2 Holomorphic vector bundles

Let E be a differentiable vector bundle of rank r over a compact complex manifold X. Let $A^{p,q}(E)$ be the space of $(p,q)$ -forms with values in E.

Definition 2.1 A semiconnection on E is a $\mathbb {C}$ -linear map $D:\;A^{0,0}(E) \rightarrow A^{0,1}(E) $ satisfying the Leibnitz rule, i.e.,

$$ \begin{align*}D(fs)=(\overline{\partial} f)s+f\cdot Ds\end{align*} $$

for $f \in C^{\infty }(X)$ and $s\in A^{0,0}(E) $ .

It is evident that each semiconnection D can be extended to a first-order differential operator $D:\;A^{p,q}(E) \rightarrow A^{p,q+1}(E) $ . Moreover, if we let $\mathcal {D}(E)$ be the space of semiconnections on E, then it is well known that for a fixed semiconnection $D_0$ , $\mathcal {D}$ can be identified with $D_0+ A^{0,1}(\operatorname {End}(E))$ where $\operatorname {End}(E)$ is the group of differentiable endomorphisms of E (inducing the identity on the base manifold) and thus an affine space.

Definition 2.2 A semiconnection D is called a holomorphic structure if

$$ \begin{align*}D\circ D=0.\end{align*} $$

This condition is called the integrability condition.

Now, let $\mathcal {H}(E)$ be the subset of $\mathcal {D}(E)$ , consisting of holomorphic semiconnections D. Then $\mathcal {H}(E)$ is nothing but the set of holomorphic bundle structure on the differentiable complex vector bundle E. If we denote the group of differentiable automorphisms of E (inducing the identity on the base manifold X) by $\operatorname {GL}(E)$ , then $\operatorname {End}(E)$ be thought of as the Lie algebra of $\operatorname {GL}(E)$ and an action of $\operatorname {GL}(E)$ on $\mathcal {D}(E)$ is given by

$$ \begin{align*}g.D=g^{-1}\circ D \circ g,\end{align*} $$

where $g \in \operatorname {Aut}(E)$ and $D\in \mathcal {D}(E)$ .

3 Deformation of holomorphic vector bundles

We first recall some basic definitions in deformation theory of holomorphic vector bundles (cf. [Reference Miyajima16] for more details). Let $\mathfrak {B}$ be the category of germs of pointed complex spaces $(B,0)$ (a complex space with a reference point) whose associated reduced complex space is a point and X a complex compact manifold. Let E be a holomorphic vector bundle over X, whose associated holomorphic semiconnection and underlying differentiable complex vector bundle will be denoted by $D_E$ and $\underline {E}$ , respectively. We fix once for all a sufficiently large integer k. Consider the Hilbert space $ A^{0,1}(\operatorname {End}(E))_k$ (resp. $ A^{0,2}(\operatorname {End}(E))_{k-1}$ ) obtained by completing $ A^{0,1}(\operatorname {End}(E)))$ (resp. $A^{0,2}(\operatorname {End}(E))))$ with respect to the Sobolev k-norm (resp. ( $k-1$ )-norm). We have an induced analytic map coming from Proposition 2.1

$$ \begin{align*} \mathcal{P}_{D_E} :A^{0,1}(\operatorname{End}(E))_k &\rightarrow A^{0,2}(\operatorname{End}(E))_{k-1}\\ \alpha &\mapsto D_E.\alpha +\alpha \wedge \alpha, \end{align*} $$

which then gives a germ of a Banach analytic space $(\mathcal {P}_{D_E}^{-1}(0), 0)$ .

If $(\pi , (B,0))$ is a local deformation of E and $f:\;(S,0)\rightarrow (B,0)$ is an analytic map of germs of complex spaces, then the pullback of $(\pi , (B,0))$ by f is defined to be the local deformation $(\pi \circ f,(S,0))$ , which we shall denote by $f^*(\pi ,(B,0))$ .

The following fundamental theorem is essentially due to Kuranishi [Reference Miyajima16, Theorem  1].

Next, let us take a moment to recall the definition of group actions on complex spaces. For the sake of completeness, we recall first that a mapping $\alpha $ from a real analytic (resp. complex) manifold W to a Fréchet space F over $\mathbb {C}$ is called real analytic (resp. holomorphic) if, for each point $w_0\in W$ , there exists an open coordinate neighborhood $N_{w_0}$ and a real analytic (resp. holomorphic) coordinate system $t_1,\ldots ,t_n $ in N such that $t_i(w_0)=0$ and for all $w\in N$ , we have that

$$ \begin{align*}\alpha(w)=\sum a_{i_1,\ldots,i_n}t_1^{i_1}(w)\ldots t_n^{i_n}(w),\end{align*} $$

where $a_{i_1,\ldots ,i_n} \in F$ and the convergence is absolute with respect to any continuous seminorm on F. Furthermore, by a $C^p$ -map, we insinuate a p-times continuously differentiable function. Let G be a real (resp. complex) Lie group, and let X be a complex space. A G-action on X is given by a group homomorphism $\Phi :\; G \rightarrow \operatorname {Aut}(X)$ , where $\operatorname {Aut}(X)$ is the group of biholomorphisms of X.

Finally, it is time for us to introduce G-equivariant deformations, which are of central interest of the article. As before, let X be a complex compact manifold over which a holomorphic vector bundle E is defined. Let G be a subgroup of the group of holomorphic automorphisms of E.

Remark 3.2 We make the following convention. Whenever we have a G-action on $(B,0)$ , the G-action on $X\times B$ in Definition 3.4 is defined by

$$ \begin{align*}g(x,b)=(x,g.b)\end{align*} $$

for $g \in G$ and $(x,b)\in X\times B$ . This is exactly the action with respect to which we want the projection $\mathcal {E}_{B} \rightarrow X\times B$ to be G-equivariant. Moreover, the restriction of the G-action on $\mathcal {E}_B$ on the $\mathcal {E}_{B,0}$ is nothing but the initial G-action on E.

As a matter of course, the main goal of this paper is to construct a real analytic (resp. holomorphic) G-equivariant semi-universal local deformation of a complex vector bundle with a real analytic (resp. holomorphic) G-action. Intuitively, the expected extended G-action on the “Kuranishi space” permutes the nearby holomorphic structures and keeps the central one untouched.

4 Existence of group operations on local moduli spaces

In this section, we shall follow strictly the construction given in [Reference Miyajima16], in which the G-action can be naturally integrated along the lines. As usual, let X be a compact complex manifold over which a holomorphic vector bundle E is defined, and let G be a subgroup of the group of holomorphic automorphisms of E. The case that G is a compact Lie group shall be treated first. It should be noted that G will induce a holomorphic G-action on the bundle $\operatorname {End}(E)$ given by

(4.1) $$ \begin{align} g.\sigma =g^{-1}\circ \sigma \circ g \end{align} $$

for $g \in G$ and $\sigma \in \operatorname {End}(E)$ . Consequently, we obtain natural G-action on $A^{0,1}(\operatorname {End}(E))$ , $A^{0,2}(\operatorname {End}(E))$ , and then on $H^{1}(X,\operatorname {End}(E))$ . The compactness of G permits us to impose a G-invariant Hermitian metric on $\operatorname {End}(E)$ , by the unitary trick. This metric induces a G-invariant metric on $A^{0,1}(\operatorname {End}(E))$ with respect to which a formal adjoint

$$ \begin{align*}D_{E}^*: A^{p,q}(\operatorname{End}(E)) \rightarrow A^{p,q-1}(\operatorname{End}(E))\end{align*} $$

of

$$ \begin{align*}D_{E}:A^{p,q-1}(\operatorname{End}(E)) \rightarrow A^{p,q}(\operatorname{End}(E))\end{align*} $$

is provided. Since the G-action is holomorphic, then the connection $D_E$ is a G-equivariant differential operator. The equivariance of $D_{E}^*$ follows from the one of $D_E$ and from the G-invariance of the imposed metric. As a matter of fact, the Laplace–Beltrami operator associated with $\operatorname {End}(E)$ ,

$$ \begin{align*}\square_{E}:= D_{E}^* D_{E}+D_{E}D_{E}^*,\end{align*} $$

is G-equivariant as well. Moreover, the principal part of $\square _{E}$ coincides with that of the usual Laplace–Beltrami operator. The latter is well known to be a strongly elliptic self-adjoint operator of second order. Hence, so is $\square _{E}$ . This is where we can make use of the Hodge theory for the bundle $\operatorname {End}(E)$ . Namely, if we denote the space of harmonic $(0,1)$ -form with coefficients in $\operatorname {End}(E)$ by $\mathcal {H}^{0,1}$ , then $\mathcal {H}^{0,1}$ can be naturally identified with the first cohomology $H^1(X,\operatorname {End}(E))$ and the following orthogonal decomposition is available:

$$ \begin{align*} A^{0,1}(\operatorname{End}(E)) &=\mathcal{H}^{0,1}\oplus \square_{E} A^{0,1}(\operatorname{End}(E)) \\ &= \mathcal{H}^{0,1}\oplus D_E A^{0,0}(\operatorname{End}(E))\oplus D_E^* A^{0,2}(\operatorname{End}(E)) \end{align*} $$

together with two linear operators:

1. (i) the Green operator: $\mathcal {G}:\; A^{0,1}(\operatorname {End}(E)) \rightarrow \square _{E} A^{0,1}(\operatorname {End}(E))$ ,

2. (ii) the harmonic projection: $\mathrm {P}_{\mathcal {H}^{0,1}}:\; A^{0,1}(\operatorname {End}(E)) \rightarrow \mathcal {H}^{0,1}$

such that

(4.2) $$ \begin{align} \mathrm{Id}_{A^{0,1}(\operatorname{End}(E))}= \mathrm{P}_{\mathcal{H}^{0,1}}+\square_{E}\mathcal{G}. \end{align} $$

Consider the map

$$ \begin{align*} \mathcal{P}_{D_E} :A^{0,1}(\operatorname{End}(E))_k &\rightarrow A^{0,2}(\operatorname{End}(E))_{k-1}\\ \alpha &\mapsto D_E.\alpha +\alpha \wedge \alpha \end{align*} $$

defined the previous section.

Proof Indeed, it suffices to prove that for $g\in G$ and $\alpha \in A^{0,1}(\operatorname {End}(E))$ , we have

$$ \begin{align*}g.(\alpha\wedge \alpha)=g.\alpha \wedge g.\alpha.\end{align*} $$

However, this follows immediately from the fact that G acts trivially on the form part and acts on the endomorphism part by the rule (4.1).

Now, we are ready to state our main result.

Proof We consider the following set:

$$ \begin{align*}Q:= \lbrace \alpha \in A^{0,1}(\operatorname{End}(E))_k\vert \left \| \alpha \right \|_k < \epsilon,\; D_E^*\alpha=0,\; D_E^*\circ \mathcal{P}_{D_E}(\alpha)=0 \rbrace .\end{align*} $$

Each $\alpha \in Q$ is a solution of the elliptic partial differential equation

(4.3) $$ \begin{align}\square_{E}\alpha + D_E^*(\alpha\wedge \alpha)=0. \end{align} $$

Thus, Q is actually a subset of $A^{0,1}(\operatorname {End}(E))$ . We claim further that Q is a direct finite-dimensional submanifold of an open neighborhood of $0$ in $ A^{0,1}(\operatorname {End}(E))_k$ . Indeed, let us take into account the following auxiliary analytic function

$$ \begin{align*} \gamma :A^{0,1}(\operatorname{End}(E))_k &\rightarrow \mathcal{H}^{0,1}\oplus D_E^* A^{0,1}(\operatorname{End}(E))_{k} \oplus D_E^* A^{0,2}(\operatorname{End}(E))_{k-1}\\ \alpha &\mapsto \left( \mathrm{P}_{\mathcal{H}^{0,1}}(\alpha),D_E^*\alpha,D_E^*\circ \mathcal{P}_{D_E}(\alpha) \right). \end{align*} $$

The differential of $\gamma $ at $0$ is

$$ \begin{align*}d\gamma_0 =(\mathrm{P}_{\mathcal{H}^{0,1}},D_E^*,D_E^*D_E)\end{align*} $$

whose inverse can be explicitly given by

$$ \begin{align*}(d\gamma_0)^{-1}(\alpha_0,\alpha_1,\alpha_2)=\alpha_0+\mathcal{G}D_E(\alpha_2)+\mathcal{G}(\alpha_2),\end{align*} $$

so that $\gamma $ is a local analytic isomorphism around $0$ due to the inverse mapping theorem for Banach manifolds. Therefore, locally,

$$ \begin{align*}Q=\gamma^{-1}(\mathcal{H}^{0,1}\times 0 \times 0),\end{align*} $$

where $\mathcal {H}^{0,1}$ is known to be a finite-dimensional vector space over $\mathbb {C}$ . This justifies the claim for $\epsilon $ small enough. Now, on the one hand, note that because each component of $\gamma $ is G-equivariant, then so is $\gamma $ . On the other hand, $\mathcal {H}^{0,1}\times 0 \times 0$ is G-invariant. Thus, after shrinking Q (if necessary), we obtain a G-action on Q. For the sake of G-equivariance of $\mathcal {P}_{D_E}$ , the germ of Banach analytic spaces $(\mathcal {P}_{D_E}^{-1}(0), 0)$ carries as well a G-action. Let $T =Q \cap \mathcal {P}_{D_E}^{-1}(0)$ . Then the germ of complex space $(T,0)$ and the inclusion

(4.4) $$ \begin{align}\omega:\; (T,0) \rightarrow (\mathcal{P}_{D_E}^{-1}(0),0) \end{align} $$

will determine a semi-universal local deformation $\mathcal {E} \rightarrow X\times (T,0)$ of E. This can be carried out in a similar way as in the rest of the proof of [Reference Miyajima16, Theorem 1]. What is new here is the fact that the analytic map $\omega $ is further G-equivariant.

Now, we would like to equip $\mathcal {E}$ with a compatible G-action so that the local deformation $\mathcal {E} \rightarrow X\times (T,0)$ becomes G-equivariant in the sense of Definition 3.4. First, as a complex space, the bundle $\mathcal {E}$ is nothing but $\underline {E} \times T$ equipped with the complex structure induced by the holomorphic semiconnection $\omega $ . In addition, each fiber $\mathcal {E}_{s}$ is exactly $\underline {E}$ equipped with a structure $\omega (s)$ of holomorphic vector bundles and in particular a structure $J_s$ of complex manifolds on the differentiable manifold $\underline {E}$ . Thus, the G-equivariance of $\omega $ implies that

(4.5) $$ \begin{align} dg.J_s =J_{gs}.dg, \end{align} $$

where $g\in G$ and $dg$ is the differential of g. At this point, it should be noted that g is a biholomorphism with respect to the complex structure $J_0$ of the initial holomorphic bundle E and that elsewhere we think of g just as a diffeomorphism of $\underline {E}$ . These discussions allow us to define the following G-action on $\underline {E} \times T$ :

$$ \begin{align*} g :\underline{E} \times T &\rightarrow \underline{E} \times T\\ (e,t) &\mapsto (ge,gt) \end{align*} $$

for each $g \in G$ . We claim that the diffeomorphism g defined in this way is actually a biholomorphism of $\mathcal {E}$ . It is the same as verifying that the differential of g at the point $(e,t)$

$$ \begin{align*}dg_{(e,t)}: \;\mathcal{T}_{(e,t)}^{\text{Zar}}\mathcal{E}=\mathcal{T}_e\underline{E}\oplus \mathcal{T}_t^{\text{Zar}} T \rightarrow \mathcal{T}_{g.(e,t)}^{\text{Zar}}\mathcal{E}=\mathcal{T}_{g.e}\underline{E}\oplus \mathcal{T}_{gt}^{\text{Zar}}T\end{align*} $$

is $\mathbb {C}$ -linear, where for each complex space S and for each point $s \in S$ , $\mathcal {T}_s^{\text {Zar}}S$ denotes the Zariski tangent space of S at s. On the one hand, $dg_{(e,t)}=(dg_e,dg_t)$ is diagonal. On the other hand, the G acts on T by biholomorphisms. Therefore, it is reduced to checking that

$$ \begin{align*}dg_e:\;(\mathcal{T}_e\underline{E},J_{e,t}) \rightarrow (\mathcal{T}_{ge}\underline{E},J_{ge,gt})\end{align*} $$

is $\mathbb {C}$ -linear. However, this follows immediately from (4.5), the fact that g is biholomorphic on the central fiber and [Reference Doan7, Lemma 3.1]. In brief, we have just defined a compatible G-action on $\mathcal {E}$ by biholomorphisms, satisfying all the conditions given in Definition 3.4.

Remark 4.1 A local chart of Q is given by the harmonic projection

$$ \begin{align*}\mathrm{P}_{\mathcal{H}^{0,1}}:\; Q \rightarrow \mathcal{H}^{0,1}\end{align*} $$

whose target can be identified with $\mathbb {C}^{\dim _{\mathbb {C}} \mathcal {H}^{0,1}}$ . This map turns out to be the restriction of the usual Kuranishi map on Q

$$ \begin{align*} \mathcal{K} :Q \subset A^{0,1}(\operatorname{End}(E))_k &\rightarrow A^{0,1}(\operatorname{End}(E))_k\\ \alpha &\mapsto \alpha -\frac{1}{2}D_E^* \mathcal{G}[\alpha,\alpha], \end{align*} $$

by the definition of $\mathcal {P}_{D_E}$ and by the decomposition (4.2).

Proof Let K be the connected real maximal compact subgroup of G such that its complexification is G. We repeat the proof of Theorem 4.1 to obtain a K-equivariant analytic maps of germs of (Banach) analytic spaces

$$ \begin{align*}\omega:\; (T,0) \rightarrow (\mathcal{P}_{D_E}^{-1}(0),0).\end{align*} $$

By [Reference Doan7, Theorem 5.1], we can equip local G-actions on $(T,0)$ and on $(\mathcal {P}_{D_E}^{-1}(0),0)$ (extending the initial K-actions), with respect to which the map $\omega $ is G-equivariant. Finally, we can use the same argument as in the proof of Theorem 4.1 to construct a holomorphic G-equivariant semi-universal local deformation $\mathcal {E}$ of E where the extended holomorphic G-actions are local.

Corollary 4.1 tells us that if the automorphism group $\operatorname {Aut}(E)$ of a holomorphic vector bundle E is reductive, then $\operatorname {Aut}(E)$ acts holomorphically on the base $(T,0)$ of its semi-universal local deformation. A somehow natural question arising along this story is to describe the local structure of the moduli space in terms of Kuranishi spaces. More precisely, if we think of E as a point in the “moduli space” $\mathcal {M}(\underline {E})$ of holomorphic complex bundle structures on $\underline {E}$ , it would be interesting to know whether a neighborhood of E in $\mathcal {M}(\underline {E})$ can be modeled on the quotient $T/\operatorname {Aut}(E)$ in some sense. We refer the curious reader to the papers [Reference Catanese2, Reference Catanese3] of Catanese and [Reference Meersseman15] of Meerssemann for an analog discussion where Teichmüller space and Kuranishi space of complex structures on a given differentiable manifold are taken into account.

5 The associated differential graded Lie algebra

Over a field of characteristic zero, a well-known theorem of Lurie in [Reference Lurie14] (obtained independently by Pridham in [Reference Pridham17]) claims that any reasonable moduli problem is controlled by a dgLa. The philosophy hidden behind this theorem is often credited to many big names in the domain: P. Deligne and V. Drinfeld first and foremost, then M. Kontsevich, J. Stasheff, M. Schlessinger, S. Barannikov, V. Schechtman, V. Hinich, and M. Manetti. In this approach, given an object X of which one wishes to study small variations (complex compact manifolds, algebraic schemes, vector bundles, isolated singularities, etc.), the philosophy suggests that there exists a dgLa $\mathfrak {g}_X$ governing deformations of X in the sense that the deformation functor, which to each local Artin algebra A associates the set of Maurer–Cartan solutions modulo the gauge action (defined by means of $\mathfrak {g}_X$ ), is isomorphic to the set of isomorphism classes of deformations of X over $\operatorname {Spec}(A)$ . One of the classic illustrations of this phenomenon is when X is a complex compact manifold. In this case, the controlling dgLa is nothing but the Dolbeault complex with values in the holomorphic tangent bundle of X. This allows us to transform a purely geometric problem to a purely algebraic one in the view that the associated dgLa gives almost all information about the initial local moduli problem: its $0{\text {th}}$ , $1{\text {st}}$ , and $2{\text {nd}}$ cohomology groups are nothing but the space of infinitesimal automorphisms, that of first-order infinitesimal deformations (or equivalently, the tangent space) and that of obstructions to the (formal) smoothness, respectively. For more historical details about this direction, the interested reader is referred to the exceptionally beautiful seminar paper [Reference Toën20] of Toën.

Continuing this spirit, in this section, we first translate the deformation problem of vector bundle in Section 3 into the language of functors of Artinian rings and then compute the dgLa associated with the local moduli problem of holomorphic vector bundle. Hence, a formal version of Corollary 4.1 follows immediately from the mechanism that we developed in [Reference Doan6]. Let us first recall some standard conventions:

1. (1) $\operatorname {\textbf {Set}}$ is the category of sets.

2. (2) $\operatorname {\textbf {Grp}}$ is the category of groups.

3. (3) $\operatorname {\mathbf {Art}_{\mathbb {C}}}$ is the category of local Artinian $\mathbb {C}$ -algebras with residue field $\mathbb {C}$ . For each $A \in \operatorname {\mathbf {Art}_{\mathbb {C}}}$ , we denote its associated germ of complex spaces and its maximal ideal by $\mathrm {Spec}(A)$ and $\mathfrak {m}_A$ , respectively.

In the language of Artinian rings, Definition 3.1 can be read as follows.

Now, consider the Dolbeault complex with values in the endomorphism bundle $\operatorname {End}(E)$ of E

$$ \begin{align*}A^{0,0}(\operatorname{End}(E)) \overset{D_E}{\longrightarrow} A^{0,1}(\operatorname{End}(E)) \overset{D_E}{\longrightarrow} A^{0,2}(\operatorname{End}(E))\overset{D_E}{\longrightarrow}\cdots\overset{D_E}{\longrightarrow}A^{0,n}(\operatorname{End}(E)),\end{align*} $$

which can be further equipped with a graded Lie structure by using the following Lie bracket:

(5.2) $$ \begin{align}[\phi d\bar{z}_I, \psi d\bar{z}_J]=(\phi\circ\psi-(-1)^{\left | I \right |.\left | J \right |} \psi\circ \phi) d\bar{z}_I \wedge\bar{z}_J, \end{align} $$

where $n:=\dim {X}$ , $I,J\subset \lbrace 1,\ldots , n \rbrace $ and $z_1,\ldots , z_n$ are local holomorphic coordinates. We denote this dgLa by $\mathfrak {g}_{*}$ . Observe that the relation (5.1) becomes

$$ \begin{align*}D_E\alpha+\frac{1}{2}[\alpha,\alpha]=0,\end{align*} $$

which is in the form of a Maurer–Cartan equation. As a matter of fact, the functor of Artinian rings corresponding to the local moduli problem of E is given by

$$ \begin{align*} \operatorname{Def}_E :\operatorname{\mathbf{Art}_{\mathbb{C}}} &\rightarrow \operatorname{\textbf{Set}} \\ A &\mapsto \left \{\alpha \in A^{0,1}(\operatorname{End}(E))\otimes \mathfrak{m}_A \mid D_E\alpha+\frac{1}{2}[\alpha,\alpha]=0 \right \}/\sim, \end{align*} $$

where the equivalence relation $\sim $ is given in Definition 5.1(ii).

Finally, for the completeness, we recall the classical deformation functor $\mathbf {\mathrm {MC}}_{\mathfrak {g}_*}$ associated with $\mathfrak {g}_*$ , defined via the Maurer–Cartan equation. We have two functors:

1. (1) The Gauge functor

   $$ \begin{align*} G_{\mathfrak{g}_*}:\; \operatorname{\mathbf{Art}_{\mathbb{C}}} & \rightarrow \operatorname{\textbf{Grp}} \\ A &\mapsto \mathrm{exp}(\mathfrak{g}_0\otimes \mathfrak{m}_A). \end{align*} $$

2. (2) The Maurer–Cartan functor $MC_{\mathfrak {g}_*}:\; \operatorname {\mathbf {Art}_{\mathbb {C}}} \rightarrow \operatorname {\textbf {Set}} $ defined as

   $$ \begin{align*} MC_{\mathfrak{g}_*}:\; \operatorname{\mathbf{Art}_{\mathbb{C}}} & \rightarrow \operatorname{\textbf{Grp}} \\ A & \mapsto \left \{ x \in \mathfrak{g}_1\otimes \mathfrak{m}_A\mid D_E x+\frac{1}{2}[x,x]=0 \right \}. \end{align*} $$

For each A, the gauge action of $G_{\mathfrak {g}_*}(A)$ on the set $MC_{\mathfrak {g}_*}(A)$ is functorial in A and gives an action of the group functor $G_{\mathfrak {g}_*}$ on $MC_{\mathfrak {g}_*}$ . This allows us to define the quotient functor

$$ \begin{align*} \mathbf{\mathrm{MC}}_{\mathfrak{g}_*}:\; \operatorname{\mathbf{Art}_{\mathbb{C}}} & \rightarrow \operatorname{\textbf{Set}} \\ A & \mapsto MC_{\mathfrak{g}_*}(A)/G_{\mathfrak{g}_*}(A). \end{align*} $$

Theorem 5.1 As a consequence, there is an isomorphism

$$ \begin{align*}\mathbf{\mathrm{MC}}_{\mathfrak{g}_*}\cong \operatorname{Def}_E\end{align*} $$

as functors of Artinian rings. As a sequence, the differential graded Lie algebra controlling the deformations of E is

$$ \begin{align*}A^{0,0}(\operatorname{End}(E)) \overset{D_E}{\longrightarrow} A^{0,1}(\operatorname{End}(E)) \overset{D_E}{\longrightarrow} A^{0,2}(\operatorname{End}(E))\overset{D_E}{\longrightarrow}\cdots\overset{D_E}{\longrightarrow}A^{0,n}(\operatorname{End}(E)),\end{align*} $$

where the differential is given by the connection $D_E$ and the Lie bracket given by the rule (5.2).

Proof The local isomorphism

$$ \begin{align*}\mathrm{exp}:\;(A^{0,0}(\operatorname{End}(E)),0) \rightarrow (A^{0,0}(\operatorname{GL}(E)),\mathrm{Id}_E)\end{align*} $$

and the fact that $(A^{0,0}(\operatorname {GL}(E)),\mathrm {Id}_E)$ acts on $A^{0,1}(\operatorname {End}(E))$ by conjugations permit us to conclude that the equivalence relation $\sim $ is given in Definition 5.1(ii) is the same as the one induced by the gauge action of $G_{\mathfrak {g}_*}(A)$ . Therefore, the desired isomorphism follows immediately.

Proof The functor $\mathbf {\mathrm {MC}}_{\mathfrak {g}_*}$ can be naturally upgraded to a derived formal moduli problem $F_{\mathfrak {g}_*}$ in Lurie’s sense (cf. [Reference Lurie14]) via a simplicial version of the Maurer–Cartan equation (see [Reference Hinich10] for such a construction). Moreover, the associated dgLa of $F_{\mathfrak {g}_*}$ is nothing but $\mathfrak {g}_*$ . Consequently, $F_{\mathfrak {g}_*}$ is a naturally extension of $\operatorname {Def}_E$ in the derived world.

Now, note that the action of G on E induces a natural G-action on E. By the same argument as in [Reference Doan6, Lemma 3.1], we can write $\mathfrak {g}_*$ as a homotopic colimit of “simple” dgLas, i.e.,

$$ \begin{align*}\mathfrak{g}_*=\mathrm{colim}_i \; \mathfrak{g}(i)_*,\end{align*} $$

where

1. (i) each $\mathfrak {g}(i)_k$ is finite-dimensional,

2. (ii) $\mathfrak {g}(i)_*$ is cohomologically concentrated in $[0,+\infty )$ ,

3. (iii) each $ \mathfrak {g}(i)_*$ carries a G-action and the colimit of these G-actions gives back the initial G-action on $\mathfrak {g}_*$ .

A remark is in order. Even in the formal aspect, to make the above G-approximation of the associated dgLa $\mathfrak {g}_*$ possible, the G-equivariant Hodge decomposition

$$ \begin{align*}A^{0,n}(\operatorname{End}(E)) =\mathcal{H}^{0,n}\oplus \square_{E} A^{0,n}(\operatorname{End}(E)),\end{align*} $$

which is purely analytic, still plays a crucial role.

By [Reference Doan6, Theorem 2.3], the semi-prorepresentable object of $F_{\mathfrak {g}_*}$ carries a G-action. Hence, the restriction of $F_{\mathfrak {g}_*}$ on $\operatorname {\mathbf {Art}_{\mathbb {C}}}$ (which is nothing but $\operatorname {Def}_E$ ) has a semi-universal element whose base is equipped with a compatible G-action. This finishes the proof.

6 Perspectives

In this final section, we summarize what we did in this paper and in [Reference Doan7] in a more general setting in terms of associated dgLas (we also refer the reader to [Reference Kosarew12] for a version without group actions).

To start, we consider the deformation problem of an analytic object $X_0$ , whose associated controlling differential graded Lie algebra is $(\mathfrak {g}_*,d)$ . As usual, the space of infinitesimal deformations and that of obstructions are the first and the second cohomology of $\mathfrak {g}_*$ , i.e., $H^1(\mathfrak {g}_*)$ and $H^2(\mathfrak {g}_*)$ , respectively. Let

$$ \begin{align*} MC_{\mathfrak{g}_*} :\mathfrak{g}_1 &\rightarrow \mathfrak{g}_2\\ \alpha &\mapsto d\alpha +\frac{1}{2}[ \alpha, \alpha] \end{align*} $$

be the Maurer–Cartan equation associated with $\mathfrak {g}_*$ . Any subgroup G of the automorphism group of $X_0$ induces a natural G-action on each component of $\mathfrak {g}_*$ compatible with the differential d. We assume further that there are good analytic structures on $\mathfrak {g}_0$ , $\mathfrak {g}_1$ , and $\mathfrak {g}_2$ where the implicit function theorem is available (for example, Banach analytic spaces) and there exists a G-invariant metric on $\mathfrak {g}_*$ , with respect to which we are able to compute the formal adjoint $d^*$ of degree $-1$ . Let us denote $\square := dd^*+d^*d$ . Supposedly, we have a decomposition

$$ \begin{align*} \mathfrak{g}_1 &=\ker\square \bigoplus \mathrm{Im}\square \end{align*} $$

together with two linear operators:

1. (i) the “Green operator”: $\mathcal {G}:\; \mathfrak {g}_1 \rightarrow \mathrm {Im}\square $ ,

2. (ii) the “harmonic projection”: $\mathrm {P}_{\ker \square }:\;\mathfrak {g}_1 \rightarrow \ker \square $

such that

$$ \begin{align*}\mathrm{Id}_{\mathfrak{g}_1}= \mathrm{P}_{\ker\square}+\square\mathcal{G}\end{align*} $$

and $\ker \square $ can be naturally identified with $H^1(\mathfrak {g}_*)$ . Consider the following “Kuranishi map”:

$$ \begin{align*} \mathcal{K} :\mathfrak{g}_1 &\rightarrow \mathfrak{g}_1\\ \alpha &\mapsto\alpha +\frac{1}{2}d^*\mathcal{G} [\alpha, \alpha]. \end{align*} $$

Proof Let us denote by N the following space:

$$ \begin{align*}\lbrace \alpha \in \mathfrak{g}_1 \mid \left( \mathcal{K}-\mathrm{P}_{\ker\square}\right)(\alpha)=0 \rbrace.\end{align*} $$

Then it can be checked that the germ of analytic space

$$ \begin{align*}(T,0):=(N,0)\cap(MC_{\mathfrak{g}}^{-1}(0),0)\end{align*} $$

is the desired “Kuranishi space” (see [Reference Kosarew12, Theorem 3.1] for such a verification). The existence of group operations on $(T,0)$ follows immediately from the G-equivariance of all the maps and of all the operators involved.

Remark 6.1 The key point here is the existence of the G-invariant metric and that of the splitting

$$ \begin{align*}\mathrm{Id}_{\mathfrak{g}_1}= \mathrm{P}_{\ker\square}+\square\mathcal{G}.\end{align*} $$

The former is assured if G is a compact Lie group by the unitary trick, whereas the latter can come from the Hodge theory if we deal with complex compact manifolds. In general, we do not have such powerful tools.

The existence of reductive group operations on the Kuranishi space of complex compact manifolds (cf. [Reference Doan7]) and that on the Kuranishi space of holomorphic vector bundles, dealt in this paper, can be thought of as living illustrations of the following philosophy.

“Reductive subgroups of the automorphism group of the analytic object under deformation can be (at least locally) analytically extended to its semi-universal deformation.”

In other words, there should be a compatible extended action on the “Kuranishi space,” which permutes nearby complex structures and the initial group action might be regarded as the stabilizer group with respect to the prescribed complex structure (corresponding to the reference point). The formal aspect of this philosophy was systematically treated in the groundbreaking work of Rim, as mentioned in the introduction, in which a formal extendability of reductive actions is guaranteed, unique up to noncanonical equivariant isomorphisms, for any homogeneous fibered category in groupoid. However, the convergence of his construction, which is necessarily required in the analytic setting, is extremely hard to prove even in simple cases. Therefore, analytically speaking, a rigorously mathematical formulation of this philosophy might be a good problem to work on.