Supersymmetry and Hodge theory on
Sasakian and Vaisman manifolds
Liviu Ornea111Liviu Ornea is partially supported by a grant of Ministry of
Research and Innovation, CNCS - UEFISCDI, project number PN-III-P4-ID-
PCE-2016-0065, within PNCDI III., Misha Verbitsky222Misha Verbitsky is
partially supported by the Russian Academic Excellence Project ’5-100”, FAPERJ
E-26/202.912/2018 and CNPq - Process 313608/2017-2.
Keywords: Lie superalgebra, Kähler manifold, Sasakian manifold, Vaisman
manifold, Reeb field, Hodge theory, transversally Kähler foliation, basic
forms. 2010 Mathematics Subject Classification: 53C55, 53C25, 17B60, 58A12.

Abstract

Sasakian manifolds are odd-dimensional counterpart to Kähler manifolds. They
can be defined as contact manifolds equipped with an invariant Kähler
structure on their symplectic cone. The quotient of this cone by the homothety
action is a complex manifold called Vaisman. We study harmonic forms and Hodge
decomposition on Vaisman and Sasakian manifolds. We construct a Lie
superalgebra associated to a Sasakian manifold in the same way as the Kähler
supersymmetry algebra is associated to a Kähler manifold. We use this
construction to produce a self-contained, coordinate-free proof of the results
by Tachibana, Kashiwada and Sato on the decomposition of harmonic forms and
cohomology of Sasakian and Vaisman manifolds. In the last section, we compute
the supersymmetry algebra of Sasakian manifolds explicitly.

###### Contents

  1. 1 Introduction
    1. 1.1 Supersymmetry in Kähler and non-Kähler geometry
    2. 1.2 Structure of this paper
  2. 2 Lie superalgebras acting on the de Rham algebra
    1. 2.1 Lie superalgebras and superderivations
    2. 2.2 Differential operators on graded commutative algebras
    3. 2.3 Supersymmetry on Kähler manifolds
  3. 3 Sasakian manifolds: definition and the basic notions
  4. 4 Hattori differentials on Sasakian manifolds
    1. 4.1 Hattori spectral sequence and associated differentials
    2. 4.2 Hattori differentials on Sasakian manifolds
  5. 5 Transversally Kähler manifolds
  6. 6 Basic cohomology and Hodge theory on Sasakian manifolds
    1. 6.1 Cone of a morphism of complexes and cohomology of Sasakian manifolds
    2. 6.2 Harmonic form decomposition on Sasakian manifolds
  7. 7 Vaisman manifolds
    1. 7.1 LCK manifolds
    2. 7.2 Vaisman manifolds
  8. 8 Hodge theory on Vaisman manifolds
    1. 8.1 Basic cohomology of Vaisman manifolds
    2. 8.2 Harmonic forms on Vaisman manifolds
  9. 9 Supersymmetry on Sasakian manifolds

##  1 Introduction

Sasakian manifolds are odd-dimensional counterpart to Kähler manifolds. They
can be defined as contact manifolds equipped with an invariant Kähler
structure on their symplectic cone. Taking a quotient of this cone by the
homothety action one obtains a complex manifold called Vaisman. Here we
develop a version of Hodge theory on the Sasakian and Vaisman manifolds.

The modern approach to Hodge theory is inspired by the supersymmetry. Given a
manifold with a geometric structure (such as Kähler, hyperkähler, HKT,
$G_{2}$-manifold and so on), one takes a bunch of natural operators on the de
Rham algebra (such as the de Rham differential, the Lefschetz
${\mathfrak{sl}}(2)$-triple etc.) and proves that these operators generate a
finite-dimensional Lie superalgebra. In most cases, the Laplacian is central
in this superalgebra. This gives an interesting geometric action on the
cohomology.

In this paper we write the natural superalgebra for a Sasakian manifold. Hodge
theory for Sasakian manifolds is well developed, but most proofs are written
in classical differential geometry style, mixing coordinate computations with
invariant ones. We wanted to give a more conceptual proof.

In the end we arrived at a very simple approach to the Hodge theory on
Sasakian and Vaisman manifolds (6.2, 8.2). The Sasakian supersymmetry algebra
is not used in this development, but it is interesting in itself, and the
relations are quite surprising.

###  1.1 Supersymmetry in Kähler and non-Kähler geometry

The connection between de Rham calculus on manifolds with special geometry
(such as Kähler and hyperkähler) and their supersymmetry appeared as early as
in 1997 ([FKS]). It is well known that extra supersymmetries of the
$\sigma$-model force the targed space to acquire extra geometric structures:
the $N=1$ supersymmetry implies Kähler structure on the target space, the
$N=2$ supersymmetry makes it hyperkähler, and so on. In [FKS], this
supersymmetry was interpreted in terms of the de Rham calculus on the target
space.

In [KiS], the connection between the supersymmetry and rational homotopy
theory is further expounded, with the constructions of rational homotopy
theory (such as Sullivan’s minimal models) interpreted in terms of quantum
mechanics.

Traditionally, the Kähler identities were obtained using the Levi-Civita
connection. Alternatively, one may show that a Kähler manifold can be
approximated up to second order by a flat space. Using supersymmetry to prove
the Kähler identities (Section 2) has many advantages over either of these
approaches. In [Ve2], supersymmetry was used to develop the Hodge theory for
the HKT manifolds (hypercomplex manifolds with torsion), where the structure
tensors are neither preserved by the Levi-Civita connection nor admit a second
order approximation. The supersymmetry approach was later used in
supersymmetric $\sigma$-models associated to HKT manifolds ([Sm, FS, FIS]).

In [Ve4], the supersymmetry approach was used to obtain the Hodge
decomposition on nearly Kähler manifolds, where the structure tensors are also
non-parallel.

In another direction, one may use the supersymmetry to obtain the superalgebra
action on manifolds with parallel differential forms ([Ve3]). The Hodge-
theoretic results obtained in this direction were used in [Hu] to study the
geometry of complete $G_{2}$\- and $\operatorname{Spin}(7)$-manifolds with
$d(\text{linear})$ structure form.

Summing up, finding a superalgebra associated with a given geometric structure
seems to be worthwhile, for mathematics as well as for mathematical physics.

###  1.2 Structure of this paper

  * •

In Section 2 we relate the Lie superalgebra approach to de Rham calculus and
obtain the Kähler identities which lie at the foundation of Hodge theory.

  * •

Section 3 introduces the Sasakian manifolds and explains the basic notions of
Sasakian geometry.

  * •

In Section 4 we explain how one employs de Rham calculus to obtain the Leray-
Serre spectral sequence; this approach was used by A. Hattori in 1960. We need
a version of the Leray-Serre spectral sequence which can be applied to smooth
foliations with fibers which are not necessarily closed or compact. We compute
the differentials of this spectral sequence for Sasakian manifolds explicitly,
for later use.

  * •

In Section 5 we deal with transversally Kähler structures on smooth
foliations. We give simple proofs of the standard results on transversal
(basic) cohomology, due to El Kacimi-Alaoui ([E]), much simplified because we
need them only for Vaisman and Sasakian manifolds.

  * •

In Section 6, we prove the standard results on cohomology for Sasakian
manifolds, in Section 7 we introduce the Vaisman manifolds, and in Section 8
we prove the standard results on cohomology for Vaisman manifolds. The
cohomology calculations for Sasakian and Vaisman are very similar and rely on
the same homological algebra argument. Throughout Section 8, we use the
1-dimensional transversally Sasakian foliation, generated by the Lee field,
and apply the superalgebra computations done in Section 6 on Sasakian
manifolds. However, the results on Vaisman manifolds cannot be deduced from
the results on Sasakian manifolds, because the Lee foliation might have non-
closed leaves.

  * •

We finish the paper with an explicit calculation of Sasakian superalgebra in
Section 9. This section depends only on Sections 2-4.

##  2 Lie superalgebras acting on the de Rham algebra

###  2.1 Lie superalgebras and superderivations

In the following, all vector spaces and algebras are considered over
${\mathbb{R}}$. Let $A$ be a ${\mathbb{Z}}/2{\mathbb{Z}}$-graded vector space,

$A=A^{\rm even}\oplus A^{\rm odd}.$

We say that $a\in A$ is pure if $a$ belongs to $A^{\rm even}$ or $A^{\rm
odd}$. For a pure element $a\in A$, we write $\tilde{a}=0$ if $a\in A^{\rm
even}$, and $\tilde{a}=1$ if $a\in A^{\rm odd}$. Consider a bilinear operator

$\\{\cdot,\cdot\\}:\;A\times A{\>\longrightarrow\>}A,$

called supercommutator. Assume that $\\{\cdot,\cdot\\}$ is graded anti-
commutative, that is, satisfies

$\\{a,b\\}=-(-1)^{\tilde{a}\tilde{b}}\\{b,a\\}$

for pure $a,b\in A$. Assume, moreover, that $\\{\cdot,\cdot\\}$ is compatible
with the grading: the commutator $\\{a,b\\}$ is even when both $a$, $b$ are
even or odd, and odd if one of these elements is odd and another is even. We
say that $(A,\\{\cdot,\cdot\\})$ is a Lie superalgebra if the following
identity (called the graded Jacobi identity, or super Jacobi identity) holds,
for all pure elements $a,b,c\in A$:

$\\{a,\\{b,c\\}\\}=\\{\\{a,b\\},c\\}+(-1)^{\tilde{a}\tilde{b}}\\{b,\\{a,c\\}\\}.$
(2.1)

Up to a sign, this is the usual Jacobi identity.

Every reasonable property of Lie algebras has a natural analogue for Lie
superalgebras, using the following rule of thumb: every time one would
exchange two elements $a$ and $b$, one adds a multiplier
$(-1)^{\tilde{a}\tilde{b}}$.

Example 2.1: Let $V=V^{\rm even}\oplus V^{\rm odd}$ be a
${\mathbb{Z}}/2{\mathbb{Z}}$-graded vector space, and $\operatorname{End}(V)$
its space of endomorphisms, equipped with the induced grading. We define a
supercommutator in $\operatorname{End}(V)$ by the formula:

$\\{a,b\\}=ab-(-1)^{\tilde{a}\tilde{b}}ba$

It is easy to check that
$\left(\operatorname{End}(V),\\{\cdot,\cdot\\}\right)$ is a Lie superalgebra.

Remark 2.2: Given a ${\mathbb{Z}}$-graded vector space $A$, one defines
$A^{\rm even}$ as the direct sum of even components, and $A^{\rm odd}$ as the
direct sum of odd components. Then a ${\mathbb{Z}}$-graded Lie superalgebra is
given by a supercommutator on $A$ satisfying $\\{A^{p},A^{q}\\}\subset
A^{p+q}$ and satisfying the graded Jacobi identity. In the sequel, all super
Lie algebras we consider are of this type. An endomorphism
$u\in\operatorname{End}(A)$ is called even if $u(A^{\rm odd})\subset A^{\rm
odd}$ and $u(A^{\rm even})\subset A^{\rm even}$, and odd if $u(A^{\rm
even})\subset A^{\rm odd}$ and $u(A^{\rm odd})\subset A^{\rm even}$. An
endomorphism which is either odd or even is called pure.

Definition 2.3: A graded algebra $A$ is called graded commutative if
$\\{a,b\\}=0$ for all $a,b\in A$.

The Grassmann algebra and de Rham algebra are clearly graded commutative.

Definition 2.4: Let ${\mathfrak{g}}$ be a graded commutative algebra. A map
$\delta:\;{\mathfrak{g}}{\>\longrightarrow\>}{\mathfrak{g}}$ is called an even
derivation if it is even and satisfies $\delta(xy)=\delta(x)y+x\delta(y)$. It
is called an odd derivation if it is odd and satisfies
$\delta(xy)=\delta(x)y+(-1)^{\tilde{x}}x\delta(y)$. It is called graded
derivation, or superderivation, if it shifts the grading by $i$ and satisfies
$\delta(ab)=\delta(a)b+(-1)^{ij}a\delta(b)$, for each $a\in A^{j}$.

Remark 2.5: The supercommutator of two super-derivations is again a
derivation. Therefore, the derivations form a Lie superalgebra.

###  2.2 Differential operators on graded commutative algebras

We need some algebraic results, which are almost trivial, and well-known for
commutative algebras. We extend these statements to graded commutative
algebras; the proofs are the same as in the commutative setting (see [Co]).

Definition 2.6:  Let $A^{*}$ be a graded commutative algebra. The algebra of
differential operators $\operatorname{Diff}(A^{*})$ is an associative
subalgebra of $\operatorname{End}(A^{*})$ generated by graded derivations and
$A^{*}$-linear self-maps. Let $\operatorname{Diff}^{0}(A^{*})$ be the space of
$A^{*}$-linear self-maps, $\operatorname{Diff}^{0}(A^{*})=A^{*}$,
$\operatorname{Diff}^{1}(A^{*})\supset\operatorname{Diff}^{0}(A^{*})$ the
subspace generated by $\operatorname{Diff}^{0}(A^{*})$ and all graded
derivations, and
$\operatorname{Diff}^{i}(A^{*}):=\operatorname{Diff}^{i-1}(A^{*})\cdot\operatorname{Diff}^{1}(A^{*})$.
This gives a multiplicative filtration
$\operatorname{Diff}^{0}(A^{*})\subset\operatorname{Diff}^{1}(A^{*})\subset\operatorname{Diff}^{2}(A^{*})\subset\cdots$
The elements of $\operatorname{Diff}^{i}(A^{*})$ are called differential
operators of order $i$ on $A^{*}$.

Claim 2.7:  Let
$D\in\operatorname{Diff}^{i}(A^{*}),D^{\prime}\in\operatorname{Diff}^{j}(A^{*})$
be differential operators on a graded commutative algebra. Then
$\\{D,D^{\prime}\\}\in\operatorname{Diff}^{i+j-1}(A^{*})$.

Proof: Since the commutator of two derivations is a derivation, one has
$\\{\operatorname{Diff}^{1}(A^{*}),\operatorname{Diff}^{1}(A^{*})\\}\subset\operatorname{Diff}^{1}(A^{*})$.
Then we use induction in $i$ and the standard commutator identities in the
associative algebra.

We shall apply this claim to geometric operations on the de Rham algebra,
obtaining differential operators of first order. Note that the “differential
operator” in the usual sense is a different notion. For example, the interior
product operator $i_{v}$ of contraction with a vector field $v$ on a manifold
$M$ is an odd derivation of the de Rham algebra, hence it is a first order
differential operator; however, $i_{v}$ is $C^{\infty}(M)$-linear, and thus it
is not a differential operator in the usual sense.

Claim 2.8:  Let $D$ be a differential operator of first order on $A^{*}$. Then
$D(x)=D(1)x+\delta(x)$, where $\delta$ is a derivation.

Proof: This is seen by defining $\delta:=D-D(1)\cdot$, then observing that
$\delta(1)=0$, which implies that $\delta\circ a-a\circ\delta=\delta(a)$ (we
identify $a\in A$ with $a\in\operatorname{End}(A)$, $a(b)=ab$). We then have

$\delta(ab)=(\delta\circ
a)(b)=(a\circ\delta)(b)+\delta(a)(b)=a\delta(b)+\delta(a)b,$

proving that $\delta$ is a derivation.

We use this formalism to compare first order differential operators on $A^{*}$
as follows.

Claim 2.9:  Let $\delta$ be a derivation on $A^{*}$. Then $\delta$ is uniquely
determined by the values it takes on any set of multiplicative generators of
$A^{*}$.

Claim 2.10:  Let $D$ be a first order differential operator on $A^{*}$. Then
$D$ is uniquely determined by $D(1)$ and the values it takes on any set of
multiplicative generators of $A^{*}$

Proof: By 2.2, $D-D(1)$ is a derivation, hence 2.2 implies 2.2.

Corollary 2.11:  Let
$d_{1},d_{2}:\;\Lambda^{*}(M){\>\longrightarrow\>}\Lambda^{*+1}(M)$ be first
order differential operators111Here, as elsewhere, “differential operators on
the de Rham algebra” are understood in the algebraic sense as above. on the de
Rham algebra of a manifold $M$, satisfying $d_{1}^{2}=d_{2}^{2}=0$, and
$V\subset\Lambda^{1}(M)$ a subspace such that the space
$d_{1}(C^{\infty}(M))+V$ generates $\Lambda^{1}(M)$ as a
$C^{\infty}(M)$-module. Suppose that
$d_{1}{\left|{}_{{\phantom{|}\\!\\!}_{C^{\infty}(M)}}\right.}=d_{2}{\left|{}_{{\phantom{|}\\!\\!}_{C^{\infty}(M)}}\right.}$
and
$d_{1}{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}=d_{2}{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}$.
Then $d_{1}=d_{2}$.

Proof: Clearly, $d_{1}=d_{2}$ on $C^{\infty}(M)\oplus d_{1}(C^{\infty}(M))$.
Then $d_{1}=d_{2}$ on the set of multiplicative generators
$C^{\infty}(M)+d_{1}(C^{\infty}(M))+V$, and 2.2 implies 2.2.

The following claim is used many times in the sequel.

Claim 2.12:  Let $d$ be an odd element in a Lie superalgebra $\mathfrak{h}$,
satisfying $\\{d,d\\}=0$. Then $\\{d,\\{d,u\\}\\}=0$ for any
$u\in{\mathfrak{h}}$.

Proof: By the super Jacobi identity,

$\\{d,\\{d,u\\}\\}=-\\{d,\\{d,u\\}\\}+\\{\\{d,d\\},u\\}=-\\{d,\\{d,u\\}\\}.$
(2.2)

This claim is a special case of the following:

Claim 2.13:  Let $d$ be an odd element in a Lie superalgebra $\mathfrak{h}$.
Then $2\\{d,\\{d,u\\}\\}=\\{\\{d,d\\},u\\}$ for any $u\in{\mathfrak{h}}$.

Proof: Follows from (2.2).

###  2.3 Supersymmetry on Kähler manifolds

One of the purposes of this paper is to obtain a natural superalgebra acting
on the de Rham algebra of a Sasakian manifold. This is modeled on the
superalgebra of a Kähler manifold, generated by the de Rham differential,
Lefschetz triple, and other geometric operators. To make the analogy more
clear, we recall the main results on the supersymmetry algebra of Kähler
manifolds. We follow [Ve2, Section 1.3].

Let $(M,I,g,\omega)$ be a Kähler manifold. Consider $\Lambda^{*}(M)$ as a
graded vector space. The differentials $d,d^{c}:=-IdId=IdI^{-1}$ can be
interpreted as odd elements in $\operatorname{End}(\Lambda^{*}(M))$, and the
Hodge operators $L,\Lambda,H$ as even elements. To avoid confusion, we denote
the supercommutator of odd elements as $\\{\cdot,\cdot\\}$. In terms of the
associative algebra, $\\{a,b\\}=ab+ba$. Let $d^{*}:=\\{\Lambda,d^{c}\\}$,
$(d^{c})^{*}:=-\\{\Lambda,d\\}$. The usual Kodaira relations can be stated as
follows

$\displaystyle\\{L,d^{*}\\}=-d^{c},\ \ \ \ \\{L,(d^{c})^{*}\\}=d,\ \ \ \
\\{d,(d^{c})^{*}\\}=\\{d^{*},d^{c}\\}=0,$ (2.3)
$\displaystyle\\{d,d^{c}\\}=\\{d^{*},(d^{c})^{*}\\}=0,\ \ \ \
\\{d,d^{*}\\}=\\{d^{c},(d^{c})^{*}\\}=\Delta,$

where $\Delta$ is the Laplace operator, commuting with $L,\Lambda,H$, and $d$,
$d^{c}$.

Definition 2.14:  Let $(M,I,g,\omega)$ be a Kähler manifold. Consider the Lie
superalgebra ${\mathfrak{a}}\subset\operatorname{End}(\Lambda^{*}(M))$
generated by the following operators:

  1. 1.

$d$, $d^{*}$, $\Delta$, constructed out of the Riemannian metric.

  2. 2.

$L(\alpha):=\omega\wedge\alpha$.

  3. 3.

$\Lambda(\alpha):=*L*\alpha$. It is easily seen that $\Lambda=L^{*}$.

  4. 4.

The Weil operator
$W{\left|{}_{{\phantom{|}\\!\\!}_{\Lambda^{p,q}(M)}}\right.}=\sqrt{-1}\>(p-q)$

This Lie superalgebra is called the algebra of supersymmetry of the Kähler
manifold.

Using 2.3 below, is easy to see that $\mathfrak{a}$ is in fact independent
from $M$.

This Lie superalgebra was studied from the physicists’ point of view in [FKS].

Theorem 2.15:  Let $M$ be a Kähler manifold, and $\mathfrak{a}$ its
supersymmetry algebra acting on $\Lambda^{*}(M)$. Then $\mathfrak{a}$ has
dimension $(5|4)$ (that is, its odd part is 4-dimensional, and its even part
is 5-dimensional). The odd part is generated by
$d,d^{c}=IdI^{-1},d^{*},(d^{c})^{*}$, the even part is generated by the
Lefschetz triple $L,\Lambda,H=[L,\Lambda]$, the Weil operator $W$ and the
Laplacian $\Delta=\\{d,d^{*}\\}$. Moreover, the Laplacian $\Delta$ is central
in $\mathfrak{a}$, hence $\mathfrak{a}$ also acts on the cohomology of $M$.
The following are the only non-zero commutator relations in $\mathfrak{a}$:

  1. 1.

${\mathfrak{sl}}(2)$-relations in $\langle L,\Lambda,H=[L,\Lambda]\rangle$:

$[H,L]=2L,\ \ [H,\Lambda]=2\Lambda,\ \ [L,\Lambda]=H.$

For any operator $D$ of grading $k$, one has $[H,D]=kD$.

  2. 2.

The Weil operator acts as a complex structure on the odd part of
$\mathfrak{a}$:

$[W,d]=d^{c},\ \ [W,d^{c}]=-d,\ \ [W,d^{*}]=-(d^{c})^{*},\ \
[W,(d^{c})^{*}]=d^{*}.$

  3. 3.

The Kähler-Kodaira relations between the differentials and the Lefschetz
operators:

$[\Lambda,d]=(d^{c})^{*},\ \ [L,d^{*}]=-d^{c},\ \ [\Lambda,d^{c}]=-d^{*},\ \
[L,(d^{c})^{*}]=d.$ (2.4)

  4. 4.

Almost all odd elements supercommute, with the only exception

$\Delta=\\{d,d^{*}\\}=\\{d^{c},(d^{c})^{*}\\},$

and $\Delta$ is central. In other words, the odd elements of $\mathfrak{a}$
generate the odd Heisenberg superalgebra, see 2.3.

Proof: These relations are standard in algebraic geometry (see e.g. [GH]), but
probably the easiest way to prove them is using the results about Lie
superalgebras collected in Subsection 2.2.

Proof of the Lefschetz ${\mathfrak{sl}}(2)$-relations: These relations would
follow if we prove that $H:=[L,\Lambda]$ acts on $p$-forms by multiplication
by $p-n$, where $n=\dim_{\mathbb{C}}M$. Since $L,\Lambda,H$ are
$C^{\infty}(M)$-linear, it would suffice to prove these relations on a
Hermitian vector space.

Let $V$ be a real vector space equipped with a scalar product, and fix an
orthonormal basis $\\{v_{1},\ldots,v_{m}\\}$. Denote by
$e_{v_{i}}:\;\Lambda^{k}V{\>\longrightarrow\>}\Lambda^{k+1}V$ the operator of
multiplication, $e_{v_{i}}(\eta)=v_{i}\wedge\eta$. Let
$i_{v_{i}}:\;\Lambda^{k}V{\>\longrightarrow\>}\Lambda^{k-1}V$ be the operator
of contraction with $v_{i}$. The following claim is clear.

Claim 2.16:  The operators $e_{v_{i}}$, $i_{v_{i}}$, $\operatorname{Id}$ form
a basis of the odd Heisenberg Lie superalgebra, with the only non-trivial
supercommutator given by the formula
$\\{e_{v_{i}},i_{v_{j}}\\}=\delta_{i,j}\operatorname{Id}$.

Let now $V$ be an even-dimensional real vector space equipped with a scalar
product, and $\\{x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}\\}$ an orthonormal
basis. Consider the complex structure operator $I$ such that $I(x_{i})=y_{i}$,
$I(y_{i})=-x_{i}$. The fundamental symplectic form is given by
$\sum_{i}x_{i}\wedge y_{i}$, hence

$L=\sum_{i}e_{x_{i}}e_{y_{i}},\ \ \ \Lambda=\sum_{i}i_{x_{i}}i_{y_{i}}.$

Clearly, for any odd elements $a,b,c,d$ such that
$\\{a,b\\}=\\{a,d\\}=\\{b,c\\}=\\{c,d\\}=0$, one has
$\\{ab,cd\\}=-\\{a,c\\}bd+ca\\{b,d\\}$. Then

$\begin{split}[L,\Lambda]&=\left[\sum_{i}e_{x_{i}}e_{y_{i}},\sum
i_{x_{i}}i_{y_{i}}\right]=\sum_{i=1}^{n}e_{y_{i}}i_{y_{i}}-\sum_{i=1}^{n}i_{x_{i}}e_{x_{i}}\\\
&=\sum_{i=1}^{n}e_{y_{i}}i_{y_{i}}-\sum_{i=1}^{n}(e_{y_{i}}i_{y_{i}}-1)\end{split}$

This term, applied to a monomial $\alpha$ of degree $d$, would give
$(d-n)\alpha$. This proves the Lefschetz ${\mathfrak{sl}}(2)$-relations.

Proof of the relations between the Weil operator $W$ and the odd part of
$\mathfrak{a}$. Clearly, it is enough to prove $[W,d]=d^{c}$, the remaining
relations follow by duality or by complex conjugation. Writing the Hodge
components of $d=d^{1,0}+d^{0,1}$, with $d^{1,0}=\frac{d+\sqrt{-1}\>d^{c}}{2}$
and $d^{0,1}=\frac{d-\sqrt{-1}\>d^{c}}{2}$ , we obtain
$[W,d]=\sqrt{-1}\>d^{1,0}-\sqrt{-1}\>d^{0,1}=d^{c}$.

Proof of the Kähler-Kodaira relations between the Lefschetz
${\mathfrak{sl}}(2)$-operators and the odd part of $\mathfrak{a}$. As before,
it is enough to prove $[L,d^{*}]=-d^{c}$, the remaining Kähler-Kodaira
relations follow by duality or by complex conjugation. The operator $L$ is
$C^{\infty}(M)$-linear, hence it is a differential operator of order 0. The
operator $d^{*}$ can be written in a frame $\\{v_{i}\\}$ of $TM$ as

$d^{*}(\eta)=\sum_{i}i_{v_{i}}\nabla_{v_{i}}\eta,$ (2.5)

where $\nabla$ is the Levi-Civita connection of the metric $g$. Since
$\nabla_{v_{i}}$ and $i_{v_{i}}$ are both derivations of the de Rham algebra,
their product is an order 2 differential operator (in the algebraic sense as
given by 2.2). From 2.2 it follows that $[L,d^{*}]$ is a first order operator.

We shall prove $[L,d^{*}]=-d^{c}$ by applying 2.2. First, let us show that
$[L,[L,d^{*}]]=0$. Clearly, $[\Lambda,d^{*}]=([L,d])^{*}=0$, and
$[H,d^{*}]=-d^{*}$, hence $d^{*}$ is the lowest weight vector in a weight 1
representation of $\mathfrak{sl}(2)$. This gives $[L,[L,d^{*}]]=0$. Now, the
super Jacobi identity gives

$\\{\\{L,d^{*}\\},\\{L,d^{*}\\}\\}=\\{L,\\{d^{*},\\{L,d^{*}\\}\\}+\\{d^{*},\\{L,\\{L,d^{*}\\}\\}.$

The first term in the RHS vanishes by 2.2, and the second term vanishes
because $[L,[L,d^{*}]]=0$. Then $([L,d^{*}])^{2}=0$. Clearly,
$d^{c}(C^{\infty}(M))$ generates $\Lambda^{1}(M)$ over $C^{\infty}(M)$. To
deduce $[L,d^{*}]=-d^{c}$ from 2.2, it remains to show that
$[L,d^{*}]{\left|{}_{{\phantom{|}\\!\\!}_{C^{\infty}(M)}}\right.}=-d^{c}{\left|{}_{{\phantom{|}\\!\\!}_{C^{\infty}(M)}}\right.}$.
This is clear for the following reason. For any function $f\in C^{\infty}(M)$,
one has $[L,d^{*}]f=-d^{*}(f\omega)$. Writing
$d^{*}=\sum_{i}i_{v_{i}}\nabla_{v_{i}}$ as in (2.5), and using
$\nabla\omega=0$, we obtain

$d^{*}(f\omega)=\sum_{i}i_{v_{i}}\nabla_{v_{i}}(f\omega)=\sum_{i}\operatorname{Lie}_{v_{i}}(f)i_{v_{i}}(\omega)=\sum_{i}-\operatorname{Lie}_{v_{i}}(f)I(v_{i})=-d^{c}f.$

This finishes the proof of $[L,d^{*}]=-d^{c}$.

Proof of the commutator relations between the odd part of $\mathfrak{a}$. We
have already shown that $d^{c}=[W,d]$. Then
$\\{d,d^{c}\\}=\\{d,\\{W,d\\}\\}=0$ by 2.2. Similarly,
$d^{*}=-[\Lambda,d^{c}]$, giving $\\{d^{c},d^{*}\\}=0$. The relation
$\\{(d^{c})^{*},d\\}=0$ is obtained by duality. Finally,
$\\{d,d^{*}\\}=\\{d^{c},(d^{c})^{*}\\}$ is obtained by applying
$[\Lambda,\cdot]$ to $\\{d,d^{c}\\}=0$. Using the Kähler-Kodaira relations
(2.4), we obtain

$0=\\{\Lambda,\\{d,d^{c}\\}\\}=\\{\\{\Lambda,d\\},d^{c}\\}+\\{d,\\{\Lambda,d^{c}\\}\\}=\\{(d^{c})^{*},d^{c}\\}-\\{d,d^{*}\\},$

giving $\\{d,d^{*}\\}=\\{d^{c},(d^{c})^{*}\\}$.

We proved all the relations in the Kähler supersymmetry algebra
$\mathfrak{a}$, finishing the proof of 2.3.

Further in this paper, we shall develop similar relations for the superalgebra
associated with a Sasakian manifold.

##  3 Sasakian manifolds: definition and the basic notions

In this section we provide the necessary background on Sasakian manifolds. For
details, please see [Bl], [BG]. The most convenient definition for our context
is the one which relates Sasakian and Kähler geometries in terms of Riemannian
cones.

Definition 3.1: Let $S$ be a contact manifold with a contact form $\eta$. The
vector field ${\vec{r}}$ defined by the equations:

$i_{\vec{r}}\,\eta=1,\qquad i_{\vec{r}}\,d\eta=0.$

is called the Reeb or characteristic field.

Definition 3.2: A Sasakian manifold is a Riemannian manifold $(S,g)$ with a
Kähler structure on its Riemannian cone
$C(S):=(S\times{\mathbb{R}}^{>0},t^{2}g+dt^{2})$, such that the homothety map
$h_{\lambda}:\;C(S){\>\longrightarrow\>}C(S)$ mapping $(m,t)$ to $(m,\lambda
t)$ is holomorphic.

Remark 3.3:  (i) A Sasakian manifold is clearly contact, because its cone is
symplectic and $h_{\lambda}$ acts by symplectic homotheties.

(ii) Let $S$ be a Sasakian manifold, $\omega$ the Kähler form on $C(S)$, and
$\xi=t\frac{d}{dt}$ the homothety vector field along the generators of the
cone (it is also called Euler field). Then $\operatorname{Lie}_{I\xi}t=\langle
dt,I\xi\rangle=0$ (where $\operatorname{Lie}_{v}$ denotes the Lie derivative
in the direction of $v$), and hence $I\xi$ is tangent to $S\subset C(S)$. Then
$\eta(I\xi)=1$ and $i_{I\xi}d\eta=0$, and thus the Reeb vector field of a
Sasakian manifold is ${\vec{r}}=I\xi$.

(iii) The function $t^{2}$ is a Kähler potential on $C(S)$. Moreover, the form
$dd^{c}\log t$ vanishes on $\langle\xi,I(\xi)\rangle$ and the rest of its
eigenvalues are positive.

Proposition 3.4:  The Reeb field has constant length and acts on a Sasakian
manifold by contact isometries, i.e. the flow of ${\vec{r}}$ contains only
isometries which preserve the contact subbundle. Moreover, its action lifts to
a holomorphic action on its cone.

Definition 3.5: A Sasakian (resp. contact) manifold is called regular if its
Reeb field generates a free action of $S^{1}$; it is called quasi-regular if
all orbits of the Reeb field are closed, and it is called irregular otherwise.

Example 3.6:  Let $S$ be a regular Sasakian manifold, and ${\vec{r}}$ its Reeb
field. Then the space of ${\vec{r}}$-orbits $X$ is Kähler. Moreover, $X$ is
equipped with a positive holomorphic Hermitian line bundle $L$ such that $S$
is the space of unit vectors in $L$. Conversely: if $X$ is a compact
projective manifold, together with an ample line bundle
$L{\>\longrightarrow\>}X$, then the space of unit vectors in $L$ is a regular
Sasakian manifold.

##  4 Hattori differentials on Sasakian manifolds

###  4.1 Hattori spectral sequence and associated differentials

Let $\pi:\;M{\>\longrightarrow\>}B$ be a smooth fibration, and
$F_{k}\subset\Lambda^{*}(M)$ be the ideal generated by
$\pi^{*}\Lambda^{k}(B)$. The Hattori spectral sequence ([Ha]; in [Br, Section
1.6] it is named after Cartan) is the spectral sequence associated with this
filtration.

The $E_{1}^{p,q}$-term of this sequence is
$\Lambda^{p}(B)\otimes_{\mathbb{R}}R^{q}\pi_{*}{\mathbb{R}}_{M}$, where
$R^{*}\pi_{*}{\mathbb{R}}_{M}$ denotes the local system of cohomology of the
fibres, and the $E_{2}^{p,q}$-term is $H^{p}(B,R^{q}\pi_{*}{\mathbb{R}}_{M})$.

The same can be done when $M$ is a manifold equipped with an integrable
distribution $F\subset TM$, giving a spectral sequence converging to
$H^{*}(M)$. This is done as follows.

Definition 4.1: Let $M$ be a manifold, and $F\subset TM$ an integrable
distribution. A $k$-form $\alpha\in\Lambda^{*}(M)$ is called basic if for any
vector field $v\in F$, one has $\operatorname{Lie}_{v}\alpha=0$ and
$i_{v}\alpha=0$.

If $F$ is the tangent bundle of the fibres of a fibration
$\pi:\;M{\>\longrightarrow\>}B$, then the space of basic forms is
$\pi^{*}\Lambda^{k}(B)$. We are going to produce a spectral sequence which
gives the standard Hattori spectral sequence when $F$ is tagent to the leaves
of a fibration.

Define the Hattori filtration associated with $F$ as $F_{n}\subset
F_{n-1}\subset\cdots$ by putting $F_{k}\subset\Lambda^{*}(M)$, where $F_{k}$
is the ideal generated by basic $k$-forms.

When $M$ is Riemannian and the metric is compatible with the foliation, this
filtered bundle is decomposed into the direct sum of subquotients. This gives
a decomposition of the de Rham differential,
$d=d_{0}+d_{1}+d_{2}+\cdots+d_{r+1}$, where $r=\operatorname{rk}F$ with each
successive piece associated with the differential in $E_{k}^{p,q}$, as
follows.

Using the metric, we split the cotangent bundle into orthogonal complements as
$\Lambda^{1}(M)=\Lambda^{1}_{\operatorname{\text{\sf
hor}}}(M)\oplus\Lambda^{1}_{\operatorname{\text{\sf vert}}}(M)$, where
$\Lambda^{1}_{\operatorname{\text{\sf hor}}}(M)$ is generated by basic
1-forms, and $\Lambda^{1}_{\operatorname{\text{\sf vert}}}(M)=F^{*}$ is its
orthogonal complement. Denote by $\Lambda^{p}_{\operatorname{\text{\sf
hor}}}(M)$, $\Lambda^{q}_{\operatorname{\text{\sf vert}}}(M)$ the exterior
powers of these bundles.

This gives the following splitting of the de Rham algebra of $M$:

$\Lambda^{m}(M)=\bigoplus_{p}\Lambda^{p}_{\operatorname{\text{\sf
hor}}}(M)\otimes\Lambda^{m-p}_{\operatorname{\text{\sf vert}}}(M)$ (4.1)

with $F_{p}/F_{p-1}=\bigoplus_{q}\Lambda^{p}_{\operatorname{\text{\sf
hor}}}(M)\otimes\Lambda^{q}_{\operatorname{\text{\sf vert}}}(M)$ where $F_{*}$
denotes the Hattori filtration.

Consider the associated decomposition of the de Rham differential,
$d=d_{0}+d_{1}+d_{2}+\cdots+d_{r+1}$, where $r=\operatorname{rk}F$, and

$d_{i}:\;\Lambda^{p}_{\operatorname{\text{\sf
hor}}}(M)\otimes\Lambda^{q}_{\operatorname{\text{\sf
vert}}}(M){\>\longrightarrow\>}\Lambda^{p+i}_{\operatorname{\text{\sf
hor}}}(M)\otimes\Lambda^{q+1-i}_{\operatorname{\text{\sf vert}}}(M)$

The terms $d_{i}$ vanish for $i>r+1$ because $F$ is $r$-dimensional, hence for
$i>r+1$ either $\Lambda^{q+1-i}_{\operatorname{\text{\sf vert}}}(M)$ or
$\Lambda^{q}_{\operatorname{\text{\sf vert}}}(M)$ is 0.

These differentials are related to the differentials in the Hattori spectral
sequence in the following way: to find $E_{1}^{p,q}$, one takes the cohomology
of $d_{0}$. Then one restricts $d_{1}$ to $E_{1}^{p,q}$, and its cohomology
gives $E_{2}^{p,q}$, and so on. In this sense, the Hattori differentials are
indeed differentials in the Hattori spectral sequence.

Remark 4.2: Each of the differentials $d_{i}$ is a derivation, because the
decomposition
$\Lambda^{*}(M)=\bigoplus_{p,q}\Lambda^{p}_{\operatorname{\text{\sf
hor}}}(M)\otimes\Lambda^{q}_{\operatorname{\text{\sf vert}}}(M)$ is
multiplicative.

###  4.2 Hattori differentials on Sasakian manifolds

Let now $Q$ be a Sasakian manifold, ${\vec{r}}$ its Reeb field, normalized in
such a way that $|{\vec{r}}|=1$, and $R\subset TQ$ the 1-dimensional foliation
generated by the Reeb field. The corresponding Hattori differentials are
written as $d=d_{0}+d_{1}+d_{2}$, because $R$ is 1-dimensional. Since
$d^{2}=0$, one has $d_{0}^{2}=d_{2}^{2}=0$ and
$\\{d_{0},d_{2}\\}=-\\{d_{1},d_{1}\\}$. The differentials $d_{0},d_{2}$ can be
described explicitly as follows.

Claim 4.3:  Let
$e_{\vec{r}}:\;\Lambda^{*}(Q){\>\longrightarrow\>}\Lambda^{*+1}(Q)$ be the
operator of multiplication by the form ${\vec{r}}^{\,\flat}=\eta$ dual to
${\vec{r}}$. Then $d_{0}=e_{\vec{r}}\operatorname{Lie}_{\vec{r}}$.

Proof: Locally, the sheaf $\Lambda^{1}(Q)$ is generated over $C^{\infty}(Q)$
by the basic 1-forms and ${\vec{r}}^{\,\flat}$. Clearly, the differential of a
basic 1-form $\alpha$ belongs to $\Lambda^{2}_{\operatorname{\text{\sf
bas}}}(Q)$, hence $d_{0}(\alpha)=0$, and
$e_{\vec{r}}\operatorname{Lie}_{\vec{r}}(\alpha)=0$. Also,
$d_{0}({\vec{r}}^{\,\flat})=e_{\vec{r}}\operatorname{Lie}_{\vec{r}}({\vec{r}}^{\,\flat})=0$.
By 2.2, to prove $d_{0}=e_{\vec{r}}\operatorname{Lie}_{\vec{r}}$ it remains to
show that these two operators are equal on $C^{\infty}(Q)$. However, on
$C^{\infty}(Q)$, we have $d_{0}=e_{\vec{r}}\operatorname{Lie}_{\vec{r}}$,
because $d_{0}f$ is the orthogonal projection of $df$ to
$\Lambda^{1}_{\operatorname{\text{\sf vert}}}(Q)=R^{*}$ generated by
$e_{\vec{r}}$, for all $f\in C^{\infty}(Q)$.

Recall that the space of leaves of $R$ on a regular Sasakian manifold is
equipped with a complex and Kähler structure, 3. The corresponding Kähler
structure can be described very explicitly.

Claim 4.4:  Let $Q$ be a regular Sasakian manifold, and ${\vec{r}}^{\,\flat}$
its contact form. Then $\omega_{0}:=d({\vec{r}}^{\,\flat})$ is basic with
respect to $R$, and defines a transversally Kähler structure, that is, the
Kähler structure on the space of leaves of $R$ (5).

Proof: Let $X=Q/{\vec{r}}$ be the space of orbits. This quotient is well
defined and smooth, because ${\vec{r}}$ is regular. Then
$X=C(Q)/{\mathbb{C}}^{*}$, where the ${\mathbb{C}}^{*}$-action is generated by
$\xi=t\frac{d}{dt}$ and ${\vec{r}}=I(\xi)$, and hence it is holomorphic.
Therefore, $X$ is a complex manifold (as a quotient of a complex manifold by a
holomorphic action of a Lie group). It is Kähler by 3 (iii).

Proposition 4.5:  Let
$L_{\omega_{0}}:\;\Lambda^{*}(Q){\>\longrightarrow\>}\Lambda^{*+2}(Q)$ be the
operator of multiplication by the transversally Kähler form
$\omega_{0}=d({\vec{r}}^{\,\flat})$, and $i_{\vec{r}}$ the contraction with
the Reeb field. Then $d_{2}=L_{\omega_{0}}i_{\vec{r}}$.

Proof: Clearly, the Hattori differentials

$d_{i}:\;\Lambda^{p}_{\operatorname{\text{\sf
hor}}}(Q)\times\Lambda^{q}_{\operatorname{\text{\sf
vert}}}(Q){\>\longrightarrow\>}\Lambda^{p+i}_{\operatorname{\text{\sf
hor}}}(Q)\otimes\Lambda^{q+1-i}_{\operatorname{\text{\sf vert}}}(Q)$

vanish on $\Lambda^{0}(Q)$ unless $i=0,1$. Therefore, the differentials
$d_{2},d_{3},...$ are always $C^{\infty}(Q)$-linear. By 2.2, it only remains
to show that $d_{2}=L_{\omega_{0}}i_{\vec{r}}$ on some set of 1-forms
generating $\Lambda^{1}(Q)$ over $C^{\infty}(Q)$.

Clearly, on $\Lambda^{1}_{\operatorname{\text{\sf hor}}}(Q)$ the differential
$d_{2}$ should act as

$d_{2}:\;\Lambda^{1}_{\operatorname{\text{\sf
hor}}}(Q){\>\longrightarrow\>}\Lambda^{3}_{\operatorname{\text{\sf
hor}}}(Q)\otimes\Lambda^{-1}_{\operatorname{\text{\sf vert}}}(Q),$

hence
$d_{2}{\left|{}_{{\phantom{|}\\!\\!}_{\Lambda^{1}_{\operatorname{\text{\sf
hor}}}(Q)}}\right.}=0$. To prove 4.2 it remains to show that
$d_{2}({\vec{r}}^{\,\flat})=L_{\omega_{0}}i_{\vec{r}}({\vec{r}}^{\,\flat})$.
However, $d_{2}({\vec{r}}^{\,\flat})$ is the
$\Lambda^{2}_{\operatorname{\text{\sf hor}}}(Q)$-part of
$d({\vec{r}}^{\,\flat})$, giving
$d_{2}({\vec{r}}^{\,\flat})=d({\vec{r}}^{\,\flat})=\omega_{0}$, and
$L_{\omega_{0}}i_{\vec{r}}({\vec{r}}^{\,\flat})=\omega_{0}$ because
$i_{\vec{r}}({\vec{r}}^{\,\flat})=1$.

The Hattori differential $d_{1}$ is, heuristically speaking, the “transversal
component” of the de Rham differential. Indeed, $d_{1}(\eta)=d(\eta)$ for any
basic form $\eta$. Since the leaf space of $R$ is equipped with a complex
structure, it is natural to expect that the Hodge components of $d_{1}$ have
the same properties as the Hodge components of the de Rham differential on a
complex manifold.

Claim 4.6:  Let $Q$ be a Sasakian manifold, and
$\Lambda^{m}(Q)=\bigoplus_{p}\Lambda^{p}_{\operatorname{\text{\sf
hor}}}(Q)\otimes\Lambda^{m-p}_{\operatorname{\text{\sf vert}}}(Q)$ the
decomposition associated with $R\subset TQ$ as in (4.1). Using the complex
structure on the basic forms, consider the Hodge decomposition
$\Lambda^{m}_{\operatorname{\text{\sf
hor}}}(Q)=\bigoplus_{p}\Lambda^{m-p,p}_{\operatorname{\text{\sf hor}}}(Q)$.
Then the differential $d_{1}:\;\Lambda^{p}_{\operatorname{\text{\sf
hor}}}(Q)\otimes\Lambda^{q}_{\operatorname{\text{\sf
vert}}}(Q){\>\longrightarrow\>}\Lambda^{p+1}_{\operatorname{\text{\sf
hor}}}(Q)\otimes\Lambda^{q}_{\operatorname{\text{\sf vert}}}(Q)$ has two Hodge
components $d_{1}^{0,1}$ and $d_{1}^{1,0}$. Moreover, the differential
$d_{1}^{c}:=d_{1}^{0,1}-d_{1}^{1,0}$ satisfies $d_{1}^{c}=Id_{1}I^{-1}$, where
$I$ acts as $\sqrt{-1}\>^{p-q}$ on $\Lambda^{p,q}_{\operatorname{\text{\sf
hor}}}(Q)\otimes\Lambda^{m}_{\operatorname{\text{\sf vert}}}(Q)$.

Proof: First, let us prove that $d_{1}$ has only two non-zero Hodge
components. A priori, $d_{1}$ could have several Hodge components,
$d_{1}=d^{-k,k+1}_{1}+d^{-k+1,k}_{1}+\cdots+d^{k,-k+1}_{1}+d^{-k,k+1}_{1}$.
This is what happens with the Hodge components of de Rham differential on an
almost complex manifolds. All these components are clearly derivations.
However,

$d(\Lambda^{0}(Q))\subset\Lambda^{1}_{\operatorname{\text{\sf
vert}}}(Q)\oplus\Lambda^{1,0}_{\operatorname{\text{\sf
hor}}}(Q)\oplus\Lambda^{0,1}_{\operatorname{\text{\sf hor}}}(Q).$

Therefore, only $d_{1}^{0,1}$ and $d_{1}^{1,0}$ are non-zero on functions, the
rest of the Hodge components are $C^{\infty}(Q)$-linear. By 2.2, it would
suffice to show that the other differentials vanish on a basis in
$\Lambda^{1}(Q)$.

Since the space of leaves of $R$ is a complex manifold, and $d=d_{1}$ on basic
forms, we have $d_{1}=d_{1}^{0,1}+d_{1}^{1,0}$ on basic forms. Since
$d({\vec{r}}^{\,\flat})=\omega_{0}$, we find $d_{1}({\vec{r}}^{\,\flat})=0$,
which gives $d_{1}=d_{1}^{0,1}+d_{1}^{1,0}$ on ${\vec{r}}^{\,\flat}$. We
proved the decomposition $d_{1}=d_{1}^{0,1}+d_{1}^{1,0}$. The relation
$d_{1}^{0,1}-d_{1}^{1,0}=Id_{1}I^{-1}$ follows in the usual way, because
$\frac{d_{1}+\sqrt{-1}\>d_{1}^{c}}{2}$ has Hodge type $(1,0)$, hence satisfies
$d_{1}^{1,0}=\frac{d_{1}+\sqrt{-1}\>d_{1}^{c}}{2}$.

##  5 Transversally Kähler manifolds

Definition 5.1:  A manifold $M$ equipped with an integrable distribution
$F\subset TM$ is called a foliated manifold. In the sequel, we shall always
assume that $F$ is orientable. Let $\omega_{0}\in\Lambda^{2}(M)$ be a closed,
basic 2-form on a foliated manifold $(M,F)$, vanishing on $F$ and non-
degenerate on $TM/F$. Let $g_{0}\in\operatorname{Sym}^{2}(T^{*}M)$ be a basic
bilinear symmetric form which is positive definite on $TM/F$. Since
$g_{0},\omega$ are basic, the operator $I:=\omega^{-1}_{0}\circ
g_{0}:\;TM/F{\>\longrightarrow\>}TM/F$ is well defined on the leaf space $L$
of $F$ (locally the leaf space always exists by Frobenius theorem). Assume
that $I$ defines an integrable complex structure on $L$ for any open set
$U\subset M$ for which the leaf space is well defined. Then $(M,F,g,\omega)$
is called transversally Kähler. A vector field $v$ such that
$\operatorname{Lie}_{v}(F)\subset F$, and $\operatorname{Lie}_{v}I=0$ is
called transversally holomorphic; it is called transversally Killing if, in
addition, $\operatorname{Lie}_{v}g_{0}=0$.

Definition 5.2: Let $F\subset TM$ be an integrable distribution and
$\Lambda_{\operatorname{\text{\sf bas}}}^{*}(M)$ the complex of basic forms.
Its cohomology algebra is called the basic, or transversal cohomology of $M$.
We denote the basic cohomology by $H_{\operatorname{\text{\sf bas}}}^{*}(M)$.

Remark 5.3: Note that $H_{\operatorname{\text{\sf bas}}}^{*}(M)$ can be
infinitely-dimensional even when $M$ is compact, [Sch].

The main result of this section is the following theorem, which is a weaker
form of the main theorem from [E]. Our result is less general, but the proof
is simple and self-contained.

Theorem 5.4:  Let $(M,F,\omega_{0},g_{0})$ be a compact, transverally Kähler
manifold. Assume, moreover that:

  * (*)

$M$ is equipped with a Riemannian metric $g$ such that the restriction of $g$
to the orthogonal complement $F^{\bot}=TM/F$ coincides with $g_{0}$, and $F$
is generated by a collection of Killing vector fields $v_{1},...,v_{r}$.

  * (**)

There exists a closed differential form $\Phi$ on $M$ which vanishes on $F$
and gives a Riemannian volume form on $TM/F$.111The assumptions (*) and (**)
hold for Sasakian manifolds (3) and Vaisman manifolds (7.2).

Then the basic cohomology $H^{*}_{\operatorname{\text{\sf bas}}}(M)$ of $M$ is
finite-dimensional and admits the Hodge decomposition and the Lefschetz
${\mathfrak{sl}}(2)$-action, as in the Kähler case.

Proof: Consider the differential graded algebra of basic forms
$\Lambda^{*}_{\operatorname{\text{\sf bas}}}(M)$. This algebra is equipped
with an action of the superalgebra of Kähler supersymmetry $\mathfrak{a}$ as
in 2.3. Indeed, define the Lefschetz ${\mathfrak{sl}}(2)$-action by taking the
Lefschetz triple $L_{\omega_{0}},\Lambda_{\omega_{0}}:=*L_{\omega_{0}}*$ and
$H_{\omega_{0}}:=[L_{\omega_{0}},\Lambda_{\omega_{0}}]$, and the transversal
Weil operator $W$ acting in the standard way on
$\Lambda^{*}_{\operatorname{\text{\sf bas}}}(M)$ and extended to
$\Lambda^{*}(M)$ by acting trivially on $\Lambda^{1}_{\operatorname{\text{\sf
bas}}}(M)^{\bot}$ (or in any other way, it does not matter). Together with the
de Rham differential $d:\;\Lambda^{*}_{\operatorname{\text{\sf
bas}}}(M){\>\longrightarrow\>}\Lambda^{*}_{\operatorname{\text{\sf bas}}}(M)$
these operators generate the Lie superalgebra
$\mathfrak{a}\subset\operatorname{End}(\Lambda^{*}_{\operatorname{\text{\sf
bas}}}(M))$, which is isomorphic to the superalgebra of Kähler supersymmetry
$\mathfrak{a}$ (2.3), because $\Lambda^{*}_{\operatorname{\text{\sf bas}}}(M)$
is locally identified with the algebra of differential forms on the leaf space
of $F$ which is Kähler.

Then 5 would follow if we identify $\ker(\Delta_{\operatorname{\text{\sf
bas}}}{\left|{}_{{\phantom{|}\\!\\!}_{\Lambda^{*}_{\operatorname{\text{\sf
bas}}}(M)}}\right.})$ with the space $H^{*}_{\operatorname{\text{\sf
bas}}}(M)$, where $\Delta_{\operatorname{\text{\sf bas}}}\in\mathfrak{a}$ is
the transversal Laplace operator, $\Delta_{\operatorname{\text{\sf
bas}}}=\\{d,d^{*}_{\operatorname{\text{\sf bas}}}\\}$, where
$d^{*}_{\operatorname{\text{\sf bas}}}$ denotes the $d^{*}$-operator on the
leaf space.

We reduced 5 to the following result.

Proposition 5.5:  Let $(M,F,g)$, $\operatorname{rk}F=r$, be a Riemannian
foliated manifold, $d=d_{0}+d_{1}+\cdots+d_{r+1}$ the Hattori decomposition of
the differential, and $\Delta_{\operatorname{\text{\sf
bas}}}:=\\{d,d_{\operatorname{\text{\sf bas}}}^{*}\\}$ the basic Laplacian
defined on basic forms. Assume that:

  * (*)

$F$ is generated by a collection of Killing vector fields $v_{1},v_{2},\ldots$

  * (**)

The Riemannian volume form $\Phi\in\Lambda^{r}_{\operatorname{\text{\sf
vert}}}(M)$ satisfies $d_{1}(\Phi)=0$.

Then there exists a natural isomorphism between the basic harmonic forms and
$H^{*}_{\operatorname{\text{\sf bas}}}(M)$.

We start from the following lemma.

Lemma 5.6: In the assumptions of 5, let
$\alpha,\beta\in\Lambda^{*}_{\operatorname{\text{\sf bas}}}(M)$ be two basic
forms. Then $g(d\alpha,\beta)=g(\alpha,d^{*}_{\operatorname{\text{\sf
bas}}}\beta)$, where $d^{*}_{\operatorname{\text{\sf bas}}}$ denotes the
$d^{*}$-operator on the leaf space of $F$.

Proof: This is where we use the assumption (**) of 5. Let $d^{*}_{h}$ be the
composition of $d^{*}$ with the orthogonal projection to the horizontal part
$\Lambda^{*}_{\operatorname{\text{\sf hor}}}(M)$ (see Subsection 4.1). We only
need to show that

$g(d^{*}_{h}\alpha,\beta)=g(d^{*}_{\operatorname{\text{\sf
bas}}}\alpha,\beta)\ \ \text{for all}\ \
\alpha,\beta\in\Lambda^{*}_{\operatorname{\text{\sf bas}}}(M).$ (5.1)

By 2.3, one has $d^{*}_{\operatorname{\text{\sf
bas}}}=-*_{\operatorname{\text{\sf bas}}}d*_{\operatorname{\text{\sf bas}}}$
where $*_{\operatorname{\text{\sf bas}}}$ is the Hodge star operator on the
leaf space. Let $r=\operatorname{rk}F$ and $\Phi\in\Lambda^{r}F$ be the
Riemannian volume form. Using the assumption (**), we obtain that
$d\Phi\in\bigoplus_{i=0}^{r-1}\Lambda^{*}_{\operatorname{\text{\sf
hor}}}(M)\otimes\Lambda^{i}_{\operatorname{\text{\sf vert}}}(M)$. Then
$*\alpha=\Phi\wedge*_{\operatorname{\text{\sf bas}}}(\alpha)$, which gives

$*d*\alpha=*d(\Phi\wedge*_{\operatorname{\text{\sf
bas}}}(\alpha))=d^{*}_{\operatorname{\text{\sf
bas}}}\alpha+*(d\Phi\wedge\alpha).$

The last term belongs to
$\bigoplus_{i=1}^{r}\Lambda^{*}_{\operatorname{\text{\sf
hor}}}(M)\otimes\Lambda^{i}_{\operatorname{\text{\sf vert}}}(M)$, hence it is
orthogonal to $\Lambda^{*}_{\operatorname{\text{\sf hor}}}(M)$. This proves
(5.1).

Now we can prove 5. By (5.1), for any basic form $\alpha$ we have

$g(\Delta_{\operatorname{\text{\sf
bas}}}\alpha,\alpha)=(d\alpha,d\alpha)+(d^{*}_{\operatorname{\text{\sf
bas}}}\alpha,d^{*}_{\operatorname{\text{\sf bas}}}\alpha),$ (5.2)

hence a basic form belongs to $\ker\Delta_{\operatorname{\text{\sf bas}}}$ if
and only if it is closed and orthogonal to all exact basic forms. This gives
an embedding

$\ker\Delta_{\operatorname{\text{\sf
bas}}}{\left|{}_{{\phantom{|}\\!\\!}_{\Lambda^{*}_{\operatorname{\text{\sf
bas}}}(M)}}\right.}\hookrightarrow H^{*}_{\operatorname{\text{\sf bas}}}(M).$
(5.3)

It remains only to show that the map (5.3) is surjective. This is where we use
the assumption (*) of 5.

Consider the Hattori decomposition $d=d_{0}+d_{1}+\cdots+d_{r+1}$ (see
Subsection 4.1). Let $v_{1},\ldots,v_{r}\in F$ be the Killing, transversally
Killing vector fields, postulated in (*), and $\Delta_{s}$ the “split
Laplacian”,
$\Delta_{s}:=\\{d_{1},d_{1}^{*}\\}-\sum_{i}\operatorname{Lie}_{v_{i}}^{2}$.
Clearly, $\Delta_{s}$ is an elliptic, second order differential operator. By
definition,
$g(e_{v_{i}}i_{v_{i}}\alpha,\alpha)=g(i_{v_{i}}\alpha,i_{v_{i}}\alpha)$. Since
$v_{i}$ are Killing, one has
$\operatorname{Lie}_{v_{i}}=-\operatorname{Lie}_{v_{i}}^{*}$. Therefore
$\Delta_{s}$ is self-adjoint and positive, with

$g(\Delta_{s}\alpha,\alpha)=g(d_{1}\alpha,d_{1}\alpha)+g(d^{*}_{1}\alpha,d^{*}_{1}\alpha)+\sum_{i}g(\operatorname{Lie}_{v_{i}}\alpha,\operatorname{Lie}_{v_{i}}\alpha).$

We obtain that each $\alpha\in\ker\Delta_{s}$ satisfies
$i_{v_{i}}\alpha=\operatorname{Lie}_{v_{i}}\alpha=0$.

Consider the orthogonal projection map $\Pi_{\operatorname{\text{\sf
hor}}}:\;\Lambda^{*}(M){\>\longrightarrow\>}\Lambda^{*}_{\operatorname{\text{\sf
hor}}}(M)$. Since the vector fields $v_{i}$ are Killing and preserve $F$, the
projection $\Pi_{\operatorname{\text{\sf hor}}}$ commutes with
$\operatorname{Lie}_{v_{i}}$. It is not hard to see that
$\Pi_{\operatorname{\text{\sf hor}}}$ commutes with $d_{1}$ and $d_{1}^{*}$.
Indeed, $d_{1}$ is the part of $d$ which maps
$\Lambda^{p}_{\operatorname{\text{\sf
hor}}}(M)\otimes\Lambda^{q}_{\operatorname{\text{\sf vert}}}(M)$ to
$\Lambda^{p+1}_{\operatorname{\text{\sf
hor}}}(M)\otimes\Lambda^{q}_{\operatorname{\text{\sf vert}}}(M)$, and
$d_{1}^{*}$ its adjoint. We obtain that $\Pi_{\operatorname{\text{\sf hor}}}$
commutes with $\Delta_{s}$.

Since $\Delta_{s}$ is a positive, self-adjoint, Fredholm operator, its
eigenvectors are dense in $\Lambda^{*}(M)$. Since
$[\Pi_{\operatorname{\text{\sf hor}}},\Delta_{s}]=0$, the eigenvectors of
$\Delta_{s}$ are dense in $\Lambda^{*}_{\operatorname{\text{\sf hor}}}(M)$.

Let $G$ be the closure of the Lie group generated by the action of
$e^{{\mathbb{R}}v_{i}}$. Since each $e^{{\mathbb{R}}v_{i}}$ acts by
isometries, the group $G$ is compact. By construction, $G$ acts on $M$
preserving the foliation $F$, the metric and the transversal Kähler structure,
hence it commutes with the Laplacian. Averaging on $G$, we obtain that the
eigenvectors of $\Delta_{s}$ are dense in the space
$\Lambda^{*}_{\operatorname{\text{\sf
hor}}}(M)^{G}=\Lambda^{*}_{\operatorname{\text{\sf bas}}}(M)$ of all basic
forms.

On basic forms, $d=d_{1}$, hence on $\Lambda^{*}_{\operatorname{\text{\sf
bas}}}(M)$ one has $[d,\Delta_{s}]=0$. Let
$\Lambda^{*}_{\operatorname{\text{\sf bas}}}(M)_{\lambda}$ be the eigenspace
of
$\Delta_{s}{\left|{}_{{\phantom{|}\\!\\!}_{\Lambda^{*}_{\operatorname{\text{\sf
bas}}}}}\right.}(M)$ corresponding to the eigenvalue $\lambda$. For any closed
$\alpha\in\Lambda^{*}_{\operatorname{\text{\sf bas}}}(M)$ we have
$\lambda\alpha=(dd^{*}+d^{*}d)(\alpha)=dd^{*}\alpha$. Therefore, any closed
form in $\Lambda^{*}_{\operatorname{\text{\sf bas}}}(M)_{\lambda}$ is exact
when $\lambda\neq 0$. We have shown that the only eigenspace of
$\Delta_{\operatorname{\text{\sf bas}}}$ which contributes to the basic
cohomology is $\Lambda^{*}_{\operatorname{\text{\sf
bas}}}(M)_{0}=\ker\Delta_{\operatorname{\text{\sf
bas}}}{\left|{}_{{\phantom{|}\\!\\!}_{\Lambda^{*}_{\operatorname{\text{\sf
bas}}}(M)}}\right.}$. This proves (5.3); we finished the proof of 5 and 5.

##  6 Basic cohomology and Hodge theory on Sasakian manifolds

###  6.1 Cone of a morphism of complexes and cohomology of Sasakian manifolds

We recall first several notions in homological algebra, see [GM]:

Definition 6.1: A complex $(C_{*},d)$ is a collection of vector spaces and
homomorphisms

$\dots\stackrel{{\scriptstyle
d}}{{{\>\longrightarrow\>}}}C_{i}\stackrel{{\scriptstyle
d}}{{{\>\longrightarrow\>}}}C_{i+1}\stackrel{{\scriptstyle
d}}{{{\>\longrightarrow\>}}}\cdots$

(more generally, a collection of objects in an abelian category) such that
$d^{2}=0$. A morphism of complexes is a collection of maps
$C_{i}{\>\longrightarrow\>}C_{i}^{\prime}$ from the vector spaces of a complex
$(C_{*},d)$ to the vector spaces of $(C^{\prime}_{*},d)$, commuting with the
differential. The cohomology groups of a complex $(C_{*},d)$ are the groups

$H^{i}(C_{*},d):=\frac{\ker
d{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}_{i}}{\operatorname{im}d{\left|{}_{{\phantom{|}\\!\\!}_{C_{i-1}}}\right.}}.$

Clearly, any morphism induces a homomorphism in cohomology. An exact sequence
of complexes is a sequence

$0{\>\longrightarrow\>}A_{*}{\>\longrightarrow\>}B_{*}{\>\longrightarrow\>}C_{*}{\>\longrightarrow\>}0$

of morphisms of complexes such that the corresponding sequences

$0{\>\longrightarrow\>}A_{i}{\>\longrightarrow\>}B_{i}{\>\longrightarrow\>}C_{i}{\>\longrightarrow\>}0$

are exact for all $i$.

The following claim is very basic.

Claim 6.2: Let
$0{\>\longrightarrow\>}A_{*}{\>\longrightarrow\>}B_{*}{\>\longrightarrow\>}C_{*}{\>\longrightarrow\>}0$
be an exact sequence of complexes. Then there is a natural long exact sequence
of cohomology

$\cdots\rightarrow H^{i-1}(C_{*},d_{C})\rightarrow
H^{i}(A_{*},d_{A})\rightarrow H^{i}(B_{*},d_{B})\rightarrow
H^{i}(C_{*},d_{C})\rightarrow\cdots\quad\text{\hbox{\vrule
width=5.0pt,height=5.0pt,depth=0.0pt}}$

Definition 6.3: Let
$(C_{*},d_{C})\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}(C^{\prime}_{*},d_{C^{\prime}})$
be a morphism of complexes. Consider the complex $C(\varphi)_{*}$, with
$C(\varphi)_{i}=C_{i+1}\oplus C^{\prime}_{i}$ and differential

$d:=d_{C}+\varphi-d_{C^{\prime}}:\;C_{i}\oplus
C^{\prime}_{i-1}{\>\longrightarrow\>}C_{i+1}\oplus C^{\prime}_{i}.$

or, explicitly:

$d(c_{i},c^{\prime}_{i-1})=\left(d_{C}(c_{i}),\varphi(c_{i})-d_{C^{\prime}}(c_{i-1}^{\prime})\right).$

The complex $(C(\varphi)_{*},d)$ is called the cone of $\varphi$.

Denote by $C_{*}[1]$ the complex $C_{*}$ shifted by one, with
$C_{i}[1]=C_{i+1}$. The exact sequence of complexes

$0{\>\longrightarrow\>}C_{*}^{\prime}{\>\longrightarrow\>}C(\varphi)_{*}{\>\longrightarrow\>}C_{*}[1]{\>\longrightarrow\>}0$

gives the long exact sequence

$\cdots{\>\longrightarrow\>}H^{i}(C)\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}H^{i}(C^{\prime}){\>\longrightarrow\>}H^{i}(C(\varphi)){\>\longrightarrow\>}\\\
{\>\longrightarrow\>}H^{i+1}(C)\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}H^{i+1}(C^{\prime}){\>\longrightarrow\>}\cdots$
(6.1)

Proposition 6.4:  Let $Q$ be a Sasakian manifold, and ${\vec{r}}$ the Reeb
field. Denote by $\Lambda^{*}_{\vec{r}}(Q)$ the differential graded algebra of
$\operatorname{Lie}_{\vec{r}}$-invariant forms, and let
$\Lambda^{*}_{\operatorname{\text{\sf
bas}}}(Q)\subset\Lambda^{*}_{\vec{r}}(Q)$ be the algebra of basic forms.
Denote by $\omega_{0}\in\Lambda^{*}_{\operatorname{\text{\sf bas}}}(Q)$ the
transversal Kähler form (4.2), and let
$L_{\omega_{0}}:\;\Lambda^{*}_{\operatorname{\text{\sf
bas}}}(Q){\>\longrightarrow\>}\Lambda^{*+2}_{\operatorname{\text{\sf
bas}}}(Q)$ be the multiplication map. Then the complex
$\Lambda^{*}_{\vec{r}}(Q)$ is naturally identified with
$C(L_{\omega_{0}})[-1]$, where $C(L_{\omega_{0}})$ is the cone of the morphism
$L_{\omega_{0}}:\;\Lambda^{*}_{\operatorname{\text{\sf
bas}}}(Q){\>\longrightarrow\>}\Lambda^{*+2}_{\operatorname{\text{\sf
bas}}}(Q)$.

Proof: Consider the Hattori decomposition of the differential in
$\Lambda^{*}(Q)$, $d=d_{0}+d_{1}+d_{2}$ (Subsection 4.2). By 4.2, $d_{0}$
vanishes on $\Lambda^{*}_{\vec{r}}(Q)$; 4.2 implies that
$d_{2}=L_{\omega_{0}}i_{\vec{r}}$. Clearly,
$\Lambda^{*}_{\vec{r}}(Q)=\Lambda_{\operatorname{\text{\sf
bas}}}^{*}(Q)\oplus{\vec{r}}^{\,\flat}\wedge\Lambda_{\operatorname{\text{\sf
bas}}}^{*}(Q)$. The operator $d_{2}\Lambda_{\omega_{0}}i_{\vec{r}}$ acts
trivially on $\Lambda_{\operatorname{\text{\sf bas}}}^{*}(Q)$ and maps
${\vec{r}}^{\,\flat}\wedge\alpha$ to $L_{\omega_{0}}(\alpha)$ for any
$\alpha\in\Lambda_{\operatorname{\text{\sf bas}}}^{*}(Q)$. Therefore, under
the natural identification

$\Lambda^{*}_{\vec{r}}(Q)=\Lambda_{\operatorname{\text{\sf
bas}}}^{*}(Q)\oplus{\vec{r}}^{\,\flat}\wedge\Lambda_{\operatorname{\text{\sf
bas}}}^{*}(Q)=\Lambda_{\operatorname{\text{\sf
bas}}}^{*}(Q)\oplus\Lambda_{\operatorname{\text{\sf bas}}}^{*}(Q)[-1].$

By 4.2, the differential $d_{1}+d_{2}$ in $\Lambda^{*}_{\vec{r}}(Q)$ gives the
same differential as in
$C(L_{\omega_{0}})[-1]=\Lambda_{\operatorname{\text{\sf
bas}}}^{*}(Q)\oplus\Lambda_{\operatorname{\text{\sf bas}}}^{*}(Q)[-1]$:

$d(\alpha\oplus{\vec{r}}^{\,\flat}\wedge\beta)=d\alpha+L_{\omega_{0}}(\beta)\oplus(-{\vec{r}}^{\,\flat}\wedge
d\beta)$

for any $\alpha,\beta\in\Lambda_{\operatorname{\text{\sf bas}}}^{*}(Q)$.

###  6.2 Harmonic form decomposition on Sasakian manifolds

We give a new proof of the main result on harmonic forms on compact Sasakian
manifolds (see [BG, Proposition 7.4.13], essentially based on a result by
Tachibana, [Ta]).

Theorem 6.5:  Let $Q$ be a $2n+1$-dimensional compact Sasakian manifold, $R$
the Reeb foliation, and $H^{*}_{\operatorname{\text{\sf bas}}}(Q)$ the
corresponding basic cohomology. Consider the Lefschetz
${\mathfrak{sl}}(2)$-triple
$L_{\omega_{0}},\Lambda_{\omega_{0}},H_{\omega_{0}}$ acting on the basic
cohomology (5). Then:

$H^{i}(Q)=\begin{cases}\ker
L_{\omega_{0}}{\left|{}_{{\phantom{|}\\!\\!}_{H^{i}_{\operatorname{\text{\sf
bas}}}(Q)}}\right.},\ \ \text{for}\ i\leqslant n,\\\
\dfrac{H^{i}_{\operatorname{\text{\sf
bas}}}(Q)}{\operatorname{im}L_{\omega_{0}}},\qquad\qquad\text{for}\
i>n.\end{cases}$

Proof. Step 1: The cohomology of the algebra $\Lambda^{*}_{\vec{r}}(Q)$ of
$\operatorname{Lie}_{\vec{r}}\ $-invariant forms is equal to the cohomology of
$\Lambda^{*}(Q)$. Indeed, let $G$ be the closure of the action of
$e^{t{\vec{r}}}$. Since ${\vec{r}}$ is Killing, $e^{t{\vec{r}}}$ acts on $Q$
by isometries, hence its closure $G$ is compact. Since $G$ is connected its
action on cohomology is trivial. The averaging map
$\operatorname{Av}_{G}:\;\Lambda^{*}(Q){\>\longrightarrow\>}\Lambda^{*}_{\vec{r}}(Q)$
induces an isomorphism on cohomology.

Step 2: Applying (6.1), 6.1, and taking into account the isomorphism
$H^{*}(\Lambda^{*}_{\vec{r}}(Q))=H^{*}(Q)$, we obtain the long exact sequence

$\cdots{\>\longrightarrow\>}H^{i-2}_{\operatorname{\text{\sf
bas}}}(Q)\xrightarrow{L_{\omega_{0}}}H^{i}_{\operatorname{\text{\sf
bas}}}(Q){\>\longrightarrow\>}H^{i}(Q){\>\longrightarrow\>}\\\
{\>\longrightarrow\>}H^{i-1}_{\operatorname{\text{\sf
bas}}}(Q)\xrightarrow{L_{\omega_{0}}}H^{i+1}_{\operatorname{\text{\sf
bas}}}(Q){\>\longrightarrow\>}\cdots$ (6.2)

Since the Lefschetz triple
$L_{\omega_{0}},\Lambda_{\omega_{0}},H_{\omega_{0}}$ induces an
$\mathfrak{sl}(2)$-action on $H^{i}_{\operatorname{\text{\sf bas}}}(Q)$ (5),
the map $H^{i-2}_{\operatorname{\text{\sf bas}}}(Q)\stackrel{{\scriptstyle
L_{\omega_{0}}}}{{{\>\longrightarrow\>}}}H^{i}_{\operatorname{\text{\sf
bas}}}(Q)$ is injective for $i\leqslant n$ and surjective for $i>n$.
Therefore, the long exact sequence (6.2) gives the short exact sequences

$0{\>\longrightarrow\>}H^{i-2}_{\operatorname{\text{\sf
bas}}}(Q)\stackrel{{\scriptstyle
L_{\omega_{0}}}}{{{\>\longrightarrow\>}}}H^{i}_{\operatorname{\text{\sf
bas}}}(Q){\>\longrightarrow\>}H^{i}(Q){\>\longrightarrow\>}0$

for $i\leqslant n$ and

$0{\>\longrightarrow\>}H^{i}(Q){\>\longrightarrow\>}H^{i-1}_{\operatorname{\text{\sf
bas}}}(Q)\stackrel{{\scriptstyle
L_{\omega_{0}}}}{{{\>\longrightarrow\>}}}H^{i+1}_{\operatorname{\text{\sf
bas}}}(Q){\>\longrightarrow\>}0$

for $i>n$.

Theorem 6.6:  Let $Q$ be a $2n+1$-dimensional compact Sasakian manifold, and
${\cal H}^{i}_{\operatorname{\text{\sf bas}}}(Q)$ the space of basic harmonic
forms. Let ${\vec{r}}^{\,\flat}$ be the contact form, dual to the Reeb field.
Denote by ${\cal H}^{i}$ the space of all $i$-forms $\alpha$ on $Q$ which
satisfy

for $i\leqslant n$:

$\alpha$ is basic harmonic (that is, belongs to the kernel of the basic
Laplacian) and satisfies $\Lambda_{\omega_{0}}(\alpha)=0$;

for $i>n$:

$\alpha=\beta\wedge{\vec{r}}^{\,\flat}$ where $\beta$ is basic harmonic and
satisfies $L_{\omega_{0}}(\beta)=0$.

Then all elements of ${\cal H}^{*}$ are harmonic, and, moreover, all harmonic
forms on $Q$ belong to ${\cal H}^{*}$.

Proof. Step 1: We prove that all $\gamma\in{\cal H}^{*}$ are harmonic. Let
$*_{\operatorname{\text{\sf bas}}}$ be the Hodge star operator on basic forms.
Then for any $\gamma\in\Lambda^{*}_{\operatorname{\text{\sf hor}}}(Q)$ one has
$*(\gamma)=*_{\operatorname{\text{\sf
bas}}}(\gamma)\wedge{\vec{r}}^{\,\flat}$. This implies that the two classes of
forms in 6.2 are exchanged by $*$, and it suffices to prove that all
$\alpha\in\Lambda^{i}(Q)$ which are basic harmonic and satisfy
$\Lambda_{\omega_{0}}(\alpha)=0$ for $i\leqslant n$ are harmonic. Such a form
$\alpha$ is closed by 5. Since $\alpha$ is basic hence
$\operatorname{Lie}_{\vec{r}}$-invariant, it satisfies $d_{0}^{*}(\alpha)=0$
(4.2). By (5.2), a basic form $\alpha$ is basic harmonic if and only if
$d_{1}(\alpha)=d_{1}^{*}(\alpha)=0$. Finally,
$d_{2}^{*}=\Lambda_{\omega_{0}}e_{{\vec{r}}}$ (4.2), hence
$d_{2}^{*}(\alpha)=0$. This implies
$d^{*}\alpha=d_{0}^{*}+d_{1}^{*}+d_{2}^{*}(\alpha)=0$.

Step 2: Now we prove that all harmonic forms on $Q$ belong to ${\cal H}^{*}$.
By 6.2, the dimension of $H^{i}(Q)$ is equal to the dimension of ${\cal
H}^{i}$, hence the embedding ${\cal H}^{i}{\>\longrightarrow\>}H^{i}(Q)$
constructed in Step 1 is also surjective for all indices $i$.

##  7 Vaisman manifolds

In this section we present, without proofs, the necessary background for LCK
and Vaisman manifolds. We refer to [DO] and to recent papers of ours for
details, e.g. [OV2].

###  7.1 LCK manifolds

Definition 7.1: Let $(M,I,g)$ be a Hermitian manifold of complex dimension
$n\geqslant 2$, with fundamental form $\omega$. It is called locally
conformally Kähler (LCK) if there exists a closed 1-form $\theta$ (the Lee
form) such that $d\omega=\theta\wedge\omega$. If $\theta$ is exact, $M$ is
called globally conformally Kähler (GCK). The vector field $\theta^{\sharp}$
metrically equivalent with $\theta$ is called the Lee field.

Remark 7.2: The definition is conformally invariant: if $(M,I,g,\theta)$ is
LCK, then $(M,I,e^{f}g,\theta+df)$ is LCK too.

Remark 7.3: (Equivalent definition) $(M,I,g)$ is LCK if and only if there
exists a cover $\tilde{M}$ admitting a Kähler metric $\tilde{g}$ such that the
deck group $\Gamma$ acts with holomorphic homotheties w.r.t. $\tilde{g}$.

Remark 7.4: (i) On a Kähler cover as above, the pull-back of the Lee form is
exact and the pull-back of the LCK metric is GCK.

(ii) Let $(\tilde{M},I,\tilde{g})$ be the universal cover of $M$. The
equivalent definition in 7.1 shows the existence of a homothety character
$\chi:\pi_{1}(M)\longrightarrow{\mathbb{R}}^{>0}$ defined by
$\chi(\gamma)=\frac{\gamma^{*}g}{g}$, for all $\gamma\in\pi_{1}(M)$.

Definition 7.5: The rank of $\operatorname{im}(\chi)\subset{\mathbb{R}}^{>0}$
is called the LCK rank of the LCK manifold $M$.

###  7.2 Vaisman manifolds

Definition 7.6: An LCK manifold $(M,I,g,\theta)$ is called Vaisman if the Lee
form is parallel w.r.t. the Levi-Civita connection of $g$.

Remark 7.7:  (i) The Lee form of a Vaisman metric is, in particular, co-
closed, and hence it is a Gauduchon metric. Therefore, on a compact LCK
manifold a Vaisman metric, if it exists, is unique in its conformal class (up
to constant multiplier).

(ii) Since $\theta$ is parallel, it has constant norm. The metric $g$ can be
rescaled such that $|\theta|=1$. The fundamental form $\omega$ of the Vaisman
metric with unit length Lee form satisfies the equality:

$d(I\theta)=\omega-\theta\wedge(I\theta).$ (7.1)

(iii) Let $(M,I,g,\theta)$ be a Vaisman manifold, $\theta^{\sharp}$ its Lee
field. Then $\theta^{\sharp}$ and $I\theta^{\sharp}$ are holomorphic
($\operatorname{Lie}_{\theta^{\sharp}}I=0$,
$\operatorname{Lie}_{I\theta^{\sharp}}I=0$) and Killing
($\operatorname{Lie}_{\theta^{\sharp}}g=0$,
$\operatorname{Lie}_{I\theta^{\sharp}}g=0$). Therefore, they generate a
foliation $\Sigma$ of real dimension 2 which is complex, Riemannian and
totally geodesic. In particular, $\Sigma$ is transversally Kähler (see 5).

(iv) Moreover, $\Sigma$ is canonical in the following sense: on a compact LCK
manifold, the Lee fields of all Vaisman metrics are proportional, and hence
the foliation $\Sigma$ is the same for all Vaisman metrics. Therefore,
$\Sigma$ is called the canonical foliation.

(v) Compact complex submanifolds of Vaisman manifolds are Vaisman ([Ve1]).

7.2 (iii) admits the following converse which is a powerful criterion for the
existence of a Vaisman metric in a conformal class (it was proven in [KO] for
compact LCK manifolds, but a careful analysis of the proof shows that it is
valid for non-compact manifolds too).

Theorem 7.8: Let $(M,I,g,\theta)$ be an LCK manifold equipped with a
holomorphic and conformal ${\mathbb{C}}$-action $\rho$ without fixed points,
which lifts to non-isometric homotheties on a Kähler covering $\widetilde{M}$.
Then $g$ is conformally equivalent with a Vaisman metric.

A Vaisman manifold $M$ can have any LCK rank between 1 and $b_{1}(M)$.
However, they always admit deformations to Vaisman structures with LCK rank 1:

Theorem 7.9: ([OV2]) Let $(M,I,g,\theta)$ be a compact Vaisman manifold,
$\alpha$ a harmonic 1-form such that the deformed 1-form
$\theta^{\prime}:=\theta+\alpha$ has rational cohomology class. Consider the
(1,1)-form $\omega^{\prime}:=d_{\theta^{\prime}}=\theta^{\prime}\wedge
I\theta^{\prime}-d^{c}\theta^{\prime}$ obtained as a deformation of
$\omega=\theta\wedge I\theta-d^{c}\theta$ (cf. (7.1)). Assume that $\alpha$ is
chosen sufficiently small in such a way that $\omega^{\prime}$ is positive
definite. Then the LCK metric associated $\omega^{\prime}$ is conformally
equivalent to a Vaisman metric.

Remark 7.10: It is known that a mapping torus of a compact contact manifold is
locally conformally symplectic. Correspondingly, a mapping torus of a compact
Sasakian manifold is a Vaisman manifold ([Va]). 7.2 can be used to prove the
following converse (the Structure Theorem for compact Vaisman manifolds):

Theorem 7.11: ([OV1]) Every compact Vaisman manifold is biholomorphic to
$C(S)/{\mathbb{Z}}$, where $S$ is Sasakian,
${\mathbb{Z}}=\bigg{\langle}(x,t)\mapsto(\varphi(x),qt)\bigg{\rangle}$, $q>1$,
$\varphi$ is a Sasakian automorphism of $S$, and $C(S)$ is the Sasakian cone
considered as a complex manifold. Moreover, the triple $(S,\varphi,q)$ is
unique.

Remark 7.12: Since $\theta$ is parallel, the de Rham splitting theorem implies
that a Vaisman manifold is locally the product of ${\mathbb{R}}$ with a
Riemannian manifold which can be shown to be Sasakian. The above Structure
Theorem shows that this local decomposition is canonical.

Example 7.13: (i) All linear Hopf manifolds $({\mathbb{C}}^{n}\setminus
0)/\langle A\rangle$, with $A\in\operatorname{GL}(n,{\mathbb{C}})$ semi-simple
are Vaisman. From 7.2 (v) it follows that all compact submanifolds of semi-
simple linear Hopf manifolds are Vaisman.

(ii) Vaisman compact surfaces are classified by Belgun (see [Be], [OVV]):
diagonal Hopf surfaces and elliptic surfaces.

(iii) Non semi-simple Hopf manifolds, Kato manifolds ([IOP]) and some
Oeljeklaus-Toma manifolds are LCK manifolds which cannot have Vaisman metrics.

##  8 Hodge theory on Vaisman manifolds

###  8.1 Basic cohomology of Vaisman manifolds

Let $M$ be a Vaisman manifold, $\theta^{\sharp}$ its Lie field, and
${\vec{r}}:=I(\theta^{\sharp})$. Using the local decomposition of $M$ as a
product of ${\mathbb{R}}$ and a Sasakian manifold (7.2), ${\vec{r}}$ can be
identified with the Reeb field on its Sasakian component. The manifold $M$ is
equipped with 2 remarkable foliations,
$\Sigma=\langle{\vec{r}},\theta^{\sharp}\rangle$ and ${\cal
L}=\langle\theta^{\sharp}\rangle$. Both of these foliations satisfy the
assumptions of 5 (see also 7.2 (iii)).

Consider the corresponding algebras of basic forms:

  * (i)

$\Lambda^{*}_{\operatorname{\text{\sf sas}}}(M)$ – forms which are basic with
respect to ${\cal L}$,

  * (ii)

$\Lambda^{*}_{\operatorname{\text{\sf kah}}}(M)$ – forms which are basic with
respect to $\Sigma$.

Denote by $H^{*}_{\operatorname{\text{\sf sas}}}(M)$,
$H^{*}_{\operatorname{\text{\sf kah}}}(M)$ the corresponding basic cohomology
algebras.

By 5, the algebra $H^{*}_{\operatorname{\text{\sf kah}}}(M)$ is equipped with
the Lefschetz ${\mathfrak{sl}}(2)$-action and the Hodge decomposition in the
same way as for Kähler manifolds.

The cohomology of Vaisman manifolds can be expressed non-ambiguously in terms
of $H^{*}_{\operatorname{\text{\sf kah}}}(M)$ and the Lefschetz
${\mathfrak{sl}}(2)$-action. The following theorem is the Vaisman analogue of
6.2.

Theorem 8.1:  Let $M$ be a compact Vaisman manifold, $\dim_{\mathbb{R}}M=2n$,
and $\theta$ its Lee form. Then

$\displaystyle H^{i}(M)$ $\displaystyle\cong H^{i}_{\operatorname{\text{\sf
sas}}}(M)\oplus\theta\wedge H^{i-1}_{\operatorname{\text{\sf sas}}}(M),$ (8.1)
$\displaystyle H^{i}_{\operatorname{\text{\sf sas}}}(M)$
$\displaystyle=\begin{cases}\displaystyle\ker
L_{\omega_{0}}{\left|{}_{{\phantom{|}\\!\\!}_{H^{i}_{\operatorname{\text{\sf
kah}}}(M)}}\right.}&\text{for}\ \ i\leqslant n-1,\\\\[14.45377pt]
\displaystyle\frac{H^{i}_{\operatorname{\text{\sf
kah}}}(M)}{\operatorname{im}L_{\omega_{0}}}&\text{for}\ \ i>n-1\end{cases}$

where $\\{L_{\omega_{0}},\Lambda_{\omega_{0}},H_{\omega_{0}}\\}$ is the
Lefschetz ${\mathfrak{sl}}(2)$-triple acting on
$H^{i}_{\operatorname{\text{\sf kah}}}(M)$.

Proof. Step 1: We start by proving (8.1). Let $G_{\theta^{\sharp}}$ be the
closure of the 1-parametric group $e^{t\theta^{\sharp}}$ in the group
$\operatorname{Iso}(M)$ of isometries of $M$. Since $\operatorname{Iso}(M)$ is
compact, the group $G_{\theta^{\sharp}}$ is also compact. Clearly, averaging
on $G_{\theta^{\sharp}}$ does not change the cohomology class of a form.
Therefore, the algebra of $G_{\theta^{\sharp}}$-invariant forms has the same
cohomology as $\Lambda^{*}(M)$. Consider the decomposition (4.1) associated
with the rank 1 foliation ${\cal L}$:

$\Lambda^{m}(M)=\bigoplus_{p}\Lambda^{p}_{\operatorname{\text{\sf
hor}}}(M)\otimes\Lambda^{m-p}_{\operatorname{\text{\sf vert}}}(M).$ (8.2)

The bundle $\Lambda^{1}_{\operatorname{\text{\sf vert}}}(M)={\cal L}^{*}$ is
1-dimensional and generated by $\theta$. Therefore, (8.2) gives

$\Lambda^{m}(M)=\Lambda^{m}_{\operatorname{\text{\sf
hor}}}(M)\oplus\theta\wedge\Lambda^{m-1}_{\operatorname{\text{\sf hor}}}(M).$

Denote by $\Lambda^{m}(M)^{G_{\theta^{\sharp}}}$ the
$G_{\theta^{\sharp}}$-invariant part of $\Lambda^{m}(M)$. Since $\theta$ is
$G_{\theta^{\sharp}}$-invariant, and the $G_{\theta^{\sharp}}$-invariant part
of $\Lambda^{m}_{\operatorname{\text{\sf hor}}}(M)$ is identified with
$\Lambda^{m}_{\operatorname{\text{\sf sas}}}(M)$, this gives

$\Lambda^{m}(M)^{G_{\theta^{\sharp}}}=\Lambda^{m}_{\operatorname{\text{\sf
sas}}}(M)\oplus\theta\wedge\Lambda^{m-1}_{\operatorname{\text{\sf sas}}}(M).$

Taking cohomology, we obtain (8.1).

Step 2: Now we shall prove $H^{i}_{\operatorname{\text{\sf sas}}}(M)=\ker
L_{\omega_{0}}{\left|{}_{{\phantom{|}\\!\\!}_{H^{i}_{\operatorname{\text{\sf
kah}}}(M)}}\right.}$ for $i\leqslant n-1$ and $H^{i}_{\operatorname{\text{\sf
sas}}}(M)=\dfrac{H^{i}_{\operatorname{\text{\sf
kah}}}(M)}{\operatorname{im}L_{\omega_{0}}}$ for $i>n-1$.

Let ${\vec{r}}\subset TM$ be the Reeb field defined as above, and
$G_{\vec{r}}$ the closure of the 1-parametric group $e^{t{\vec{r}}}$. Then (as
in Step 1 and in the proof of 6.2), the $G_{\vec{r}}$-invariant part of
$\Lambda^{*}_{\operatorname{\text{\sf sas}}}(M)$ is written as

$\Lambda^{i}_{\operatorname{\text{\sf
sas}}}(M)^{G_{\vec{r}}}=\Lambda^{i}_{\operatorname{\text{\sf
kah}}}(M)\oplus{\vec{r}}^{\,\flat}\wedge\Lambda^{i-1}_{\operatorname{\text{\sf
kah}}}(M)$ (8.3)

with the differential acting (see 6.1) as

$d(\alpha\oplus{\vec{r}}^{\,\flat}\wedge\beta)=d\alpha+L_{\omega_{0}}(\beta)\oplus(-{\vec{r}}^{\,\flat}\wedge
d\beta)$ (8.4)

for any $\alpha,\beta\in\Lambda^{*}_{\operatorname{\text{\sf kah}}}(M)$.

Using Cartan’s formula, we notice that $G_{\vec{r}}$ acts trivially on the
cohomology of the complex $(\Lambda^{*}_{\operatorname{\text{\sf
sas}}}(M),d)$, hence the cohomology of $(\Lambda^{*}_{\operatorname{\text{\sf
sas}}}(M)^{G_{\vec{r}}},d)$ is identified with $H^{*}_{\operatorname{\text{\sf
sas}}}(M)$. From (8.3) and (8.4), we obtain that
$\Lambda^{i}_{\operatorname{\text{\sf sas}}}(M)^{G_{\vec{r}}}$ is the cone of
the morphism of complexes
$L_{\omega_{0}}:\;\Lambda^{*}_{\operatorname{\text{\sf
kah}}}(M)[-1]{\>\longrightarrow\>}\Lambda^{*}_{\operatorname{\text{\sf
kah}}}(M)[1]$. From the long exact sequence (8.4) we obtain a long exact
sequence identical to (6.2):

$\cdots{\>\longrightarrow\>}H^{i-2}_{\operatorname{\text{\sf
kah}}}(M)\xrightarrow{L_{\omega_{0}}}H^{i}_{\operatorname{\text{\sf
kah}}}(M){\>\longrightarrow\>}H^{i}_{\operatorname{\text{\sf
sas}}}(M){\>\longrightarrow\>}\\\
{\>\longrightarrow\>}H^{i-1}_{\operatorname{\text{\sf
kah}}}(M)\xrightarrow{L_{\omega_{0}}}H^{i+1}_{\operatorname{\text{\sf
kah}}}(M){\>\longrightarrow\>}\cdots$ (8.5)

Since $L_{\omega_{0}}:\;H^{i-2}_{\operatorname{\text{\sf
kah}}}(M){\>\longrightarrow\>}H^{i}_{\operatorname{\text{\sf kah}}}(M)$ is
injective for $i\leqslant n-1$ and surjective for $i>n-1$, this long exact
sequence breaks into short exact sequences of the form

$0{\>\longrightarrow\>}H^{i-2}_{\operatorname{\text{\sf
kah}}}(M)\stackrel{{\scriptstyle
L_{\omega_{0}}}}{{{\>\longrightarrow\>}}}H^{i}_{\operatorname{\text{\sf
kah}}}(M){\>\longrightarrow\>}H^{i}_{\operatorname{\text{\sf
sas}}}(M){\>\longrightarrow\>}0$

for $i\leqslant n-1$ and

$0{\>\longrightarrow\>}H^{i}(M)_{\operatorname{\text{\sf
sas}}}{\>\longrightarrow\>}H^{i-1}_{\operatorname{\text{\sf
kah}}}(M)\stackrel{{\scriptstyle
L_{\omega_{0}}}}{{{\>\longrightarrow\>}}}H^{i+1}_{\operatorname{\text{\sf
kah}}}(M)\ {\>\longrightarrow\>}0$

for $i>n$. We finished the proof of 8.1.

###  8.2 Harmonic forms on Vaisman manifolds

It turns out that (just like it happens in the Sasakian case) the cohomology
decomposition obtained in 8.1 gives a harmonic form decomposition. Together
with the Hodge decomposition of the basic cohomology of transversally Kähler
structure this allows us to represent certain cohomology classes by forms of a
given Hodge type. This theorem was obtained in [KaS], see also [Va] and [Ts].

Recall that on a compact Vaisman manifold $(M,I,g,\theta)$ with fundamental
form $\omega$, the canonical foliation $\Sigma$ is transversally Kähler (7.2
(iii)). We denoted ${\vec{r}}$ the vector field $g$-equivalent with $I\theta$,
$\Lambda^{*}_{\operatorname{\text{\sf kah}}}(M)$ the space of basic forms with
respect to $\Sigma$, and $\Delta_{\operatorname{\text{\sf
kah}}}:\;\Lambda^{*}_{\operatorname{\text{\sf
kah}}}(M){\>\longrightarrow\>}\Lambda^{*}_{\operatorname{\text{\sf kah}}}(M)$
the transversal Laplacian (5). From 5 we obtain that the space ${\cal
H}^{i}_{\operatorname{\text{\sf kah}}}(M)$ of basic harmonic forms is equipped
with the Lefschetz ${\mathfrak{sl}}(2)$-action by the operators
$L_{\omega_{0}},\Lambda_{\omega_{0}},H_{\omega_{0}}$. The main result of this
section is:

Theorem 8.2:  Let $(M,I,g,\theta)$ be a compact Vaisman manifold of complex
dimension $n$, with fundamental form $\omega$, and canonical foliation
$\Sigma$. Denote by ${\cal H}^{i}$ the space of all basic $i$-forms
$\alpha\in\Lambda^{*}_{\operatorname{\text{\sf kah}}}(M)$ which satisfy:

for $i\leqslant n$:

$\alpha$ is basic harmonic (i.e. $\Delta_{\operatorname{\text{\sf
kah}}}(\alpha)=0$) and satisfies $\Lambda_{\omega_{0}}(\alpha)=0$;

for $i>n$:

$\alpha=\beta\wedge I\theta$ where $\beta$ is basic harmonic and satisfies
$L_{\omega_{0}}(\beta)=0$.

Then all elements of ${\cal H}^{*}\oplus\theta\wedge{\cal H}^{*}$ are harmonic
and, moreover, all harmonic forms on $M$ belong to ${\cal
H}^{*}\oplus\theta\wedge{\cal H}^{*}$.

Proof. Step 1: This statement is similar to 6.2, and the proof is essentially
the same. We start by proving that all $\gamma\in{\cal
H}^{*}\oplus\theta\wedge{\cal H}^{*}$ are harmonic. Notice that a product of a
parallel form $\rho$ and a harmonic form is again harmonic (see e.g. [Ve3,
Proposition 2.7]). This result is proven in the same way as 2.3, by
considering the Lie superalgebra generated by $d,d^{*}$, and multiplication by
$\rho$. Since $\theta$ is parallel, it suffices only to show that all elements
in ${\cal H}^{*}$ are harmonic:

${\cal H}^{*}\subset\ker\Delta.$ (8.6)

Consider the foliation ${\cal L}=\langle\theta^{\sharp}\rangle$. Since $M$ is
locally a product of a Sasakian manifold and a line (7.2), all ${\cal
L}$-basic, transversally harmonic forms are harmonic. Therefore, (8.6) would
follow if we prove

${\cal H}^{*}\subset\ker\Delta_{\operatorname{\text{\sf sas}}}.$ (8.7)

where $\Delta_{\operatorname{\text{\sf sas}}}$ is the transversal Laplacian of
the foliation ${\cal L}$.

It would suffice to prove (8.7) locally in $M$. However, locally ${\cal L}$
admits a leaf space $Q$, which is Sasakian, and we can regard elements from
${\cal H}^{*}$ as forms on $Q$ and $\Delta_{\operatorname{\text{\sf sas}}}$ as
the usual Laplacian on $\Lambda^{*}(Q)$. Then (8.7) follows from 6.2, step
1.111The statement of 6.2 is global, however, the proof of 6.2, step 1 is
local, and this statement is essentially identical to (8.7).

Step 2: We have shown that all elements of ${\cal
H}^{*}\oplus\theta\wedge{\cal H}^{*}$ are harmonic; this gives a natural
linear map

$\Psi:\;{\cal H}^{*}\oplus\theta\wedge{\cal
H}^{*}{\>\longrightarrow\>}H^{*}(M).$

To show that all harmonic form are obtained this way, it would suffice to
prove that $\Psi$ is surjective. This follows from 8.1 by dimension count.

By 8.1, the cohomology of $M$ is isomorphic to $H^{i}_{\operatorname{\text{\sf
sas}}}(M)\oplus\theta\wedge H^{i-1}_{\operatorname{\text{\sf sas}}}(M)$. All
forms in ${\cal H}^{*}$ are closed and belong to
$\Lambda^{*}_{\operatorname{\text{\sf sas}}}(M)$, which gives a map
$\Psi_{\operatorname{\text{\sf sas}}}:\;{\cal
H}^{*}{\>\longrightarrow\>}H^{i}_{\operatorname{\text{\sf sas}}}(M)$. To prove
that $\Psi$ is surjective, it remains to show that
$\Psi_{\operatorname{\text{\sf sas}}}$ is surjective.

However, dimension of $H^{i}_{\operatorname{\text{\sf sas}}}(M)$ is equal to
$\dim\ker
L_{\omega_{0}}{\left|{}_{{\phantom{|}\\!\\!}_{H^{i}_{\operatorname{\text{\sf
kah}}}(M)}}\right.}$ for $i\leqslant n-1$ and to
$\dim\dfrac{H^{i}_{\operatorname{\text{\sf
kah}}}(M)}{\operatorname{im}L_{\omega_{0}}}$ for $i>n-1$. The space ${\cal
H}^{i}$ has the same dimension by the transversal Hodge decomposition (5).
This finishes the proof of 8.2.

##  9 Supersymmetry on Sasakian manifolds

Let $Q$ be a Sasakian manifold. In this section we describe the Lie
superalgebra $\mathfrak{q}\subset\operatorname{End}(\Lambda^{*}(Q))$
reminiscent of the supersymmetry algebra of a Kähler manifold (2.3). Unlike
the Kähler supersymmetry algebra, the algebra $\mathfrak{q}$ is infinitely-
dimensional; however, it has a simple and compact description, independent of
the choice of $Q$.

When this project was started, we expected to use 9 to give a more conceptual
proof of the classical results on Hodge decomposition of the cohomology of
Sasakian and Vaisman manifolds (6.2, 8.2). However, the current version of the
proof of 8.2 is independent from 9.

Remark 9.1: Some of the relations we derive can be found in P.A.-Nagy’s
doctoral thesis, in a more general setting, for a compact Riemannian manifold
endowed with a unitary vector field (see [N]).

We first recall the notations. Let $Q$ be a Sasakian manifold, ${\vec{r}}$ its
Reeb field, and $R\subset TM$ the corresponding rank 1 distribution. Let

$\Lambda^{*}(Q):=\bigoplus\Lambda^{p,q}_{\operatorname{\text{\sf
hor}}}(Q)\times\Lambda^{m}_{\operatorname{\text{\sf vert}}}(Q)$

be the corresponding decomposition (Hodge and Hattori) of the de Rham algebra,
and $d=d_{0}+d_{1}+d_{2}$ the Hattori differentials (see Subsection 4.2).
Denote by $W$ the Weil operator acting as multiplication by $\sqrt{-1}\>(p-q)$
on $\bigoplus\Lambda^{p,q}_{\operatorname{\text{\sf
hor}}}(Q)\times\Lambda^{m}_{\operatorname{\text{\sf vert}}}(Q)$. Let
$d_{1},d_{1}^{*},d_{1}^{c}$ and $(d_{1}^{c})^{*}$ be the differentials defined
in 4.2. Let
$L_{\omega_{0}},\Lambda_{\omega_{0}}:=L_{\omega_{0}}^{*},H_{\omega_{0}}:=[L_{\omega_{0}},\Lambda_{\omega_{0}}]$
be the Lefschetz ${\mathfrak{sl}}(2)$-triple associated with the transversally
Kähler form $\omega_{0}$. For an operator $A\subset\operatorname{End}(Q)$,
commuting with $\operatorname{Lie}_{\vec{r}}$, we denote by $A(k)$ the
composition $A\circ(\operatorname{Lie}_{\vec{r}})^{k}$. Let $i_{\vec{r}}$ be
the contraction with ${\vec{r}}$ and $e_{\vec{r}}$ the dual operator (above
denoted $e_{{\vec{r}}^{\,\flat}}$).

Denote by $\mathfrak{q}$ the Lie superalgebra generated by all operators

$L_{\omega_{0}}(i),\ \Lambda_{\omega_{0}}(i),\ H_{\omega_{0}}(i),\ d_{1}(i),\
e_{\vec{r}}(i),\ i_{\vec{r}}(i),\ \operatorname{Id}(i),W(i)\ \text{for all
$i\in{\mathbb{Z}}^{\geqslant 0}$}.$

Since the vector field ${\vec{r}}$ acts by Sasakian isometries, $\mathfrak{q}$
commutes with $\operatorname{Lie}_{\vec{r}}$; in other words,
$\operatorname{Lie}_{\vec{r}}$ is central in $\mathfrak{q}$. We consider
$\mathfrak{q}$ as an ${\mathbb{R}}[t]$-module, with $t$ mapping $A(i)$ to
$A(i+1)$. Then the Lie superalgebra $\mathfrak{q}$ is a free
${\mathbb{R}}[t]$-module of rank $(6|6)$ (with 6 even and 6 odd generators)
over ${\mathbb{R}}[t]$. Its even generators are

$L_{\omega_{0}},\ \Lambda_{\omega_{0}},\ H_{\omega_{0}},\ W,\
\Delta_{1}:=\\{d_{1},d_{1}^{*}\\},\ \operatorname{Id},$

and the odd generators are

$d_{1},\ d_{1}^{*},\ d_{1}^{c},\ (d_{1}^{c})^{*},\ e_{\vec{r}},\ i_{\vec{r}}.$

Notice that $d_{0}=e_{\vec{r}}(1)$ and $d_{0}^{*}=i_{\vec{r}}(1)$ (4.2).

The main result of this section is:

Theorem 9.2: The only non-zero commutator relations in $\mathfrak{q}$ can be
written as follows.

(i)

The Lefschetz’s ${\mathfrak{sl}}(2)$-action: the even elements
$L_{\omega_{0}},\Lambda_{\omega_{0}},H_{\omega_{0}}$ satisfy the usual
${\mathfrak{sl}}(2)$-relations (2.3) and commute with

$W,\Delta_{1},\Delta_{0},d_{0},d_{0}^{*}.$

The operator $H_{\omega_{0}}$ acts as multiplication by $p-n$ on
$\Lambda^{p}_{\operatorname{\text{\sf
hor}}}(Q)\times\Lambda^{m}_{\operatorname{\text{\sf vert}}}(Q)$, where
$\dim_{\mathbb{R}}Q=2n+1$.

(ii)

The Weil operator satisfies

$[W,d]=d_{1}^{c},\ [W,d^{c}_{1}]=-d_{1},\ [W,d_{1}^{*}]=-(d_{1}^{c})^{*},\
[W,(d^{c}_{1})^{*}]=d_{1}.$

Also, $W$ commutes with the rest of the generators of $\mathfrak{q}$.

(iii)

The differentials $d_{1},d_{1}^{*},d_{1}^{c},(d_{1}^{c})^{*}$ have non-zero
square:

$\\{d_{1},d_{1}\\}=\\{d_{1}^{c},d_{1}^{c}\\}=-L_{\omega_{0}}(1),\ \
\\{d_{1}^{*},d_{1}^{*}\\}=\\{(d_{1}^{c})^{*},(d_{1}^{c})^{*}\\}=\Lambda_{\omega_{0}}(1)$

Moreover, $\\{d_{1},d_{1}^{c}\\}=\\{d_{1}^{*},(d_{1}^{c})^{*}\\}=0$.

(iv)

The usual Kodaira relations still hold:

$\begin{split}[\Lambda_{\omega_{0}},d_{1}]&=(d_{1}^{c})^{*},\ \ \ \ \
[L_{\omega_{0}},d_{1}^{*}]=-d_{1}^{c},\\\
[\Lambda_{\omega_{0}},d_{1}^{c}]&=-d_{1}^{*},\ \ \ \ \
[L_{\omega_{0}},(d^{c}_{1})^{*}]=d_{1}.\end{split}$ (9.1)

(v)

Unlike it happens in the Kähler case, the differentials $d_{1}^{*},d_{1}^{c}$,
etc. do not (super-)commute:

$\\{d_{1}^{*},d_{1}^{c}\\}=\\{d_{1},(d_{1}^{c})^{*}\\}=-\frac{1}{2}H_{\omega_{0}}(1).$
(9.2)

(vi)

The only non-zero commutator between
$e_{\vec{r}},i_{\vec{r}},\operatorname{Id}$ is
$\\{e_{\vec{r}},i_{\vec{r}}\\}=\operatorname{Id}$. These elements commute with
the rest of $\mathfrak{q}$.

(vii)

The Laplacian $\Delta_{1}:=\\{d_{1},d_{1}^{*}\\}$ satisfies
$\Delta_{1}=\\{d_{1}^{*},(d_{1}^{c})^{*}\\}$ and commutes with all even
generators in $\mathfrak{q}$ and with $e_{\vec{r}},i_{\vec{r}}$. Its
commutators with the other 4 odd generators are expressed as follows

$\begin{split}\\{d_{1},\Delta_{1}\\}&=-\frac{1}{2}d_{1}^{c}(1),\ \ \ \ \ \ \ \
\ \\{d_{1}^{c},\Delta_{1}\\}=\frac{1}{2}d_{1}(1),\\\
\\{d_{1}^{*},\Delta_{1}\\}&=-\frac{1}{2}(d_{1}^{c})^{*}(1),\ \
\\{(d_{1}^{c})^{*},\Delta_{1}\\}=\frac{1}{2}(d_{1})^{*}(1).\end{split}$ (9.3)

Proof of 9 (i): The ${\mathfrak{sl}}(2)$-relations and the expression for
$H_{\omega_{0}}$ are proven in Subsection 2.3; the proof in the Sasakian case
is literally the same. Also, from the definition it is clear that
$L_{\omega_{0}},\Lambda_{\omega_{0}},H_{\omega_{0}}$ commute with $W$,
$i_{\vec{r}}$ and $e_{\vec{r}}$. Since $d_{0}=e_{\vec{r}}(1)$ and
$d_{0}^{*}=i_{\vec{r}}(1)$ (4.2), these operators commute with the
$\mathfrak{sl}(2)$-action and satisfy
$\\{d_{0},d_{0}^{*}\\}=\operatorname{Lie}^{2}_{\vec{r}}$. We postpone the
commutator relation for $\Delta_{1}$ until we proved 9 (iv).

Proof of 9 (ii): Same as 2.3, part 2.

Proof of 9 (iii): Start from $\\{d_{1},d_{1}\\}=-L_{\omega_{0}}(1)$.

Since $d^{2}=0$, one has $d_{0}^{2}=d_{2}^{2}=0$ and
$\\{d_{0},d_{2}\\}=-\\{d_{1},d_{1}\\}$. However, $d_{0}=e_{{\vec{r}}}(1)$ and
$d_{2}=L_{\omega_{0}}i_{\vec{r}}$, hence
$\\{d_{0},d_{2}\\}=\\{e_{{\vec{r}}},i_{\vec{r}}\\}L_{\omega_{0}}(1)=L_{\omega_{0}}(1)$.
The squares of the rest of the differentials $d_{1}^{c}$, etc., are obtained
by duality and complex conjugation.

To show that $\\{d_{1},d_{1}^{c}\\}=0$, we use $d_{1}^{c}=[W,d_{1}]$. By 2.2,

$\\{d_{1},d_{1}^{c}\\}=\\{d_{1},\\{d_{1},W\\}\\}=\frac{1}{2}\\{\\{d_{1},d_{1}\\},W\\}.$

Since $\\{d_{1},d_{1}\\}=-L_{\omega_{0}}(1)$ and $W$ commutes with
$L_{\omega_{0}}$, this gives $\\{d_{1},d_{1}^{c}\\}=0$. The equation
$\\{d_{1}^{*},(d_{1}^{c})^{*}\\}=0$ is obtained by duality. We finished 9
(iii).

Proof of 9 (iv), the Kähler-Kodaira relations: Again, it suffices to prove
$[L_{\omega_{0}},d_{1}^{*}]=-d_{1}^{c}$, the rest is obtained by duality and
complex conjugation. We prove it applying the same argument as used in 2.3. As
an operator on $\Lambda^{*}(Q)$, the commutator $[L_{\omega_{0}},d_{1}^{*}]$
has first order, because $L_{\omega_{0}}$ is zero order, and $d_{1}^{*}$ is
order 2 (2.2).

The differential operators $[L_{\omega_{0}},d_{1}^{*}]$ and $-d_{1}^{c}$ are
equal on functions for the same reasons as in 2.3.

Clearly, $d_{1}^{c}(C^{\infty}(Q))$ generates
$\Lambda^{1}_{\operatorname{\text{\sf hor}}}(Q)$. Therefore, to prove
$[L_{\omega_{0}},d_{1}^{*}]=-d_{1}^{c}$ on
$\Lambda^{1}_{\operatorname{\text{\sf hor}}}(Q)$, we need only to show that

$([L_{\omega_{0}},d_{1}^{*}])^{2}=(-d_{1}^{c})^{2}.$ (9.4)

This is implied by graded Jacobi identity, applied as follows. First, we
notice that $[\Lambda_{\omega_{0}},d_{1}^{*}]=0$, because
$d_{1}(\omega_{0})=0$. Therefore, the $\mathfrak{sl}(2)$-representation
generated by the Lefschetz triple $\langle
L_{\omega_{0}},\Lambda_{\omega_{0}},H_{\omega_{0}}\rangle$ from $d_{1}^{*}$
has weight 1, and $[L_{\omega_{0}},[L_{\omega_{0}},d_{1}^{*}]]=0$. Applying
the graded Jacobi identity, and using
$[L_{\omega_{0}},[L_{\omega_{0}},d_{1}^{*}]]=0$, we obtain

$\begin{split}\\{[L_{\omega_{0}},d_{1}^{*}],[L_{\omega_{0}},d_{1}^{*}]\\}&=[L_{\omega_{0}},\\{d_{1}^{*},[L_{\omega_{0}},d_{1}^{*}]\\}]\\\\[7.22743pt]
&=\frac{1}{2}[L_{\omega_{0}}(1),[L_{\omega_{0}}(1),\\{d_{1}^{*},d_{1}^{*}\\}]]\quad\text{by
\ref{_d^2neq0_super_Claim_}}\\\
&=-\frac{1}{2}[L_{\omega_{0}}(1),[L_{\omega_{0}}(1),\Lambda_{\omega_{0}}]]\quad\text{by
\ref{_susy_Sasa_Theorem_} (iii)}\\\ &=-L_{\omega_{0}}(1).\end{split}$

However, $(-d_{1}^{c})^{2}=-L_{\omega_{0}}(1)$ by 9 (iii), which proves (9.4).

This implies that $[L_{\omega_{0}},d_{1}^{*}]=-d_{1}^{c}$ on
$\Lambda^{1}_{\operatorname{\text{\sf hor}}}(Q)$ . The space $\Lambda^{1}(Q)$
is generated over $C^{\infty}(Q)$ by $\Lambda^{1}_{\operatorname{\text{\sf
hor}}}(Q)$ and $V=\langle{\vec{r}}^{\,\flat}\rangle$. Applying 2.2, we obtain
that $[L_{\omega_{0}},d_{1}^{*}]=-d_{1}^{c}$ if
$[L_{\omega_{0}},d_{1}^{*}]({\vec{r}}^{\,\flat})=-d_{1}^{c}({\vec{r}}^{\,\flat})$.
However, $d({\vec{r}}^{\,\flat})=\omega_{0}$, hence
$d_{1}^{c}({\vec{r}}^{\,\flat})=0$. Then,
$[L_{\omega_{0}},d_{1}^{*}]({\vec{r}}^{\,\flat})=d_{1}^{*}({\vec{r}}^{\,\flat}\wedge\omega_{0})$.
It is not hard to see that
$*({\vec{r}}^{\,\flat}\wedge\omega_{0})=\frac{1}{(n-1)!}\omega_{0}^{n-1}$,
where $\dim_{\mathbb{R}}Q=2n+1.$ Therefore,

$d^{*}_{1}({\vec{r}}^{\,\flat}\wedge\omega_{0})=\pm*d_{1}*(\omega_{0}\wedge{\vec{r}}^{\,\flat})=\pm*d_{1}\frac{1}{(n-1)!}\omega_{0}^{n-1}=0$

because $d\omega_{0}^{n-1}=0$. We proved that
$[L_{\omega_{0}},d_{1}^{*}]({\vec{r}}^{\,\flat})=-d_{1}^{c}({\vec{r}}^{\,\flat})$
and finished the proof of the Kähler-Kodaira relations.

Proof of 9 (v), the commutators of $d_{1}^{*},d^{c}$: The commutator
$\\{d_{1}^{*},d^{c}\\}$ is obtained from the Kähler-Kodaira relations. Indeed,
$\\{d_{1}^{*},d^{c}\\}=\\{d_{1}^{*},\\{d_{1}^{*},L_{\omega_{0}}\\}\\}$. Then
2.2 gives

$\\{d_{1}^{*},\\{d_{1}^{*},L_{\omega_{0}}\\}=\frac{1}{2}\\{\\{d_{1}^{*},d_{1}^{*}\\},L_{\omega_{0}}\\}=\frac{1}{2}[\Lambda_{\omega_{0}}(1),L_{\omega_{0}}]=-\frac{1}{2}H_{\omega_{0}}.$

The relation $\\{d_{1},(d_{1}^{c})^{*}\\}=-\frac{1}{2}H_{\omega_{0}}(1)$ is
dual to $\\{d_{1}^{*},d_{1}^{c}\\}=-\frac{1}{2}H_{\omega_{0}}(1)$.

Proof of 9 (vi), the commutators of $e_{\vec{r}}$ and $i_{\vec{r}}$: The
equation $\\{e_{\vec{r}},i_{\vec{r}}\\}=1$ is standard. Vanishing of the
commutators between $e_{\vec{r}}$, $i_{\vec{r}}$ and
$L_{\omega_{0}},\Lambda_{\omega_{0}},H_{\omega_{0}},W$ is standard linear
algebra. The only commutators for which we have to prove the vanishing is
between $e_{\vec{r}}$, $i_{\vec{r}}$ and $d_{1},d_{1}^{*},d_{1}^{c}$,
$(d_{1}^{c})^{*}$. Using duality and complex conjugation, we reduce the
vanishing of these commutators to only two of them:
$\\{e_{\vec{r}},d_{1}\\}=0$ and $\\{e_{\vec{r}},d_{1}^{*}\\}=0$. As $d^{2}=0$
and $\\{d_{0},d_{1}\\}$ is the grading 1 part of $d^{2}$, one has
$\\{d_{0},d_{1}\\}=0$. Since $d_{0}=e_{\vec{r}}(1)$, this also implies
$\\{e_{\vec{r}},d_{1}\\}=0$. Twisting with $I$, we obtain
$\\{e_{\vec{r}},d_{1}^{c}\\}=0$. Applying the graded Jacobi identity to
$\\{e_{\vec{r}},d_{1}^{*}\\}=-\\{e_{\vec{r}},\\{\Lambda_{\omega_{0}},d_{1}^{c}\\}\\}$
(9 (iv)) and using $\\{e_{\vec{r}},L_{\omega_{0}}\\}=0$, we obtain

$\\{e_{\vec{r}},d_{1}^{*}\\}=-\\{e_{\vec{r}},\\{\Lambda_{\omega_{0}},d_{1}^{c}\\}\\}=\\{\\{e_{\vec{r}},\Lambda_{\omega_{0}}\\},d_{1}^{c}\\}+\\{\Lambda_{\omega_{0}},\\{e_{\vec{r}},d_{1}^{c}\\}\\}=0.$

This finishes the proof of 9 (vi).

Proof of 9 (vii). The equation

$\\{d_{1},d_{1}^{*}\\}=\\{(d_{1}^{c}),(d_{1}^{c})^{*}\\}$

follows from $\\{d_{1},d_{1}^{c}\\}=0$ (9 (iii)) because

$0=\\{\Lambda,\\{d_{1},d_{1}^{c}\\}\\}=\\{\\{\Lambda,d_{1}\\},d_{1}^{c}\\}+\\{d_{1},\\{\Lambda,d_{1}^{c}\\}\\}=\\{(d_{1}^{c})^{*},d_{1}^{c}\\}-\\{d_{1},d_{1}^{*}\\}.$

This implies, in particular, that $[W,\\{d_{1},d_{1}^{*}\\}]=0$. The
commutators between the Lefschetz operators and $\Delta_{1}$ follow from the
Kähler-Kodaira relations:

$\\{L_{\omega_{0}},\\{d_{1},d_{1}^{*}\\}\\}=\\{\\{L_{\omega_{0}},d_{1}\\},d_{1}^{*}\\}+\\{d_{1},\\{L_{\omega_{0}},d_{1}^{*}\\}\\}=-\\{d_{1},d_{1}^{c}\\}=0.$

We proved that $\Delta_{1}$ commutes with the even part of $\mathfrak{q}$. By
duality and complex conjugation, to prove (9.3) it would suffice to prove only
one of these relations, say,
$\\{d_{1},\Delta_{1}\\}=-\frac{1}{2}d_{1}^{c}(1)$. This equation follows from
(9.1), (9.2) (i.e. (iv) and (v) of this theorem) and 2.2:

$\\{d_{1},\\{d_{1},d_{1}^{*}\\}\\}=\frac{1}{2}\\{\\{d_{1},d_{1}\\},d_{1}^{*}\\}=-\frac{1}{2}\\{L_{\omega_{0}}(1),d_{1}^{*}\\}=-\frac{1}{2}(d_{1}^{c})^{*}.$

We finished the proof of 9.

Acknowledgment: L.O. thanks IMPA (Rio de Janeiro) and HSE (Moscow) for
financial support and excellent research environment during the preparation of
this paper. Both authors thank P.-A. Nagy for useful discussions.

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Liviu Ornea
University of Bucharest, Faculty of Mathematics,
14 Academiei str., 70109 Bucharest, Romania, and:
Institute of Mathematics ”Simion Stoilow” of the Romanian Academy,
21, Calea Grivitei Str. 010702-Bucharest, Romania
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
also:
Laboratory of Algebraic Geometry,
Faculty of Mathematics, National Research University Higher School of
Economics, 7 Vavilova Str. Moscow, Russia
<EMAIL_ADDRESS><EMAIL_ADDRESS>