# Homotopy BV-algebras in Hermitian geometry

Joana Cirici  Departament de Matemàtiques i Informàtica, Universitat de
Barcelona
Gran Via 585
08007 Barcelona, Spain<EMAIL_ADDRESS>and  Scott O. Wilson  Department of
Mathematics, Queens College, City University of New York, 65-30 Kissena Blvd.,
Flushing, NY 11367<EMAIL_ADDRESS>
###### Abstract.

We show that the de Rham complex of any almost Hermitian manifold carries a
natural commutative $\mathrm{BV}_{\infty}$-algebra structure satisfying the
degeneration property. In the almost Kähler case, this recovers Koszul’s BV-
algebra, defined for any Poisson manifold. As a consequence, both the
Dolbeault and the de Rham cohomologies of any compact Hermitian manifold are
canonically endowed with homotopy hypercommutative algebra structures, also
known as formal homotopy Frobenius manifolds. Similar results are developed
for (almost) symplectic manifolds with Lagrangian subbundles.

J. Cirici acknowledges the Serra Húnter Program, the Spanish State Research
Agency (EUR2023-143450, PID2020-117971GB-C22 and CEX2020-001084-M), the French
National Research Agency (ANR-20-CE40-0016 HighAGT) and Govern de Catalunya
(2021-SGR-00697). S. O. Wilson acknowledges support provided by the Simons
Foundation Award program for Mathematicians (#963783) and a PSC-CUNY Award,
jointly funded by The Professional Staff Congress and The City University of
New York (TRADB # 66358-00 54).

##  1\. Introduction

The notion of Batalin–Vilkovisky algebra plays an important role in geometry,
topology and mathematical physics. Initially developed in the context of
quantum field theory as a tool to compute path integrals in the presence of
symmetry, the BV formalism was subsequently translated to the context of
supermanifolds equipped with symplectic or Poisson structures. In algebraic
topology, BV-algebras have proven to be extremely useful, with applications to
string topology, the study of cohomological operations, and to deformation
quantization, amongst others.

Motivated by the mirror symmetry program, Barannikov and Kontsevich [BK98]
gave a recipe which, out of a BV algebra satisfying a certain degeneration
property, one obtains a hypercommutative algebra structure in cohomology.
Hypercommutative algebras are governed by the homology operad of the Deligne-
Mumford-Knudsen compactification of the moduli space of genus zero surfaces
with marked points, and lead to formal Frobenious manifold structures, which
in turn are related to quantum cohomology. The approach via deformation theory
of Barannikov and Kontsevich has been recently refined and generalized by
means of operadic machinnery [KMS13], [DCV13], [DSV15]. In its higher
homotopical generalization, any $\mathrm{BV}_{\infty}$-algebra satisfying the
so-called degeneration property gives rise to a homotopy hypercommutative
algebra structure.

Various geometries are known to encode BV-algebra structures satisfying the
degeneration property: Koszul [Kos85] showed that the de Rham algebra of any
Poisson manifold carries a BV-operator, defined using the exterior
differential and the contraction operator by the Poisson bi-vector field. This
in particular applies to symplectic manifolds. The degeneration property was
noticed by Dotsenko, Shadrin and Vallette [DSV15], who also extended the
theory to Jacobi manifolds and their associated commutative
$\mathrm{BV}_{\infty}$-algebras. Similarly, Braun and Lazarev [BL13] prove
that generalized Poisson structures (given by multivector fields) give rise to
commutative $\mathrm{BV}_{\infty}$-algebras satisfying degeneration. Here we
enlarge this list to including Hermitian, and almost Hermitian manifolds.

A useful result that will guide all our geometric examples is the fact that,
given a commutative dg-algebra together with an operator $\Lambda$ of order
$\leq 2$, one always obtains a commutative $\mathrm{BV}_{\infty}$-algebra by
conjugating the differential with $\exp(\Lambda\xi)$, where $\xi$ is a formal
parameter (see Proposition 2.9). The underlying multicomplex satisfies the
degeneration property, by construction. The $\mathrm{BV}_{\infty}$-operators
are in this case given by

$\Delta_{k}={1\over
k!}\underbrace{[\Lambda,[\Lambda,\cdots[\Lambda}_{k},d]]].$

For Hermitian manifolds, the main observation is that a suitable
generalization of the Kähler identities, first due to Demailly in the
Hermitian case [Dem86], and derived independently by Tomassini and Wang [TW18]
and Fernandez and Hosmer in the almost Hermitian case [FH22], allows for an
explicit description of the commutative $\mathrm{BV}_{\infty}$-algebra
obtained after conjugation by $\exp(\Lambda\xi)$, where $\Lambda$ is the
formal adjoint to the Lefschetz operator. In short, for any Hermitian
manifold, the tuple

$(\mathcal{A}_{\mathbb{C}},\wedge,\,\Delta_{0}={\bar{\partial}}\,,\,\Delta_{1}=-i(\partial^{*}+\tau^{*})\,,\,\Delta_{2}=i\lambda^{*}\,)$

is a commutative $\mathrm{BV}_{\infty}$-algebra extending the Dolbeault
algebra (see Theorem 3.1). Here $(\mathcal{A}_{\mathbb{C}},\wedge)$ is the
complex de Rham algebra of the manifold, $\Delta_{2}$ is the adjoint of
wedging with the $3$-form $\lambda=\partial\omega$, where $\omega$ is the
fundamental form associated to the Hermitian metric, and
$\tau:=[\Lambda,\lambda]$ is the torsion operator. Note that both $\tau$ and
$\lambda$ vanish in the Kähler case.

Similarly, there is a real de Rham commutative $\mathrm{BV}_{\infty}$-algebra
for any almost Hermitian manifold,

$\left(\mathcal{A},\wedge,d,\,\Delta_{1}=-(d_{c}^{*}+T_{c}^{*})\,,\,\Delta_{2}=d\omega_{c}^{*}\,\right)$

where $d_{c}^{*}:=-\star J^{-1}dJ\star$ is the formal adjoint of
$d_{c}:=J^{-1}dJ$, $d\omega_{c}^{*}$ is the adjoint of
$d\omega_{c}=J^{-1}d\omega J$, and $T_{c}^{*}$ is the adjoint of
$T_{c}:=J^{-1}[\Lambda,d\omega]J$. This generalizes Koszul’s
$\mathrm{BV}$-algebra on the de Rham forms of an almost Kähler (and hence
symplectic) manifold, since in that case $d\omega=0$ and
$T:=[\Lambda,d\omega]=0$ (see Theorem 4.1). Using canoncial deformation
retracts given by Hodge theory one obtains, in the compact case, canonical
hypercommutative algebra structures that are non-trivial in general.

By similar methods, an additional geometric set-up we consider is that of
(almost) symplectic manifolds equipped with a Lagrangian subbundle. Indeed,
there is real-de Rham version, as well as a “Dolbeault-type” version in the
case that the Lagrangian subbundle is integrable (Theorem 4.7). Such
Lagrangian-type structures have garnered significant attention recently in
mathematical considerations of Type II-A data [LTY15], [HK24], [FPPZ21]. It is
natural to wonder if the commutative $\mathrm{BV}_{\infty}$-algebras obtained
here, and its induced homotopy hypercommutative algebras, play a role in a
theory of mirror symmetry, beyond the Kähler realm.

Let us briefly explain the structure of this paper. Section 2 includes
algebraic preliminaries on $\mathrm{BV}$ and hypercommutative algebras.
Hermitian manifolds are treated in Section 3. We then treat the almost
Hermitian setting in Section 4, where we also study (almost) symplectic
manifolds with integrable polarizations. Lastly, Section 5 is devoted to
examples, using Chevalley-Eilenberg algebras in nilmanifold theory. We have
tried to make this paper accessible to both geometers and homotopy theorists,
keeping the operadic machinery to a minimum and recalling the main geometric
structures needed, in each section. In this sense, we intend the last section
on examples to be quite pedagogical.

### Acknowledgments

The authors would like to thank Vladimir Dotsenko, Geoffroy Horel, Fernando
Muro and Bruno Vallette for many useful discussions.

##  2\. Algebraic preliminaries

In this purely algebraic section, we collect the main results on
$\mathrm{BV}$-algebras and their relation to hypercommutative algebras needed
later. Most of these appear in or are inspired by the works of Khoroshkin,
Shadrin and Markarian [KMS13] and Dotsenko, Shadrin and Vallette [DSV15]. We
will use cohomological grading. For the reader that is mainly interested in
the applications to complex and symplectic geometry, yielding (homotopy)
$\mathrm{BV}$-algebras, the more operadic methods of this section might be
skipped until it is clear they are necessary, in particular for development of
homotopy hypercommutative algebras.

###  2.1. Multicomplexes and the gauge equation

We review how any multicomplex satisfying the degeneration property determines
an explicit solution to the gauge equation in terms of the data given by any
deformation retract.

######  Definition 2.1.

A _multicomplex_ $(A,\Delta_{0}=d,\Delta_{1},\Delta_{2},\ldots)$ is a
$\mathbb{Z}$-graded vector space $A$ with a family of linear operators
$\Delta_{i}$ of degree $1-2i$ satisfying

$\sum_{i=0}^{n}\Delta_{i}\Delta_{n-i}=0\quad\text{for all }\quad n\geq 0.$

In particular, $(A,d)$ is a cochain complex, and we denote its cohomology by
$H(A)$. Any multicomplex with $\Delta_{i}=0$ for $i\geq 2$, is a double
complex with $|d|=+1$ and $|\Delta_{1}|=-1$. For a multicomplex of the form
$(A,\Delta_{0}=d,\Delta_{1},\Delta_{2},\ldots,\Delta_{n})$, the defining
equation is equivalent to

$(d+\Delta_{1}+\cdots+\Delta_{n})^{2}=0.$

Multicomplex structures may be transferred to cohomology using deformation
retracts as follows:

######  Definition 2.2.

A deformation retract of a cochain complex $(A,d)$ is given by a diagram

${(A,d)}$${(H(A),0),}$$\scriptstyle{\rho}$$\scriptstyle{\iota}$$\scriptstyle{h}$

where $\iota$ and $\rho$ are morphisms of complexes such that $\rho\iota=1$
and $dh+hd=\iota\rho-1$.

Given a multicomplex $(A,\Delta_{0}=d,\Delta_{1},\Delta_{2},\ldots)$, a choice
of a deformation retract $(\iota,\rho,h)$ for $(A,d)$ defines a multicomplex
structure
$(H(A),\Delta_{0}^{\prime}=0,\Delta_{1}^{\prime},\Delta_{2}^{\prime},\ldots)$
on cohomology, where

$\Delta_{n}^{\prime}:=\sum_{\begin{subarray}{c}j_{1}+\cdots+j_{k}=n\\\
j_{i}\geq
1\end{subarray}}\rho\Delta_{j_{1}}(h\Delta_{j_{2}})\cdots(h\Delta_{j_{k}})\iota\text{
for all }n\geq 1.$

An important special class of multicomplexes arises when imposing the so-
called degeneration property, for which $\Delta_{i}^{\prime}=0$. This was
introduced in [Ter08], [KKP08] for $\mathrm{BV}$-algebras but, in fact, the
degeneration property only affects the underlying double complex. A
generalization to multicomplexes appears in [BL13] and [DSV15]. We refer to
the latter reference for the various equivalences in the definition below.

######  Definition-Proposition 2.3.

A multicomplex $(A,\Delta_{0}=d,\Delta_{1},\Delta_{2},\ldots)$ is said to have
the degeneration property if the following equivalent conditions are
satisfied:

  1. (1)

There exists a solution

$\varphi(\xi)=\sum_{i\geq 1}\varphi_{i}\xi^{i}\in\mathrm{End}(A)[[\xi]]$

to the gauge equation

$e^{\varphi(\xi)}de^{-\varphi(\xi)}=d+\Delta_{1}\xi+\Delta_{2}\xi^{2}+\cdots.$

  2. (2)

There is a deformation retract $(\iota,\rho,h)$ for $(A,d)$ such that
$\Delta_{i}^{\prime}=0$ on $H(A)$, for all $i\geq 1$.

  3. (3)

Any deformation retract $(\iota,\rho,h)$ for $(A,d)$ satisfies
$\Delta_{i}^{\prime}=0$ on $H(A)$, for all $i\geq 1$.

  4. (4)

The spectral sequence associated to the multicomplex degenerates at the first
stage.

An expression for a solution $\varphi(\xi)$ to the gauge equation in terms of
a deformation retract $(\rho,\iota,h)$ is given in [DSV15] for the case of
bicomplexes. Following the same strategy, we give below a formula in the case
of multicomplexes. In order to not unnerve the reader, we point out that only
the first-order term’s explicit formula,

$\varphi_{1}=h\Delta_{1}-\iota\rho\Delta_{1}h,$

is needed in the remainder of the paper, when we will compute examples of
hypercommutative operations on nilmanifolds.

######  Proposition 2.4.

Let $(A,d,\Delta_{1},\Delta_{2},\ldots)$ be a multicomplex satisfying the
degeneration property and let $(\iota,\rho,h)$ be a deformation retract. A
solution to the gauge equation is given by

$\varphi(\xi)=-\log\left(1-\sum_{n\geq
1}h\Delta_{n}\xi^{n}+\sum_{\begin{subarray}{c}n\geq 1\\\
j_{1}+\cdots+j_{k}=n\\\ j_{i}\geq
1\end{subarray}}(\iota\rho\Delta_{j_{1}}h\Delta_{j_{2}}h\cdots\Delta_{j_{k}}h)\xi^{n}\right).$

###### Proof.

A solution to the gauge equation of the form

$\varphi(\xi)=-\log(1+\sum_{n\geq 1}f_{n}\xi^{n})$

is equivalent to a collection of maps $f=\\{f_{n}:A\to A\\}$ for $n\geq 1$
satisfying

$[d,f_{n}]=\sum_{i+j=n;i,j>0}f_{i}\Delta_{j}.$

In turn, this is equivalent to having an $\infty$-isotopy of multicomplexes
$f:(A,d,\Delta_{i})\rightsquigarrow(A,d,0)$ (so $f$ is an $\infty$-morphism
with $f_{0}=1$). By homotopy transfer theory of multicomplexes (see for
instance Proposition 1.4 of [DSV15]), the deformation retract $(\iota,\rho,h)$
gives an $\infty$-morphism
$\rho_{\infty}:(A,d,\Delta_{i})\rightsquigarrow(H(A),0,0)$ by letting

$\rho_{0}=\rho\text{ and
}\rho_{n}=\sum_{\begin{subarray}{c}j_{1}+\cdots+j_{k}=n\\\ j_{i}\geq
1\end{subarray}}\rho\Delta_{j_{1}}h\Delta_{j_{2}}h\cdots\Delta_{j_{k}}h\text{
for }n\geq 1.$

Also, we have an $\infty$-quasi-isomorphism
$q_{\infty}:(A,d,\Delta_{i})\rightsquigarrow(A,d,0)$ given by

$q_{0}=1-\iota\rho\text{ and }q_{n}=-h\Delta_{n}\text{ for }n\geq 1.$

We then obtain the required $\infty$-isotopy by letting
$f=\iota\rho_{\infty}+q_{\infty}$. ∎

###  2.2. Batalin-Vilkovisky and hypercommutative algebras

We first recall the definition of the algebraic order of linear operators on a
graded commutative unital algebra $(A,\wedge)$, due to Grothendieck [Gro67].
Let $\mathcal{D}_{0}^{k}$ be the space of $A$-linear endomorphisms of $A$ of
degree $k$, which is isomorphic to $A^{k}$ as the space of (left)
multiplication operators, $a\mapsto L_{a}$. Inductively, the degree $k$
operators of algebraic order $\leq r$, denoted $\mathcal{D}_{r}^{k}$, is the
space of linear homomorphisms $f$ of degree $k$ such that
$[L_{a},f]\in\mathcal{D}_{r-1}^{k+j}$ for all $L_{a}\in\mathcal{D}_{0}^{j}$.
The bracket always indicates the graded-commutator,

$[f_{1},f_{2}]=f_{1}f_{2}-(-1)^{|f_{1}|\cdot|f_{2}|}f_{2}f_{1}.$

Using the fact that $(\mathrm{End}(A),\circ,[\,,\,])$ is a graded Poisson
algebra, i.e. the bracket satisfies the Jacobi identity and $[f_{1},-]$ is a
derivation of composition, it is immediate to show that

$\mathcal{D}_{r}^{k}\cdot\mathcal{D}_{s}^{\ell}\subset\mathcal{D}_{r+s}^{k+\ell},\quad\quad\quad[\mathcal{D}_{r}^{k},\mathcal{D}_{s}^{\ell}]\subset\mathcal{D}_{r+s-1}^{k+\ell}.$

######  Definition 2.5.

A commutative $\mathrm{BV}_{\infty}$-algebra is a tuple
$(A,\wedge,d,\Delta_{1},\Delta_{2},\ldots)$ such that $(A,\wedge,d)$ is a
commutative dg-algebra, $(A,d,\Delta_{1},\Delta_{2},\ldots)$ is a
multicomplex, and each $\Delta_{n}$ has order $\leq n+1$.

Note that (differential graded) $\mathrm{BV}$-algebras are precisely those
commutative $\mathrm{BV}_{\infty}$-algebras such that $\Delta_{n}=0$ for all
$n\geq 2$.

Let $c\mathrm{BV}_{\infty}$ denote the operad governing commutative
$\mathrm{BV}_{\infty}$-algebras. The obvious map of operads
$c\mathrm{BV}_{\infty}\to\mathrm{BV}$ is a quasi-isomorphism. One may view
$c\mathrm{BV}_{\infty}$ as a partial cofibrant resolution of $\mathrm{BV}$,
where only the operation $\Delta$ is resolved (but not the commutative
structure). Algebras over a cofibrant (resp. minimal) resolution of
$\mathrm{BV}$ are studied in [GTV12] and [DCV13], respectively, and are called
$\mathrm{BV}_{\infty}$ (resp. skeletal $\mathrm{BV}_{\infty}$) algebras. For
our purposes, it will suffice to work with the more restrictive notion of
commutative $\mathrm{BV}_{\infty}$-algebra introduced above.

The last ingredient required to state the main result of this section are
hypercommutative algebras. These are defined as algebras over the operad

$\mathrm{Hycom}(n):=H_{*}(\overline{\mathcal{M}}_{0,n+1}),$

the Deligne-Mumford-Knudsen moduli space of stable genus 0 curves. The
following algebraic description appears in [Get95].

######  Definition 2.6.

A hypercommutative algebra is a cochain complex $(A,d)$ with a sequence of
graded symmetric $n$-ary operations

$m_{n}:A^{\otimes n}\to A$

of degree $2(2-n)$, compatible with the differential and satisfying the
following generalized associativity condition: for all $n\geq 0$ and
$a,b,c,x_{j}\in A$,

$\sum_{S_{1}\cup S_{2}=\\{1,\cdots,n\\}}\pm
m_{*}(m_{*}(a,b,x_{S_{1}}),c,x_{S_{2}})=\sum_{S_{1}\cup
S_{2}=\\{1,\cdots,n\\}}\pm m_{*}(a,m_{*}(b,c,x_{S_{1}}),x_{S_{2}}),$

where $\pm$ is the Koszul sign rule and $x_{S}$ denotes
$x_{s_{1}},\cdots,x_{s_{m}}$ for $S=\\{s_{1},\cdots,s_{m}\\}$.

The operad $\mathrm{Hycom}$ is Koszul, with Koszul dual cooperad
$H^{*+1}({\mathcal{M}}_{0,*+1})$, so that the cobar construction yields a
cofibrant resolution

$\Omega H^{*+1}({\mathcal{M}}_{0,*+1})\to
H_{*}(\overline{\mathcal{M}}_{0,*+1}).$

By definition then, a homotopy hypercommutative algebra is an algebra over the
operad

$\mathrm{Hycom}_{\infty}:=\Omega H^{*+1}({\mathcal{M}}_{0,*+1}).$

The following result can be found in Proposition 10 of [DSV13] (see also
Proposition 4.4 of [DSV15]). The proofs therein use homotopy transfer
techniques. We give below an alternative proof which follows from a quasi-
isomorphism of operads $\mathrm{Hycom}\to\mathrm{BV}/\Delta$, given by
Khoroshkin, Markarian and Shadrin in [KMS13]. Here $\mathrm{BV}/\Delta$ is a
model of the homotopy quotient of $\mathrm{BV}$ by $\Delta$ and will be
explained in the proof.

######  Theorem 2.7.

Let $(A,\wedge,d,\Delta_{1},\Delta_{2},\ldots)$ be a commutative
$\mathrm{BV}_{\infty}$-algebra. Any solution

$\varphi(\xi)=\sum_{i\geq 1}\varphi_{i}\xi^{i}\in\mathrm{End}(A)[[\xi]]$

to the gauge equation

$e^{\varphi(\xi)}de^{-\varphi(\xi)}=d+\Delta_{1}\xi+\Delta_{2}\xi^{2}+\cdots$

defines a unique homotopy hypercommutative algebra structure on $(A,d)$
extending the commutative product and such that its homotopy type determines
the homotopy type of the initial commutative $\mathrm{BV}_{\infty}$-algebra.

###### Proof.

A commutative $\mathrm{BV}_{\infty}$-algebra equipped with the trivialization
data $\varphi(\xi)$ is a representation of the homotopy quotient of
$c\mathrm{BV}_{\infty}$ by the operators $\Delta_{i}$, which we denote by
$c\mathrm{BV}_{\infty}/\\{\Delta_{i}\\}_{i>0}$. A model for this homotopy
quotient may be defined as follows: we add, to $c\mathrm{BV}_{\infty}$, a
family of generators $\\{\varphi_{i}\\}$ of arity 1 and degree $-2i$. The
differential $\delta$ of $\varphi_{i}$ is defined by writing
$\varphi(\xi)=\sum\varphi_{i}\xi^{i}$ and imposing the gauge equation. This
gives the various terms

$\begin{array}[]{l}\Delta_{1}=[d,\varphi_{1}],\\\
\Delta_{2}=[d,\varphi_{2}]+{1\over 2}[[d,\varphi_{1}],\varphi_{1}],\\\
\Delta_{3}=[d,\varphi_{3}]+[[d,\varphi_{1}],\varphi_{2}]+[[d,\varphi_{2}],\varphi_{1}]+{1\over
6}[[[d,\varphi_{1}],\varphi_{1}],\varphi_{1}],\\\ \cdots\end{array}$

We then let $\delta\varphi_{i}:=[d,\varphi_{i}]$. The lower terms are:

$\begin{array}[]{l}\delta\varphi_{1}=\Delta_{1},\\\ \delta\varphi_{2}=-{1\over
2}[\Delta_{1},\varphi_{1}]+\Delta_{2},\\\
\delta\varphi_{3}=-[\Delta_{1},\varphi_{2}]-[\Delta_{2},\varphi_{1}]+{1\over
3}[[\Delta_{1},\varphi_{1}],\varphi_{1}]+\Delta_{3},\\\ \cdots\end{array}$

The map of operads $c\mathrm{BV}_{\infty}\to\mathrm{BV}$ defined by sending
$\Delta_{1}\mapsto\Delta$ and $\Delta_{i}\mapsto 0$ for $i>1$ descends to a
surjective quasi-isomorphism,
$\pi:c\mathrm{BV}_{\infty}/\\{\Delta_{i}\\}_{i>0}\to\mathrm{BV}/\Delta$, of
the respective homotopy quotients, where the homotopy quotient
$\mathrm{BV}/\Delta$ is defined exactly as above but with $\Delta_{i}=0$ for
$i>1$ (see also [GM23]). By [KMS13], there is a quasi-isomorphism of operads
$\theta:\mathrm{Hycom}\to\mathrm{BV}/\Delta$. Therefore, by the lifting
property, the dotted arrow in the diagram

$\textstyle{c\mathrm{BV}_{\infty}/\\{\Delta_{i}\\}_{i>0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathrm{Hycom}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{\prime}}$$\textstyle{\mathrm{Hycom}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{\mathrm{BV}/\Delta}$

exists and the diagram strictly commutes. Moreover, all maps are quasi-
isomorphisms.

Finally, the composition

$\mathrm{BV}_{\infty}\to c\mathrm{BV}_{\infty}\to
c\mathrm{BV}_{\infty}/\\{\Delta_{i}\\}_{i>0}$

has a lift

$\textstyle{\mathrm{Hycom}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{\prime}}$$\textstyle{\mathrm{BV}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c\mathrm{BV}_{\infty}/\\{\Delta_{i}\\}_{i>0},}$

which makes the diagram commute up to homotopy. ∎

As is customary, a $c\mathrm{BV}_{\infty}/\\{\Delta_{i}\\}$-algebra is an
algebra over the operad $c\mathrm{BV}_{\infty}/\\{\Delta_{i}\\}$.

The following will be useful to make explicit computations on geometric
examples. It follows by combining Proposition 2.4 and Theorem 2.7.

######  Corollary 2.8.

Let $(A,\wedge,d,\Delta_{1},\Delta_{2},\ldots)$ be a commutative
$\mathrm{BV}_{\infty}$-algebra whose underlying multicomplex satisfies the
degeneration property. Then any deformation retract $(\rho,\iota,h)$ for
$(A,d)$ determines a unique homotopy hypercommutative algebra on $A$. The
strict arity-3 hypercommutative product induced on $A$ is given by

$\begin{array}[]{ll}m_{3}(a,b,c)=&\varphi_{1}(a\wedge b\wedge
c)+\varphi_{1}(a)\wedge b\wedge c+a\wedge\varphi_{1}(b)\wedge c+a\wedge
b\wedge\varphi_{1}(c)\\\ &-\varphi_{1}(a\wedge b)\wedge
c-(-1)^{|b|\cdot|c|}\varphi_{1}(a\wedge c)\wedge b-a\wedge\varphi_{1}(b\wedge
c).\end{array}$

where $\varphi_{1}=h\Delta_{1}-\iota\rho\Delta_{1}h$.

Note that the operation $m_{3}$ is a strict hypercommutative operation, rather
than a higher homotopy operation. In particular, whenever $m_{3}$ is non-zero
in cohomology, the homotopy hypercommutative structure induced on $H(A)$ is
not $\infty$-isomorphic to a homotopy commutative algebra. On the other hand,
an initial commutative $\mathrm{BV}_{\infty}$-algebra may be
$\infty$-isomorphic to a homotopy commutative algebra, or even a commutative
dg-algebra, as in the geometric cases below, but still admit non-trivial
hypercommutative structures. We state a general fact to this effect here.

######  Proposition 2.9.

Let $(A,\wedge,d)$ be a commutative dg-algebra and suppose
$\Lambda\in\mathrm{End}(A)$ is an operator of degree $-2$ and order $\leq 2$.
For all $k\geq 1$, let

$\Delta_{k}={1\over
k!}\underbrace{[\Lambda,[\Lambda,\cdots[\Lambda}_{k},d]]].$

Then:

  1. (1)

The tuple $(A,\wedge,d,\Delta_{i})$ is a commutative
$\mathrm{BV}_{\infty}$-algebra, which is quasi-isomorphic to the commutative
dg-algebra $(A,\wedge,d)$ as commutative $\mathrm{BV}_{\infty}$-algebras.

  2. (2)

The choice of gauge solution $\varphi(\xi)=\Lambda\xi$ yields a homotopy
hypercommutative algebra which is quasi-isomorphic to its underlying homotopy
commutative algebra.

###### Proof.

The fact that the $(A,d,\Delta_{i})$ is a multicomplex follows from the
identity

$\mathbb{D}:=e^{\Lambda\xi}de^{-\Lambda\xi}=Ad_{e^{\Lambda\xi}}(d)=e^{ad_{{\Lambda\xi}}}(d)=\sum_{k\geq
0}{1\over k!}(ad_{\Lambda\xi})^{k}d=\sum_{k\geq 0}{[\Lambda,-]^{k}\xi^{k}\over
k!}d=d+\sum_{k\geq 1}\Delta_{k}\xi^{k}.$

Indeed, $\mathbb{D}^{2}=0$ is the multicomplex condition. Since $d$ has order
$\leq 1$ and $\Lambda$ has order $\leq 2$, one inductively shows that
$\Delta_{i}$ has order $\leq i+1$, using that the commutator of operators
lowers the order by at least 1. This proves that $(A,\wedge,d,\Delta_{i})$ is
a commutative $\mathrm{BV}_{\infty}$-algebra.

Consider now the commutative dg-algebra $(A(t,dt),\wedge,d)$ and define
$\Lambda_{t}:=\Lambda\otimes t\in\mathrm{End}(A(t,dt))$, so that

$\Lambda_{t}(\sum
a_{i}t^{i}+b_{i}t^{i}dt):=\sum\Lambda(a_{i})t^{i+1}+\Lambda(b_{i})t^{i+1}dt.$

This is again an operator of degree $-2$ and order $\leq 2$ and so the same
argument as before shows that $(A(t,dt),\wedge,d,\Delta(t)_{i})$ is a
commutative $\mathrm{BV}_{\infty}$-algebra with

$\Delta(t)_{k}={1\over
k!}\underbrace{[\Lambda_{t},[\Lambda_{t},\cdots[\Lambda_{t}}_{k},d]]].$

The evaluation map $\delta^{\lambda}:A(t,dt)\to A$ given by $t\mapsto\lambda$
and $dt\mapsto 0$ satisfies $\delta^{1}\Delta(t)_{k}=\Delta_{k}$ and
$\delta^{0}\Delta(t)_{k}=0$ for all $k\geq 1$. Therefore we obtain quasi-
isomorphisms of commutative $\mathrm{BV}_{\infty}$-algebras

$(A,\wedge,d,\Delta_{i})\stackrel{{\scriptstyle\delta^{1}}}{{\longleftarrow}}(A(t,dt),\wedge,d,\Delta(t)_{i})\stackrel{{\scriptstyle\delta^{0}}}{{\longrightarrow}}(A,\wedge,d,0).$

This proves the first claim.

The gauge solution $\varphi(\xi)=\Lambda_{t}\xi$ gives a
$\mathrm{BV}_{\infty}/\\{\Delta_{i}\\}$-algebra structure on $A(t,dt)$ with
$\varphi_{1}=\Lambda_{t}$ and $\varphi_{i}=0$ for $i>0$. Since
$\delta^{k}\Lambda_{t}=\Lambda_{k}\delta^{k}$, the above string of quasi-
isomorphisms promotes to a string of quasi-isomorphisms of
$\mathrm{BV}_{\infty}/\\{\Delta_{i}\\}$-algebras. Therefore the corresponding
homotopy hypercommutative algebras are also quasi-isomorphic. ∎

######  Remark 2.10.

The above result recovers in particular the hypercommutative triviality
results of Guan and Muro [GM23] for Poisson and Jacobi manifolds. In the
geometric examples below, the first case of Proposition 2.9 will always occur,
and so we will not choose the trivial gauge solution mentioned in the second
case. Rather, in the geometric situations throughout this paper, the gauge
solutions will be obtained from deformation retracts that are induced
canonically by the underlying structure, e.g. the choice of metric.

##  3\. Hermitian manifolds

Let $(M,J,g)$ be a Hermitian $2n$-manifold (not necessarily compact) with
fundamental $(1,1)$-form $\omega(X,Y)=g(JX,Y)$. We will denote by
$\mathcal{A}=\mathcal{A}_{\mathrm{dR}}(M)$ its de Rham algebra and by
$\mathcal{A}_{\mathbb{C}}=\mathcal{A}\otimes_{\mathbb{R}}\mathbb{C}$ its
complexification. The almost complex structure $J:TM\to TM$ induces a bi-
degree decomposition by forms of type $(p,q)$

$\mathcal{A}^{k}_{\mathbb{C}}=\bigoplus_{p+q=k}\mathcal{A}^{p,q},$

and the exterior differential $d$ decomposes into two components
$d=\partial+{\bar{\partial}}$, where $\partial$ has bidegree $(1,0)$ and
${\bar{\partial}}$ is its complex conjugate, of bidegree $(0,1)$.

Let $d^{*}:=-\star d\star$ be the formal adjoint of $d$ with respect the
Hodge-star operator
$\star:{\mathcal{A}}^{k}_{\mathbb{C}}\to{\mathcal{A}}^{2n-k}_{\mathbb{C}}$
which satisfies $\star^{2}=(-1)^{k}$ on ${\mathcal{A}}^{k}$. Consider the real
differential operator

$d_{c}:=J^{-1}\circ d\circ J=i({\bar{\partial}}-\partial)$

and denote by $d_{c}^{*}:=-\star d_{c}\star$ its formal adjoint. It satisfies

$d_{c}^{*}=i(\partial^{*}-{\bar{\partial}}^{*}),$

where $\partial^{*}:=-\star{\bar{\partial}}\star$ and
${\bar{\partial}}^{*}:=-\star\partial\star$ are the formal adjoints to
$\partial$ and ${\bar{\partial}}$, respectively.

Recall there is the $\mathfrak{sl}_{2}$-representation
$\\{L,\Lambda,H=[L,\Lambda]\\}$ on ${\mathcal{A}}$, where
$L\eta:=\omega\wedge\eta$ is the Lefschetz operator,
$\Lambda:=L^{*}=\star^{-1}L\star$ is the metric adjoint of $L$, and $H$ acts
on ${\mathcal{A}}^{k}$ by $(k-n)$, where $n$ is the complex dimension of $M$.
We will also consider the operators

$\lambda:=[\partial,L]=\partial w\quad\text{ and
}\quad\tau:=[\Lambda,\lambda]$

of bidegrees $(2,1)$ and $(1,0)$ respectively, as well as their adjoints
$\lambda^{*}$ of bidgree $(-2,-1)$ and $\tau^{*}$ of bidegree $(-1,0)$.

For a Kähler manifold we have $\lambda=\tau=0$, and the tuples

$(\mathcal{A}_{\mathbb{C}},\wedge,\Delta_{0}={\bar{\partial}},\Delta_{1}=-i\partial^{*})\quad\text{
and }\quad(\mathcal{A},\wedge,\Delta_{0}=d,\Delta_{1}=-d_{c}^{*})$

are commutative differential graded $\mathrm{BV}$-algebras over $\mathbb{C}$
and $\mathbb{R}$, respectively (see [Kos85] and [CZ00]), and both satisfy the
degeneration property (in fact, they satisfy the stronger so-called
$\Delta_{0}\Delta_{1}$-condition). Note, in both cases, the fundamental Kähler
identity

$[\Lambda,{\bar{\partial}}]=-i\partial^{*}$

is needed for the bicomplex relation
$\Delta_{0}\Delta_{1}+\Delta_{1}\Delta_{0}=0$ to hold.

For Hermitian manifolds, the operators ${\bar{\partial}}$ and $\partial^{*}$
do not anti-commute, but the fundamental Hermitian identity

$[\Lambda,{\bar{\partial}}]=-i(\partial^{*}+\tau^{*}),$

due to Demailly [Dem86], generalizes the Kähler identity above, and is proved
by a local calculation. We remark that Demailly used this relation to prove
several vanishing theorems for Hermitian vector bundles [Dem12], and more
recently it established a Lefschetz duality for Hermitian manifolds [Wil20].
With this identity at hand, we have:

######  Theorem 3.1.

For any Hermitian manifold the tuple

$(\mathcal{A}_{\mathbb{C}},\wedge,\,\Delta_{0}={\bar{\partial}}\,,\,\Delta_{1}=-i(\partial^{*}+\tau^{*})\,,\,\Delta_{2}=i\lambda^{*}\,)$

is a commutative $\mathrm{BV}_{\infty}$-algebra. The total differential

$\mathbb{D}={\bar{\partial}}-i\left(\partial^{*}+\tau^{*}\right)\xi+i\lambda^{*}\xi^{2}\in\mathrm{End}(\mathcal{A}_{\mathbb{C}})[[\xi]]$

satisfies

$\mathbb{D}=e^{\Lambda\xi}{\bar{\partial}}e^{-\Lambda\xi}.$

###### Proof.

By (1) of Proposition 2.9 it suffices to show that

$\Delta_{1}=-i(\partial^{*}+\tau^{*}),\quad\Delta_{2}=i\lambda^{*},\quad\Delta_{3}=0.$

The first equation is Demailly’s identity,
$[\Lambda,{\bar{\partial}}]=-i(\partial^{*}+\tau^{*})$, and for the second we
compute

$\Delta_{2}={1\over 2}[\Lambda,[\Lambda,{\bar{\partial}}]]=-{i\over
2}([\Lambda,\partial^{*}]+[\Lambda,\tau^{*}]).$

First, one checks using $\partial^{*}=-\star{\bar{\partial}}\star$, that
$[\Lambda,\partial^{*}]=[\partial,L]^{*}=\lambda^{*}$. On the other hand,
since $\tau=[\Lambda,\lambda]$, we have

$[\Lambda,\tau^{*}]=[\Lambda,[\lambda^{*},L]]=[\lambda^{*},[\Lambda,L]]=[H,\lambda^{*}]=-3\lambda^{*}.$

To explain, in the second equality we used the Jacobi identity, together with
the fact that $[\Lambda,\lambda^{*}]=[\lambda,L]^{*}=0$, since the graded
commutator of wedging with $\partial\omega$ and $\omega$ vanishes. Since
$\lambda^{*}$ has degree $-3$, we have $[H,\lambda^{*}]=-3\lambda^{*}$.
Altogether, this implies $\Delta_{2}=i\lambda^{*}$. Finally,
$\Delta_{3}=[\Lambda,i\lambda^{*}]=0$, as already noted. ∎

There is also a real commutative $\mathrm{BV}_{\infty}$-algebra with
underlying differential $d$, simply by adding the previous structure to its
complex conjugate. Indeed, since $\Lambda$ is a real operator, we can define
the real operator $T:=[\Lambda,d\omega\wedge-]$, which decomposes as
$T=\tau+{\bar{\tau}}$, while $[d,L]=d\omega=\lambda+{\bar{\lambda}}$.
Conjugating by $J$ we have

$T_{c}=i({\bar{\tau}}-\tau)\quad\quad
d\omega_{c}=i({\bar{\lambda}}^{*}-\lambda^{*}),$

and

$T_{c}^{*}=i(\tau^{*}-{\bar{\tau}}^{*})\quad\quad
d\omega_{c}^{*}=i(\lambda^{*}-{\bar{\lambda}}^{*}).$

Recall $d_{c}:=J^{-1}dJ$ and $d_{c}^{*}:=-\star d_{c}\star$. We then have:

######  Theorem 3.2.

For any Hermitian manifold the tuple

$(\mathcal{A},\wedge,\,\Delta_{0}=d\,,\,\Delta_{1}=-(d_{c}^{*}+T_{c}^{*})\,,\,\Delta_{2}=d\omega_{c}^{*})\,$

is a commutative $\mathrm{BV}_{\infty}$-algebra. The total differential

$\mathbb{D}=d-i\left(d_{c}^{*}+T_{c}^{*}\right)\xi+d\omega_{c}^{*}\,\xi^{2}\in\mathrm{End}(\mathcal{A}_{\mathbb{C}})[[\xi]]$

satisfies

$\mathbb{D}=e^{\Lambda\xi}{\bar{\partial}}e^{-\Lambda\xi}.$

Theorems 3.1 and 3.2 imply that the underlying multicomplexes satisfy the
degeneration property. In order to obtain non-trivial and canonical homotopy
hypercommutative structures, we will use the following deformation retracts,
which are functorial for maps of Hermitian manifolds.

By Hodge theory any de Rham (resp. Dolbeault) cohomology class admits a unique
$d$-harmonic (resp. ${\bar{\partial}}$-harmonic) representative. Moreover,
there are projection maps

$\pi_{d}:\mathcal{A}^{k}\to\mathcal{H}_{d}^{k}\text{ and
}\pi_{\bar{\partial}}:\mathcal{A}_{\mathbb{C}}^{p,q}\to\mathcal{H}_{\bar{\partial}}^{p,q}$

to $d$-harmonic and ${\bar{\partial}}$-harmonic forms respectively, as well as
Green operators

$G_{d}:\mathcal{A}^{k}\to\mathcal{A}^{k}\text{ and
}G_{\bar{\partial}}:\mathcal{A}_{\mathbb{C}}^{p,q}\to\mathcal{A}_{\mathbb{C}}^{p,q}$

defined by zero on harmonic forms and by the inverse of the corresponding
Laplacian on their orthogonal complement. We recall these deformation
retractions here, which were also used in [CH23]). We note that the choice of
sign here in defining $h$ is to simplify the formulas in Proposition 2.4.

######  Lemma 3.3 (De Rham deformation retract).

For a compact Riemannian manifold there is a canonical deformation retract
$(\iota,\rho,h)$ for the de Rham complex $(\mathcal{A},d)$ where $\iota$ and
$\rho$ are given by taking $d$-harmonic representatives, and projecting to
$d$-harmonic forms, respectively, and

$h:=-d^{*}G_{d}.$

Explicitly, using the isomorphism $d:\mathrm{Im}\,d^{*}\to\mathrm{Im}\,d$, we
have

$h=-d^{-1}\circ\pi_{|\mathrm{Im}d}:\mathcal{A}\to{\mathrm{Im}\,d^{*}}.$

######  Lemma 3.4 (Dolbeault deformation retract).

For a compact Hermitian manifold there is a canonical deformation retract
$(\iota,\rho,h)$ for the de Dolbeault complex $(\mathcal{A},{\bar{\partial}})$
where the maps $\iota$ and $\rho$ are given by taking
${\bar{\partial}}$-harmonic representatives, and projecting to
${\bar{\partial}}$-harmonic forms, respectively, and

$h:=-{\bar{\partial}}^{*}G_{\bar{\partial}}=-{\bar{\partial}}^{-1}\circ\pi_{|\mathrm{Im}{\bar{\partial}}}:\mathcal{A}_{\mathbb{C}}\to{\mathrm{Im}\,{\bar{\partial}}^{*}}.$

Using these canonical deformation retracts, we obtain:

######  Corollary 3.5.

The de Rham and Dolbeault cohomologies of any compact Hermitian manifold carry
canonical homotopy hypercommutative algebra structures extending the
commutative product. Such homotopy hypercommutative algebras encode the
homotopy type of the initial commutative $\mathrm{BV}_{\infty}$-algebras of
Theorems 3.1 and 3.2, respectively.

###### Proof.

By Theorems 3.1 and 3.2 respectively we have commutative
$\mathrm{BV}_{\infty}$-algebras on the Dolbeault (resp. de Rham) algebra of
forms, and both satisfy the degeneration property. The result follows from
Theorem 2.7 together the canonical deformation retracts of Lemmas 3.4 and 3.3,
respectively. ∎

Note that even if the Frölicher spectral sequence of a complex manifold
degenerates at the first stage, the Dolbeault and de Rham cohomologies of the
manifold are not in general isomorphic as commutative algebras. In particular,
in general, the above two homotopy hypercommutative algebra structures will
not be homotopy equivalent. In contrast, in the case of Kähler manifolds, in
[CH23] it is shown that the Dolbeault and de Rham algebras are in fact quasi-
isomorphic as $\mathrm{BV}$-algebras, and the associated homotopy
hypercommutative algebra structures are in this case formal. We will give
examples of non-formal and non-trivial hypercommutative structures on
Hermitian manifolds in Section 5.

##  4\. Almost Hermitian and almost symplectic manifolds

In this section we extend the previous results to the case of almost Hermitian
and almost symplectic manifolds. In the particular case of almost Kähler (and
thus symplectic) manifolds, one recovers the $\mathrm{BV}$ and
hypercommutative structures defined by [Kos85] and [DSV15], defined for any
Poisson manifold. Therefore, the results may be viewed as a natural extension
of the theory of hypercommutative structures for Poisson manifolds to (almost)
Hermitian manifolds. Finally, the (almost) symplectic case is shown to have
bigraded refinement, given the data of an integrable Lagrangian subbundle of
the real tangent bundle. We view this symplectic-Lagrangian version, on the
real differential forms, as the symplectic analogue of the Hermitian case of
the previous section.

###  4.1. Almost Hermitian manifolds

With small modifications, the results of Section 3 generalize easily to the
case of almost Hermitian manifolds, so we will be brief. This requires a
generalization of the Kähler identities which, to our knowledge, was first
deduced by Tomassini and Wang [TW18]. These identities were independently
derived, via a different method using algebraic calculations in the Clifford
algebra, by Fernandez and Hosmer [FH22], and we follow the notation given
therein. The fundamental relation is

$[\Lambda,d]=-(d_{c}^{*}+T_{c}^{*})$

where

$T=[\Lambda,[d,L]]=[\Lambda,d\omega\wedge-].$

where $d\omega\wedge-$ is the operator of degree $3$ given by wedging with
$d\omega$, we let

$T_{c}=J^{-1}TJ\text{ and }d\omega_{c}=J^{-1}d\omega J,$

and $T_{c}^{*}$ and $d\omega_{c}^{*}$ are the adjoints.

An argument similar to the proof of Theorem 3.1 yields:

######  Theorem 4.1.

For any almost Hermitian manifold, the tuple

$\left(\mathcal{A},\wedge,d,\,\Delta_{1}=-(d_{c}^{*}+T_{c}^{*})\,,\,\Delta_{2}=d\omega_{c}^{*}\,\right)$

is a commutative $\mathrm{BV}_{\infty}$-algebra on the real differential
forms. The total differential

$\mathbb{D}=d+\Delta_{1}\xi+\Delta_{2}\xi^{2}\in\mathrm{End}(\mathcal{A})[[\xi]]$

satisfies

$\mathbb{D}=e^{\Lambda\xi}de^{-\Lambda\xi}.$

###### Proof.

By (1) of Proposition 2.9 it suffices to show that

$\Delta_{1}=[\Lambda,d]=-(d_{c}^{*}+T_{c}^{*}),\,\Delta_{2}={1\over
2}[\Lambda,[\Lambda,d]]=d\omega_{c}^{*}\text{ and }\Delta_{3}={1\over
6}[\Lambda,[\Lambda,[\Lambda,d]]]=0.$

The expression for $\Delta_{1}$ is cited above. Next, note that

$[\Lambda,[\Lambda,d]]=-[\Lambda,d_{c}^{*}]-[\Lambda,T_{c}^{*}].$

One checks using $d^{*}=-\star d\star$ that
$[\Lambda,d_{c}^{*}]=d\omega_{c}^{*}$. Next, we have

$[\Lambda,T_{c}^{*}]=J^{-1}[\Lambda,T^{*}]J=J^{-1}[T,L]^{*}J=-3d\omega_{c}^{*}$

where we used

$[T,L]=[[\Lambda,d\omega],L]=-[H,d\omega]=-3d\omega,$

using the Jacobi identity. This gives $\Delta_{2}=d\omega_{c}^{*}$ as desired.
Finally $\Delta_{3}=0$ since
$[\Lambda,d\omega_{c}^{*}]=[d\omega_{c},L]^{*}=0$. ∎

Note in the integrable case, the relation $[\Lambda,d]=-(d_{c}^{*}+T_{c}^{*})$
reduces to the fundamental identity of Demailly for Hermitian manifolds, so we
recover Theorem 3.2. Moreover, the above result, together with the canonical
deformation retract of Lemma 3.3 allow to generalize Corollary 3.5 to almost
Hermitian manifolds. We obtain:

######  Corollary 4.2.

The de Rham cohomology of any compact almost Hermitian manifold carries a
canonical homotopy hypercommutative algebra structure extending the
commutative product.

As in the case of complex manifolds, the complex de Rham algebra of any almost
Hermitian manifold admits a bidegree decomposition induced by the $\pm
i$-eigenspaces of $J$. In this general case, the exterior differential
decomposes as

$d={\bar{\mu}}+{\bar{\partial}}+\partial+\mu$

with bidegrees given by $|{\bar{\mu}}|=(-1,2)$, $|{\bar{\partial}}|=(0,1)$,
$|{\bar{\partial}}|=(0,1)$ and $|\mu|=(2,-1)$. Note that $\mu$ and
${\bar{\mu}}$ vanish if and only if $J$ is integrable. However, there is no
obvious bigraded version of Corollary 4.2. While Dolbeault cohomology may
still be defined as in [CW21], there is no Hodge theory available in general,
and so we don’t have canonical deformation retracts for Dolbeault cohomology
in the non-integrable case. As we will later see, a bigraded version may be
defined for the case of symplectic manifolds admitting real polarizations.

###  4.2. Almost symplectic and almost Kähler manifolds

In this subsection we indicate an almost-symplectic version and observe that
whenever the fundamental form is closed, $d\omega=0$, the induced commutative
$\mathrm{BV}_{\infty}$-algebra of Theorem 4.1 is the $\mathrm{BV}$-algebra
induced by Koszul [Kos85], for symplectic manifolds. In particular, it is
metric independent. Therefore, the induced hypercommutative structures agree
as well.

Recall from Koszul [Kos85], on a Poisson manifold $(M,\pi)$ there is a
differential graded $\mathrm{BV}$-algebra

$(\mathcal{A},\wedge,d,\Delta=[i_{\pi},d]),$

where $i_{\pi}:\mathcal{A}^{*}\to\mathcal{A}^{*-2}$ is defined by contracting
forms with the Poisson bi-vector field. Moreover, the gauge equation

$e^{i_{\pi}\xi}de^{-i_{\pi}\xi}=d+\Delta\xi$

is satisfied. In particular, this fits into the setting of Proposition 2.9.
Recall, the choice of solution $\varphi(\xi)=i_{\pi}\xi$ to the gauge equation
leads to trivial hypercommutative structures, as first noted in [GM23].
However, by Theorem 2.7 and choosing a Riemannian metric, the canonical
deformation retract for the de Rham algebra of Lemma 3.3 gives homotopy
hypercommutative structures for Poisson manifolds, which are non-trivial in
general (as shown in Example 5.2 below).

Recall that given a symplectic manifold $(M,\omega)$, there is a symplectic
Hodge-star operator

$\star_{\omega}:{\mathcal{A}}^{k}\to{\mathcal{A}}^{2n-k}\quad\text{ defined by
}\quad\alpha\wedge\star_{\omega}\beta=\langle\alpha,\beta\rangle_{\omega}\frac{\omega^{n}}{n!},$

where $\langle-,-\rangle_{\omega}$ is the pairing on ${\mathcal{A}}^{*}$
induced by $\omega$. Explicitly, if $\alpha=a_{1}\wedge\cdots\wedge a_{k}$ and
$\beta=b_{1}\wedge\cdots\wedge b_{k}$ then
$\langle\alpha,\beta\rangle_{\omega}=\det(\omega^{-1}(a_{i},b_{j}))$. One can
verify that $\star_{\omega}^{2}=1$, and
$\star_{\omega}(1)=\frac{\omega^{n}}{n!}$ is the volume form, and define the
pairing

$(\alpha,\beta)_{\omega}=\int_{M}\langle\alpha,\beta\rangle_{\omega}\frac{\omega^{n}}{n!}$

in the case that $M$ is closed.

Letting $L\eta=\omega\wedge\eta$, the symplectic adoint of $L$, denoted by
$\Lambda=L^{\omega}$, is defined by
$\langle\Lambda\alpha,\beta\rangle_{\omega}=\langle\alpha,L\beta\rangle_{\omega}$,
and satisfies $\Lambda=\star_{\omega}L\star_{\omega}$. More generally, we can
similarly define the symplectic adjoints of any operators $\phi_{1}$ and
$\phi_{2}$ of algebraic order zero, which satisfies
$[\phi_{1},\phi_{2}]^{\omega}=[\phi_{2}^{\omega},\phi_{1}^{\omega}]$. The
symplectic adjoint of such a zeroth-order operator $\phi$, of degree $|\phi|$,
equals $\phi^{\omega}=(-1)^{|\phi|(k-\phi)}\star_{\omega}\phi\star_{\omega}$
on ${\mathcal{A}}^{k}$. In particular, for the degree $3$ operator $[d,L]$, of
algebraic order zero, we have

$[d,L]^{\omega}=(-1)^{k+1}\star_{\omega}[d,L]\star_{\omega}\quad\textrm{on
${\mathcal{A}}^{k}$}.$

The symplectic adjoint of $d$ is defined by

$d^{\omega}:=(-1)^{k}\star_{\omega}d\star_{\omega}\quad\textrm{on
${\mathcal{A}}^{k}$},$

and satisfies $(d\alpha,\beta)_{\omega}=(\alpha,d^{\omega}\beta)_{\omega}$
when $M$ is compact.

To connect with the Poisson case, a non-degenerate Poisson bi-vector field
$\pi$ defines a contraction operator $i_{\pi}$, and in the symplectic case,
$i_{\pi}=-\Lambda$ (c.f. [Bry88] p. 96) and the relation $[i_{\pi},d]=\Delta$
of Koszul [Kos85] becomes

$[\Lambda,d]=d^{\omega}.$

We can now state an almost-symplectic analogue.

######  Theorem 4.3.

Let $(M,\omega)$ be an almost symplectic manifold. The tuple

$(\mathcal{A},\wedge,d,\Delta_{1}:=d^{\omega}+[\Lambda,[d,L]]^{\omega},\Delta_{2}=-[d,L]^{\omega})$

is a commutative $\mathrm{BV}_{\infty}$-algebra. The total differential

$\mathbb{D}=d+(d^{\omega}+[\Lambda,[d,L]]^{\omega})\xi-([d,L]^{\omega})\xi^{2}\in\mathrm{End}(\mathcal{A})[[\xi]]$

satisfies

$\mathbb{D}=e^{\Lambda\xi}de^{-\Lambda\xi}.$

###### Proof.

In [TW18], Tomassini and Wang show that, for any almost symplectic manifold,

$[d,\Lambda]=d^{\Lambda}+[[\Lambda,d^{\Lambda}],L],$

where the sign convention
$d^{\Lambda}:=(-1)^{k+1}\star_{\omega}d\star_{\omega}$ on ${\mathcal{A}}^{k}$
is used, so that $d^{\omega}=-d^{\Lambda}$ . We claim this is equivalent to

$[\Lambda,d]=d^{\omega}+[\Lambda,[d,L]]^{\omega}.$

To justify the right-hand summand we compute

$[[\Lambda,d^{\omega}],L]=[[d,L]^{\omega},L]=[\Lambda,[d,L]]^{\omega},$

where the first equality is justified by the formal adjoint calculation

$[\Lambda,d^{\omega}]=\Lambda(-1)^{k}\star_{\omega}d\star_{\omega}-(-1)^{k}\star_{\omega}d\star_{\omega}\Lambda=(-1)^{k}\star_{\omega}[L,d]\star_{\omega}=[d,L]^{\omega}.$

The remainder of the proof is similar to the previous cases. From [TW18], we
have

$\Delta_{1}=[\Lambda,d]=d^{\omega}+[\Lambda,[d,L]]^{\omega}.$

Then computing we have

$2\Delta_{2}=[\Lambda,d^{\omega}+[\Lambda,[d,L]]^{\omega}]=[\Lambda,d^{\omega}]+[\Lambda,[\Lambda,[d,L]]^{\omega}]$

where $[\Lambda,d^{\omega}]=[d,L]^{\omega}$ and

$[\Lambda,[\Lambda,[d,L]]^{\omega}]=-[L,[\Lambda,[d,L]]]^{\omega}=-[H,[d,L]]^{\omega}=-3[d,L]^{\omega},$

so that $\Delta_{2}=-[d,L]^{\omega}$. Finally, $\Delta_{k}=0$ for $k\geq 0$
since $[\Lambda,[d,L]^{\omega}]=[L,[d,L]]^{\omega}=0$. ∎

In the symplectic case, with $[d,L]=d\omega=0$, this recovers Koszul’s
$\mathrm{BV}$-algebra as

$(\mathcal{A},\wedge,d,\Delta=d^{\omega}).$

Now, suppose we have an almost Hermitian manifold, i.e. an almost symplectic
structure $\omega$, with compatible almost complex structure $J$, so that
$g(X,Y)=\omega(X,JY)$ is a Riemannian metric. Then $g$ determines a metric
$\langle-,-\rangle$ on each space ${\mathcal{A}}^{k}$, and the (Riemannian)
Hodge star is

$\star:{\mathcal{A}}^{k}\to{\mathcal{A}}^{2n-k}\quad\text{ defined by
}\quad\alpha\wedge\star\beta=\langle\alpha,\beta\rangle\Omega.$

where $\Omega$ is a volume form. Recall that $\star^{2}=(-1)^{k}$ on
${\mathcal{A}}^{k}$, and $\star=\star_{\omega}J=J\star_{\omega}$, where $J$ is
the extension to ${\mathcal{A}}^{*}$ as an algebra automorphism.

######  Lemma 4.4.

For any almost Hermitian manifold $(M,g,J,\omega)$ we have

$\displaystyle d^{*}_{c}$ $\displaystyle=-d^{\omega}$ $\displaystyle
d\omega_{c}^{*}$ $\displaystyle=-[d,L]^{\omega}$ $\displaystyle T_{c}^{*}$
$\displaystyle=-[\Lambda,[d,L]]^{\omega}.$

###### Proof.

First, the metric adjoint of a zeroth order operator $\phi$ of degree $|\phi|$
equals to

$\phi^{*}=(-1)^{|\phi|(k-\phi)}\star^{-1}\phi\star\quad\textrm{on
${\mathcal{A}}^{k}$.}$

In particular, $[d,L]^{*}=(-1)^{k+1}\star^{-1}[d,L]\star$ and
$d\omega_{c}^{*}=(-1)^{k+1}\star^{-1}d\omega_{c}\star$ on ${\mathcal{A}}^{k}$.
Then, using the fact that $J\star=\star J=(-1)^{k}\star_{\omega}$ on
${\mathcal{A}}^{k}$, we compute on ${\mathcal{A}}^{k}$ that

$\displaystyle d^{*}_{c}=-\star J^{-1}dJ\star$
$\displaystyle=-(-1)^{k+1}\star^{-1}J^{-1}dJ\star$
$\displaystyle=(-1)^{k}(J\star)^{-1}d(J\star)$
$\displaystyle=(-1)^{k}(-1)^{k+1}\star_{\omega}d(-1)^{k}\star_{\omega}=-d^{\omega},$

and

$\displaystyle d\omega_{c}^{*}=(-1)^{k+1}\star^{-1}J^{-1}d\omega J\star$
$\displaystyle=(-1)^{k+1}(-1)^{k+3}\star_{\omega}^{-1}d\omega(-1)^{k}\star_{\omega}$
$\displaystyle=(-1)^{k}\star_{\omega}[d,L]\star_{\omega}=-[d,L]^{\omega},$

and

$T_{c}^{*}=[\Lambda,d\omega_{c}]^{*}=[d\omega_{c}^{*},L]=-[[d,L]^{\omega},L]=-[\Lambda,[d,L]]^{\omega}.$

∎

######  Corollary 4.5.

The commutative $BV_{\infty}$-structures of Theorem 4.1 and Theorem 4.3
coincide for almost Hermitian manifolds, and in particular depend only on
$\omega$, not $J$ and the metric individually. In the almost Kähler case,
$d\omega=0$, the commutative $\mathrm{BV}_{\infty}$-algebra of Theorem 4.1 is

$\left(\mathcal{A},\wedge,d,\,\Delta_{1}=-(d_{c}^{*}+T_{c}^{*})\,,\,\Delta_{2}=d\omega_{c}^{*}\,\right)=\left(\mathcal{A},\wedge,d,\,\Delta_{1}=d^{\omega}\,,\Delta_{2}=0\,\right),$

which is Koszul’s $\mathrm{BV}$-algebra.

###  4.3. Symplectic manifolds with real polarizations

An almost complex manifold $(M,J)$ has a complex splitting, or polarization,
$T_{\mathbb{C}}M=T^{0,1}M\oplus T^{0,1}M$ inducing a decomposition of
differential forms. In the integrable case, this leads to the bigraded
commutative $\mathrm{BV}_{\infty}$ and hypercommutative structures on
Dolbeault cohomology, as we have seen in the previous sections. In the case of
an (almost) symplectic manifold $(M,\omega)$ of dimension $2n$, we can
similarly ask for a splitting of the real tangent bundle into subbundles
$TM=\mathcal{L}\oplus\mathcal{L}^{\prime}$ where $\mathcal{L}$ and
$\mathcal{L}^{\prime}$ are Lagrangian, i.e. $\mathrm{rank}\,\mathcal{L}=n$ and
$\omega|_{\mathcal{L}}=\omega|_{\mathcal{L}^{\prime}}=0$. Sometimes this is
referred to a real polarization.

Note that for any Lagrangian sub-bundle $\mathcal{L}$, and any almost complex
structure $J$ on $M$ compatible with $\omega$, we have that
$\mathcal{L}^{\prime}=J\mathcal{L}$ is also Lagrangian, and $\mathcal{L}$ are
$\mathcal{L}^{\prime}$ are orthogonal. So, in particular such polarizations
exist for every Lagrangian subbundle $\mathcal{L}$. The prototypical non-
compact example is the cotangent bundle of a manifold, with tautological
symplectic form.

A real polarization $(\mathcal{L},\mathcal{L}^{\prime})$ for the tangent
bundle of a symplectic manifold $(M,\omega)$ induces a decomposition of de
Rham differential forms

${\mathcal{A}}^{k}=\bigoplus_{p+q=k}{\mathcal{A}}^{p,q}\quad\text{ with
}\quad{\mathcal{A}}^{p,q}=\Gamma(M,\wedge^{p}(\mathcal{L}^{\prime})^{*}\otimes\wedge^{q}\mathcal{L}^{*}).$

This is an orthogonal direct sum decomposition with respect to the induced
pairing

$\langle\alpha,\beta\rangle=\det(\omega^{-1}(a_{i},b_{j}))$

on ${\mathcal{A}}^{k}$.

With respect to this decomposition, the symplectic form $\omega$ has bidegree
$(1,1)$, since $\omega=\sum_{i=1}^{n}e^{i}\wedge{f^{i}}$, where $\\{e^{i}\\}$
is some basis of $\mathcal{L}^{*}$, and $\\{f^{i}\\}$ of
$(\mathcal{L}^{\prime})^{*}$. Additionally,
$\star_{\omega}:{\mathcal{A}}^{p,q}\to{\mathcal{A}}^{n-q,n-p}$.

As in the case of almost complex manifolds, the exterior differential
decomposes into $4$ components, which by analogy we denote by

$d={\bar{\mu}}+{\bar{\partial}}+\partial+\mu,$

where the bidegrees of each component are given by

$|{\bar{\mu}}|=(-1,2),\,|{\bar{\partial}}|=(0,1),\,|\partial|=(1,0),\text{ and
}\,|\mu|=(2,-1).$

To prove this, it suffices to note that the algebra of forms is generated by
degrees $0$ and $1$, the claim holds in these degrees, and $d$ is a
derivation. In particular, there is a spectral sequence associated to this
multicomplex decomposition, which is the real-symplectic analogue of the
Fröhlicher-type spectral sequence in [CW21] defined for almost complex
manifolds, and which converges to de Rham cohomology.

For any almost symplectic manifold we have symplectic adjoints
$\delta^{\omega}$, for any of
$\delta={\bar{\mu}},{\bar{\partial}},\partial,\mu$. One can check from
$d^{\omega}=(-1)^{k}\star_{\omega}d\star_{\omega}$ on ${\mathcal{A}}^{k}$, and
the bidegrees, that

$\partial^{\omega}=(-1)^{k}\star_{\omega}{\bar{\partial}}\star_{\omega},$

and similarly for other $\delta$.

A Lagrangian subbundle $\mathcal{L}$ is said to be integrable if it is closed
under the Lie bracket of vector fields. We have:

######  Lemma 4.6.

Let $(M,\omega)$ be an almost symplectic manifold with real polarization
$(\mathcal{L},\mathcal{L}^{\prime})$.

  1. (1)

The Lagrangian sub-bundle $\mathcal{L}$ is integrable if and only if
${\bar{\mu}}=0$. In this case, ${\bar{\partial}}^{2}=0$ and so
$({\mathcal{A}}^{*,*},{\bar{\partial}})$ is a cochain complex.

  2. (2)

If both $\mathcal{L}$ and $\mathcal{L}^{\prime}$ are integrable then
${\bar{\mu}}=\mu=0$ and $({\mathcal{A}}^{*,*},{\bar{\partial}},\partial)$ is a
double complex.

###### Proof.

Using Cartan’s formula, relating $d$ and the Lie bracket, we compute for any
$(1,0)$-form $\eta$

$({\bar{\mu}}\eta)(X,Y)=(\pi^{0,2}\circ
d\eta)(X,Y)=d\eta(\pi_{(0,2)}(X,Y))=-\eta([\pi_{0,1}X,\pi_{0,1}Y]),$

which vanishes for all $X,Y$ if and only if $\mathcal{L}$ is closed under Lie
bracket. In this case, since $d^{2}=0$ always implies
${\bar{\partial}}^{2}+{\bar{\mu}}\partial+\partial{\bar{\mu}}=0$, we obtain
${\bar{\partial}}^{2}=0$. The second statement follows similarly. ∎

We emphasize that only the integrability of $\mathcal{L}$ is required (not
$\mathcal{L}^{\prime}$) to define the complex
$({\mathcal{A}}^{*,*},{\bar{\partial}})$. This occurs in many natural examples
(see for instance Example 5.7 below). It is an open problem in some cases to
determine if an integrable Lagrangian has an integrable Lagrangian complement
(see [HK24]).

Note that if $\mathcal{L}$ is integrable, then it integrates to the Lagrangian
submanifold $L\subset M$ with tangent bundle $\mathcal{L}$, and we have
$({\mathcal{A}}(L),d)\cong\left(\bigwedge^{*}\mathcal{L}^{*},{\bar{\partial}}\right)$
as commutative dg-algebras, so in particular
$H_{\bar{\partial}}\left(\bigwedge^{*}\mathcal{L}^{*}\right)\cong H_{d}(L)$.

The following is the symplectic-Lagrangian analogue of the (integrable)
Dolbeault case:

######  Theorem 4.7.

Let $(M,\omega)$ be a symplectic manifold with a real polarization
$(\mathcal{L},\mathcal{L}^{\prime})$. Assume $\mathcal{L}$ is integrable. The
tuple

$(\mathcal{A},\wedge,{\bar{\partial}},\Delta:=\partial^{\omega})$

is a differential graded $\mathrm{BV}$-algebra. The total differential

$\mathbb{D}={\bar{\partial}}+\partial^{\omega}\xi\in\mathrm{End}(\mathcal{A})[[\xi]]$

satisfies

$\mathbb{D}=e^{\Lambda\xi}{\bar{\partial}}e^{-\Lambda\xi}.$

More generally, if $(M,\omega)$ is an almost symplectic with real polarization
$(\mathcal{L},\mathcal{L}^{\prime})$ and $\mathcal{L}$ is integrable then

$(\mathcal{A},\wedge,{\bar{\partial}},\Delta_{1}:=\partial^{\omega}+[\Lambda,[\partial,L]]^{\omega},\Delta_{2}=-[\partial,L]^{\omega})$

is a commutative $BV_{\infty}$-algebra.

###### Proof.

The last statement is simply one bi-graded component of Theorem 4.3, and it
implies the first in the case $\partial\omega=0$. ∎

In order to obtain canonical homotopy hypercommutative structures we shall
choose a metric that makes the Lagrangian sub-bundles orthogonal. In this
case, the Laplacian
$\Delta_{\bar{\partial}}:={\bar{\partial}}{\bar{\partial}}^{*}+{\bar{\partial}}^{*}{\bar{\partial}}$
is elliptic for compact manifolds and so Hodge theory is available. We have:

######  Lemma 4.8 (Symplectic-Lagrangian deformation retract).

Let $(M,\omega)$ be a compact symplectic manifold with a real polarization
$(\mathcal{L},\mathcal{L}^{\prime})$ such that $\mathcal{L}$ is integrable.
Given a metric making $\mathcal{L}$ and $\mathcal{L}^{\prime}$ orthogonal,
there is a canonical deformation retract $(\iota,\rho,h)$ for the de Rham
complex $(\mathcal{A},{\bar{\partial}})$ where the maps $\iota$ and $\rho$ are
given by taking ${\bar{\partial}}$-harmonic representatives, and projecting to
${\bar{\partial}}$-harmonic forms, respectively, and
$h:=-{\bar{\partial}}^{*}G_{\bar{\partial}}$.

Using the above deformation retract, we obtain:

######  Corollary 4.9.

Let $(M,\omega)$ be a compact symplectic manifold with a real polarization
$(\mathcal{L},\mathcal{L}^{\prime})$, such that $\mathcal{L}$ is integrable.
Given a metric making $\mathcal{L}$ and $\mathcal{L}^{\prime}$ orthogonal,
there is a canonical homotopy hypercommutative algebra structure extending the
commutative product.

##  5\. Chevalley-Eilenberg algebras and examples

In order to compute examples, in this section we will resort to the theory of
nilmanifolds. Let $G$ be a nilpotent Lie group with Lie algebra
$\mathfrak{g}$. If $\mathfrak{g}=\mathfrak{g}_{\mathbb{Q}}\otimes\mathbb{R}$
has a rational structure, then by a well-known result of Malcev there exists a
discrete subgroup $\Gamma$ such that the quotient $M=\Gamma\backslash G$ is a
compact manifold. Recall that the Chevalley-Eilenberg algebra associated to
$\mathfrak{g}$ is given by

$\mathcal{A}^{*}_{\mathfrak{g}}:=\bigoplus_{k\geq
0}\Lambda^{k}(\mathfrak{g}^{\vee}).$

The differential on $\mathfrak{g}^{\vee}$ is defined as the negative of the
dual of the Lie bracket and is extended uniquely to a derivation of
$\mathcal{A}^{*}_{\mathfrak{g}}$. The commutative dg-algebra
$\mathcal{A}^{*}_{\mathfrak{g}}$ is isomorphic to the algebra of left-
invariant forms on $M$. This gives an inclusion of commutative dg-algebras
$\mathcal{A}^{*}_{\mathfrak{g}}\hookrightarrow\mathcal{A}^{*}_{\mathrm{dR}}(M)$
which by Nomizu’s Theorem, induces an isomorphism in cohomology
$H^{*}(\mathfrak{g})\cong H^{*}(M,\mathbb{R})$. In particular
$\mathcal{A}^{*}_{\mathfrak{g}}$ encodes the real homotopy type of the
nilmanifold $M$ (and is actually a minimal model of $M$ in the sense of
Sullivan). Additional geometric structures on $M$, such a Poisson, symplectic,
almost-Hermitian or Hermitian structures, may be defined at the Lie algebra
level, allowing for finite-dimensional linear algebra computations of the
homotopical structures that arise.

###  5.1. Canonical deformation retracts

Consider a real inner product $\langle-,-\rangle$ on $\mathfrak{g}$. This
defines a Hodge star operator

$\star:\mathcal{A}_{\mathfrak{g}}^{k}\to\mathcal{A}_{\mathfrak{g}}^{n-k}\quad\text{
by }\quad\alpha\wedge\star\beta=\langle\alpha,\beta\rangle\ \Omega,$

where $\Omega$ is a volume form determined by the metric and $n$ is the
dimension of $\mathfrak{g}$. The space of $d$-harmonic forms is then defined
by

$\mathcal{H}^{k}(\mathfrak{g}):=\mathrm{Ker}(d)\cap\mathrm{Ker}(d^{*})\cap\mathcal{A}_{\mathfrak{g}}^{k},$

where
$d^{*}:\mathcal{A}_{\mathfrak{g}}^{k}\to\mathcal{A}_{\mathfrak{g}}^{k-1}$ is
given by $d^{*}=-\star d\star$. Under the assumption that
$H^{n}(\mathfrak{g})\cong\mathbb{R}$ (which is the algebraic analogue of
having a closed manifold), $d^{*}$ is the formal adjoint to $d$. In this case,
we obtain an orthogonal direct sum Hodge decomposition

$\mathcal{A}_{\mathfrak{g}}^{k}\cong\mathcal{H}^{k}(\mathfrak{g})\oplus
d(\mathcal{A}^{k-1}_{\mathfrak{g}})\oplus
d^{*}(\mathcal{A}^{k+1}_{\mathfrak{g}})$

and there are isomorphisms $\mathcal{H}^{k}(\mathfrak{g})\cong
H^{k}(\mathfrak{g})$.

Taking the standard inner product we obtain a canonical deformation retract
$(\iota,\rho,h)$ for the Chevalley-Eilenberg algebra of $\mathfrak{g}$, where
$\iota$ and $\rho$ are defined by projecting to harmonic forms and
$h=-d^{*}G_{d}$, where $G_{d}$ denotes the Green operator. This is just a
Chevalley-Eilenberg version of Lemma 3.3 for compact Riemannian manifolds.

###  5.2. Poisson structures

A Poisson structure on $\mathfrak{g}$ is a bi-vector
$\pi\in\Lambda^{2}\mathfrak{g}$ such that $[\pi,\pi]=0$. This makes
$\mathcal{A}_{\mathfrak{g}}$ into a differential graded $\mathrm{BV}$-algebra
with $\Delta=[i_{\pi},d]$, where
$i_{\pi}:\mathcal{A}_{\mathfrak{g}}^{*}\to\mathcal{A}_{g}^{*-2}$ is the
contraction operator by $\pi$. Moreover, the equation

$e^{i_{\pi}\xi}de^{-i_{\pi}\xi}=d+\Delta\xi$

is satisfied. By Theorem 2.7 and using the canonical deformation retract
defined above, we get:

######  Corollary 5.1.

Let $\mathfrak{g}$ be a real Lie algebra of dimension $n$ such that
$H^{n}(\mathfrak{g})\cong\mathbb{R}$. Then $\mathcal{A}_{\mathfrak{g}}$
carries a canonical hypercommutative algebra structure extending the
commutative product.

In the case when $M=G/\Gamma$ is a nilmanifold with $Lie(G)=\mathfrak{g}$, any
Poisson structure on $\mathfrak{g}$ gives a (left-invariant) Poisson structure
on $M$. The same formula $\Delta=[i_{\pi},d]$ makes the de Rham algebra of $M$
into a $\mathrm{BV}$-algebra and the inclusion
$\mathcal{A}^{*}_{\mathfrak{g}}\hookrightarrow\mathcal{A}^{*}_{\mathrm{dR}}(M)$
is a quasi-isomorphism of $\mathrm{BV}$-algebras.

######  Example 5.2 (Poisson structures on the Kodaira-Thurston manifold).

The Kodaira-Thurston manifold is a compact nilmanifold of dimension 4 whose
defining nilpotent Lie algebra is given by
$\mathfrak{g}=\mathbb{R}\\{X,Y,Z,T\\}$ with the only non-trivial Lie bracket
given by $[X,Y]=-Z$. This is the simplest example of a manifold admitting both
symplectic and complex structures, but no Kähler structure. Its algebra of
left-invariant forms is given by
$\mathcal{A}_{\mathfrak{g}}\cong\Lambda(x,y,z,t)$ with $dz=xy$, and so its
cohomology ring is given by

$H^{1}\cong\mathbb{R}\\{[x],[y],[t]\\},H^{2}\cong\mathbb{R}\\{[xz],[xt],[yz],[yt]\\},H^{3}\cong\mathbb{R}\\{[xyz],[xzt],[yzt]\\},H^{4}\cong\mathbb{R}\\{[xyzt]\\}.$

Any Poisson structure on $\mathfrak{g}$ has the form

$\pi=aX\wedge Z+bX\wedge T+cY\wedge Z+eY\wedge T+fZ\wedge T.$

Computing $\Delta=[i_{\pi},d]$, we obtain

$\Delta(xzt)=b(xy),\Delta(yzt)=e(xy)\text{ and }\Delta(zt)=ex-by,$

and $\Delta=0$ for the remaining cochain complex generators. The maps $\rho$
and $\iota$ of the canonical deformation retract described above are obvious.
The homotopy is given by $h(xy)=-z$, $h(xyt)=-zt$ and is zero otherwise. We
may now compute $\varphi_{1}=h\Delta-\iota\rho\Delta h$. Its only non-trivial
values are

$\varphi_{1}(xyt)=ex-by,\,\varphi_{1}(xzt)=-bz\text{ and
}\varphi_{1}(yzt)=-ez.$

Using the expression for $m_{3}$ of Corollary 2.8, we get

$m_{3}(x,y,t)=\varphi_{1}(xyt)=ex-by,$

which induces a non-trivial strict hypercommutative operation in cohomology
whenever $e$ or $b$ are non-zero.

###  5.3. Almost Hermitian structures

A linear (almost) complex structure on $\mathfrak{g}$ is given by an
endomorphism $J$ of $\mathfrak{g}$ such that $J\circ J=-\mathrm{Id}$. The
complexified Chevalley-Eilenberg algebra then becomes a bigraded algebra
$\mathcal{A}^{*}_{\mathfrak{g}_{\mathbb{C}}}=\bigoplus\mathcal{A}^{p,q}_{\mathfrak{g}}$
with

$\mathcal{A}^{p,q}_{\mathfrak{g}}:=\Lambda^{p}\left((\mathfrak{g^{\vee}_{\mathbb{C}}})^{1,0}\right)\wedge\Lambda^{q}\left((\mathfrak{g^{\vee}_{\mathbb{C}}})^{0,1}\right).$

A real inner product $\langle-,-\rangle$ on $\mathfrak{g}$ satisfying the
compatibility condition $\langle JX,JY\rangle=\langle X,Y\rangle$ extends to a
Hermitian inner product on
$\mathfrak{g}_{\mathbb{C}}=\mathfrak{g}\otimes\mathbb{C}$. The pointwise
version of Theorem 4.1 gives:

######  Corollary 5.3.

Let $(\mathfrak{g},J,\langle-,-\rangle)$ be a real Lie algebra of dimension
$2n$ together with an almost complex structure $J$ and a compatible metric.
Then $\mathcal{A}_{\mathfrak{g}}$ carries the structure of a commutative
$\mathrm{BV}_{\infty}$-algebra extending the commutative product. Moreover, if
$H^{2n}(\mathfrak{g})\cong\mathbb{R}$ then $H^{*}(\mathfrak{g})$ carries a
canonical homotopy hypercommutative algebra.

For a general almost complex Lie algebra, the exterior differential on
$\mathcal{A}_{\mathfrak{g}}^{*}$ decomposes into four components
$d={\bar{\mu}}+{\bar{\partial}}+\partial+\mu$. The integrability of $J$ is
measured by the Nuijenhuis tensor

$N_{J}(X,Y):=[X,Y]+J[X,JY]+J[X,JY]-[JX,JY]$

and $N_{J}\equiv 0$ if and only if $d={\bar{\partial}}+\partial$. In this
case, the pointwise version of Theorem 3.1 gives a Dolbeault version of the
above corollary:

######  Corollary 5.4.

Let $(\mathfrak{g},J,\langle-,-\rangle)$ be a real Lie algebra of dimension
$2n$ together with an integrable almost complex structure $J$ and a compatible
metric. Then $(\mathcal{A}_{\mathfrak{g}_{\mathbb{C}}},{\bar{\partial}})$
carries the structure of a commutative $\mathrm{BV}_{\infty}$-algebra
extending the commutative product. Moreover, if
$H^{n,n}(\mathfrak{g}_{\mathbb{C}})\cong\mathbb{C}$ then
$H^{*}_{\bar{\partial}}(\mathfrak{g}_{\mathbb{C}})$ carries a canonical
homotopy hypercommutative algebra.

######  Example 5.5 (Complex structure on the Kodaira-Thurston manifold).

Let us now consider the complex structure $J$ on
$\mathfrak{g}=\mathbb{R}\\{X,Y,Z,T\\}$ defined by $J(X)=Y$ and $J(Z)=-T$. Let
$A=X-iJX$ and $B=Z-iJZ$. The only-non trivial Lie bracket of $\mathfrak{g}$ is
$[X,Y]=-Z$, and we get

$[A,\overline{A}]=-i(B+\overline{B}).$

Therefore we may write
$\mathcal{A}^{*}_{\mathfrak{g}_{\mathbb{C}}}\cong\Lambda(a,b,\overline{a},\overline{b})$
with $db={\bar{\partial}}b=ia\overline{a}$ and
$d\overline{b}=\partial\overline{b}=ia\overline{a}$. The Dolbeault cohomology
reads:

$H_{\bar{\partial}}^{*,*}\cong\begin{array}[]{|c|c|c|c|}\hline\cr[\overline{a}\overline{b}]&[b\overline{a}\overline{b}]&[ab\overline{a}\overline{b}]\\\
\hline\cr[\overline{a}],[\overline{b}]&[a\overline{b}],[b\overline{a}]&[ab\overline{a}],[ab\overline{b}]\\\
\hline\cr 1&[a]&[ab]\\\ \hline\cr\end{array}$

Consider the standard inner product on $\mathfrak{g}$. This makes the obvious
representatives of the Dolbeault cohomology classes harmonic. The fundamental
$(1,1)$-form determined by this metric is

$\omega=\frac{i}{2}\left(a\bar{a}+b\bar{b}\right).$

The canonical deformation retract gives

$h(a\bar{a})=-ib\text{ and }h(a\bar{a}\bar{b})=-ib\bar{b}$

and zero otherwise. We may now compute

$\varphi_{1}=h\Delta-\iota\rho\Delta h,\text{ where
}\Delta=[\Lambda,{\bar{\partial}}]=-i(\partial^{*}+\tau^{*}).$

We obtain

$\varphi_{1}(a\overline{a})=-2i,\,\varphi_{1}(ab\overline{a})=4ib,\,\varphi_{1}(a\overline{a}\overline{b})=-4i\overline{b}\text{
and }\varphi_{1}(ab\overline{a}\overline{b})=-8ib\overline{b}.$

Using the expression for $m_{3}$ of Corollary 2.8, we get

$m_{3}(a,\overline{a},\overline{b})=\varphi_{1}(a\overline{a}\overline{b})-\varphi_{1}(a\overline{a})\overline{b}=-2i\overline{b},$

which induces a non-trivial strict hypercommutative operation in Dolbeault
cohomology.

###  5.4. Symplectic-Lagrangian structures

A non-degenerate Poisson structure on a real Lie algebra $\mathfrak{g}$ of
even dimension gives a symplectic form $\omega$ on its Chevalley-Eilenberg
algebra $\mathcal{A}_{\mathfrak{g}}$. A Lagrangian polarization for
$(\mathfrak{g},\omega)$ is just a decomposition
$\mathfrak{g}=\mathcal{L}\oplus\mathcal{L}^{\prime}$ such that
$\dim\mathcal{L}=\dim\mathcal{L}^{\prime}$ and
$\omega|_{\mathcal{L}}=\omega|_{\mathcal{L}^{\prime}}=0$. The subspace
$\mathcal{L}$ is integrable if it is closed under the Lie bracket of
$\mathfrak{g}$.

######  Example 5.6 (Kodaira-Thurston manifold: symplectic-Lagrangian case).

Taking $a=e=1$ in Example 5.2 and zero otherwise, we obtain left-invariant
symplectic form $\omega=yt+xz$ on the Kodaira-Thurston manifold, which has an
integrable Lagrangian subspace $L=\mathbb{R}\\{X,T\\}$, in bi-degree $(0,1)$.
In this case, we have $d={\bar{\partial}}$ since the only nonzero differential
is $dz={\bar{\partial}}z=xy\in{\mathcal{A}}^{1,1}$. Thus, the operation
$m_{3}$ obtained in Example 5.2 is also a nontrivial hypercommutative
operation in ${\bar{\partial}}$-cohomology of the symplectic-Lagrangian case.

The above example is somewhat trivial, as the hypercommutative algebra defined
on the Lagrangian structure does not refine the one defined on the underlying
Poisson structure. A more interesting example is the following:

######  Example 5.7 (Iwasawa manifold: symplectic-Lagrangian case).

The Iwasawa manifold is a 6-dimensional compact nilmanifold with real
nilpotent Lie algebra
$\mathfrak{g}=\mathbb{R}\\{X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}\\}$, defined by
the non-trivial Lie brackets

$[X_{1},X_{3}]=X_{5},\,[X_{2},X_{4}]=-X_{5},\,[X_{1},X_{4}]=X_{6}\text{ and
}[X_{2},X_{3}]=X_{6}.$

The symplectic form $\omega=x_{1}x_{6}+x_{2}x_{5}+x_{3}x_{4}$ with
$\mathcal{L}=\mathbb{R}\\{X_{1},X_{3},X_{5}\\}$ is Lagrangian and integrable,
while $\mathcal{L}^{\prime}=\mathbb{R}\\{X_{2},X_{4},X_{6}\\}$ is Lagrangian
but not integrable.

With this bigrading, $x_{1},x_{3},x_{5}\in{\mathcal{A}}^{0,1}$,
$x_{2},x_{4},x_{6}\in{\mathcal{A}}^{1,0}$ and the only nonzero differentials
are

$\displaystyle dx_{5}$
$\displaystyle=x_{3}x_{1}-x_{4}x_{2},\quad\textrm{i.e.}\quad\mu
x_{5}=-x_{4}x_{2}\quad\textrm{and}\quad{\bar{\partial}}x_{5}=x_{3}x_{1},$
$\displaystyle dx_{6}$
$\displaystyle=x_{3}x_{2}+x_{4}x_{1},\quad\textrm{i.e.}\quad\quad{\bar{\partial}}x_{6}=x_{3}x_{2}+x_{4}x_{1}.$

Note that ${\bar{\mu}}=0$ and therefore ${\bar{\partial}}^{2}=0$, so
Proposition 4.7 applies.

Clearly,
$H^{1}_{{\bar{\partial}}}\cong\mathbb{R}\\{[x_{1}],[x_{2}],[x_{3}],[x_{4}]\\}$,
and for classes $a,b,c\in H^{1}_{{\bar{\partial}}}$

$m_{3}(a,b,c)=\varphi_{1}(abc)\quad\textrm{where}\quad\varphi_{1}=h\Delta_{1}-\iota\rho\Delta_{1}h,$

where $\Delta_{1}=\partial^{\omega}=[\Lambda,{\bar{\partial}}]$. For the
examples below, only the second summand $\iota\rho\Delta_{1}h$ will be non-
zero. We compute

$m_{3}(x_{2},x_{3},x_{4})=\varphi_{1}(x_{2}x_{3}x_{4})=-\iota\rho\Delta_{1}h(x_{2}x_{3}x_{4})=\iota\rho\Delta_{1}(x_{4}x_{6})=\iota\rho(x_{2})=x_{2}.$

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