# Formality of hypercommutative algebras of Kähler
and Calabi-Yau manifolds

Joana Cirici  and  Geoffroy Horel  Departament de Matemàtiques i Informàtica,
Universitat de Barcelona
Gran Via 585
08007 Barcelona, Spain / Centre de Recerca Matemàtica, Edifici C, Campus
Bellaterra, 08193 Bellaterra, Spain<EMAIL_ADDRESS>Université Sorbonne Paris
Nord, Laboratoire Analyse, Géométrie et Applications, CNRS (UMR 7539), 93430,
Villetaneuse, France<EMAIL_ADDRESS>
###### Abstract.

Any Batalin-Vilkovisky algebra with a homotopy trivialization of the BV-
operator gives rise to a hypercommutative algebra structure at the cochain
level which, in general, contains more homotopical information than the
hypercommutative algebra introduced by Barannikov and Kontsevich on
cohomology. In this paper, we use the purity of mixed Hodge structures to show
that the canonical hypercommutative algebra defined on any compact Calabi-Yau
manifold is formal. We also study related hypercommutative algebras associated
to compact Kähler and Hermitian manifolds.

J. Cirici acknowledges the Serra Húnter Program, the Spanish State Research
Agency (CEX2020-001084-M and PID2020-117971GB-C22) and the Govern de Catalunya
(2021-SGR-00697). G. Horel acknowledges support from a grant of the Romanian
Ministry of Education and Research, CNCS - UE- FISCDI, project number PN-
III-P4-ID-PCE-2020-2798. Both authors are funded by the ANR-20-CE40-0016
HighAGT and the CNRS IEA00979.

##  1\. Introduction

Motivated by the mirror symmetry program, Barannikov and Kontsevich defined a
canonical hypercommutative algebra structure on the cohomology of any compact
Calabi-Yau manifold [BK98]. More generally, they gave a recipe for obtaining
such a structure on the cohomology of any Batalin-Vilkovisky algebra
satisfying the so-called $d\Delta$-condition. This recipe, described in detail
by Manin [Man99a], [Man99b], arises as an abstract interpretation of the Tian-
Todorov lemma leading to unobstructedness of the deformation theory of Calabi-
Yau manifolds. It has been successfully applied to various geometric settings,
such as symplectic manifolds satisfying Hard Lefschetz [Mer98] and compact
Kähler manifolds [CZ00].

More recently, Drummond-Cole and Vallette [DCV13] refined the above recipe by
obtaining a homotopy hypercommutative structure on cohomology which allows one
to recover the homotopy type of the original Batalin-Vilkovisky algebra. Its
truncation to a strict hypercommutative algebra is the one defined by
Barannikov and Kontsevich via deformation theory. The scope of this refined
theory is wider, in the sense that the $d\Delta$-condition on the initial
Batalin-Vilkovisky algebra is replaced by a weaker condition called Hodge-to-
de-Rham degeneration. This turns out to be a sufficient condition for the
existence of a homotopy trivialization of the Batalin-Vilkovisky operator. The
emergence of a hypercommutative algebra structure from this data is a
reflection of the fact that the homotopy quotient of the framed little discs
operad by rotations is weakly equivalent to the operad of moduli spaces of
stable genus 0 curves [DC14b]. In [KMS13], Khoroshkin, Markarian and Shadrin
proved an algebraic counterpart of this result, providing an explicit quasi-
isomorphism between the operad
${\mathrm{Hycom}}=\\{H_{*}(\overline{M}_{0,\bullet+1})\\}$ governing
hypercommutative algebras and the homotopy quotient ${\mathrm{BV}}/\Delta$
governing Batalin-Vilkovisky algebras enhanced with a trivialization of the
BV-operator. In particular, the hypercommutative structure associated to a
Batalin-Vilkovisky algebra exists at the cochain level, rather than on
cohomology.

As shown in [DCV13], even under the stronger $d\Delta$-condition, the
hypercommutative structure at the cochain level is not formal in general. The
authors ask whether, in the case of Calabi-Yau manifolds, the hypercommutative
structure obtained by Barannikov and Kontsevich on its cohomology, contains
all the information of the homotopy type of the original Batalin-Vilkovisky
algebra, thus giving formality. Drummond-Cole gave an affirmative answer in
the case of low-dimensional Calabi-Yau manifolds [DC14a]. In this paper, we
give a positive answer in the general case of arbitrary dimension.

The standard proof of the well-known Formality Theorem for the de Rham algebra
of compact Kähler manifolds [DGMS75] uses the fact that the
$\partial{\overline{\partial}}$-condition gives quasi-isomorphisms of
commutative algebras

$(A,{\overline{\partial}},\wedge)\hookleftarrow(\mathrm{Ker}(\partial),{\overline{\partial}},\wedge)\twoheadrightarrow(H,0,\wedge)$

In the case of a Batalin-Vilkovisky algebra $(A,d,\cdot,\Delta)$ satisfying
the $d\Delta$-condition one is led to consider an analogous string of quasi-
isomorphisms

$(A,d,\cdot,\Delta)\hookleftarrow(\mathrm{Ker}(\Delta),d,\cdot,0)\twoheadrightarrow(H,0,\cdot,0).$

However, since $\mathrm{Ker}(\Delta)$ is not a commutative algebra with
respect to the commutative product, the above maps do not lead to formality of
the BV-algebra, but only of the associated Lie algebra, whose Lie bracket
measures the failure of $\Delta$ to be a derivation. The formality of such Lie
algebra immediately leads to the unobstructedness of its associated
deformation theory.

A more abstract approach to the formality of Kähler manifolds uses the fact
that their cohomology carries pure Hodge structures, and that the filtrations
defining such Hodge structures exist at the cochain level. More generally, the
theory of mixed Hodge structures also leads to formality results in certain
situations. Both the operad of Batalin-Vilkovisky algebras and the
hypercommutative operad may be described as homologies of geometric objects
and as such, they carry natural mixed Hodge structures satisfying certain
purity properties. Moreover, the quasi-isomorphism of operads
${\mathrm{Hycom}}\longrightarrow{\mathrm{BV}}/\Delta$, relating
hypercommutative algebras with Batalin-Vilkovisky algebras with a
trivialization of the BV-operator, is compatible with such mixed Hodge
structures. It therefore makes sense to talk about hypercommutative algebras
in mixed Hodge structures and, more generally, in mixed Hodge complexes. We
actually consider $\mathbb{C}$-mixed Hodge objects, for which there is no real
structure and there are two (possibly unrelated) filtrations $F$ and
$\overline{F}$ that are $n$-opposed in each $n$-graded piece of the weight
filtration. In this paper, we adapt the “purity implies formality” machinery
of [CH20] to show that any hypercommutative algebra in $\mathbb{C}$-mixed
Hodge complexes whose Hodge structure is pure in cohomology, is formal as a
hypercommutative algebra in complexes of vector spaces. We study three
geometric instances where this situation applies:

### I. Kähler manifolds

Consider the complex de Rham algebra
$\mathcal{A}:=\mathcal{A}^{*}_{\mathrm{dR}}(M)\otimes_{\mathbb{R}}\mathbb{C}$
of a complex manifold $M$. Given a Hermitian metric one obtains a
decomposition $d^{*}={\overline{\partial}}^{*}+\partial^{*}$ of the formal
adjoint to $d={\overline{\partial}}+\partial$. Note that
${\overline{\partial}}^{*}$ and $\partial^{*}$ are not derivations in general.
Instead, as shown by Cao-Zhou in [CZ00] and [CZ99], in the case of a Kähler
manifold, the tuples

$\mathcal{A}_{\mathrm{Dol}}:=(\mathcal{A},d={\overline{\partial}},\wedge,\Delta=-i\partial^{*})\text{
and
}\mathcal{A}_{\mathrm{dR}}:=(\mathcal{A},d={\overline{\partial}}+\partial,\wedge,\Delta=i({\overline{\partial}}^{*}-\partial^{*}))$

are BV-algebras which satisfy the $d\Delta$-condition. Furthermore, there is a
canonical trivialization of the operator $\Delta$, given in terms of the Green
operator. This allows one to talk about canonically induced hypercommutative
structures in cohomology.

The following result promotes the Formality Theorem of [DGMS75] of compact
Kähler manifolds to the hypercommutative setting:

######  Theorem 4.4.

For any compact Kähler manifold, the canonical hypercommutative structures
defined on $\mathcal{A}_{\mathrm{Dol}}$ and $\mathcal{A}_{\mathrm{dR}}$ are
formal and quasi-isomorphic.

In particular, the canonically induced hypercommutative algebra structures on
Dolbeault and complex de Rham cohomology on any compact Kähler manifold are
isomorphic and carry no higher hypercommutative operations.

### II. Calabi-Yau manifolds

Let now $M$ be a compact Calabi-Yau manifold of complex dimension $m$ and
consider the complex bigraded algebra given by
$\mathcal{L}^{p,q}:=\Gamma(M,\Lambda^{p}T\otimes\Lambda^{q}\overline{T}^{*}),$
where $T$ denotes the holomorphic tangent bundle of $M$. The algebra structure
is determined by the exterior product $\wedge$. There is an isomorphism of
bigraded vector spaces $\eta:\mathcal{L}^{p,q}\to\mathcal{A}^{m-p,q}$ given by
the contraction $\alpha\mapsto\alpha\vdash\Omega$ with the nowhere vanishing
holomorphic $m$-form $\Omega$. Note that such isomorphism does not preserve
the product structures. In fact, the exterior product on $\mathcal{A}$ has
different bidegree. Consider on $\mathcal{L}$ the operators
${\overline{\partial}}_{\eta}:=\eta^{-1}\circ{\overline{\partial}}\circ\eta$
and $\partial_{\eta}:=\eta^{-1}\circ\partial\circ\eta$. Barannikov and
Kontsevich showed in [BK98] that the tuple

$\mathcal{L}_{\mathrm{Dol}}:=(\mathcal{L},d={\overline{\partial}}_{\eta},\wedge,\Delta=\partial_{\eta})$

is a Batalin-Vilkovisky algebra satisfying the $d\Delta$-condition. As
conjectured by Cao-Zhou in [CZ00], this algebra should correspond, via mirror
symmetry, to the algebra $\mathcal{A}_{\mathrm{Dol}}$ introduced earlier. With
this idea in mind, and with the objective to obtain an object in mixed Hodge
complexes, we introduce a “de Rham version” of this Batalin-Vilkovisky algebra

$\mathcal{L}_{\mathrm{dR}}:=(\mathcal{L}^{*,*},d:={\overline{\partial}}_{\eta}+\partial^{*}_{\eta},\wedge,\Delta:=\partial_{\eta}-{\overline{\partial}}^{*}_{\eta})$

which, conjecturally, corresponds to $\mathcal{A}_{\mathrm{dR}}$ through
mirror symmetry. The incorporation of the extra terms in both the differential
and the BV-operator allow for the spectral sequences associated to both the
column and row filtrations to degenerate at the first stage. This is a
necessary condition for obtaining a hypercommutative algebra in
$\mathbb{C}$-mixed Hodge complexes. We show:

######  Theorem 4.9.

For any compact Calabi-Yau manifold, the canonical hypercommutative structures
defined on $\mathcal{L}_{\mathrm{Dol}}$ and $\mathcal{L}_{\mathrm{dR}}$ are
formal and quasi-isomorphic.

### III. Hermitian manifolds

A third geometric situation arises by considering, on a complex manifold, the
tuple $(\mathcal{A},{\overline{\partial}},\wedge,\partial)$ as a BV-algebra.
In this case, $\partial$ is obviously a derivation and so this is a BV-algebra
whose associated Lie bracket is trivial. In the compact Kähler case, the
$\partial{\overline{\partial}}$-condition immediately implies that such BV-
algebra is formal and the induced hypercommutative structure is trivial. The
interesting situation is therefore outside the Kähler setting. We show that
for a compact Hermitian manifold, the induced hypercommutative algebra
structure is not formal in general, and contains more information than the
purely commutative structure. We do this by computing higher operations on the
Kodaira-Thurston manifold.

The paper is organized as follows. In Section 2 we collect preliminaries on
Batalin-Vilkovisky and hypercommutative algebras. In particular, we recall,
following [KMS13], the procedure for obtaining a hypercommutative algebra
structure at the cochain level, from a Batalin-Vilkovisky algebra with a
trivialization of $\Delta$. Section 3 concerns mixed Hodge structures and
formality. Mainly, we adapt the “purity implies formality” result of [CH20] to
the setting of complex mixed Hodge structures (rather than real mixed Hodge
structures) and prove a general result of formality for operadic algebras with
pure cohomology, whose governing operad is also pure. Lastly, in Section 4 we
study the three main geometric situations mentioned above.

### Acknowledgments

We would like to thank Vladimir Dotsenko and Fernando Muro for answering our
questions and Gabriel Drummond-Cole for introducing us to the problem that led
to this work.

This project started during the authors’s research stay at the Institut
Mittag-Leffler, within the program Higher algebraic structures in algebra,
topology and geometry and continued during the research stay of the second
author at the Institut de Matemàtiques de la Universitat de Barcelona (IMUB).
We thank both institutes for the great working conditions and hospitality.

##  2\. From Batalin-Vilkovisky to hypercommutative algebras

In this preliminary section, we collect some results on double complexes,
Batalin-Vilkovisky algebras and hypercommutative algebras.

###  2.1. Double complexes and the $d\Delta$-condition

Let $(A,d,\Delta)$ be a double complex of vector spaces, with $d$ and $\Delta$
linear operators of degree $+1$ and $-1$ respectively and satisfying

$d^{2}=\Delta^{2}=d\Delta+\Delta d=0.$

######  Definition 2.1.

The $d\Delta$-condition is satisfied if and only if

$\mathrm{Ker}(d)\cap\mathrm{Im}(\Delta)=\mathrm{Ker}(\Delta)\cap\mathrm{Im}(d)=\mathrm{Im}(d\Delta).$

Associated to a double complex there are various cohomologies: aside from the
$d$\- and $\Delta$-cohomologies

$H_{d}:={{\mathrm{Ker}(d)}\over{\mathrm{Im}(d)}}\quad\text{ and }\quad
H_{\Delta}:={{\mathrm{Ker}(\Delta)}\over{\mathrm{Im}(\Delta)}}$

we also have the Bott-Chern $H_{BC}$ and Aeppli $H_{A}$ cohomologies, given
respectively by

$H_{BC}:={{\mathrm{Ker}(d)\cap\mathrm{Ker}(\Delta)}\over{\mathrm{Im}(d\Delta)}}\quad\text{
and }\quad
H_{A}:={{\mathrm{Ker}(d\Delta)}\over{\mathrm{Im}(d)+\mathrm{Im}(\Delta)}}.$

The identity induces maps

$\textstyle{H_{BC}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{\Delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{A}}$

which are isomorphisms if and only if the $d\Delta$-condition is satisfied
(see Lemma 5.15 of [DGMS75]). In particular, in this case there is “only one
cohomology”, which we just denote by $H$. The following is straightforward:

######  Lemma 2.2.

Let $(A,d,\Delta)$ be a double complex satisfying the $d\Delta$-condition.
Then:

  1. (i)

The inclusions of complexes $(\mathrm{Ker}(\Delta),d)\hookrightarrow(A,d)$ and
$(\mathrm{Ker}(d),\Delta)\hookrightarrow(A,\Delta)$ are quasi-isomorphisms.

  2. (ii)

The surjections of complexes $(\mathrm{Ker}(\Delta),d)\twoheadrightarrow(H,0)$
and $(\mathrm{Ker}(d),\Delta)\twoheadrightarrow(H,0)$ are quasi-isomorphisms.

######  Definition 2.3.

A homotopy transfer diagram for a cochain complex $(A,d)$ is given by a
diagram

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

where $\iota$ and $\rho$ are morphisms of complexes such that
$\rho\iota=\mathrm{Id}$ and $h$ is a homotopy of morphisms of cochain
complexes from $\iota\rho$ to the identity, so that
$dh+hd=\mathrm{Id}-\iota\rho$.

######  Lemma 2.4.

Let $(A,d,\Delta)$ be a double complex satisfying the $d\Delta$-condition.
Then there is homotopy transfer diagram $(\iota,\rho,h)$ such that
$\Delta\iota=0$, $\rho\Delta=0$ and $h\Delta+\Delta h=0.$

###### Proof.

By [DGMS75, Proposition 3.17], the $d\Delta$-condition allows one to write $A$
in each degree as a direct sum decomposition

$A^{n}=H^{n}\oplus S^{n}\oplus dS^{n-1}\oplus\Delta S^{n+1}\oplus d\Delta
S^{n},$

where $H^{n}$ is the cohomology of $A$ in degree $n$ and the differentials $d$
and $\Delta$ are given by

$d(x,y,dz,\Delta w,d\Delta t)=(0,0,dy,0,d\Delta w)\text{ and
}\Delta(x,y,dz,\Delta w,d\Delta t)=(0,0,0,\Delta y,-d\Delta z).$

Define $\iota:H\to A$ and $\rho:A\to H$ respectively by inclusion and
projection to the first component. Define $h:A^{p,q}\to A^{p,q-1}$ by
$h(x,y,dz,\Delta w,d\Delta t):=(0,z,0,\Delta t,0).$ ∎

###  2.2. Batalin-Vilkovisky algebras

A Batalin-Vilkovisky algebra is a Gerstenhaber algebra together with a
compatible unary operation squaring to zero. While Gerstenhaber algebras may
be identified with algebras over the homology of the little disks operad, the
operad ${\mathrm{BV}}$ encoding Batalin-Vilkovisky algebras is given by the
homology of the framed little disks operad [Get94].

######  Definition 2.5.

A BV-algebra structure on a cochain complex $(A,d)$ is given by a symmetric
binary product $\cdot:A\times A\longrightarrow A$ of degree 0, together with a
unary operator $\Delta:A\to A$ of degree -1, such that:

  1. (1)

The triple $(A,d,\Delta)$ is a double complex:

$d^{2}=d\Delta+\Delta d=\Delta^{2}=0.$

  2. (2)

The triple $(A,d,\cdot)$ is a dg-commutative algebra:

$x\cdot y=(-1)^{|x|\cdot|y|}y\cdot x\quad\text{ and }\quad d(x\cdot y)=dx\cdot
y+(-1)^{|x|}x\cdot dy.$

  3. (3)

The binary operation $[-,-]:A\times A\longrightarrow A$ defined by

$[x,y]:=(-1)^{|x|}(\Delta(x\cdot y)-(\Delta(x)\cdot
y)-(-1)^{|x|}x\cdot\Delta(y))$

defines, for all $x$, a derivation $[x,-]$ of degree $|x|-1$ with respect to
the commutative product:

$[x,(y\cdot z)]=[x,y]\cdot z+(-1)^{(|x|-1)\cdot|y|}y\cdot[x,z].$

######  Remark 2.6.

Note that the last condition is equivalent to imposing the relation

$\begin{array}[]{ll}\Delta(x\cdot y\cdot z)=&\Delta(x\cdot y)\cdot
z+(-1)^{|x|}x\cdot\Delta(y\cdot z)+(-1)^{(|x|-1)\cdot|y|}y\cdot\Delta(x\cdot
z)-\\\ &-\Delta(x)\cdot y\cdot z-(-1)^{|x|}x\cdot\Delta(y)\cdot
z-(-1)^{|x|+|y|}x\cdot y\cdot\Delta(z)\end{array}$

appearing in [Get94] to define a Batalin-Vilkovisky algebra. In particular, by
Proposition 1.2 of [Get94], given a BV-algebra $(A,d,\cdot,\Delta)$ with
associated bracket $[-,-]$, then $(A,d,\cdot,[-,-])$ is a Gerstenhaber algebra
and so $(A,d,[-,-])$ is a dg-Lie algebra.

######  Remark 2.7.

The $d\Delta$-condition implies in particular that the associated dg-Lie
algebra $(A,[-,-],d)$ is both formal and quasi-abelian. Indeed, we have quasi-
isomorphisms of dg-Lie algebras

$(A,[-,-],d)\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}(\mathrm{Ker}(\Delta),[-,-],d)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(H,0,0).$

Note however that the above are not maps of BV-algebras: the Lie bracket is an
obstruction for $\Delta$ to be a derivation, and so in general
$(\mathrm{Ker}(\Delta),\wedge)$ is not a commutative algebra. Therefore these
maps do not give formality of the BV-algebra.

###  2.3. Hypercommutative algebras

Following [KMS13], we review the equivalence between the category of Batalin-
Vilkovisky algebras enhanced with the trivialization of $\Delta$ and the
category of hypercommutative algebras.

######  Definition 2.8.

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 $S=\\{s_{1},\cdots,s_{m}\\}$.

######  Remark 2.9.

We may think of the operations in a hypercommutative algebra as the Taylor
coefficients of a formal deformation of the commutative product: the above
relations are equivalent to asking that the binary operation

$(a,b)\mapsto\sum{\hbar^{n}\over n!}m_{n+2}(a,b,x,\cdots,x)\in A[[\hbar]]$

is associative for any $x$ (see [Get95]).

The operad ${\mathrm{Hycom}}$ encoding hypercommutative algebras is a cyclic
operad which may be identified with the homology
$H_{*}(\overline{\mathcal{M}}_{0,\bullet+1})$ of the moduli spaces of stable
genus zero curves and marked points [Get95], [KM97]. With this identification,
the arity $n$ generator in $\mathrm{Hycom}$ corresponds to the fundamental
class

$[\overline{\mathcal{M}}_{0,n+1}]\in
H_{2(n-2)}(\overline{\mathcal{M}}_{0,n+1}).$

In [KMS13], Khoroshkin, Markarian and Shadrin give a quasi-isomorphism of
operads

$\mathrm{Hycom}\longrightarrow{\mathrm{BV}}/\Delta,$

where ${\mathrm{BV}}/\Delta$ denotes the homotopy quotient of the operad
${\mathrm{BV}}$ by the generator $\Delta$.

The operad ${\mathrm{BV}}/\Delta$ is defined by adding, to ${\mathrm{BV}}$,
generators $\varphi_{i}$ of degree $-2i$, for $n\geq 1$ together with a
differential $\delta$ defined as follows: write $\varphi(z):=\sum_{n\geq
1}\varphi_{i}z^{n}$, where $z$ is a formal parameter. The expansion of the
equation

(1) $\exp(\varphi(z))d=(d+\Delta z)\exp(\varphi(z))$

gives the various terms

$\begin{array}[]{l}\Delta=[d,\varphi_{1}],\\\ 0=[d,\varphi_{2}]+{1\over
2}[[d,\varphi_{1}],\varphi_{1}],\\\
0=[d,\varphi_{3}]+[[d,\varphi_{1}],\varphi_{2}]+[[d,\varphi_{2}],\varphi_{1}]+{1\over
6}[[[d,\varphi_{1}],\varphi_{1}],\varphi_{1}],\\\ \cdots\end{array}$

The differerential $\delta$ on $\varphi_{i}$ is then defined by the equation
$\delta\varphi_{i}:=[d,\varphi_{i}]$. The lower terms are:

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

The map of operads $\mathrm{Hycom}\longrightarrow{\mathrm{BV}}/\Delta$ is
defined by assigning, to each generator $m_{n}\in{\mathrm{Hycom}}(n)$, an
element $\theta_{i}$ in ${\mathrm{BV}}/\Delta(n)$ represented by a finite sum
of combinations of the arity $2$ product of ${\mathrm{BV}}$ together with
generators $\varphi_{j_{1}},\cdots,\varphi_{j_{k}}$ with
$j_{1}+\cdots+j_{k}=n-2$ (we refer to [KMS13, Section 1.2] for details).

######  Example 2.10.

The lowest terms are $\theta_{2}(x,y)=x\cdot y$ and

$\begin{array}[]{ll}\theta_{3}(x,y,z)=&\varphi_{1}(x\cdot y\cdot
z)+\varphi_{1}(x)\cdot y\cdot z+x\cdot\varphi_{1}(y)\cdot z+x\cdot
y\cdot\varphi_{1}(z)-\\\ &-\varphi_{1}(x\cdot y)\cdot
z-(-1)^{|y|\cdot|z|}\varphi_{1}(x\cdot z)\cdot y-x\cdot\varphi_{1}(y\cdot
z).\end{array}$

The above description gives a method for producing a hypercommutative algebra
from any BV-algebra $(A,d,\wedge,\Delta)$ enhanced with a homotopy that
trivializes the BV-operator: assume there is a homotopy transfer diagram
$(\iota,\rho,h)$ for the complex $(A,d)$, satisfying the Hodge-to-de-Rham
degeneration condition

$\rho(\Delta h)^{k-1}\Delta\iota=0\quad\text{ for all }\quad k\geq 1.$

Then a solution $\varphi(z):=\sum_{n\geq 0}\varphi_{n}z^{n}$ to Equation (1)
is given by

$\varphi_{n}={(h\Delta)^{n}\over
n}-n\sum_{\ell=1}^{n}{(h\Delta)^{\ell-1}\iota\rho(\Delta
h)^{n-\ell+1}\over\ell}$

(see Remark 2.2 of [DSV15]). In particular, we get the following proposition.

######  Proposition 2.11.

Let $(A,d,\wedge,\Delta)$ be a BV-algebra together with homotopy transfer
diagram $(\iota,\rho,h)$ for the complex $(A,d)$, satisfying
$\Delta\iota=\rho\Delta=h\Delta+\Delta h=0.$ Then $(A,d)$ carries a
hypercommutative structure, where the operation $m_{n}$ of degree $4-2n$ is
given by a finite composition of the commutative product $\wedge$ together
with copies of the unary operation $h\Delta$.

###### Proof.

It suffices to check that, in this case, we have $\varphi_{1}=h\Delta$ and
$\varphi_{n}=0$ for all $n>1$. ∎

##  3\. Weights and formality

We explain in this section how to obtain formality using the theory of
weights. We adapt the theory developed in [CH20] to the setting of complex
mixed Hodge structures, and prove a result of formality for operadic algebras
whose cohomology is pure.

###  3.1. Complex mixed Hodge structures

Classically, Hodge structures are defined on a vector space $V$ defined over a
subfield of the real numbers. Then, on the complexification
$V_{\mathbb{C}}:=V\otimes\mathbb{C}$, there is a notion of complex conjugation
and having a pure Hodge structure of weight $n$ on $V$ is equivalent to having
a filtration $F$ on $V_{\mathbb{C}}$ which is $n$-opposed to its complex
conjugate filtration. We will instead be considering complex Hodge structures.
These are given by pairs of $n$-opposed filtrations on a complex vector space,
which are not necessarily complex conjugate to each other.

######  Definition 3.1.

Let $n$ be an integer. Two decreasing filtrations $F$ and $\overline{F}$ on a
vector space $V$ are said to be $n$-opposed if

$Gr^{p}_{F}Gr^{q}_{\overline{F}}V=0\text{ for all }p+q\neq n.$

This is equivalent to asking that $F^{p}V\oplus\overline{F}{}^{q}V\cong V$ for
all $p,q$ such that $p+q=n+1$.

######  Definition 3.2.

A $\mathbb{C}$-mixed Hodge structure is a complex vector space $V$ with an
increasing filtration $W$ and two decreasing filtrations $F$ and
$\overline{F}$ such that, for all $n$, the filtrations $F$ and $\overline{F}$
induced on $Gr_{n}^{W}V:=W_{n}V/W_{n-1}V$ are $n$-opposed:

$Gr^{p}_{F}Gr^{q}_{\overline{F}}Gr_{n}^{W}V=0\text{ for all }p+q+n\neq 0.$

Denote by ${\mathsf{MHS}}_{\mathbb{C}}$ the category of $\mathbb{C}$-mixed
Hodge structures. It is an abelian symmetric monoidal category. Morphisms in
this category, defined as morphisms of complex vector spaces compatible with
all filtrations, are in fact strictly compatible with each filtration. This
follows from Deligne’s splitting lemma, which is central in the study of mixed
Hodge structures:

######  Lemma 3.3 (Deligne’s splitting lemma ([Del71], Lemma 1.2.11)).

Let $(V,W,F,\overline{F})$ be a $\mathbb{C}$-mixed Hodge structure. There are
functorial decompositions

$V=\bigoplus I^{p,q}_{i}\quad\text{ for }\quad i=0,1$

such that

$F^{p}V=\bigoplus_{p^{\prime}\geq
p}I^{p^{\prime},q^{\prime}}_{0},\quad\overline{F}{}^{q}V=\bigoplus_{q^{\prime}\geq
q}I^{p^{\prime},q^{\prime}}_{1}\quad\text{ and }\quad W_{m}V=\bigoplus_{m\leq
p+q}I^{p,q}_{i}.$

In order to obtain mixed Hodge structures on the cohomology of complex
algebraic varieties, Deligne [Del74] introduced mixed Hodge complexes as
objects somewhat more flexible than complexes of mixed Hodge structures. The
following definition is a shifted version of the corresponding notion in the
setting of $\mathbb{C}$-mixed Hodge structures.

######  Definition 3.4.

A $\mathbb{C}$-mixed Hodge complex is a cochain complex $(A,d)$ over
$\mathbb{C}$ with three filtrations $W$, $F$ and $\overline{F}$ satisfying:

  1. $(H_{0})$

The cohomology $H^{*}(A)$ is of finite type.

  2. $(H_{1})$

The differential is strictly compatible with the three filtrations $W$, $F$
and $\overline{F}$.

  3. $(H_{2})$

For all $p$, the filtrations induced by $F$ and $\overline{F}$ on
$H^{n}(Gr_{p}^{W}A)$ are $p$-opposed.

Denote by ${\mathrm{MHC}}_{\mathbb{C}}$ the category $\mathbb{C}$-mixed Hodge
complexes. Morphisms are given by morphisms of cochain complexes compatible
with all filtrations. The cohomology of any $\mathbb{C}$-mixed Hodge complex
is a graded $\mathbb{C}$-mixed Hodge structure. Also, there is an inclusion
$\mathrm{Ch^{*}({{\mathsf{MHS}}_{\mathbb{C}}})}\hookrightarrow{\mathrm{MHC}}_{\mathbb{C}}$.
Note that given a mixed Hodge complex in the sense of Deligne, one may always
get a $\mathbb{C}$-mixed Hodge complex by taking the décalage of the weight
filtration on its complex part.

Beilinson gave an equivalence of categories between the derived category of
mixed Hodge structures and the homotopy category of mixed Hodge complexes. In
[CH20] we gave a symmetric monoidal enhancement of this result in terms of
$\infty$-categories. We prove an analogous result for $\mathbb{C}$-mixed Hodge
complexes.

######  Proposition 3.5.

The inclusion
$\mathrm{Ch^{*}({{\mathsf{MHS}}_{\mathbb{C}}})}\hookrightarrow{\mathrm{MHC}}_{\mathbb{C}}$
induces an equivalence of symmetric monoidal $\infty$-categories.

###### Proof.

Since both $\infty$-categories are stable and the inclusion is exact and
symmetric monoidal, it suffices to prove that the inclusion induces an
equivalence of homotopy categories

$\mathcal{D}({\mathsf{MHS}}_{\mathbb{C}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Ho}({\mathrm{MHC}}_{\mathbb{C}}).$

This follows analogously to Beilinson’s equivalence for the case of real mixed
Hodge structures (see [Beĭ86], see also [CG16]). The only minor difference is
the proof that every $\mathbb{C}$-mixed Hodge complex $(A,d,W,F,\overline{F})$
is quasi-isomorphic to its cohomology, giving that the inclusion induces an
essentially surjective functor at the homotopy level. We explain this below.

Since $H^{n}(A)$ is a $\mathbb{C}$-mixed Hodge structure, for every $n\geq 0$
there are decompositions

$H^{n}(A)=\bigoplus H^{p,q}_{i}\text{ for }i=0,1$

with the properties of Lemma 3.3. Take sections $\sigma:H^{p,q}_{0}\to
Z^{n}(A)$ and $\overline{\sigma}:H^{p,q}_{1}\to H^{n}(A)$. Since the spectral
sequence associated to any of the filtrations $W$, $F$ and $\overline{F}$
degenerates at $E_{1}$, we obtain a morphism of complexes $\sigma:H^{n}(A)\to
A$ compatible with $W$ and $F$ and a morphism of complexes
$\overline{\sigma}:H^{n}(A)\to A$ compatible with $W$ and $\overline{F}$. Now,
for any cohomology class $x\in H^{n}(A)$, the difference
$(\sigma-\overline{\sigma})(x)$ is a coboundary. The strictness of $d$ with
respect to $W$ ensures that if $x\in W_{p}$, then there is $y\in W_{p}$ with
$(\sigma-\overline{\sigma})(x)=dy$. Therefore the map $\Sigma(x):=y$ gives a
homotopy $\Sigma:\sigma\simeq\overline{\sigma}$ compatible with $W$. We next
strictify the triple $(\sigma,\overline{\sigma},\Sigma)$ using the mapping
cylinders of $\sigma$ and $\overline{\sigma}$. The mapping cylinder of
$\sigma$ carries compatible filtrations $W$ and $F$, given by

$W_{m}F^{p}\mathrm{Cyl}^{n}(\sigma):=W_{m}F^{p}H^{n+1}(A)\oplus
W_{m}F^{p}A^{n}\oplus W_{m}F^{p}H^{n}(A)$

with differential $d(x,a,x^{\prime})=(0,da-\sigma x,0)$. The mapping cylinder
for $\overline{\sigma}$ is defined analogously, and carries filtrations $W$
and $\overline{F}$. There is an isomorphism of complexes compatible with $W$

$\varphi:\mathrm{Cyl}(\sigma)\to\mathrm{Cyl}(\overline{\sigma})\quad\text{given
by}\quad(x,a,x^{\prime})\mapsto(x,a-\Sigma(x),x^{\prime}).$

The filtration $\overline{F}$ defined on $\mathrm{Cyl}(\sigma)$ by

$\overline{F}{}^{p}\mathrm{Cyl}(\sigma):=\varphi^{-1}(\overline{F}{}^{p}\mathrm{Cyl}(\overline{\sigma}))$

is compatible with the differential and so
$(\mathrm{Cyl}(\sigma),d,W,F,\overline{F})$ is a trifiltered complex. The
inclusions $i:H^{*}(A)\to\mathrm{Cyl}(\sigma)\leftarrow A:j$ are compatible
with the three filtrations and induce isomorphisms in cohomology. In
particular, the axioms $(H_{0})-(H_{2})$ for a $\mathbb{C}$-mixed Hodge
complex are verified by $\mathrm{Cyl}(\sigma)$ and so $A$ and $H^{*}(A)$ are
isomorphic in $\mathrm{Ho}({\mathrm{MHC}}_{\mathbb{C}})$. ∎

###  3.2. Purity implies formality

On the category of graded vector spaces over a field $\mathbb{K}$, there are
two distinct choices of braiding isomorphisms that make it into a symmetric
monoidal category. In one of them, the isomorphism

$A\otimes B\to B\otimes A$

sends an elementary tensor of homogeneous elements $a\otimes b$ to $b\otimes
a$ in the other $a\otimes b$ is sent to $(-1)^{|a||b|}b\otimes a$. We shall
denote these symmetric monoidal structures $Gr\mathrm{Vect}_{\mathbb{K}}$ and
$gr\mathrm{Vect}_{\mathbb{K}}$ respectively. In the first category, the source
for gradings will be the weights of mixed Hodge theory, while in the second
category gradings are cohomological.

We recall the definition of formality for algebras over graded operads. There
is an obvious symmetric monoidal functor

$T:gr\mathrm{Vect}_{\mathbb{K}}\to\mathrm{Ch^{*}({\mathbb{K}})}$

giving a graded vector space the trivial differential. Let $\mathcal{O}$ be an
operad in $gr\mathrm{Vect}_{\mathbb{K}}$. We shall often omit $T$ in the
notation and view a graded object as a chain complex via $T$. The functor $T$
induces a functor

$T:\mathrm{Alg}_{\mathcal{O}}(gr\mathrm{Vect}_{\mathbb{K}})\to\mathrm{Alg}_{\mathcal{O}}(\mathrm{Ch^{*}({\mathbb{K}})})$

######  Definition 3.6.

An $\mathcal{O}$-algebra is said to be formal if it is in the homotopical
essential image of the functor $T$.

######  Remark 3.7.

There are two ways of interpreting this definition depending on whether we
consider the model category of $\mathcal{O}$-algebras or the $\infty$-category
of $\mathcal{O}$-algebras. It turns out that these two ways are equivalent
thanks to a rigidification theorem due to Haugseng [Hau19, Theorem 4.10]. The
$\infty$-categorical interpretation will be more convenient to work with in
the following.

######  Definition 3.8.

Let $\alpha\neq 0$ be a rational number. A $\mathbb{C}$-mixed Hodge complex
$(A,d,W,F,\overline{F})$ is said to be $\alpha$-pure if

$Gr_{p}^{W}H^{n}(A)=0\text{ for all }p\neq\alpha n.$

The full subcategory ${\mathrm{MHC}}_{\mathbb{C}}^{\alpha\text{-pure}}$ of
$\mathbb{C}$-mixed Hodge complexes that are $\alpha$-pure is a symmetric
monoidal category. The main result of [CH20] gives formality for algebraic
varieties whose cohomology is $\alpha$-pure, where $W$ is Deligne’s weight
filtration.

We shall now adapt the theory to the case of interest to us. We consider the
$\infty$-category $\mathrm{Ch^{*}({Gr\mathrm{Vect}_{\mathbb{K}}})}$ of cochain
complexes of graded vector spaces with quasi-isomorphisms inverted. We denote
by $\mathrm{Ch^{*}({Gr\mathrm{Vect}_{\mathbb{K}}})}^{\alpha\text{-pure}}$ the
full subcategory whose objects are the chain complexes such that

$H^{n}(C)_{p}=0\text{ for all }p\neq\alpha n.$

Clearly the symmetric monoidal structure on
$\mathrm{Ch^{*}({Gr\mathrm{Vect}_{\mathbb{K}}})}$ restricts to one on
$\mathrm{Ch^{*}({Gr\mathrm{Vect}_{\mathbb{K}}})}^{\alpha\text{-pure}}$.

######  Proposition 3.9.

Let $\alpha=r/s$ with $r$ and $s$ two coprime integers. The cohomology functor

$\mathrm{Ch^{*}({Gr\mathrm{Vect}_{\mathbb{K}}})}^{\alpha\text{-pure}}\to
gr\mathrm{Vect}_{\mathbb{K}}$

induces a symmetric monoidal equivalence of $\infty$-categories onto the full
subcategory of $gr\mathrm{Vect}_{\mathbb{K}}$ spanned by graded vector spaces
that are zero except in degree divisible by $s$.

###### Proof.

This functor is clearly essentially surjective. In order to prove that it is
fully faithful, it suffices to observe the following formula for the homotopy
groups of the mapping spaces in
$\mathrm{Ch^{*}({Gr\mathrm{Vect}_{\mathbb{K}}})}$

$\pi_{k}\mathrm{Map}_{\mathrm{Ch^{*}({Gr\mathrm{Vect}_{\mathbb{K}}})}}(C,D)=\bigoplus_{i\in\mathbb{Z}}[C_{i},D_{i}[k]]$

in particular, we see that if $C$ and $D$ are both $\alpha$-pure, all the
higher homotopy groups are trivial and we have

$\pi_{0}\mathrm{Map}_{D(Gr\mathrm{Vect}_{\mathbb{K}})}(C,D)=\bigoplus_{j\in\mathbb{Z}}\mathrm{Hom}(H^{sj}(C_{rj}),H^{sj}(D_{rj}))$

This shows that the cohomology functor is fully faithful. ∎

######  Theorem 3.10.

Let $\mathcal{O}$ be an operad in graded mixed Hodge structures which is
$\alpha$-pure. Every $\mathcal{O}$-algebra in $\alpha$-pure mixed Hodge
complexes is formal as an algebra in $\mathrm{Ch^{*}({\mathbb{C}})}$.

###### Proof.

An immediate consequence of Deligne’s splitting gives a homotopy commutative
diagram of symmetric monoidal $\infty$-categories

$\textstyle{\mathrm{Ch^{*}({{\mathsf{MHS}}_{\mathbb{C}}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{G}$$\textstyle{{\mathrm{MHC}}_{\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Ch^{*}({Gr\mathrm{Vect}_{\mathbb{C}}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Ch^{*}({\mathbb{C}})}}$

where the top map is the equivalence of Proposition 3.5, $G$ is the associated
graded of the weight filtration and the unlabelled maps are the obvious
forgetful maps. This gives us a homotopy commutative square of
$\infty$-categories

$\textstyle{\mathrm{Alg}_{\mathcal{O}}(\mathrm{Ch^{*}({{\mathsf{MHS}}_{\mathbb{C}}})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{G}$$\textstyle{\mathrm{Alg}_{\mathcal{O}}({\mathrm{MHC}}_{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Alg}_{\mathcal{O}}(\mathrm{Ch^{*}({Gr\mathrm{Vect}_{\mathbb{C}}})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Alg}_{\mathcal{O}}(\mathrm{Ch^{*}({\mathbb{C}})})}$

(in each case $\mathcal{O}$ should be considered as an operad in the relevant
$\infty$-category but we have kept the same notation everywhere for
simplicity).

We can restrict to $\alpha$-pure objects and we obtain a homotopy commutative
square

$\textstyle{\mathrm{Alg}_{\mathcal{O}}(\mathrm{Ch^{*}({{\mathsf{MHS}}_{\mathbb{C}}})}^{\alpha\text{-pure}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{G}$$\textstyle{\mathrm{Alg}_{\mathcal{O}}({\mathrm{MHC}}_{\mathbb{C}}^{\alpha\text{-pure}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Alg}_{\mathcal{O}}(\mathrm{Ch^{*}({Gr\mathrm{Vect}_{\mathbb{C}}})}^{\alpha\text{-pure}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Alg}_{\mathcal{O}}(\mathrm{Ch^{*}({\mathbb{C}})})}$

From this diagram, we see that it suffices to check that the image of the
bottom map consists of formal $\mathcal{O}$-algebras.

By the previous proposition, we have a homotopy commutative triangle of
symmetric monoidal $\infty$-categories

$\textstyle{\mathrm{Ch^{*}({Gr\mathrm{Vect}_{\mathbb{K}}})}^{\alpha\text{-pure}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H^{*}}$$\textstyle{gr\mathrm{Vect}_{\mathbb{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\textstyle{\mathrm{Ch^{*}({\mathbb{K}})}}$

which induces a homotopy commutative square

$\textstyle{\mathrm{Alg}_{\mathcal{O}}(\mathrm{Ch^{*}({Gr\mathrm{Vect}_{\mathbb{K}}})}^{\alpha\text{-pure}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H^{*}}$$\textstyle{\mathrm{Alg}_{\mathcal{O}}(gr\mathrm{Vect}_{\mathbb{K}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\textstyle{\mathrm{Alg}_{\mathcal{O}}(\mathrm{Ch^{*}({\mathbb{K}})})}$

Therefore the essential image of the diagonal map consists of formal
$\mathcal{O}$-algebras as desired. ∎

######  Remark 3.11.

The same proof of Theorem 3.10 is also of course valid for classical mixed
Hodge complexes, whose weight filtration exists on a cochain complex defined
over a field $\mathbb{K}\subseteq\mathbb{R}$. In this case, the formality of
the underlying algebra is obtained over $\mathbb{K}$.

######  Remark 3.12.

One of the main results of [CH20] gives formality of the forgetful functor

${\mathrm{MHC}}_{\mathbb{C}}^{\alpha\text{-pure}}\longrightarrow\mathrm{Ch^{*}({\mathbb{K}})}$

as a symmetric monoidal functor. This result, although closely related to
Theorem 3.10 above does not seem to imply it. Indeed, given an
$\mathcal{O}$-algebra $A$ in pure $\mathbb{C}$-mixed Hodge complexes, the
formality of this functor gives formality of the pair $(\mathcal{O},A)$ in the
fibred category of algebras over all operads: assuming that $\mathcal{O}$ has
trivial differential, this means that there are quasi-isomorphisms of
$\mathcal{O}_{\infty}$-algebras $F^{*}A\leftarrow\bullet\rightarrow
G^{*}H(A)$, where $F,G:\mathcal{O}_{\infty}\to\mathcal{O}$ are (possibly
distinct) quasi-isomorphisms of operads. Therefore in general, one does not
obtain formality of $A$ as an $\mathcal{O}$-algebra from the formality of
functors.

###  3.3. Mixed Hodge theory of hypercommutative algebras

The moduli spaces $\overline{\mathcal{M}}_{0,\bullet}$ are smooth complex
projective schemes and as such, their homology carries pure Hodge structures.
Therefore we can view $\mathrm{Hycom}$ as an operad in graded
$\mathbb{C}$-mixed Hodge structures which is 1-pure. The generator $m_{i}$ of
$\mathrm{Hycom}(n)$, which corresponds to the fundamental class
$[\overline{\mathcal{M}}_{0,n+1}]\in
H_{2(n-2)}(\overline{\mathcal{M}}_{0,n+1})$, has pure weight $4-2i$, so that:

$m_{n}\in W_{4-2n}F^{2-n}\overline{F}{}^{2-n}\mathrm{Hycom}(n)^{4-2n}.$

In particular, a hypercommutative algebra in $\mathbb{C}$-mixed Hodge
complexes is just a mixed Hodge complex $(A,d,W,F,\overline{F})$ together with
the structure of a hypercommutative algebra $\\{m_{n}\\}$ with the above
compatibility condition with respect to the filtrations $W$, $F$ and
$\overline{F}$, so that

$\sum_{\sum m_{i}=m,\sum p_{i}=p,\sum
q_{i}=q}W_{m_{1}}F^{p_{1}}\overline{F}{}^{q_{1}}A\otimes\cdots\otimes
W_{m_{n}}F^{p_{n}}\overline{F}{}^{q_{n}}A\subseteq
W_{m+4-2n}F^{p+2-n}\overline{F}{}^{q+2-n}A.$

The operad ${\mathrm{BV}}$ can be viewed as an operad in graded mixed Hodge
structures. This can be done in an ad-hoc manner by declaring that
${\mathrm{BV}}_{k}(n)$ is of pure Hodge type $(-k,-k)$ for any value of $n$.
This can be given a geometric interpretation as the homology of a specific
model of the framed little $2$-disks operad (see [Vai19]). In particular, the
arity 1 generator $\Delta$ has pure weight $-2$, so that

$\Delta\in W_{-2}F^{-1}\overline{F}{}^{-1}\mathrm{BV}(1)^{-1}.$

We can also make ${\mathrm{BV}}/\Delta$ into a $\mathbb{C}$-mixed Hodge
complex by setting the generators $\varphi_{n}$ of degree $-2n$, to be of pure
weight $-2n$, so that

$\varphi_{n}\in
W_{-2n}F^{-n}{\overline{F}}{}^{-n}{\mathrm{BV}}/\Delta(1)^{-2n}.$

With this definition, the differential $\delta$ of ${\mathrm{BV}}/\Delta$ has
weight zero and is strictly compatible with respect to all filtrations. Since
$\Delta$ is trivial in homology, ${\mathrm{BV}}/\Delta$ is an operad in
${\mathrm{MHC}}_{\mathbb{C}}$ which is $1$-pure. Note as well that the map
$\mathrm{Hycom}\to{\mathrm{BV}}/\Delta$ is compatible with all filtrations and
so it is a map of 1-pure $\mathbb{C}$-mixed Hodge complexes.

The following is now an immediate consequence of Theorem 3.10:

######  Corollary 3.13.

Let $A$ be a hypercommutative algebra in 1-pure $\mathbb{C}$-mixed Hodge
complexes. Then $A$ is formal as a hypercommutative algebra in cochain
complexes of $\mathbb{C}$-vector spaces.

######  Remark 3.14.

Of course, there is an analogous corollary for BV-algebras in
$\mathbb{C}$-mixed Hodge complexes that are 2-pure. The main geometric
situations that we will consider in the following section start with a BV-
algebra which can be made into a $\mathbb{C}$-mixed Hodge in a compatible way.
Since the underlying objects are compact Kähler manifolds, the cohomology is
1-pure (rather than 2-pure). Therefore one does not obtain formality of the
BV-algebra, but we will obtain formality of the associated hypercommutative
algebra, whose governing operad is 1-pure.

##  4\. Hypercommutative algebra structures on complex manifolds

In this last section we prove the main results of this paper. We consider
three geometric situations that lead to BV-algebras whose associated
hypercommutative algebras are in fact algebras in $\mathbb{C}$-mixed Hodge
complexes. Using the “purity implies formality” theory, we deduce that such
hypercommutative structures are formal.

###  4.1. Canonical homotopy transfer on a Hermitian manifold

Let $M$ be a compact complex manifold. Its complex de Rham algebra
$\mathcal{A}^{*}:=\mathcal{A}^{*}_{\mathrm{dR}}(M)\otimes_{\mathbb{R}}\mathbb{C}$
admits a bidegree decomposition by forms of type $(p,q)$:

$\mathcal{A}^{n}=\bigoplus_{p+q=n}\mathcal{A}^{p,q},$

and the exterior differential decomposes into two components
$d=\partial+{\overline{\partial}}$, where $\partial$ has bidegree $(1,0)$ and
${\overline{\partial}}$ is its complex conjugate. Given a Hermitian metric
$\langle-,-\rangle$ there is an associated Hodge-star operator

$\star:\mathcal{A}^{p,q}\longrightarrow\mathcal{A}^{m-q,m-p}\text{ defined by
}\alpha\wedge\star\overline{\beta}:=\langle\alpha,\beta\rangle\mathrm{vol},$

where $\mathrm{vol}$ is the volume form determined by the metric and $m$ is
the complex dimension of $M$. Let $\delta$ denote either $\partial$ or
${\overline{\partial}}$. The operator

$\delta^{*}:=-\star\overline{\delta}\star$

is the $\mathcal{L}_{2}$-adjoint of $\delta$, and we have
$d^{*}={\overline{\partial}}^{*}+\partial^{*}$. Consider the associated
Laplacian $\Box_{\delta}:=\delta\delta^{*}+\delta^{*}\delta$. Hodge theory
gives orthogonal direct sum decompositions

$\mathcal{A}^{p,q}=\mathcal{H}_{\delta}^{p,q}\oplus\Box_{\delta}(\mathcal{A}^{p,q})=\mathcal{H}_{\delta}^{p,q}\oplus\mathrm{Im}(\delta)\cap\mathcal{A}^{p,q}\oplus\mathrm{Im}(\delta^{*})\cap\mathcal{A}^{p,q}.$

where
$\mathcal{H}_{\delta}^{p,q}:=\mathrm{Ker}(\Box_{\delta})\cap\mathcal{A}^{p,q}$
denotes the space of $\delta$-harmonic forms in bidegree $(p,q)$. Also, we
have isomorphisms

$\mathcal{H}_{\delta}^{p,q}\cong
H_{\delta}^{p,q}:={\mathrm{Ker}(\delta)\over\mathrm{Im}(\delta)}|_{(p,q)}$

In particular, any $\delta$-cohomology class admits a unique $\delta$-harmonic
representative. Denote by
$\pi_{\delta}:\mathcal{A}^{p,q}\to\mathcal{H}_{\delta}^{p,q}$ the projection
and by $G_{\delta}:\mathcal{A}^{p,q}\to\mathcal{A}^{p,q}$ the Green operator
defined by zero on $\delta$-harmonic forms and by the inverse of
$\Box_{\delta}$ on their orthogonal complement.

We have $G_{\delta}\Box_{\delta}=\Box_{\delta}G_{\delta}$ and any linear
operator commuting with $\Box_{\delta}$ also commutes with $G_{\delta}$. In
particular, $G_{\delta}$ commutes with $\delta$ and $\delta^{*}$. The
following is a matter of verification.

######  Lemma 4.1.

For a compact Hermitian manifold there is homotopy transfer diagram
$(\iota,\rho,h)$ for the complex $(\mathcal{A},{\overline{\partial}})$, where
$\iota:H_{\overline{\partial}}\to\mathcal{A}$ and $\rho:\mathcal{A}\to
H_{\overline{\partial}}$ are given by taking ${\overline{\partial}}$-harmonic
representatives and projecting to ${\overline{\partial}}$-harmonic forms
respectively and $h:={\overline{\partial}}^{*}G_{\overline{\partial}}$.

Assume now that $M$ is a compact Kähler manifold. We will be using the
following main properties for the differential forms of Kähler manifolds (see
for instance [GH94]).

  1. (1)

The Laplacian identities
$\Box_{d}=2\Box_{\overline{\partial}}=2\Box_{\partial}$ identify all spaces of
harmonic forms
$\mathcal{H}_{d}^{p,q}=\mathcal{H}_{\overline{\partial}}^{p,q}=\mathcal{H}_{\partial}^{p,q}$
and so all cohomology vector spaces are canonically isomorphic:

$H_{d}\cong\bigoplus H_{\overline{\partial}}^{p,q}\cong\bigoplus
H_{\partial}^{p,q}.$

  2. (2)

We have the Kähler identity
$\partial{\overline{\partial}}^{*}+{\overline{\partial}}^{*}\partial=0$.

  3. (3)

The double complex $(\mathcal{A},\partial,{\overline{\partial}})$ satisfies
the $\partial{\overline{\partial}}$-condition:

$\mathrm{Ker}(\partial)\cap\mathrm{Im}({\overline{\partial}})=\mathrm{Ker}({\overline{\partial}})\cap\mathrm{Im}(\partial)=\mathrm{Im}(\partial{\overline{\partial}}).$

Lemma 4.1 above gives a canonical homotopy transfer diagram for the Dolbeault
complex of a compact Hermitian manifold. There is an obvious de Rham version
of the above result which is valid for any compact Riemannian manifold. In
general, given a homotopy transfer diagram for the Dolbeault complex of a
complex manifold, one may obtain a homotopy transfer diagram for the de Rham
complex by perturbation of the Dolbeault data (see Theorem 2.7 of [CSG22] and
the remark thereafter). In the Kähler case, such perturbation turns out to be
trivial:

######  Lemma 4.2.

For a compact Kähler manifold the data $(\iota,\rho,h)$ of Lemma 4.1 is also a
homotopy transfer diagram for the complex $(\mathcal{A},d)$.

###### Proof.

Since
$\mathcal{H}_{\overline{\partial}}=\mathcal{H}_{\partial}=\mathrm{Ker}(\partial)\cap\mathrm{Ker}(\partial^{*})$
we have $\partial\iota=\rho\partial=0$. This proves that $d\iota=\iota d$ and
$d\rho=\rho d$. Let us prove that $\partial h+h\partial=0$, where
$h={\overline{\partial}}^{*}G_{\overline{\partial}}$. We have

$\partial{\overline{\partial}}^{*}G_{\overline{\partial}}+{\overline{\partial}}^{*}G_{\overline{\partial}}\partial=(\partial{\overline{\partial}}^{*}+{\overline{\partial}}^{*}\partial)G_{\overline{\partial}}=0,$

where we used the Kähler identity
$\partial{\overline{\partial}}^{*}+{\overline{\partial}}^{*}\partial=0$ and
the fact that, in the Kähler case, the operators $\partial$ and
$G_{\overline{\partial}}$ commute (since any linear operator commuting with
$\square_{\overline{\partial}}$ also commutes with $G_{\overline{\partial}}$,
and for Kähler manifolds we have
$\square_{\overline{\partial}}=\square_{\partial}$). This gives
$dh+hd={\overline{\partial}}h+h{\overline{\partial}}=\mathrm{Id}-\iota\rho$. ∎

###  4.2. Kähler manifolds: the BV-algebras of Cao-Zhou

On any Hermitian manifold, while $\partial$ and ${\overline{\partial}}$ are
derivations with respect to wedge product of differential forms, their
respective formal adjoints $\partial^{*}$ and ${\overline{\partial}}^{*}$ do
not satisfy the Leibniz rule in general. Instead, on a compact Kähler manifold
the tuples

$\mathcal{A}_{\mathrm{Dol}}:=(\mathcal{A},d={\overline{\partial}},\wedge,\Delta=-i\partial^{*})\text{
and
}\mathcal{A}_{\mathrm{dR}}:=(\mathcal{A},d={\overline{\partial}}+\partial,\wedge,\Delta=i({\overline{\partial}}^{*}-\partial^{*}))$

are BV-algebras, as proven in [CZ00] and [CZ99]. Note that in both cases the
Kähler condition is necessary already for the relation $d\Delta+\Delta d=0$ to
be satisfied. The following is straightforward:

######  Lemma 4.3.

Let $\Delta$ denote the BV-operator of either of the BV-algebras
$\mathcal{A}_{\mathrm{Dol}}$ and $\mathcal{A}_{\mathrm{dR}}$. The homotopy
transfer diagram $(\iota,\rho,h)$ of Lemmas 4.1 and 4.2 satisfies
$\Delta\iota=\rho\Delta=h\Delta+\Delta h=0$.

The above lemma together with Proposition 2.11 defines hypercommutative
algebra structures on the Dolbeault and de Rham algebras of any compact Kähler
manifold. Such structures are canonical, in the sense that they only depend on
the Hermitian structure of the manifold (the complex structure together with
the chosen compatible Kähler metric).

As is well known, on a Kähler manifold the commutative algebras
$(\mathcal{A},{\overline{\partial}},\wedge)$ and $(\mathcal{A},d,\wedge)$ are
quasi-isomorphic and formal. In particular, we have a ring isomorphism
$H_{\overline{\partial}}\cong H_{\mathrm{dR}}$ and the induced
$C_{\infty}$-structure in cohomology is the trivial one. We promote this
result to the hypercommutative case:

######  Theorem 4.4.

For any compact Kähler manifold, the canonical hypercommutative structures
defined on $\mathcal{A}_{\mathrm{Dol}}$ and $\mathcal{A}_{\mathrm{dR}}$ are
formal and quasi-isomorphic.

###### Proof.

We will make $\mathcal{A}_{\mathrm{dR}}$ into a hypercommutative algebra in
pure $\mathbb{C}$-mixed Hodge complexes. Let $F$ and $\overline{F}$ denote the
column and row filtrations respectively:

$F^{p}\mathcal{A}^{n}:=\bigoplus_{q\geq p}\mathcal{A}^{q,n-q}\quad\text{ and
}\quad\overline{F}{}^{q}\mathcal{A}^{n}:=\bigoplus_{p\geq
q}\mathcal{A}^{n-p,p}.$

Since the manifold is Kähler, the double complex
$(\mathcal{A},\partial,{\overline{\partial}})$ satisfies the
$\partial{\overline{\partial}}$-condition and so by Proposition 5.17 of
[DGMS75] the spectral sequences associated to $F$ and $\overline{F}$
degenerate at $E_{1}$ and induce $n$-opposed filtrations on $H^{n}$. Let $W$
be the canonical filtration

$W_{p}\mathcal{A}^{n}=\left\\{\begin{array}[]{ll}0&,p<n\\\
\mathrm{Ker}(d:\mathcal{A}^{n}\to\mathcal{A}^{n+1})&,p=n\\\
\mathcal{A}^{n}&,p>n\end{array}\right..$

Its associated spectral sequence degenerates at $E_{1}$ and
$H^{n}(Gr_{p}^{W}\mathcal{A})\cong H^{n}(\mathcal{A})$ if $p=n$. This proves
that the tuple $(\mathcal{A},d,F,\overline{F},W)$ is a pure $\mathbb{C}$-mixed
Hodge complex. Since

$h\Delta=i{\overline{\partial}}^{*}G_{\overline{\partial}}({\overline{\partial}}^{*}-\partial^{*})=-i{\overline{\partial}}^{*}G_{\overline{\partial}}\partial^{*}$

it follows that the hypercommutative operations $m_{n}$ on
$\mathcal{A}_{\mathrm{dR}}$ have bidegree $(2-n,2-n)$. This gives

$m_{n}\in W_{4-2n}\cap F^{2-n}\cap\overline{F}{}^{2-n}$

which agrees with the mixed Hodge structure on the operad ${\mathrm{Hycom}}$
given in Section 3.3. Therefore the operations $m_{n}$ are compatible with all
filtrations, making $\mathcal{A}_{\mathrm{dR}}$ into an hypercommutative
algebra in pure $\mathbb{C}$-mixed Hodge complexes. By Corollary 3.13, it
follows that $\mathcal{A}_{\mathrm{dR}}$ is a formal hypercommutative algebra.
This is equivalent to the formality of the
${\mathrm{Hycom}}_{\infty}$-structure induced on $H_{d}$, which is isomorphic
to the one induced on $H_{\overline{\partial}}$. Indeed, both structures are
defined using canonical homotopy transfer diagrams which coincide by Lemma
4.2. In particular, $H_{\overline{\partial}}$ is a formal
${\mathrm{Hycom}}_{\infty}$-algebra, which is equivalent to
$\mathcal{A}_{\mathrm{Dol}}$ being a formal hypercommutative algebra. Since
both $\mathcal{A}_{\mathrm{dR}}$ and $\mathcal{A}_{\mathrm{Dol}}$ are formal
hypercommutative algebras and the hypercommutative structures on $H_{d}$ and
$H_{\overline{\partial}}$ are isomorphic, it follows that
$\mathcal{A}_{\mathrm{dR}}$ and $\mathcal{A}_{\mathrm{Dol}}$ are quasi-
isomorphic. ∎

We end this section with an example exhibiting that, while the above
hypercommutative structure on a Kähler manifold is formal, it is still non-
trivial. As we could not find any previous computations of these
hypercommutative structures in the literature, we believe that it may be of
interest. For this example, we actually consider a non-Kähler orbifold that
satisfies the $\partial{\overline{\partial}}$-condition. The above theory
applies to this broader setting as well.

######  Example 4.5.

Consider the Iwasawa manifold $M=H_{\mathbb{Z}}\setminus H_{\mathbb{C}}$,
defined as the quotient of the complex Heisenberg $H_{\mathbb{C}}$ group by
its integral subgroup. Its complex algebra of left-invariant differential
forms is isomorphic to the Chevalley-Eilenberg complex of the Lie algebra of
$H_{\mathbb{C}}$, and so it is given by the free commutative dg-algebra

$\mathcal{A}=\Lambda(a,b,c,\overline{a},\overline{b},\overline{c})\text{ with
}d(c)=-ab,d(\overline{c})=-\overline{a}\overline{b}.$

Here $a,b,c$ have bidegree $(1,0)$ and
$\overline{a},\overline{b},\overline{c}$ have bidegree $(0,1)$. The inclusion
of this algebra into the complex de Rham algebra of $M$ preserves bidegrees
and induces an isomorphism on Dolbeault cohomology. The Iwasawa manifold is a
non-formal compact complex 3-fold and so it does not satisfy the
$\partial{\overline{\partial}}$-condition. We will however produce a
$\partial{\overline{\partial}}$-manifold by constructing, out of $M$, an
orbifold of global quotient type. Consider the action
$\sigma:\mathbb{C}^{3}\to\mathbb{C}^{3}$ by a finite group of biholomorphisms
of order 4, given by $\sigma(z_{1},z_{2},z_{3})=(iz_{1},iz_{2},-z_{3})$. This
descends to a well-defined action on $M$. The resulting quotient
$\widehat{M}=M/\langle\sigma\rangle$ is a compact orbifold with 16 isolated
singular points. As shown in Theorem 7.1 of [ST22], the orbifold $\widehat{M}$
admits a resolution into a smooth $\partial{\overline{\partial}}$-compact
manifold. The sub-algebra of $M$ of $\sigma$-invariant differential forms
$\mathcal{A}_{\sigma}(\widehat{M})$ is generated by the elements

$a\overline{a},b\overline{b},c\overline{c},a\overline{b},b\overline{a},c\overline{a}\overline{b},ab\overline{c},abc,\overline{a}\overline{b}\overline{c}.$

The only non-trivial differentials being
$\partial(c\overline{c})=-ab\overline{c}$,
$\partial(c\overline{a}\overline{b})=-ab\overline{a}\overline{b}$ and their
complex conjugates
${\overline{\partial}}(c\overline{c})=c\overline{a}\overline{b}$,
${\overline{\partial}}(ab\overline{c})=-ab\overline{a}\overline{b}$. One
easily checks that this algebra satisfies the
$\partial{\overline{\partial}}$-lemma. In particular, its Hodge-to de Rham
spectral sequence degenerates at $E_{1}$. We obtain:

$H_{\overline{\partial}}^{*,*}(\widehat{M})\cong\begin{array}[]{|c|c|c|c|}\hline\cr[\overline{a}\overline{b}\overline{c}]&0&0&[abc\overline{a}\overline{b}\overline{c}]\\\
\hline\cr
0&0&[ac\overline{a}\overline{c}],[ac\overline{b}\overline{c}],[bc\overline{b}\overline{c}],[bc\overline{a}\overline{c}]&0\\\
\hline\cr
0&[a\overline{a}],[b\overline{b}],[a\overline{b}],[b\overline{a}]&0&0\\\
\hline\cr 1&0&0&[abc]\\\ \hline\cr\end{array}$

Considering the standard hermitian metric on
$\mathcal{A}_{\sigma}(\widehat{M})$ we obtain operators
${\overline{\partial}}^{*}$ and $\partial^{*}$. We have
$\partial^{*}(ab\overline{a}\overline{b})=-c\overline{a}\overline{b}$ and
$\partial^{*}(ab\overline{c})=c\overline{c}$. One easily checks that
${\overline{\partial}}\partial^{*}+\partial^{*}{\overline{\partial}}=0$ and
that the $d\Delta$-condition is satisfied, with $d={\overline{\partial}}$ and
$\Delta=-i\partial^{*}$. Therefore the tuple
$(\mathcal{A}_{\sigma}(\widehat{M}),d={\overline{\partial}},\wedge,\Delta=-i\partial^{*})$
is a BV-algebra (since $d\Delta+\Delta d=0$) and we may use Proposition 2.11
to compute
$\varphi_{1}=h\Delta=-i{\overline{\partial}}^{*}G_{{\overline{\partial}}}\partial^{*}$.
We obtain $\varphi_{1}(ab\overline{a}\overline{b})=-ic\overline{c}$ and
$\varphi_{1}=0$ otherwise. Using the formula of Example 2.10 we compute a non-
trivial operation of arity 3 and bidegree $(-1,-1)$ on cohomology:

$m_{3}([a\overline{a}],[b\overline{b}],[b\overline{b}])=-2[\varphi_{1}(a\overline{a}b\overline{b})\cdot
b\overline{b}]=2i[bc\overline{b}\overline{c}].$

###  4.3. Calabi-Yau manifolds: the BV-algebra of Barannikov-Kontsevich

We recall the BV-algebra considered by Barannikov and Kontsevich in [BK98]
(see also Section III.9.12 of [Man99a]). Let $M$ be a compact Calabi-Yau
manifold of complex dimension $m$. This is a compact Kähler manifold $M$ with
trivial canonical bundle. In particular, we may choose a nowhere vanishing
holomorphic $m$-form $\Omega$. Consider the complex bigraded algebra
$(\mathcal{L}^{*,*},\wedge)$ given by

$\mathcal{L}^{p,q}:=\Gamma(M,\Lambda^{p}T\otimes\Lambda^{q}\overline{T}^{*}),$

where $T$ denotes the holomorphic tangent bundle of $M$. The algebra structure
is determined by the exterior product $\wedge$. There is an isomorphism of
bigraded vector spaces $\eta:\mathcal{L}^{p,q}\to\mathcal{A}^{m-p,q}$ given by
the contraction $\alpha\mapsto\alpha\vdash\Omega$ with the holomorphic
$m$-form $\Omega$. Note that such isomorphism does not preserve the product
structures. Given an operator
$\delta:\mathcal{A}^{*,*}\to\mathcal{A}^{*+p,*+q}$ of pure bidegree $(p,q)$,
we will denote

$\delta_{\eta}:=\eta^{-1}\circ\delta\circ\eta:\mathcal{L}^{*,*}\to\mathcal{L}^{*-p,*+q}.$

the operator of bidegree $(-p,q)$ induced on $\mathcal{L}$ via the isomorphism
$\eta$. On the other hand, the Hermitian metric gives an isomorpism
$T\cong\overline{T}^{*}$ inducing, for each $(p,q)$, a $\mathbb{C}$-linear
isomorphism $g:\mathcal{L}^{p,q}\to\mathcal{L}^{q,p}$.

######  Lemma 4.6 (c.f. [CZ00], Lemma 3.1).

The following identities are satisfied:

$\begin{array}[]{ll}{\overline{\partial}}_{\eta}=\mathrm{Id}\otimes{\overline{\partial}}\,\,,&\partial^{*}_{\eta}=-g\circ(\mathrm{Id}\otimes{\overline{\partial}})\circ
g=-g\circ{\overline{\partial}}_{\eta}\circ g\\\
{\overline{\partial}}^{*}_{\eta}=\mathrm{Id}\otimes{\overline{\partial}}^{*},&\partial_{\eta}=-g\circ(\mathrm{Id}\otimes{\overline{\partial}}^{*})\circ
g=-g\circ{\overline{\partial}}^{*}_{\eta}\circ g.\end{array}$

###### Proof.

The two identities on the left are straightforward. We prove the identity
involving $\partial_{\eta}$. Since both the Kähler form $\omega$ and the
holomorphic $m$-form $\Omega$ are parallel with respect to the Levi-Civita
connection, on an open subset $U\subseteq M$ there is a local orthonormal
frame field $\\{e_{1},\cdots,e_{m}\\}$ of the holomorphic tangent bundle
$T=T^{1,0}M$ (so that for all $x\in U$, $\\{e_{1}(x),\cdots,e_{m}(x)\\}$ is an
orthonormal basis of $T_{p}^{1,0}M$) such that, up to a constant,
$\Omega=e^{1}\wedge\cdots\wedge e^{m}$, and $\omega=\sum
e^{i}\wedge\overline{e}^{i}$, where $\\{e^{1},\cdots,e^{m}\\}$ denotes the
dual coframe, defined by $e^{j}(e_{i})=\delta^{j}_{i}$. Also, at the point
$x$, we have

$\nabla_{e_{i}}e_{j}=\nabla_{\overline{e}_{i}}e_{j}=\nabla_{e_{i}}\overline{e}_{j}=\nabla_{\overline{e}_{i}}\overline{e}_{j}=0$

A standard computation shows that the operators $\partial$,
${\overline{\partial}}$ and their adjoints may be written locally as:

${\overline{\partial}}=\overline{e}^{i}\wedge\nabla_{\overline{e}_{i}}\quad,\quad\partial^{*}=-{e}_{i}\vdash\nabla_{\overline{e}_{i}}\quad,\quad\partial={e}^{i}\wedge\nabla_{{e}_{i}}\quad\text{
and }\quad{\overline{\partial}}^{*}=-\overline{e}_{i}\vdash\nabla_{{e}_{i}},$

where $(X\vdash\alpha)(X_{1},\cdots,X_{k-1}):=\alpha(X,X_{1},\cdots,X_{k-1})$
is the contraction of a vector field $X$ and a $k$-form $\alpha$ (see for
instance Lemma 3.3.4 of [Jos17]).

Near $x$ any element in $\mathcal{L}^{p,q}$ may be written as a sum of
elements of the form $\alpha=fe_{I}\otimes\overline{e}^{J}$, where $I$ and $J$
denote subsets of $\\{1,\cdots,m\\}$ of cardinalities $p$ and $q$
respectively. We have

$g(\mathrm{Id}\otimes{\overline{\partial}}^{*})g\alpha=g(\mathrm{Id}\otimes{\overline{\partial}}^{*})(fe_{J}\otimes\overline{e}^{I})=-(\nabla_{e_{i}}f)g(e_{J}\otimes\overline{e}_{i}\vdash\overline{e}^{I})=-(\nabla_{e_{i}}f)({e}^{i}\vdash{e}_{I})\otimes\overline{e}^{J}.$

On the other hand, we have

$\partial_{\eta}(\alpha)=\eta^{-1}\partial\eta(\alpha)=\eta^{-1}\partial(f(e_{I}\vdash\Omega)\otimes\overline{e}^{J})=\eta^{-1}(e^{i}\wedge\nabla_{e_{i}}f(e_{I}\vdash\Omega)\otimes\overline{e}^{J}=(\nabla_{e_{i}}f)({e}^{i}\vdash{e}_{I})\otimes\overline{e}^{J}.$

The identity involving $\partial^{*}_{\eta}$ follows analogously.

∎

Barannikov and Kontsevich noted in [BK98] that the complex
$(\mathcal{L}^{*,*},{\overline{\partial}}_{\eta})$ carries a BV-algebra
structure with the wedge product $\wedge$ and where the $\Delta$ operator is
given by $\partial_{\eta}$. Note that ${\overline{\partial}}_{\eta}$ has
bidegree $(0,1)$ while $\partial_{\eta}$ has bidegree $(-1,0)$. We next define
a “de Rham version” of this BV-algebra, in the sense that we add, to the
differential and the $\Delta$ operator, a component of bidegree $(1,0)$ and
$(0,-1)$ respectively, both defined using the original operators together with
the involution $g:\mathcal{L}^{p,q}\to\mathcal{L}^{q,p}$.

######  Proposition 4.7.

The following tuples are all BV-algebras:

$\begin{array}[]{ll}\mathcal{L}_{\mathrm{Dol}}:=(\mathcal{L}^{*,*},{\overline{\partial}}_{\eta},\wedge,\partial_{\eta}),\\\
\mathcal{L}_{\mathrm{Dol}^{*}}:=(\mathcal{L}^{*,*},\partial^{*}_{\eta},\wedge,-{\overline{\partial}}^{*}_{\eta}),\\\
\mathcal{L}_{\mathrm{dR}}:=(\mathcal{L}^{*,*},d:={\overline{\partial}}_{\eta}+\partial^{*}_{\eta},\wedge,\Delta:=\partial_{\eta}-{\overline{\partial}}^{*}_{\eta}).\end{array}$

###### Proof.

The first is the BV-algebra of [BK98]. It is clear that
$(\mathcal{L}^{*,*},\partial^{*}_{\eta},-{\overline{\partial}}^{*}_{\eta})$ is
a double complex. Moreover, by Lemma 4.6 we have

$\partial^{*}_{\eta}=-g\circ{\overline{\partial}}_{\eta}\circ g\quad\text{ and
}\quad\partial_{\eta}=-g\circ{\overline{\partial}}^{*}_{\eta}\circ g,$

where $g:\mathcal{L}^{p,q}\to\mathcal{L}^{q,p}$ denotes the involution induced
by $T\cong\overline{T}^{*}$. Since this involution preserves algebra
structures and $\mathcal{L}_{\mathrm{Dol}}$ is a BV-algebra, it follows that
the tuple
$(\mathcal{L}^{*,*},-{\overline{\partial}}^{*}_{\eta},\wedge,-\partial^{*}_{\eta})$
is a BV-algebra and so
$\mathcal{L}_{\mathrm{Dol^{*}}}=(\mathcal{L}^{*,*},{\overline{\partial}}^{*}_{\eta},\wedge,-\partial^{*}_{\eta})$
is also a BV-algebra. We now show that $\mathcal{L}_{\mathrm{dR}}$ is a BV-
algebra. The conditions $d^{2}=\Delta^{2}=0$ are straightforward. Also, we
have

$d\Delta+\Delta
d=\eta^{-1}(\Box_{\partial}-\Box_{\overline{\partial}})\eta=0.$

Note that for this last property, the minus sign in
$\Delta=\partial_{\eta}-{\overline{\partial}}^{*}_{\eta}$ is important.
Therefore $(\mathcal{L},d,\Delta)$ is a double complex. It now suffices to
note that given BV-algebras $(A,d_{i},\wedge,\Delta_{i})$ for $i=0,1$ then
$(A,d_{0}+d_{1},\wedge,\Delta_{0}+\Delta_{1})$ is a BV-algebra whenever
$(A,d_{0}+d_{1},\Delta_{0}+\Delta_{1})$ is a double complex. ∎

In analogy with Lemma 4.2 we have:

######  Lemma 4.8.

The homotopy transfer diagram $(\iota,\rho,h)$ of Lemma 4.1 induces a homotopy
transfer diagram $(\iota_{\eta},\rho_{\eta},h_{\eta})$ for both complexes
$(\mathcal{L},{\overline{\partial}}_{\eta})$ and
$(\mathcal{L},{\overline{\partial}}_{\eta}+\partial^{*}_{\eta})$. Moreover, we
have

$\Delta\iota_{\eta}=\rho_{\eta}\Delta=h_{\eta}\Delta+\Delta h_{\eta}=0,$

where $\Delta$ denotes the BV-operator of either $\mathcal{L}_{\mathrm{Dol}}$
or $\mathcal{L}_{\mathrm{dR}}$.

The above lemma together with Proposition 2.11 defines canonical
hypercommutative algebra structures on $\mathcal{L}_{\mathrm{Dol}}$ and
$\mathcal{L}_{\mathrm{dR}}$ respectively. The main result of this paper is the
following:

######  Theorem 4.9.

For any compact Calabi-Yau manifold, the canonical hypercommutative structures
defined on $\mathcal{L}_{\mathrm{Dol}}$ and $\mathcal{L}_{\mathrm{dR}}$ are
formal and quasi-isomorphic.

###### Proof.

The proof is analogous to that of Theorem 4.4. We first make
$\mathcal{L}_{\mathrm{dR}}$ into a hycommutative algebra in $\mathbb{C}$-mixed
Hodge complexes that is pure. Note first that the components of the
differential $d={\overline{\partial}}_{\eta}+\partial^{*}_{\eta}$ of
$\mathcal{L}^{*,*}$ have bidegrees $(0,1)$ and $(1,0)$ respectively. Let $F$
and $\overline{F}$ be the column and row filtrations of $\mathcal{L}^{*,*}$
respectively. The double complex
$(\mathcal{L},{\overline{\partial}}_{\eta},\partial^{*}_{\eta})$ satisfies the
${\overline{\partial}}_{\eta}\partial^{*}_{\eta}$-condition. This follows from
the fact that the Laplacians

$\Box_{{\overline{\partial}}_{\eta}}:={\overline{\partial}}_{\eta}{\overline{\partial}}_{\eta}^{*}+{\overline{\partial}}_{\eta}^{*}{\overline{\partial}}_{\eta}=(\Box_{\overline{\partial}})_{\eta}\quad\text{
and
}\quad\Box_{\partial_{\eta}}:=\partial_{\eta}\partial_{\eta}^{*}+\partial_{\eta}^{*}\partial_{\eta}=(\Box_{\partial})_{\eta}$

coincide, together with the Kähler identity
${\overline{\partial}}\partial^{*}+\partial^{*}{\overline{\partial}}=0$, which
makes ${\overline{\partial}}_{\eta}$ and $\partial^{*}_{\eta}$ anticommute.
Therefore by [DGMS75, Proposition 5.17] the spectral sequences associated to
$F$ and $\overline{F}$ degenerate at $E_{1}$ and induce $n$-opposed
filtrations on $H^{n}_{d}(\mathcal{L})$, where
$d={\overline{\partial}}_{\eta}+\partial^{*}_{\eta}$. Let $W$ be the canonical
filtration. Its associated spectral sequence degenerates at $E_{1}$ and
$H^{n}(Gr_{p}^{W}\mathcal{L})\cong H^{n}(\mathcal{L})$ if $p=n$. This proves
that the tuple $(\mathcal{L},d,F,\overline{F},W)$ is a pure $\mathbb{C}$-mixed
Hodge complex. The bidegree of $h_{\eta}\Delta$ is $(-1,-1)$. Indeed, we have

$h_{\eta}\Delta=\eta^{-1}(i{\overline{\partial}}^{*}G_{\overline{\partial}})\eta(\partial_{\eta}-{\overline{\partial}}^{*}_{\eta})=i\eta^{-1}({\overline{\partial}}^{*}\partial
G_{\overline{\partial}}-{\overline{\partial}}^{*}{\overline{\partial}}^{*}G_{\overline{\partial}})\eta=i\eta^{-1}{\overline{\partial}}^{*}\partial
G_{\overline{\partial}}\eta.$

It follows that the hypercommutative operations $m_{n}$ have bidegree
$(2-n,2-n)$. This gives

$m_{n}\in W_{4-2n}\cap F^{2-n}\cap\overline{F}{}^{2-n}$

which agrees with the mixed Hodge structure on the operad ${\mathrm{Hycom}}$
of Section 3.3. This makes $\mathcal{L}_{\mathrm{dR}}$ into a hypercommutative
algebra in pure $\mathbb{C}$-mixed Hodge complexes, and so
$\mathcal{L}_{\mathrm{dR}}$ is a formal hypercommutative algebra by Corollary
3.13. This proves formality of $\mathcal{L}_{\mathrm{dR}}$. The proof now
follows exactly as in Theorem 4.4. ∎

###  4.4. Hermitian manifolds: when $\Delta$ is a derivation

We end this paper by considering a version of a BV-algebra
$(\mathcal{A},d,\wedge,\Delta)$ for which $\Delta$ is a derivation and so its
associated Lie bracket is trivial. Such algebras are called
${\mathrm{BV}}_{1}$-algebras in [DST22], where $\Delta$ is understood as an
operator of order one.

######  Proposition 4.10.

Assume that a ${\mathrm{BV}}_{1}$-algebra $(\mathcal{A},d,\wedge,\Delta)$
satisfies the $d\Delta$-condition. Then:

  1. (1)

$(\mathcal{A},d,\wedge,\Delta)$ is formal as a ${\mathrm{BV}}$-algebra.

  2. (2)

The induced ${\mathrm{Hycom}}_{\infty}$-structure on $H$ is trivial.

###### Proof.

By Lemma 2.2 the $d\Delta$-condition gives quasi-isomorphisms of
${\mathrm{BV}}$-algebras

$(\mathcal{A},d,\Delta)\hookleftarrow(\mathrm{Ker}\Delta,d,0)\twoheadrightarrow(H,0,0).$

By Lemma 2.4 there is homotopy transfer diagram $(\iota,\rho,h)$ such that

$\Delta i=p\Delta=h\Delta+\Delta h=0,$

giving a hypercommutative algebra structure on $(\mathcal{A},d,\wedge)$.
Therefore the above zig-zag of quasi-isomorphisms extends to a zig-zag of
homotopy transfer data and so it gives trivial data on the complex $(H,0)$. In
particular, the hypercommutative algebra structure on $(\mathcal{A},d,\wedge)$
is homotopically trivial and so the induced
${\mathrm{Hycom}}_{\infty}$-structure on $H$ is trivial. ∎

######  Example 4.11.

The Dolbeault algebra $(\mathcal{A},{\overline{\partial}},\wedge)$ of any
compact Kähler manifold, with the operator $\Delta=\partial$, fits in the
setting of Proposition 4.10. In fact, only the
$\partial{\overline{\partial}}$-condition is needed, so the proposition
applies as well to any compact complex manifold which can be blown-up to a
Kähler manifold.

Assuming the weaker condition of Hodge-to-de-Rham degeneration data
$(\iota,\rho,h)$ on a ${\mathrm{BV}}_{1}$-algebra, there is still an induced
hypercommutative algebra structure, in which the arity-$n$ operation $m_{n}$
has bidegree $(n-2,2-n)$ (and so total degree 0). This structure is not formal
in general, as shown in the following example.

######  Example 4.12.

The Kodaira-Thurston manifold is the $4$-dimensional compact nilmanifold given
by the quotient $\mathrm{KT}:=H_{\mathbb{Z}}\times\mathbb{Z}\setminus
H_{\mathbb{R}}\times\mathbb{R}$ where $H_{\mathbb{R}}$ is the Heisenberg real
Lie group of dimension 3 and $H_{\mathbb{Z}}$ is the integral subgroup. This
manifold admits a left-invariant complex structure, making it into a non-
Kähler primary Kodaira surface. Its complex homotopy type may be described as
follows: consider the free bigraded commutative algebra

$\mathcal{A}:=\Lambda(a,\overline{a},b,\overline{b})\quad\text{ with
}|a|=|b|=(1,0)\text{ and }|\overline{a}|=|\overline{b}|=(0,1).$

Define differentials $\partial$ and ${\overline{\partial}}$ on $\mathcal{A}$
by extending multiplicatively the table

$\begin{tabular}[]{c|ccccc}&$a$&$b$&$\overline{a}$&$\overline{b}$\\\
\hline\cr$\partial$&0&0&0&$-ia\overline{a}$\\\
${\overline{\partial}}$&0&$ia\overline{a}$&0&0\\\ \end{tabular}.$

There is an inclusion of $\mathcal{A}$ into the complex de Rham algebra of
$\mathrm{KT}$ which preserves the bidegrees and induces an isomorphism in
Dolbeault cohomology (see Example 4.3 of [CSG22] for details). We obtain:

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

Since $\mathrm{KT}$ is a compact complex surface, its Frölicher spectral
sequence degenerates at the first page, and so the above gives the complex de
Rham cohomology of $\mathrm{KT}$. Considering the standard hermitian metric on
$\mathcal{A}$ the homotopy transfer diagram $(\iota,\rho,h)$ of Lemma 4.1
gives

$h(a\overline{a}):={\overline{\partial}}^{*}G_{\overline{\partial}}(a\overline{a})=-*{\overline{\partial}}*(a\overline{a})=-*{\overline{\partial}}(b\overline{b})=-i*a\overline{a}\overline{b}=-ib,$
$h(a\overline{a}\overline{b}):={\overline{\partial}}^{*}G_{\overline{\partial}}(a\overline{a}\overline{b})=-*{\overline{\partial}}*(a\overline{a}\overline{b})=-*{\overline{\partial}}(b)=-i*a\overline{a}=-ib\overline{b},$

and $h=0$ otherwise. Moreover, this data satisfies $\rho\Delta\iota=0$ and
$\Delta h\Delta\iota=0$ and so the Hodge-to-de-Rham degeneration condition is
satisfied. We have

$\varphi_{1}(\overline{b})=-b\text{ and }\varphi_{1}=0\text{ otherwise},$

while $\varphi_{n}=0$ for all $n>1$. We obtain operations of arity 3 and
bidegree $(-1,1)$ which induce the following non-trivial operations in
cohomology:

$m_{3}([a],[\overline{b}],[\overline{b}])=[a\cdot\overline{b}\cdot\varphi_{1}(\overline{b})]=[ab\overline{b}]\quad\text{
and }\quad
m_{3}([a],[\overline{a}],[\overline{b}])=[a\cdot\overline{a}\cdot\varphi_{1}(\overline{b})]=[ab\overline{a}].$

There are also non-trivial $C_{\infty}$-operations of arity 3 and bidegree
$(0,-1)$:

$\mu_{3}([a],[a],[\overline{a}])=i[ab]\quad\text{ and
}\quad\mu_{3}([a],[\overline{a}],\overline{a}])=-i[b\overline{a}].$

This shows that the ${\mathrm{Hycom}}_{\infty}$-structure in cohomology is not
formal and carries more information than the usual $C_{\infty}$-structure.

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