t_{k+1}}+...+\overline{t_{n}})^{n-k}$ respectively, which is
$\frac{(2k)!}{2^{k}}((n-k)!)^{2}$. Since there are $\frac{n!}{k!(n-k)!}$
different choices for $n-k$ indexes, we obtain

$(\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2})^{n}=\sum_{k=0}^{n}\frac{(n!)^{2}}{(k!)^{2}}(2k)!4^{n-k}\operatorname{Vol}$

and $ii)$ follows.

To prove $iii)$ we notice that $Sp(n)$ acts transitively on complex
coisotropic subspaces of fixed dimension. Then we choose the coisotropic
subspace $L$ spanned by
$e_{1},Ie_{1},....,e_{n},Ie_{n},Je_{1},Ke_{1},....,Je_{k},Ke_{k}$. Let
$\Omega_{K}=\omega_{I}+\sqrt{-1}\omega_{J}$, $\alpha\in L$ a subspace spanned
by $2n$ vectors and $\beta$ be a subspace generated by $2k$ vectors among
$e_{1},Ie_{1},....,e_{n},Ie_{n},Je_{1},Ke_{1},....,Je_{k},Ke_{k}$. Since
$\Psi^{n}=Re(\Omega_{K})|^{n,n}_{I}$, then $\Psi^{n}|_{\alpha}=0$ if $\alpha$
contains a quaternionic subspace or is not $I$-invariant. Similarly,
$\omega_{I}^{k}\bigg{|}_{\beta}=0$ if $\beta$ is not $I$-invariant as follows
from Lemma 5.4.

From the calculations in [HL] p. 88, we have

$\Psi^{n}(e_{1},Ie_{1},...,e_{n},Ie_{n})=n!Re(dz_{1}\wedge....\wedge
dw_{n})(e_{1},Ie_{1},...e_{n},Ie_{n})=n!,$

and from 5.4 above, $\omega_{I}^{K}(Je_{1},Ke_{1},...,Je_{k},Ke_{k})=k!$. Then
in the expression for
$\Psi^{n}\wedge\omega_{I}^{k}(e_{1},Ie_{1},....Je_{k},Ke_{k})$ the only non-
vanishing summands are $\omega_{I}^{K}(Je_{1},Ke_{1},...,Je_{k},Ke_{k})$ and
the terms where one or more pairs $e_{i},Ie_{i}$ are interchanged with
$Je_{i},Ke_{i}$. If we have exactly $s$ pairs interchanged, then there will be
$\binom{l}{s}$ terms each with value $n!k!$. So

$\Psi^{n}\wedge\omega_{I}^{k}(e_{1},Ie_{1},....Je_{k},Ke_{k})=n!k!\left(1+k+\binom{k}{2}+...+\binom{k}{k}\right)=2^{n}n!k!,$

which proves the Lemma. Note that for $k=n$ the result fits with the case $i)$
and the calculations in 5.4.

###  5.5 Holomorphic Lagrangian calibrations of degree two

The calibration 4-forms with constant coefficients in ${\mathbb{R}}^{8}$ were
studied systematically in [DHM]. Also various 4-forms which are calibrations
in ${\mathbb{H}}^{n}$ or any hyperkähler manifold are considered in [BrH]. We
want to relate our results to these works.

If $p=2$, from 5.4 we obtain

$\displaystyle\Psi^{2,2}_{I}=$
$\displaystyle\left.\frac{1}{2}\operatorname{Re}\left(\omega_{K}+\sqrt{-1}\omega_{I}\right)^{2}\right|_{I}^{2,2}$
$\displaystyle=$
$\displaystyle\left.\left(-\frac{1}{2}\omega_{I}^{2}+\frac{1}{2}\omega^{2}_{K}\right)\right|_{I}^{2,2}=-\frac{1}{2}\omega^{2}_{I}+\frac{1}{4}(\omega_{J}^{2}+\omega_{K}^{2}).$

In [BrH] R.Bryant and R. Harvey considered the forms
$\Psi_{\lambda,\mu,\nu}=\frac{\lambda}{2}\omega_{I}^{2}+\frac{\mu}{2}\omega_{J}^{2}+\frac{\nu}{2}\omega_{K}^{2}$
and showed that they are calibrations iff $-1\leqslant\nu,\lambda,\mu\leqslant
1$ and $-1\leqslant\nu+\lambda+\mu\leqslant 1$. We show here that the
”generic” form of this type calibrates either quaternionic or complex
isotropic subspaces.

Proposition 5.13:  For the forms $\Psi_{\lambda,\mu,\nu}$ the following is
valid:

i) If $\lambda,\mu,\nu\geqslant 0$ and $\lambda+\mu+\nu=1$ with at least two
of $\lambda,\mu,\nu$ non-zero, the form $\Psi_{\lambda,\mu,\nu}$ has comass 1
and the faces are the quaternionic ones.

ii) If $\mu,\nu\leqslant 0$ and $\mu+\nu\geqslant-1$ with at least two of the
inequalities being strict, then $\Psi_{1,\mu,\nu}$ has comass 1 and the faces
are the I-complex $\Omega_{I}$-isotropic subspaces of
${\mathbb{H}}^{n}={\mathbb{C}}^{2n}$.

Proof: First we note that a convex hull of calibrations is a calibration. In
case $i)$, for any unit 4-vector $\psi$,

$\Psi_{\lambda,\mu,\nu}(\psi)=\frac{\lambda}{2}\omega_{I}^{2}(\psi)+\frac{\mu}{2}\omega_{J}^{2}(\psi)+\frac{\nu}{2}\omega_{K}^{2}(\psi)\leqslant(\lambda+\mu+\nu)|\psi|=|\psi|,$

and the equality is achieved only when $\psi$ spans a subspace which is
invariant with respect to at least two of $I,J$ and $K$, hence quaternionic.

For $ii)$ we note that

$\frac{1}{2}\omega_{I}^{2}+\frac{\mu}{2}\omega_{J}^{2}+\frac{\nu}{2}\omega_{K}^{2}=\frac{1+\mu+\nu}{2}\omega_{I}^{2}-\frac{\mu}{2}(\omega_{I}^{2}-\omega_{J}^{2})-\frac{\nu}{2}(\omega_{I}^{2}-\omega_{K}^{2})$

Then according to [BrH], Theorem 2.38,
$\frac{1}{2}(\omega_{I}^{2}-\omega_{J}^{2})$ and
$\frac{1}{2}(\omega_{I}^{2}-\omega_{K}^{2})$ are calibrations with comass 1
and faces which are $\omega_{K}$ or $\omega_{J}$ isotropic and contained in
2-dimensional quaternionic subspaces. So as in $i)$ if $\psi$ is a unit
4-vector, then $\Psi_{1,\mu,\nu}(\psi)\leqslant|\psi|$ with equality if and
only if $\psi$ is a face for all terms with nonvanishing coefficients on the
right-hand-side above. If the span of $\psi$ satisfies at least two of the
following:

i) $\psi$ is I-complex

ii) $\psi$ is $\omega_{J}$ isotropic and

iii) $\psi$ is $\omega_{K}$ isotropic

then $\psi$ satisfies also the third one and the Proposition follows.

In [BrH, Theorem 6.4], 5.5 is implicit. We note also that in String Theory,
the holomorphic Lagrangian submanifolds in 8-dimensional manifolds were
related to the notion of intersecting branes [F].

###  5.6 Examples

Examples of complex Lagrangian submanifolds in hyper-Kähler manifolds are
given by many authors. In [Vo], C. Voisin has proven a result about the
stability of such submanifolds under small deformation of the complex
structure of the ambient space; she gave also several classes of examples. N.
Hitchin noticed the fact that such subspaces are coming in complete families
([Hit]). In [M], D. Matsushita has shown that the families of holomorphic
Lagrangian fibrations on a hyperkaehler manifold always deform with a
deformation of a manifold, if the cohomology class of a fiber remains of Hodge
type $(n,n)$. Existence of such families is postulated by a conjecture called
“SYZ conjecture”, or, sometimes, the “Huybrechts-Sawon conjecture”. It is also
known as a hyperkähler version of abundance conjecture, related to the minimal
model program. For a survey of related questions, please see [Saw]. Recently
in String Theory the holomorphic Lagrangian submanifolds were related to
3-dimensional topological field theory with target hyperkähler manifold [KRS].

In this section we provide examples of complex Lagrangian submanifolds of
hypercomplex manifolds with holonomy $SL(n,{\mathbb{H}})$.

The known examples of manifolds with holonomy $SL(n,{\mathbb{H}})$ are either
nilmanifolds ([BDV]) or obtained via the twist construction of A. Swann [S],
which is based on previous examples by D. Joyce. The later construction
provides also simply-connected examples. We describe briefly a simplified
version of it.

Let $(X,I,J,K,g)$ be a compact hyper-Kähler manifold. By definition, an anti-
self-dual 2-form on it is a form which is of type (1,1) with respect to $I$
and $J$ and hence with respect to all complex structures of the hypercomplex
family. Let $\alpha_{1},...,\alpha_{4k}$ be anti-self-dual closed 2-forms
representing integral cohomology classes on $X$ (instatons). Consider the
principal $T^{4k}$-bundle $\pi:M\rightarrow X$ with characteristic classes
determined by $\alpha_{1},...,\alpha_{4k}$. It admits an instanton connection
$A$ given by $4k$ 1-forms $\theta_{i}$ s.t. $d\theta_{i}=\pi^{*}(\alpha_{i})$.
Then $M$ carries a hypercomplex structure determined in the following way: on
the horizontal spaces of $A$ we have the pull-backs of $I,J,K$ and on the
vertical spaces we fix a linear hypercomplex structure of the $4k$-torus. The
structures $\cal{I},\cal{J},\cal{K}$ on $M$ are extended to act on the
cotangent bundle $T^{*}M$ using the following relations:

$\displaystyle{\cal I}(\theta_{4i+1})=\theta_{4i+2},$ $\displaystyle{\cal
I}(\theta_{4i+3})=\theta_{4i+4},$ $\displaystyle{\cal
J}(\theta_{4i+1})=\theta_{4i+3},$ $\displaystyle{\cal
J}(\theta_{4i+2})=-\theta_{4i+4},$ $\displaystyle{\cal
I}(\pi^{*}\alpha)=\pi^{*}(I\alpha),$ $\displaystyle{\cal
J}(\pi^{*}\alpha)=\pi^{*}(J\alpha)$

for any 1-form $\alpha$ on $X$ and $i=0,1,...k-1$. Similarly one can define a
hyperhermitian (or quaternion-Hermitian) metric on $M$ from $g$ and a fixed
hyper-Kähler metric on $T^{4k}$ using the splitting of $TM$ in horizontal and
vertical subspaces. As A. Swann [S] has shown the structure is HKT and has a
holonomy $SL(n,{\mathbb{H}})$.

Suppose now that $Y$ is a complex Lagrangian subspace in $X$ with respect to
$I$. Consider the $T^{2k}$-bundle over $X$ determined by
$\alpha_{4i+1},\alpha_{4i+3}$. Suppose that $N$ is its restriction to $Y$ i.e
$N$ is a principal $T^{2k}$-subbundle of $M$ over $Y$ determined by
$\alpha_{4i+1}|_{Y},\alpha_{4i+3}|_{Y}$. Then $N$ is naturally embedded in $M$
and by the definiton above $N$ is $\cal{J}$-invariant and Lagrangian with
respect to the fundamental 2-form of $\cal{I}$. Notice that in general the
complex Lagrangian subspace could be Kähler or non-Kähler depending on whether
$\alpha_{1}|Y$ and $\alpha_{3}|Y$ define zero classes or not.

As a particular case assume $X$ to be a $K3$ surface with large enough Picard
group such that there are 4 independent anti-self-dual integral classes
defining a principal $T^{4}$-bundle $M$ over $X=K3$ with finite fundamental
group. After passing to a finite cover we may assume that $M$ is simply-
connected. Now if $vol$ denotes the volume form on $X$, then we can choose
representatives $\alpha_{1},...,\alpha_{4}$ in the characteristic classes of
$M$ such that $\alpha_{i}^{2}=-F\operatorname{Vol}$ where $F$ is a function
and $F>0$ almost everywhere. We want to see what is the structure of an
arbitrary complex Lagrangian subspace $N$ of $M$. Since $N$ is 4-dimensional
and ${\cal J}$-complex, we claim that its intersection with a generic fiber of
$\pi:M{\>\longrightarrow\>}X$ is at least complex 1-dimensional. Indeed,
otherwise $N$ would be a multisection of $M$ and will intersect a generic
fiber transversally. However then $\int_{N}\pi^{*}(\alpha_{1}^{2})<0$ since
$\alpha_{1}^{2}=-vol$ on one hand, and $\int_{N}\pi^{*}(\alpha_{1}^{2})=0$
since $\pi^{*}(\alpha_{1})=d\theta_{1}$ for some connection form $\theta_{1}$
on the other. The contradiction proves the claim and we have:

Proposition 5.14: If $M$ is a principal instanton $T^{4}$-bundle over a $K3$
surface then any complex Lagrangian subspace is fibered by complex Lagrangian
curves of the fibers of $M$ over a Lagrangian curve of the base $K3$.

Remark 5.15: Notice that any complex curve is a priori Lagrangian in a K3
surface.

In general one can use a similar construction to obtain complex isotropic and
coisotropic subspaces of the instanton bundle $M$.

##  6 Calibrations on $SL(n,{\mathbb{H}})$-manifolds

Let $(M,I,J,K,\Phi_{I})$ be an $SL(n,{\mathbb{H}})$-manifold, that is, a
hypercomplex manifold with $\Phi_{I}$ a holomorphic volume form on $(M,I)$
preserved by the Obata connection. Clearly, $\overline{\Phi}_{I}$ is
proportional to $J(\Phi_{I})$. After a rescaling to $e^{\sqrt{-1}t}\Phi_{I}$
if necessary, we can assume that $\Phi_{I}$ is ${\mathbb{H}}$-real, i.e.
$J(\Phi_{I})=\overline{\Phi}_{I}$, and ${\mathbb{H}}$-positive (Subsection
4.2). A number of interesting calibrations can be constructed in this
situation.

Theorem 6.1:  Let $(M,I,J,K,\Phi_{I})$ be an $SL(n,{\mathbb{H}})$-manifold,
and $(\Phi_{I})_{J}^{n,n}$ the $(n,n)$-part of $\Phi_{I}$ taken with respect
to $J$. Pick a quaternionic Hermitian metric on $M$. Using a conformal change,
we may assume that $|\Phi_{I}|_{g}=2^{n}$. Then $Re((\Phi_{I})_{J}^{n,n})$ is
a calibration, and it calibrates complex subvarieties of $(M,J)$ which are
Lagrangian with respect to the $(2,0)$-form
$\omega_{K}+\sqrt{-1}\>\omega_{I}$.

Proof: It follows from the assumptions of 6 that

$\Phi_{I}=\lambda\frac{(\omega_{J}+\sqrt{-1}\omega_{K})^{n}}{n!}.$

Since both forms are real and ${\mathbb{H}}$-positive, $\lambda$ is real and
positive. It is easy to check that in local quaternionic Hermitian frame
$(dz_{1},dw_{1},...,dz_{n},dw_{n})$ the norm is calculated as

$\left|\frac{(\omega_{J}+\sqrt{-1}\omega_{K})^{n}}{n!}\right|^{2}=|dz_{1}|^{2}|dw_{1}|^{2}...|dz_{n}|^{2}|dw_{n}|^{2}=4^{n}.$

Then
$\left|\frac{(\omega_{J}+\sqrt{-1}\omega_{K})^{n}}{n!}\right|=2^{n}=|\Phi_{I}|$
and $\lambda=1$. Now the proof follows from the fact that $Re(\Phi_{I})$ and
$Re(\Phi_{I})_{J}^{n,n}$ are both closed,111The form $(\Phi_{I})_{J}^{n,n}$ is
parallel with respect to the Obata connection, which is torsion-free. and 5.3.

Theorem 6.2:  Let $(M,I,J,K,\Phi_{I})$ be an $SL(n,{\mathbb{H}})$-manifold,
and $(\Phi_{I})_{J}^{n,n}$ the $(n,n)$-part of $\Phi_{I}$ taken with respect
to $J$. Assume that $(M,I,J,K)$ is equipped with an HKT metric $g$ which is
balanced and $|\Phi_{I}|=2^{n}$. Then
$V_{n+i,n+i}:=\frac{1}{2^{i}i!}Re((\Phi_{I})_{J}^{n,n}\wedge\omega_{J}^{i})$
is a calibration, which calibrates complex subvarieties of $(M,J)$ which are
coisotropic with respect to the $(2,0)$-form
$\omega_{K}+\sqrt{-1}\>\omega_{I}$.

Proof: As in the previous proof,
$\Phi_{I}=\frac{(\omega_{J}+\sqrt{-1}\omega_{K})^{n}}{n!}$, so the form
$V_{n+i,n+i}$ is a pre-calibration by 5.4. It is closed, as follows from 4.4.

Remark 6.3: Notice that the form $V_{n+i,n+i}$ is not parallel with respect to
any torsion-free connection on $M$ (6), unless $M$ is hyperkähler.

Existence of a balanced HKT metric is a hard problem, which is equivalent to a
quaternionic version of a Calabi-Yau theorem ([V7]). However, even if $g$ is
not balanced, an analogue of the calibration $V_{n+i,n+i}$ is possible to
construct.

Theorem 6.4:  Let $(M,I,J,K,\Phi_{I})$ be an $SL(n,{\mathbb{H}})$-manifold,
and $(\Phi_{I})_{J}^{n,n}$ the $(n,n)$-part of $\Phi_{I}$ taken with respect
to $J$, and $g$ an HKT metric. Then there exists a function $c_{i}(m)$ on $M$,
such that $V_{n+i,n+i}:=(\Phi_{I})_{J}^{n,n}\wedge\omega_{J}^{i}$ is a
calibration with respect to the conformal metric $\widetilde{g}=c_{i}g$,
calibrating complex subvarieties of $(M,J)$ which are coisotropic with respect
to the $(2,0)$-form
$\widetilde{\omega}_{K}+\sqrt{-1}\>\widetilde{\omega}_{I}$.

Proof: Since $\Phi_{I}$ is ${\mathbb{H}}$-positive and Obata parallel, the
form $(\Phi_{I})^{n,n}_{J}$ is closed. Then 4.4 implies that $V_{n+i,n+i}$ is
also closed. If we denote by $\widetilde{\omega_{J}}$ and
$\widetilde{\Omega_{I}}$ the corresponding forms after the conformal change
$\widetilde{g}=c_{i}(m)g$, then we can find the function $c_{i}(m)$ such that

$V_{n+i,n+i}=\frac{1}{2^{i}n!i!}(\widetilde{\Omega}_{I}^{n})^{n,n}_{J}\wedge\widetilde{\omega}_{J}^{i}.$

6 then follows from 5.4.

Remark 6.5: Similarly to the hyperkähler case, it is a natural question to ask
whether the complex isotropic submanifolds are also calibrated in
$SL(n,{\mathbb{H}})$-manifolds with an HKT structure. However we can see in
the examples from Section 4.6 that this is not the case. Consider again a
toric bundle $M$ over $K3$-surface which has 4-dimensional fiber and is
simply-connected. Such fiber contains a 2-torus which will be a complex
isotropic curve with respect to some of the structures. By a spectral sequence
argument as in Lemma 4.7 of [S], one can see that all second cohomology
classes of $M$ are pull-backs from classes on the base $K3$-surface. Then such
a torus is homologous to zero, since the integral of any closed 2-form on it
vanishes. Therefore, it can not be calibrated by any form.

Claim 6.6:  Let $M$ be an $SL(n,{\mathbb{H}})$-manifold, $\Omega$ an HKT-form,
and $V_{n+i,n+i}$ the corresponding calibration, constructed above. Assume
that $\Omega$ is not hyperkähler. Then, the form $V_{n+i,n+i}$ is not
preserved by any torsion-free connection, for any $0<i<n$.

Proof: It is easy to check that the stabilizer
$St_{GL(4n,{\mathbb{R}})}(V_{n+i,n+i})$ is equal to the group $Sp(n)$ of
quaternionic Hermitian matrices. Therefore, any connection preserving
$V_{n+i,n+i}$ would also preserve an $Sp(n)$-structure. However, a torsion-
free connection preserving $Sp(n)$-structure is hyperkähler.

Acknowledgements: We are grateful to the referee for the careful reading and
many suggestions which improved the presentation of the paper.

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Misha Verbitsky
Laboratory of Algebraic Geometry, SU-HSE,
7 Vavilova Str. Moscow, Russia, 117312
<EMAIL_ADDRESS><EMAIL_ADDRESS>

Gueo Grantcharov
Department of Mathematics and Statistics
Florida International University
Miami Florida, 33199, USA
<EMAIL_ADDRESS>