2. Review of formality and minimal models of DG Lie algebras

We work over a field $\mathbb {K}$ of characteristic $0$ for the algebraic part and over the field $\mathbb {C}$ of complex numbers for the geometric applications. Every complex of vector spaces is intended as a cochain complex.

By definition a DG Lie algebra $L$ is formal if it is quasi-isomorphic to its cohomology DG Lie algebra $H^*(L)$, equipped with the trivial differential and the induced bracket. In order to avoid possible mistakes, it is useful to keep in mind that not every DG Lie algebra is formal and that if $L$ is formal, then in general there does not exist any direct quasi-isomorphism of DG Lie algebras $H^*(L)\to L$. However, since the category of DG Lie algebras admits a model structure where the fibrations (respectively, the weak equivalences) are the surjective maps (respectively, the quasi-isomorphisms) it follows that a DG Lie algebra $L$ is formal if and only if there exists a span of surjective quasi-isomorphisms of DG Lie algebras $L\xleftarrow {}M\xrightarrow {}H^*(L)$.

Since two DG Lie algebras are quasi-isomorphic if and only if they are weak equivalent as $L_{\infty }$ algebras, we also have that a DG Lie algebra $L$ is formal if and only if there exist an $L_{\infty }$ algebra $H$ and a span of $L_{\infty }$ weak equivalences

(2.1)\begin{equation} L\xleftarrow{\quad}H\xrightarrow{\quad}H^*(L). \end{equation}

We assume that the reader is familiar with the notion and basic properties of $L_{\infty }$ algebras; see, for example, [Reference GetzlerGet09, Reference KontsevichKon03, Reference Lada and MarklLM95, Reference Lada and StasheffLS93, Reference ManettiMan20]. For the reader's convenience and to fix the sign convention, we briefly recall here the definition of $L_{\infty }$ algebra in the version used for the explicit computations that we shall perform in § 4.

Let $V$ be a graded vector space. Given homogeneous vectors $v_1,\ldots ,v_n$ of $V$ and a permutation $\sigma$ of $\{1,\ldots ,n\}$, we denote by $\chi (\sigma ;v_1,\ldots ,v_n)=\pm 1$ the antisymmetric Koszul sign, defined by the relation

\[ v_{\sigma(1)}\wedge\cdots\wedge v_{\sigma(n)}=\chi(\sigma;v_1,\ldots,v_n)\, v_{1}\wedge\cdots\wedge v_{n} \]

in the $n$th exterior power $V^{\wedge n}$. We shall simply write $\chi (\sigma )$ instead of $\chi (\sigma ;v_1,\ldots ,v_n)$ when the vectors $v_1,\ldots ,v_n$ are clear from the context. For instance, if $\sigma$ is the transposition exchanging 1 and 2 we have $\chi (\sigma )=-(-1)^{|v_1|\,|v_2|}$ where $|v|$ denotes the degree of the homogeneous vector $v$. Notice that if every $v_i$ has odd degree, then $\chi (\sigma )=1$ for every $\sigma$.

Because of the universal property of wedge powers, we shall constantly interpret every linear map $V^{\wedge p}\to W$ as a graded skew-symmetric $p$-linear map $V\times \cdots \times V\to W$.

In the above definition we used the sign convention of [Reference GetzlerGet09, Reference KontsevichKon03, Reference ManettiMan20], while in [Reference Lada and MarklLM95, Reference Lada and StasheffLS93] the maps $\{\cdots \}_k$ differ by the sign $(-1)^{k(k-1)/2}$. Every DG Lie algebra $(L,d,[-,-])$ is an $L_{\infty }$ algebra where $\{\cdot \}_1=d$, $\{\cdot \,\cdot \}_2=[-,-]$ and $\{\cdots \}_n=0$ for every $n>2$. If $\{\cdot \}_1=0$ the $L_{\infty }$ algebra is called minimal.

There exists a general notion of $L_{\infty }$ morphism (see, for example, [Reference ManettiMan20]), but for simplicity of exposition we only recall here the case of morphisms from an $L_{\infty }$ algebra to a DG Lie algebra; this particular case will be sufficient for our purposes.

Definition 2.2 Let $(V,\{\cdot \}_1,\{\cdot \,\cdot \}_2,\{\cdots \}_3,\ldots )$ be an $L_{\infty }$ algebra and $(L,d,[-,-])$ a DG Lie algebra. An $L_{\infty }$ morphism $g\colon V\to L$ is a sequence of maps $g_n\colon V^{\wedge n}\to L$, $n\geqslant 1$, with $g_n$ of degree $1-n$ such that, for every $n$ and every $v_1,\ldots ,v_n\in V$ homogeneous, we have

\begin{align*} &\frac{1}{2}\sum_{p=1}^{n-1} \!\!\!\!\!\sum_{\quad\sigma\in S(p,n-p)}\chi(\sigma) (-1)^{(1-n+p)(|v_{\sigma(1)}|+\cdots+ |v_{\sigma(p)}|-p)} [g_p(v_{\sigma(1)},\ldots, v_{\sigma(p)}),\vphantom{\sum} g_{n-p}(v_{\sigma(p+1)},\ldots,v_{\sigma(n)})]\\ &\quad +dg_n(v_1,\ldots,v_n)= \sum_{k=1}^{n}(-1)^{n-k}\sum_{\sigma\in S(k,n-k)}\chi(\sigma)g_{n-k+1}(\{v_{\sigma(1)},\ldots,v_{\sigma(k)}\}_k,\ldots,v_{\sigma(n)}).\end{align*}

An $L_{\infty }$ morphism $g$ as in Definition 2.2 is called a weak equivalence or a quasi-isomorphism if $g_1\colon (V,\{\cdot \}_1)\to (L,d)$ is a quasi-isomorphism of cochain complexes.

By homotopy classification of $L_{\infty }$ algebras [Reference KontsevichKon03], for every DG Lie algebra $L$ there exist a minimal $L_{\infty }$ algebra $H$ and an $L_{\infty }$ weak equivalence $\imath \colon H\to L$. The algebra $H$ is called the $L_{\infty }$ minimal model of $L$ and it is unique up to isomorphism, while the $L_{\infty }$ morphism $i$ is unique up to homotopy. By homological perturbation theory, every splitting of the complex $(L,d)$ induces canonically a morphism $\imath \colon H\to L$ as above.

Recall that a splitting of $(L,d)$ is a direct sum decomposition $L=H\oplus d(K)\oplus K$ such that $H,K$ are graded vector subspaces of $L$ and the restrictions of the differential $d$ to $H$ and $K$ are respectively zero and injective; see [Reference WeibelWei94, § 1.4]. In particular, $d(L)=d(K)$, $Z(L)=H\oplus d(K)$ and the natural map $H\to H^*(L)$ is an isomorphism of graded vector spaces. Denoting by $\hookrightarrow$ and $\twoheadrightarrow$ the inclusions and the projections given by the splitting $L=H\oplus d(K)\oplus K$, we define the maps

\[ \imath_1\colon H\hookrightarrow L,\quad \pi\colon L\twoheadrightarrow H, \quad h\colon L\twoheadrightarrow d(K) \xrightarrow{\,-d^{-1}} K \hookrightarrow L, \]

that satisfy the contraction identities

\[ d\imath_1 = 0,\quad \pi d =0,\quad \pi\imath_1 = \operatorname{id}_{H},\quad dh + hd = \imath_1\pi - \operatorname{id}_L,\quad h\imath_1 = 0,\quad \pi h = 0,\quad h^2=0. \]

Then a minimal $L_{\infty }$ algebra $(H,0,\{\cdot \,\cdot \}_2,\{\cdots \}_3,\ldots )$ and an extension of $\imath _1$ to an $L_{\infty }$ quasi-isomorphism $\imath \colon H\to L$ are defined by the recursive equations

(2.3)\begin{align} \imath_p(\xi_1,\ldots,\xi_p) = \frac{1}{2}\sum_{k=1}^{p-1}\sum_{\sigma\in S(k,p-k)}\chi(\sigma)(-1)^{\alpha(\sigma)} h[\imath_k(\xi_{\sigma(1)},\ldots),\imath_{p-k}(\ldots,\xi_{\sigma(p)})],\quad p\geqslant2, \end{align}

(2.4)\begin{align} \{\xi_1,\ldots,\xi_p\}_p = \frac{1}{2}\sum_{k=1}^{p-1}\sum_{\sigma\in S(k,p-k)} \chi(\sigma)(-1)^{\alpha(\sigma)} \pi[\imath_k(\xi_{\sigma(1)},\ldots),\imath_{p-k}(\ldots,\xi_{\sigma(p)})], \quad p\geqslant2, \end{align}

where

\[ \alpha(\sigma)=(1-p+k)\bigg(k+\sum_{i=1}^k|\xi_{\sigma(i)}|\bigg). \]

Notice that for every $\xi ,\eta \in H$ we have

\[ \imath_2(\xi,\eta)=h[\imath_1(\xi),\imath_1(\eta)],\quad \{\xi,\eta\}_2=\pi[\imath_1(\xi),\imath_1(\eta)], \]

the integer $\alpha (\sigma )$ is even for $p=2$ and $\chi (\sigma )(-1)^{\alpha (\sigma )}=1$ if $|\xi _i|$ is odd for every $i$. Formulas (2.3) and (2.4) are well known and essentially date back to Kadeishvili's paper [Reference KadeishviliKad82]: the choice of signs comes from standard décalage isomorphisms applied to the explicit formulas used in [Reference Bandiera and ManettiBM18, Theorem 3.7] and [Reference ManettiMan20].

In [Reference ManettiMan15] the second named author proved a series of formality criteria for DG Lie algebras. As a consequence of these criteria we have the following formality transfer theorem, where $H_{CE}^*(A,B)$ denotes the Chevalley–Eilenberg cohomology of the graded Lie algebra $A$ with coefficient in the $A$-module $B$.

It should be noted that for $L=0$ the above theorem reduces to the classical criterion for intrinsic formality of graded Lie algebras.

3. Cyclic and quasi-cyclic DG Lie algebras

The general notion of cyclic (DG) algebra [Reference Getzler and KapranovGK95] specialized to DG Lie algebras gives the following definition; see also [Reference Lazarev and SchedlerLS12].

Definition 3.1 Let $n$ be an integer. A cyclic DG Lie algebra $(L,d,[-,-],(-,-))$ of degree $n$ is a finite-dimensional DG Lie algebra $(L,d,[-,-])$ equipped with a degree $-n$ non-degenerate graded symmetric bilinear form $(-,-)\colon L^{\odot 2}\to \mathbb {K}[-n]$ such that

\[ ({{d} x},y)=(-1)^{|x|+1}(x,{{d} y}),\quad ([x,y],z) = (x,[y,z]),\quad\forall\,x,y,z\in L. \]

The condition $({{d} x},y)=\pm (x,{{d} y})$ implies in particular that $d(L)^{\perp }=\ker (d)$; since $L$ is finite-dimensional we have $d(L)=\ker (d)^{\perp }$, and this implies that also the induced bilinear form in the cohomology $H^*(L)$ is non-degenerate.

Example 3.2 Symplectic representations Let $(V,\omega )$ be a finite-dimensional symplectic vector space and let $\mathfrak {g}$ be a finite-dimensional Lie algebra. Recall that a left action

\[ \mathfrak{g}\times V\to V,\quad (g,v)\mapsto gv, \]

is called symplectic if for every $v,w\in V$ and every $g\in \mathfrak {g}$ we have

\[ \omega(gv,w)+\omega(v,gw)=0. \]

There exists a natural correspondence between (isomorphism classes of) symplectic representations and (isomorphism classes of) cyclic DG Lie algebras of degree 2 with trivial differential and without elements of negative degree: given a symplectic action as above, consider the graded Lie algebra $L=H(L)=L^0\oplus L^1\oplus L^2$ whose cyclic Lie structure is defined as follows.

1. (i) $L^0=\mathfrak {g}$, $L^1=V$, and $L^2=\mathfrak {g}^{\vee }=\operatorname {Hom}_{\mathbb {K}}(\mathfrak {g},\mathbb {K})$.

2. (ii) The Lie bracket is defined by

   • – $[g,v]=gv$ for every $g\in L^0$, $v\in L^1$,

   • – $[v,w] \colon h\mapsto \omega (hv,w)$ for every $v,w\in L^1$, $h\in \mathfrak {g}$,

   • – $[g,y]\colon h\mapsto y([h,g]_{\mathfrak {g}})$ for every $g,h\in \mathfrak {g}$, $y\in \mathfrak {g}^{\vee }$.

3. (iii) The pairing is defined by

   • – $(-,-)\colon L^0\times L^2\to \mathbb {K}$ is the natural pairing,

   • – $(v,w)=\omega (v,w)$ for every $v,w\in L^1$.

The relations below easily follow from the above conditions:

\begin{gather*} (h,[g,y]_L)=([h,g]_{\mathfrak{g}},y) \quad \mbox{for every } h,g\in L^0,\, y\in L^2,\\(g,[v,w]_L)=\omega(gv,w)=\omega([g,v]_L,w) \quad \mbox{for every } g\in L^0,\, v,w\in L^1. \end{gather*}

Moreover, the equalities

\[ \omega(gv,w) + \omega(v,gw) = (gv, w) -\omega(gw,v) = ([g,v]_L,w) - (g,[v,w]_L), \enspace \mbox{for every } g\in\mathfrak{g}, \, v,w\in L^1, \]

show that the symplectic condition $\omega (gv,w)+\omega (v,gw)=0$ is equivalent to the cyclicity condition $(g,[v,w])=([g,v],w)$. The proof that the data $(L,0,[-,-],(-,-))$ defined above provides an example of a cyclic DG Lie algebra of degree 2 is now straightforward.

Notice that the Maurer–Cartan functional $\frac {1}{2}[v,v]$ coincides by definition with the moment map $\mu \colon V\to \mathfrak {g}^{\vee }$ of the symplectic representation.

Example 3.3 Consider the following complex of vector spaces in degrees 0,1,2:

\[ L\colon\quad \operatorname{Span}(a,b)\xrightarrow{d} \operatorname{Span}(x,y,p,db)\xrightarrow{d}\operatorname{Span}(z,dp) \]

equipped with the bilinear form $(-,-)\colon L^{\odot 2}\to \mathbb {K}[-2]$, where the only nontrivial products between basis vectors are

\[ (x,y)=-(y,x)=-1,\quad (db,p)=-(p,db)=-1,\quad (a,z)=(z,a)=1,\quad (b,dp)=(dp,b)=1. \]

Next consider the bracket $[-,-]\colon L^{\wedge 2}\to L$, where the only nontrivial brackets between basis vectors are

\[ [a,x]=db,\quad [a,p]=y,\quad [x,x]=dp,\quad [p,x]=z,\quad [b,x]=y. \]

The next proposition summarizes the properties or the above example that are relevant for this paper.

Proposition 3.4 In the above setup:

1. (i) $L$ is a cyclic DG Lie algebra of degree 2;

2. (ii) $L$ is not a formal DG Lie algebra;

3. (iii) there does not exist any splitting $L=H\oplus d(K)\oplus K$ such that $[H^0,H^1]\subset H^1$.

Proof. The first item is a tedious but straightforward computation. For the second item we observe that the triple Massey power of $x$ is non-trivial since $dp=[x,x]$ and $[p,x]=z$. The third item is clear since for every splitting there exists $\alpha \in \mathbb {K}$ such that $x+\alpha db\in H^1$ and therefore $[a,x+\alpha db]=[a,x]=db\not \in H^1$.

Example 3.5 Consider the following complex of vector spaces in degrees 1,2:

\[ L:\quad \operatorname{Span}(a,b)\xrightarrow{d} \operatorname{Span}(x,db) \]

equipped with the closed bilinear form $(-,-)\colon L^{\odot 2}\to \mathbb {K}[-3]$, where the only non-trivial products between basis vectors are $(a,x)=(b,db)=1$. Next consider the bracket $[-,-]\colon L^{\wedge 2}\to L$, where the only non-trivial brackets between basis vectors are $[a,a]=db$, $[a,b]=x$. The same argument used in the proof of Proposition 3.4 shows that $L$ is a cyclic non-formal DG Lie algebra of degree 3.

It is useful to enlarge the class of cyclic DG Lie algebras by removing the assumption that $L$ is finite-dimensional, which is not satisfied in most geometrical situations. The same weakening of assumption was considered by Kontsevich [Reference KontsevichKon94] in the associative case.

Definition 3.6 A quasi-cyclic DG Lie algebra $(L,d,[-,-],(-,-))$ of degree $n$ is a DG Lie algebra $(L,d,[-,-])$ with finite-dimensional cohomology, together with a degree $-n$ symmetric bilinear form $(-,-)\colon L^{\odot 2}\to \mathbb {K}[-n]$ which satisfies

\[ ({{d} x},y)=(-1)^{|x|+1}(x,{{d} y}),\quad ([x,y],z) = (x,[y,z]),\quad\forall\,x,y,z\in L, \]

and such that the induced form $(-,-)\colon H(L)^{\odot 2} \to \mathbb {K}[-n]$ is non-degenerate.

If $(L,d,[-,-],(-,-))$ is a quasi-cyclic DG Lie algebra then its cohomology $H^*(L)$ is naturally endowed with a structure of cyclic graded Lie algebra of the same degree.

Example 3.7 Vector bundles on manifolds with trivial canonical bundle Let $\mathcal {E}$ be a locally free sheaf of a smooth complex projective manifold $X$ of dimension $n$ with trivial canonical bundle, and denote by $\omega _X$ be a holomorphic volume form. Then the Dolbeault complex

\[ L=A^{0,*}_X({{\mathcal{H}}om}(\mathcal{E},\mathcal{E})) \]

of the sheaf of endomorphisms of $\mathcal {E}$ is a quasi-cyclic DG Lie algebra of degree $n$, where

\[ (f,g)=\int_X \omega_X\wedge \operatorname{Tr}(fg). \]

We have $H^i(L)=\operatorname {Ext}^i_X(\mathcal {E},\mathcal {E})$ and by Serre duality the induced pairing

\[ (-,-)\colon \operatorname{Ext}^i_X(\mathcal{E},\mathcal{E})\times \operatorname{Ext}^{n-i}_X(\mathcal{E},\mathcal{E})\to \mathbb{C} \]

is non-degenerate. In § 5 we extend this construction to coherent sheaves.

We are now ready to state one of the main results of this paper, namely a sufficient condition for formality of quasi-cyclic DG Lie algebras of degree at most $2$.

Theorem 3.8 Let $(L,d,[-,-],(-,-))$ be a quasi-cyclic DG Lie algebra of degree $n\leqslant 2$. Assume that there exists a splitting $L=H\oplus d(K)\oplus K$ such that:

1. (i) $H^i=0$ for $i<0$ (and hence also $H^i=0$ for $i>n$);

2. (ii) $H^0\subset L^0$ is closed with respect to the bracket $[-,-]$;

3. (iii) $H^i,K^i\subset L^i$ are $H^0$-submodules (with respect to the adjoint action) for all $i>0$.

Then the DG Lie algebra $(L,d,[-,-])$ is formal.

For $n\leqslant 0$ the above theorem is trivial since $H^i=0$ for every $i>0$ and then the embedding $H^0\to L$ is a quasi-isomorphism of DG Lie algebras. The next section will be entirely devoted to the (long) proof in the case $n=2$, whose first step also provides a complete proof for $n=1$. Examples 3.5 and 3.3 show that formality fails if either $n>2$ or without the assumption (iii), even for cyclic DG Lie algebras.

4. Proof of Theorem 3.8

Let $(L,d,[-,-],(-,-))$ be as in Theorem 3.8. If $A,B$ are two subsets of $L$ we shall write $A\perp B$ if $(x,y)=0$ for every $x\in A$, $y\in B$. For instance, it follows immediately from the relation $({{d} x},y)=(-1)^{|x|+1}(x,{{d} y})$ that $d(K)\perp d(K)$ and $H\perp d(K)$.

Proof. Since $H^i=0$ for every $i<0$ the DG Lie subalgebra

\[ H^0\oplus (H^1\oplus K^1)\oplus (H^2\oplus d(K^1)\oplus K^2)\oplus \cdots \]

is quasi-cyclic and quasi-isomorphic to $L$. This proves that, up to a possible restriction to a quasi-isomorphic DG Lie subalgebra, it is not restrictive to assume the validity of condition (iii). Next, for every integer $i$ consider the vector subspace

\[ C^i=\{x\in H^i\oplus K^i\mid (x,y)=0\quad\forall\ y\in H^{n-i}\}. \]

Since $(-,-)\colon H^i\times H^{n-i}\to \mathbb {K}$ is a perfect pairing, the map

\[ \begin{aligned} H^i\oplus C^i & \to H^i\oplus K^i \\ (h_1,h_2,k) & \mapsto (h_1-h_2,k) \end{aligned} \]

is an isomorphism. If $x\in C^i$ and $a\in H^0$, then $[a,x]\in H^i\oplus K^i$; for every $y\in H^{n-i}$ we have $(y,[a,x])=([y,a],x)=0$ and therefore $[a,x]\in C^i$. Finally, replacing $K^i$ with $C^i$, we may assume $H\perp K$.

For later use it should be pointed out that the non-degeneracy of $(-,-)\colon H^{\odot 2}\to \mathbb {K}$ immediately implies

(4.1)\begin{equation} x\in K\oplus d(K)\iff (x,y)=0\quad\text{for every}\ y\in H. \end{equation}

From now on we assume that $(L,d,[-,-],(-,-))$ is a quasi-cyclic DG Lie algebra of degree $n\leqslant 2$ equipped with a splitting $L=H\oplus d(K)\oplus K$ satisfying the conditions of Lemma 4.1. The fist step is to use such a splitting in order to produce a minimal $L_{\infty }$ model of $L$. Following the recipe described in § 2, we introduce the maps

\[ \imath_1\colon H\hookrightarrow L,\quad \pi\colon L\twoheadrightarrow H, \quad h\colon L\twoheadrightarrow d(K) \xrightarrow{-d^{-1}} K \hookrightarrow L \]

that satisfy the relations

\[ (\imath_1(x),\imath_1(y)) = (x,y),\quad (h(l),\imath_1(x)) = 0,\quad(\pi(l),x) = (l,\imath_1(x)),\quad\forall\,x,y\in H,\;l\in L. \]

The first one is obvious and the second one follows from the orthogonality condition $H\perp K$. Since $\operatorname {Im}(\imath _1)=H$, $\operatorname {Im}(h)=K$ and $H\perp d(K)\oplus K$, the first two imply the third:

\[ (\pi(l),x)= (\imath_1\pi(l),\imath_1(x))=((\operatorname{id}_L+dh+hd)(l),\imath_1(x))=(l,\imath_1(x)). \]

The maps $\imath _1,\pi ,h$ induce via homotopy transfer a minimal $L_\infty$-algebra structure on $H$, together with an $L_\infty$ quasi-isomorphism $\imath \colon H(L)\to L$ of $L_\infty$-algebras with linear part $\imath _1$. The quadratic components are given by

\[ \imath_2(\xi_1,\xi_2)=h[\imath(\xi_1),\imath(\xi_2)],\qquad \{\xi_1,\xi_2\}_2= \pi[\imath(\xi_1),\imath(\xi_2)]\,, \]

while the higher brackets $\{\cdots \}_p\colon H^{\wedge p}\to H[2-p]$ and the higher Taylor coefficients $\imath _p\colon H^{\wedge p}\to L[1-p]$, $p\geqslant 2$, are explicitly (and recursively) defined by

(4.2)\begin{align} \imath_p(\xi_1,\ldots,\xi_p) = \frac{1}{2}\sum_{k=1}^{p-1}\sum_{\sigma\in S(k,p-k)} \pm\, h[\imath_k(\xi_\sigma(1),\ldots),\imath_{p-k}(\ldots,\xi_{\sigma(p)})], \end{align}

(4.3)\begin{align} \{\xi_1,\ldots,\xi_p\}_p = \frac{1}{2}\sum_{k=1}^{p-1}\sum_{\sigma\in S(k,p-k)} \pm\, \pi[\imath_k(\xi_\sigma(1),\ldots),\imath_{p-k}(\ldots,\xi_{\sigma(p)})], \end{align}

where $\pm$ is the appropriate Koszul sign described explicitly in (2.3) and (2.4). These signs will simplify in our specific case, for instance $\pm 1=+1$ whenever $\xi _i\in H^1$ for every $i$, and we do not need to make them explicit.

Notice that $\{a,b\}_2=[a,b]$ for $a,b\in H^0$ and under the natural isomorphism $H\cong H^*(L)$, the quadratic bracket $\{x,y\}_2=\pi [\imath _1(x),\imath _1(y)]$ on $H$ is just the bracket induced by $[-,-]$ in cohomology.

Lemma 4.2 In the above setup, for every $p\geqslant 2$ and every $g\in H^0$ we have

\[ \imath_p(g,\ldots)=0,\quad \{g,\ldots\}_{p+1}=0 . \]

Proof. If $g\in H^0$, then $[\imath _1(g),\imath _1(\xi )]\in H \subset \operatorname {Ker}(h)$ for all $\xi \in H$, since $H$ is an $H^0$-submodule of $L$, thus $\imath _2(g,\xi )=0$ for all $g\in H^0$ and $\xi \in H$. In general, by formulas (4.2), (4.3) and induction on $p$, for all $p\geqslant 2$, $g\in H^0$ and $\xi _1,\ldots ,\xi _p\in H$, we have

\[ \{\xi_1,\ldots,\xi_p,g\}_{p+1}= \pm\pi[\imath_p(\xi_1,\ldots,\xi_p),\imath_1(g)],\quad \imath_{p+1}(\xi_1,\ldots,\xi_p,g)=\pm h[\imath_p(\xi_1,\ldots,\xi_p),\imath_1(g)]. \]

Finally, notice that for $p\geqslant 2$ we have $\operatorname {Im}(\imath _p)\subset K\subset \operatorname {Ker}(h)\bigcap \operatorname {Ker}(\pi )$, and that $K$ is by hypothesis an $H^0$-submodule of $L$. This implies that $[\imath _p(\xi _1,\ldots ,\xi _p),\imath _1(g)]\in K$ and therefore

\[ \{\xi_1,\ldots,\xi_p,g\}_{p+1}=\imath_{p+1}(\xi_1,\ldots,\xi_p,g)=0. \]

Lemma 4.2 provides a complete proof of formality for $n=1$, since $H^i=0$ for every $i\not =0,1$ and therefore for degree reasons $\{\cdots \}_{p+1}=0$ for every $p\geqslant 2$. From now on we assume that the degree of the quasi-cyclic DG Lie algebra $L$ of Theorem 3.8 is equal to $n=2$.

Lemma 4.3 In the above situation, for every $\xi _1,\ldots ,\xi _p\in H^1$, $p\geqslant 3$, we have

(4.4)\begin{gather} \pi[\imath_p(\xi_1,\ldots,\xi_{p-1}),\imath_1(\xi_{p})]=0, \end{gather}

(4.5)\begin{gather} \{\xi_1,\ldots,\xi_p\}_p = \frac{1}{2}\sum_{k=2}^{p-2}\ \sum_{\sigma\in S(k,p-k)} \pi[\imath_k(\xi_\sigma(1),\ldots),\imath_{p-k}(\ldots,\xi_{\sigma(p)})], \end{gather}

and therefore $\{\xi _1,\xi _2,\xi _3\}_3=0$ for every $\xi _1,\xi _2,\xi _3\in H^1$.

For degree reasons, Lemma 4.2 implies that for $p\geqslant 2$ we have $\{\xi _1,\ldots ,\xi _{p+1}\}_{p+1}=0$ unless $\xi _1,\ldots ,\xi _{p+1}\in H^1$, and then by Lemma 4.3 we have $\{\cdots \}_3\equiv 0$. However, it should be noted that in general the higher brackets $\{\cdots \}_p$ will not vanish for $p\geqslant 4$ and therefore the proof of Theorem 3.8 is still very far from concluded.

Now we notice that item (iii) in the hypotheses of Lemma 4.1 implies that the maps $\imath \colon H\to L$, $\pi \colon L\to H$ and $h:L\to L[-1]$ are equivariant with respect to the induced $\mathfrak {g}$-module structures.

Since $\imath =(\imath _1,\imath _2,\ldots )$ is a morphism of $L_\infty$ algebras we have

\begin{align*} &\sum_{k=1}^{p}\sum_{\sigma\in S(p+2-k,k-1)} \pm \imath_k(\{\xi_{\sigma(1)},\ldots\}_{p+2-k},\ldots,\xi_{\sigma(p+1)}) \\ &\quad = \pm d\imath_{p+1}(\xi_1,\ldots,\xi_{p+1})+\frac{1}{2}\sum_{j=1}^{p}\sum_{ \sigma\in S(j,p+1-j)}\pm[ \imath_j(\xi_{\sigma(1)},\ldots),\imath_{p+1-j}(\ldots,\xi_{\sigma(p+1)}) ], \end{align*}

and taking $p\geqslant 2$, $\xi _1,\ldots ,\xi _p\in H^1$, $\xi _{p+1}=g\in \mathfrak {g}$, by Lemma 4.2 the above expression reduces to

(4.6)\begin{equation} [\imath_p(\xi_1,\ldots,\xi_p),\imath_1(g)] = \imath_p(\{\xi_1,g \}_2,\ldots,\xi_p)+\cdots+\imath_{p}(\xi_1,\ldots,\{\xi_p,g\}_2). \end{equation}

Notice that formula (4.6) is also trivially satisfied for $p=1$. For later use it is useful to introduce, for every $0 < j < p$, the function

\[ I_j^p\colon (H^1)^{\odot j}\otimes (H^1)^{\odot p-j}\to \mathbb{K},\quad I_j^p(\xi_1,\ldots,\xi_p)=(\imath_j(\xi_1,\ldots,\xi_j),\imath_{p-j}(\xi_{j+1},\ldots,\xi_p)). \]

Then for every $0 < j < p$, $\xi _1,\ldots ,\xi _p\in H^1$ and $g\in \mathfrak {g}$, we have

(4.7)\begin{equation} \sum_{i=1}^pI_j^p(\xi_1,\ldots,\{\xi_i,g\}_2,\ldots,\xi_p)=0. \end{equation}

The proof of (4.7) is an immediate consequence of (4.6) together with the identity $([l_1,\imath _1(g)],l_2)+(l_1,[l_2,\imath _1(g)])=0$ for all $g\in \mathfrak {g}$, $l_1,l_2\in L^1$. Moreover, the orthogonality condition $H\perp K$ implies that for every $p\geqslant 2$ we have $I^{p+1}_1=I^{p+1}_p=0$.

By homotopy classification of DG Lie and $L_\infty$ algebras, in order to prove the formality of $L$ it is enough to exhibit an $L_\infty$ isomorphism

\[ f\colon(H,0,\{-,-\},0,\{\cdots\}_4,\{\cdots\}_5,\ldots)\to (H^*(L),0,\{-,-\},0,0,\ldots) \]

between $H$ with the transferred $L_\infty$ algebra structure and $H^*(L)$ with the induced graded Lie algebra structure. Denoting by $f_p\colon H^{\wedge p}\to H[1-p]$ the Taylor coefficients of $f$, the necessary relations these have to satisfy in order for $f$ to be an $L_\infty$ morphism read

(4.8)\begin{align} &\sum_{k=1}^{p}\sum_{\sigma\in S(p+2-k,k-1)} \pm f_k(\{\xi_{\sigma(1)},\ldots\},\ldots,\xi_{\sigma(p+1)}) \nonumber\\ &\quad = \frac{1}{2}\sum_{j=1}^{p}\sum_{\sigma\in S(j,p+1-j)}\pm\{ f_j(\xi_{\sigma(1)},\ldots),f_{p+1-j}(\ldots,\xi_{\sigma(p+1)}) \} \end{align}

for all $p\geqslant 2$ and $\xi _1,\ldots ,\xi _{p+1}\in H$. If these are satisfied, for $f$ to be an isomorphism of $L_\infty$ algebras it is necessary and sufficient that its linear part $f_1\colon H\to H$ is an isomorphism of graded spaces. We look for an $L_\infty$ isomorphism $f$ as above such that moreover $f_1=\operatorname {id}_H$ and $f_p(\xi _1,\ldots ,\xi _p)=0$ for $p\geqslant 2$ unless $p\geqslant 3$ and $\xi _1,\ldots ,\xi _p\in H^1$. With these hypotheses, many of the previous relations (4.8) become trivial, and the only non-trivial ones we are left to verify are

(4.9)\begin{gather} \big\{ f_p\big(\xi_1,\ldots,\xi_p\big),g \big\} = f_p\big(\big\{ \xi_1,g\big\},\ldots,\xi_p\big)+\cdots +f_p\big(\xi_1,\ldots,\big\{\xi_p,g\big\}\big), \end{gather}

(4.10)\begin{gather} \big\{ \xi_1,\ldots,\xi_{p+1} \big\}_{p+1} = \frac{1}{2}\sum_{j=1}^{p}\sum_{\sigma\in S(j,p+1-j)}\big\{ f_j(\xi_{\sigma(1)},\ldots),f_{p+1-j}(\ldots,\xi_{\sigma(p+1)})\big\}, \end{gather}

for all $p\geqslant 2$, $\xi _1,\ldots ,\xi _{p+1}\in H^1$ and $g\in \mathfrak {g}$ (as in the case of transfer formulas, Koszul signs have disappeared since $|\xi _1|=\cdots =|\xi _{p+1}|=1$). Since $f_2=0$ by definition and we already know that $\{\cdots \}_3=0$, relations (4.9) and (4.10) are trivially satisfied for $p=2$.

For every $p\geqslant 3$ and every $1<j<p$, we define recursively the linear maps

\[ f_p\colon(H^1)^{\odot p}\to H^1,\quad F^{p+1}_j\colon (H^1)^{\odot j}\otimes (H^1)^{\odot p-j+1}\to \mathbb{K}, \]

by the formulas

(4.11)\begin{gather} F^{p+1}_j(\xi_1,\ldots,\xi_{p+1})=(f_j(\xi_{1},\ldots,\xi_j),f_{p-j+1}(\xi_{j+1},\ldots,\xi_{p+1})), \end{gather}

(4.12)\begin{gather} \big(f_p(\xi_1,\ldots,\xi_p),\xi_{p+1}\big)= \frac{1}{2} \sum_{j=2}^{p-1}\sum_{\sigma\in S(j,p-j)} (I^{p+1}_j-F^{p+1}_j)(\xi_{\sigma(1)},\ldots,\xi_{\sigma(p)},\xi_{p+1}), \end{gather}

for all $\xi _1,\ldots ,\xi _{p+1}\in H^1$. The validity of (4.9) is proved in the following lemma.

Lemma 4.4 In the above situation, for every $p\geqslant 2$, every $1 < j < p$, every $\xi _1,\ldots ,\xi _{p+1}\in H^1$ and every $g\in \mathfrak {g}$, we have

(4.13)\begin{gather} \sum_{i=1}^{p}f_{p}(\xi_1,\ldots,\{\xi_i,g\},\ldots,\xi_{p})=\{ f_p(\xi_1,\ldots,\xi_p),g \}, \end{gather}

(4.14)\begin{gather} \sum_{i=1}^{p+1}F_j^{p+1}(\xi_1,\ldots,\{\xi_i,g\},\ldots,\xi_{p+1})=0. \end{gather}

Proof. The above formula are trivially satisfied for $p=2$, since $f_2=0$ and (4.14) is empty. Assuming (4.13) valid for all integers smaller than $p$, we have

\begin{align*} &\sum_{i=1}^{p+1}F_j^{p+1}(\xi_1,\ldots,\{\xi_i,g\},\ldots,\xi_{p+1})\\ &\quad=(\{f_j(\xi_1,\ldots,\xi_j),g\},f_{p-j-1}(\xi_1,\ldots,\xi_j))+ (f_j(\xi_1,\ldots,\xi_j),\{f_{p-j-1}(\xi_1,\ldots,\xi_j),g\})=0, \end{align*}

where the second equality follows from the cyclic condition $(\{x,g\},y)+(x,\{y,g\})=0$ for all $g\in \mathfrak {g}$ and $x,y\in H^1$. For the same reason we have

\begin{align*} \big(\big\{f_p(\xi_1,\ldots,\xi_p),g\big\},\xi_{p+1}\big)&= - \big(f_p(\xi_1,\ldots,\xi_p),\big\{\xi_{p+1},g\big\}\big)\\ &= \Big(- \frac{1}{2}\Big) \sum_{j=2}^{p-1}\sum_{\sigma\in S(j,p-j)} (I^{p+1}_j-F^{p+1}_j)\big(\xi_{\sigma(1)},\ldots,\xi_{\sigma(p)},\big\{\xi_{p+1},g\big\}\big)\\ &=\frac{1}{2}\sum_{i=1}^p\sum_{j=2}^{p-1}\sum_{\sigma\in S(j,p-j)} (I^{p+1}_j-F^{p+1}_j)\big(\xi_{\sigma(1)},\ldots,\big\{\xi_{\sigma(i)},g\big\},\ldots,\xi_{p+1}\big)\\ &=\sum_{h=1}^p \big(f_p(\xi_1,\ldots,\{\xi_h,g\},\ldots,\xi_p),\xi_{p+1}\big), \end{align*}

where in the second and fourth equalities we have used the defining formula (4.12), while the third equality is a consequence of (4.7) and (4.14). Since $(-,-)$ is non-degenerate in $H^1$, the formula

\[ \big(\big\{f_p(\xi_1,\ldots,\xi_p),g\big\},\xi_{p+1}\big)= \sum_{h=1}^p \big(f_p(\xi_1,\ldots,\{\xi_h,g\},\ldots,\xi_p),\xi_{p+1}\big) \]

is completely equivalent to (4.13).

Finally, we prove (4.10), or equivalently that

\[ 2\big(\big\{ \xi_1,\ldots,\xi_{p+1} \big\}_{p+1},g\big) = \sum_{j=1}^{p}\sum_{\sigma\in S(j,p+1-j)}\big(\big\{ f_j(\xi_{\sigma(1)},\ldots),f_{p+1-j}(\ldots,\xi_{\sigma(p+1)})\big\},g\big), \]

for every $\xi _1,\ldots ,\xi _{p+1}\in H^1$ and $g\in \mathfrak {g}$. By (4.5) we have

\begin{align*} 2\big(\big\{ \xi_1,\ldots,\xi_{p+1} \big\}_{p+1}, g \big) &= \sum_{j=2}^{p-1}\sum_{\sigma\in S(j,p+1-j)}\big(\pi\big[ \imath_j\big(\xi_{\sigma(1)},\ldots\big) ,\imath_{p+1-j} \big(\ldots,\xi_{\sigma(p+1)}\big) \big],g\big)\\ &= \sum_{j=2}^{p-1}\sum_{\sigma\in S(j,p+1-j)}\big(\big[ \imath_j\big(\xi_{\sigma(1)},\ldots\big) ,\imath_{p+1-j} \big(\ldots,\xi_{\sigma(p+1)}\big) \big],\imath_1(g)\big) . \end{align*}

By using the cyclic relation $([l_1,l_2],l_3)=(l_1,[l_2,l_3])$, $\forall \;l_1,l_2,l_3\in L$, and (4.6) we get

\begin{align*} 2\big(\big\{ \xi_1,\ldots,\xi_{p+1} \big\}_{p+1}, g \big) &= \sum_{j=2}^{p-1}\sum_{\sigma\in S(j,p+1-j)}\big(\imath_j\big(\xi_{\sigma(1)},\ldots\big) ,[\imath_{p+1-j} \big(\ldots,\xi_{\sigma(p+1)}\big),\imath(g)]\big) \\ &= \sum_{j=2}^{p-1}\sum_{\sigma\in S(j,p-j,1)}\big( \imath_j\big(\xi_{\sigma(1)}, \ldots\big),\imath_{p+1-j}\big(\ldots,\xi_{\sigma(p)}, \big\{ \xi_{\sigma(p+1)},g \big\}\big) \big)\\ &=\sum_{j=2}^{p-1}\sum_{\sigma\in S(j,p-j,1)}I^{p+1}_j\big(\xi_{\sigma(1)},\ldots, \xi_{\sigma(p)}, \big\{ \xi_{\sigma(p+1)},g \big\}\big), \end{align*}

where $S(j,p-j,1)$ is the set of permutations $\sigma$ of $1,\ldots ,p+1$ such that

\[ \sigma(1)<\cdots<\sigma(j),\quad \sigma(j+1)<\cdots<\sigma(p). \]

On the other hand,

\begin{align*} &\sum_{j=1}^{p}\sum_{\sigma\in S(j,p+1-j)}\big( \big\{ f_j(\xi_{\sigma(1)},\ldots),f_{p+1-j}(\ldots, \xi_{\sigma(p+1)})\big\},g\big)\\ &\quad =\sum_{j=1}^{p}\sum_{\sigma\in S(j,p+1-j)} \big( f_j(\xi_{\sigma(1)},\ldots),\{f_{p+1-j}(\ldots,\xi_{\sigma(p+1)}),g\}\big)\\ &\quad = \sum_{j=2}^{p-1}\sum_{\sigma\in S(j,p-j,1)} \big( f_j(\xi_{\sigma(1)},\ldots), \{f_{p+1-j}(\ldots,\xi_{\sigma(p+1)}),g\}\big)\\ &\qquad+\sum_{\sigma\in S(p,1)} \big( f_p\big(\xi_{\sigma(1)},\ldots,\xi_{\sigma(p)}\big), \big\{\xi_{\sigma(p+1)},g\big\} \big). \end{align*}

Reasoning as before and using the already proved (4.9), we have

\begin{align*} &\sum_{j=1}^{p}\sum_{\sigma\in S(j,p+1-j)}\big( \big\{ f_j(\xi_{\sigma(1)},\ldots),f_{p+1-j}(\ldots,\xi_{\sigma(p+1)})\big\},g\big)\\ &\quad = \sum_{j=2}^{p-1}\sum_{\sigma\in S(j,p-j,1)} \big( f_j\big(\xi_{\sigma(1)},\ldots \big), f_{p+1-j}\big(\ldots, \xi_{\sigma(p)},\big\{ \xi_{\sigma(p+1)},g\big\}\big) \big)\\ &\qquad+\sum_{\sigma\in S(p,1)} \big( f_p\big(\xi_{\sigma(1)},\ldots,\xi_{\sigma(p)}\big), \big\{\xi_{\sigma(p+1)},g\big\} \big) \\ &\quad = \sum_{j=2}^{p-1}\sum_{\sigma\in S(j,p-j,1)} F^{p+1}_j\big(\xi_{\sigma(1)},\ldots,\xi_{\sigma(p)}, \big\{ \xi_{\sigma(p+1)},g\big\}\big) \\ &\qquad+\sum_{\sigma\in S(p,1)} \big( f_p\big(\xi_{\sigma(1)},\ldots,\xi_{\sigma(p)}\big), \big\{\xi_{\sigma(p+1)},g\big\} \big) \\ &\quad =\sum_{j=2}^{p-1}\sum_{\sigma\in S(j,p-j,1)} I^{p+1}_j\big(\xi_{\sigma(1)},\ldots, \xi_{\sigma(p)}, \big\{ \xi_{\sigma(p+1)},g \big\}\big), \end{align*}

where in the last equality we used the recursive definition (4.12) of $f_p$. The proof of Theorem 3.8 is now complete.

5. Derived endomorphisms and their formality

For every coherent sheaf $\mathcal {F}$ on a smooth complex projective manifold $X$ there is a well-defined homotopy class of DG Lie algebras denoted by $R\mspace {-2mu}\operatorname {Hom}_X(\mathcal {F},\mathcal {F})$ and called, with a little abuse of language, the DG Lie algebra of derived endomorphisms of $\mathcal {F}$. There exist several possible (quasi-isomorphic) representatives for $R\mspace {-2mu}\operatorname {Hom}_X(\mathcal {F},\mathcal {F})$, and we refer to [Reference MeazziniMea18] for an explicit and concrete description of many of them. The importance of the DG Lie algebra of derived endomorphisms relies on the fact that it controls the deformation theory of $\mathcal {F}$ in the usual way via Maurer–Cartan equation modulus gauge action; cf. [Reference Arbarello and SaccàAS18, Reference Budur and ZhangBZ18, Reference Iacono and ManettiIM19, Reference MeazziniMea18]. Moreover, $H^i(R\mspace {-2mu}\operatorname {Hom}_X(\mathcal {F},\mathcal {F}))=\operatorname {Ext}^i(\mathcal {F},\mathcal {F})$ for every $i$.

Since the notion of quasi-cyclic DG Lie algebra is not stable under general quasi-isomorphisms, in view of a possible application of Theorem 3.8 it is useful to consider the Dolbeault representatives for $R\mspace {-2mu}\operatorname {Hom}_X(\mathcal {F},\mathcal {F})$. Consider a finite locally free resolution $\mathcal {E}^*=\{\cdots \mathcal {E}^{-1}\to \mathcal {E}^0\}\to \mathcal {F}$ and denote by

\[ {{\mathcal{H}}om}^*_{\mathcal{O}_X}(\mathcal{E}^*,\mathcal{E}^*)=\bigoplus_d{{\mathcal{H}}om}^d_{\mathcal{O}_X}(\mathcal{E}^*,\mathcal{E}^*)=\bigoplus_d\bigoplus_{p}{{\mathcal{H}}om}_{\mathcal{O}_X}\big(\mathcal{E}^p,\mathcal{E}^{d+p}\big) \]

the (DG) sheaf of endomorphisms of $\mathcal {E}^*$. Then ${{\mathcal {H}}om}^*_{\mathcal {O}_X}(\mathcal {E}^*,\mathcal {E}^*)$ is a sheaf of DG Lie algebras over $X$. It is important to notice that the bracket $[f,g]=fg-(-1)^{|f|\,|g|}gf$ is $\mathcal {O}_X$-bilinear and therefore it can be extended naturally to Dolbeault's resolution

\[ L=A^{0,*}_X({{\mathcal{H}}om}^*_{\mathcal{O}_X}(\mathcal{E}^*,\mathcal{E}^*))=\bigoplus_{p,q,r} A^{0,p}_X({{\mathcal{H}}om}_{\mathcal{O}_X}(\mathcal{E}^q,\mathcal{E}^{r})), \]

where $A^{0,p}_X(\mathcal {G})$ denotes the space of global differential forms of type $(0,q)$ with values in the locally free sheaf $\mathcal {G}$. Similarly, the usual trace map (see, for example, [Reference Iacono and ManettiIM19] and references therein)

\[ \operatorname{Tr}\colon {{\mathcal{H}}om}^*_{\mathcal{O}_X}(\mathcal{E}^*,\mathcal{E}^*)\to \mathcal{O}_X,\quad \operatorname{Tr}(f)=\sum_{i}(-1)^i\operatorname{Tr}(f_i^i),\quad \text{where}\ f=\sum_{i,j}f_{i}^{j},\quad f_{i}^{j}\colon \mathcal{E}^i\to \mathcal{E}^j, \]

is a morphism of sheaves of DG Lie algebras, and extends to a morphism of DG Lie algebras

\[ \operatorname{Tr}\colon L=A^{0,*}_X({{\mathcal{H}}om}^*_{\mathcal{O}_X}(\mathcal{E}^*,\mathcal{E}^*))\to A^{0,*}_X. \]

Since the bracket on $A^{0,*}_X$ is trivial we have

\[ \operatorname{Tr}([f,g])=0,\quad \operatorname{Tr}(df)=\bar{\partial}\operatorname{Tr}(f), \]

for every $f,g\in L$, and this immediately implies that $\operatorname {Tr}([f,g]h)=\operatorname {Tr}(f[h,g])$ for every $f,g,h\in L$. If $\omega$ is a non-trivial section of the canonical bundle of $X$, the graded symmetric bilinear form

(5.1)\begin{equation} (-,-)\colon L^{\odot 2}\to \mathbb{C}[-\dim X],\quad (f,g)=\int_X\omega\wedge \operatorname{Tr}(fg), \end{equation}

is a cyclic bilinear form, where this means that it satisfies the conditions $(df,g)+(-1)^{|f|}(f,dg)=0$ and $([f,g],h)=(f,[g,h])$. Finally, if $\omega$ is a holomorphic volume form, by Serre duality the above bilinear form is non-degenerate in cohomology and therefore $(L,(-,-))$ is a quasi-cyclic DG Lie algebra of degree $\dim X$.

From now on we consider only coherent sheaves on projective surfaces with torsion canonical bundle. According to the Enriques–Kodaira classification of surfaces (see, for example, [Reference Barth, Hulek, Peters and van de VenBHPV04, Reference BeauvilleBea94]), a smooth projective surface has torsion canonical bundle $K$ if and only if it is minimal of Kodaira dimension $0$. According to the values of irregularity $q$ and geometric genus $p_g$, these surfaces are classified into four (non-empty) distinguished classes:

• – projective K3 surfaces, with $q=0$, $p_g=1$ and $K=0$;

• – Enriques surfaces, with $q=0$, $p_g=0$ and $2K=0$;

• – bielliptic surfaces, with $q=1$, $p_g=0$ and $nK=0$ for some $n=2,3,4,6$;

• – Abelian surfaces, with $q=2$, $p_g=1$ and $K=0$.

We are now ready to prove the Kaledin–Lehn formality conjecture for the above surfaces, namely that $R\mspace {-2mu}\operatorname {Hom}_X(\mathcal {F},\mathcal {F})$ is formal whenever $\mathcal {F}$ is polystable with respect to any (possibly non-generic) polarization; see, for example, [Reference Huybrechts and LehnHL10, Chapter 1]. It is useful and instructive to give first a separate proof for the cases of K3 and Abelian surfaces.

Proof. Let us denote by $G$ the linearly reductive group of automorphisms of $\mathcal {F}$. Since $X$ is smooth projective it is not difficult to see that there exists a $G$-equivariant finite locally free resolution $\mathcal {E}^*=\{0\to \mathcal {E}^{-2}\to \mathcal {E}^{-1}\to \mathcal {E}^0\}\to \mathcal {F}$; a detailed proof is given, for instance, in [Reference Bandiera, Manetti and MeazziniBMM20]. We claim that the DG Lie algebra $L=A^{0,*}_X({{\mathcal {H}}om}^*_{\mathcal {O}_X}(\mathcal {E}^*,\mathcal {E}^*))$ satisfies the condition of Theorem 3.8, when equipped with the cyclic non-degenerate structure (5.1).

Assume for the moment that the induced action of $G$ on $L^i$ is rational for every $i$; since the action of $G$ commutes with the differential of the resolution $\mathcal {E}^*$. we have a natural inclusion $G\subset Z^0(L)$ and we can take $H^0=T_{\operatorname {Id}}G\subset Z^0(L)\subset L^0$ as the Lie algebra of $G$. Then, since $G$ is assumed to be linearly reductive we may extend $H^0$ to a $G$-equivariant splitting of $L$ that clearly satisfies the hypotheses of Theorem 3.8.

It remains to be shown that $L^i$ is a rational representation of $G$ for every $i$. This follows immediately from the results of [Reference Bandiera, Manetti and MeazziniBMM20], and we give here only a sketch of the proof. The key point is that if $G$ acts on a coherent sheaf $\mathcal {G}$ then, for every open affine subset $U$, the space $\mathcal {G}(U)$ is a rational finitely supported representation of $G$ [Reference Bandiera, Manetti and MeazziniBMM20, Lemma 3.5]. Recall that a representation is finitely supported if it is isomorphic to a finite direct sum ${\bigoplus _{i=1}^nH_i\otimes W_i}$, for some irreducible rational (hence finite-dimensional) representations $H_i$ and some trivial representations $W_i$; every subrepresentation and every quotient of a rational finitely supported representation remains finitely supported [Reference Bandiera, Manetti and MeazziniBMM20, Lemma 2.7 and Remark 2.8].

Let $X=\bigcup _j U_j$ be a finite open affine cover such that $\mathcal {E}^*$ is free over $U_j$ for every $j$. Then $\Gamma \big (U_j,{{\mathcal {H}}om}^*_{\mathcal {O}_X}(\mathcal {E}^*,\mathcal {E}^*)\big )$ is rational and finitely supported for every $j$, therefore

\[ L\subset \bigoplus_{j} A^{0,*}_{U_j}\big({{\mathcal{H}}om}^*_{\mathcal{O}_X}(\mathcal{E}^*,\mathcal{E}^*)\big) \subset \bigoplus_{j}A^{0,*}_{U_j}\otimes_{\mathbb{C}} \Gamma\big(U_j, {{\mathcal{H}}om}^*_{\mathcal{O}_X}(\mathcal{E}^*,\mathcal{E}^*)\big) \]

is also a rational finitely supported representation of $G$.

Let us now return to our initial situation, namely with $\mathcal {F}$ a polystable sheaf on a smooth projective surface $X$ with torsion canonical bundle $K_X=\Omega ^2_X$. We denote by $n$ be the smallest positive integer such that $K_X^{\otimes n}\simeq \mathcal {O}_X$ (we already know that $n=1,2,3,4,6$).

Every choice of an isomorphism $K_X^{\otimes n}\xrightarrow {\simeq }\mathcal {O}_X$ induces naturally a structure of commutative $\mathcal {O}_X$-algebra on the locally free sheaf of rank $n$,

\[ \mathcal{C}:=\mathcal{O}_X\oplus K_X\oplus K^{\otimes 2}_X\oplus \cdots \oplus K^{\otimes n-1}_X. \]

Since $K_X$ is a torsion line bundle we have that $\mathcal {F}\otimes \mathcal {C}$ is also polystable.

Let $\mathcal {E}^*=\{\cdots \mathcal {E}^{-1}\to \mathcal {E}^0\}\to \mathcal {F}$ be any finite locally free resolution. Then $\mathcal {E}^*\otimes \mathcal {C}$ is a finite locally free resolution of $\mathcal {F}\otimes \mathcal {C}$. Moreover,

\[ {{\mathcal{H}}om}^*_{\mathcal{O}_X}(\mathcal{E}^*\otimes\mathcal{C},\mathcal{E}^*\otimes\mathcal{C})=\bigoplus_{i,j=0}^{n-1} {{\mathcal{H}}om}^*_{\mathcal{O}_X}\big(\mathcal{E}^*\otimes K_X^{\otimes i},\mathcal{E}^*\otimes K_X^{\otimes j}\big), \]

and every direct summand is a ${{\mathcal {H}}om}^*_{\mathcal {O}_X}(\mathcal {E}^*,\mathcal {E}^*)$-module via the adjoint action. The trace map extends naturally to a morphism of sheaves

\begin{align*} \widetilde{\operatorname{Tr}}\colon &{{\mathcal{H}}om}^*_{\mathcal{O}_X}(\mathcal{E}^*\otimes\mathcal{C},\mathcal{E}^*\otimes\mathcal{C})\to \mathcal{C}, \end{align*}

with components

\begin{align*}\widetilde{\operatorname{Tr}}\colon &{{\mathcal{H}}om}^*_{\mathcal{O}_X}\big(\mathcal{E}^*\otimes K_X^{\otimes i},\mathcal{E}^*\otimes K_X^{\otimes j}\big)= {{\mathcal{H}}om}^*_{\mathcal{O}_X}\big(\mathcal{E}^*,\mathcal{E}^*\big)\otimes K_X^{\otimes j-i}\xrightarrow{\operatorname{Tr}\otimes\operatorname{Id}} K_X^{\otimes j-i}. \end{align*}

The DG Lie algebra $L=A^{0,*}_X({{\mathcal {H}}om}^*_{\mathcal {O}_X}(\mathcal {E}^*\otimes \mathcal {C},\mathcal {E}^*\otimes \mathcal {C}))$ is quasi-cyclic of degree 2, when equipped with the pairing

\[ (f,g)=\int_X p_{K}\widetilde{\operatorname{Tr}}(fg), \]

where $p_K\colon A^{0,*}_X(\mathcal {C})\to A^{0,2}_X(K_X)=A^{2,2}_X$ is the projection. In fact, for every $0\leqslant i,j < n$ the above pairing induces the Serre duality isomorphism

\[ \operatorname{Ext}^h_X\big(\mathcal{F}\otimes K^{\otimes i}_X, \mathcal{F}\otimes K^{\otimes j}_X\big)\simeq \operatorname{Ext}^{2-h}_X\big(\mathcal{F}\otimes K^{\otimes j}_X, \mathcal{F}\otimes K^{\otimes i+1}_X\big)^\vee \]

so that it is non-degenerate in cohomology.

Proof. By assumption $\mathcal {F}$ is a pure coherent sheaf that is a direct sum of stable sheaves with the same reduced Hilbert polynomial:

\[ \mathcal{F}=\mathcal{F}_1\oplus \cdots\oplus \mathcal{F}_n. \]

In particular, $\operatorname {Hom}_{\mathcal {O}_X}(\mathcal {F}_i,\mathcal {F}_j)=0$ for every $i\not =j$ and $\operatorname {Hom}_{\mathcal {O}_X}(\mathcal {F}_i,\mathcal {F}_i)=\mathbb {C}$ for every $i$. Consider the following equivalence relation on the set of direct summands

\[ \mathcal{F}_i\sim \mathcal{F}_j\iff \mathcal{F}_i\otimes \mathcal{C}\cong \mathcal{F}_j\otimes \mathcal{C}. \]

Equivalently, $\mathcal {F}_i\sim \mathcal {F}_j$ if and only if $\mathcal {F}_i$ is isomorphic to $\mathcal {F}_j\otimes K^{\otimes h}$ for some $h$. Up to permutation of indices we may assume that $\mathcal {F}_1,\ldots ,\mathcal {F}_r$ are a set or representatives for this equivalence relation. We may write

\[ \mathcal{F}=\bigoplus_{i=1}^r \mathcal{F}_i\otimes \mathcal{W}_i, \]

where every $\mathcal {W}_i$ is a direct sum of line bundles of type $K_X^{\otimes h}$. We have $\mathcal {F}\otimes \mathcal {C}=\bigoplus _{i=1}^r \mathcal {F}_i\otimes \mathcal {C}^{\oplus w_i}$ where $w_i$ is the rank of $\mathcal {W}_i$. Every non-trivial endomorphism of $\mathcal {F}_i$ is a scalar multiple of the identity and then the group of automorphisms of $\mathcal {F}\otimes \mathcal {C}$ is the product of $n$ copies of $\prod _{i=1}^r{\rm GL}_{w_i}(\mathbb {C})$.

Choose $r$ a finite locally free resolution $\mathcal {E}_i^*\to \mathcal {F}_i$: every endomorphism of $\mathcal {F}_i$ is a scalar multiple of the identity and then lifts canonically to $\mathcal {E}_i^*$. It is now easy to verify that

\[ \mathcal{E}^*=\bigoplus_{i=1}^r \mathcal{E}^*_j\otimes \mathcal{W}_i \]

is resolution of $\mathcal {F}$ with the required properties.

Proof. Let $\mathcal {E}\to \mathcal {F}$ be a resolution as in Lemma 5.2 and consider the quasi-cyclic DG Lie algebra

\[ L=A^{0,*}_X({{\mathcal{H}}om}^*_{\mathcal{O}_X}(\mathcal{E}^*\otimes \mathcal{C},\mathcal{E}^*\otimes \mathcal{C})) \]

as a representative in the homotopy class of $R\operatorname {Hom}(\mathcal {F}\otimes \mathcal {C},\mathcal {F}\otimes \mathcal {C})$. The same arguments used in the proof of Theorem 5.1 imply that $L$ is a rational and finitely supported representation of the linearly reductive group of automorphisms of $\mathcal {F}\otimes \mathcal {C}$.

By assumption there exists a natural inclusion of Lie algebras

\[ \operatorname{Hom}_X(\mathcal{F}\otimes\mathcal{C},\mathcal{F}\otimes\mathcal{C})\simeq H^0\subset \operatorname{Hom}_X(\mathcal{E}^*\otimes\mathcal{C},\mathcal{E}^*\otimes\mathcal{C})=Z^0(L) \]

that induces an isomorphism $H^0\simeq H^0(L)$. The adjoint action of $\operatorname {Hom}_X(\mathcal {F}\otimes \mathcal {C},\mathcal {F}\otimes \mathcal {C})$ on $L$ is induced by a rational action of a linearly reductive algebraic group, hence the action of $H^0$ on $L$ is completely reducible and the formality of $L$ follows from Corollary 4.5.

Taking

\[ M=A^{0,*}_X({{\mathcal{H}}om}^*_{\mathcal{O}_X}(\mathcal{E}^*,\mathcal{E}^*)) \]

as a representative in the homotopy class of $R\operatorname {Hom}(\mathcal {F},\mathcal {F})$, we have already observed that there exists a natural inclusion of DG Lie algebra $M\subset L$ together a decomposition of $L$ as a direct sum of $M$-modules:

\[ L=\bigoplus_{i,j=0}^{n-1} A_X^{0,*}\big({{\mathcal{H}}om}^*_{\mathcal{O}_X}\big(\mathcal{E}^*\otimes K_X^{\otimes i},\mathcal{E}^*\otimes K_X^{\otimes j}\big)\big). \]

Now the formality of $M$ is a direct consequence of the formality of $L$ and of the formality transfer theorem (Theorem 2.3).