2.1 Differential graded Lie algebras, deformation and Deligne functors

In this subsection, we introduce the basic definitions and properties of differential graded Lie algebras, together with their associated deformation and Deligne functors.

For full details, we refer the reader to [Reference ManettiMan99, Reference ManettiMan04, Reference ManettiMan09, Reference ManettiMan22].

Definition 2.3 The Thom–Whitney homotopy fibre product of two morphisms of dgLas $h:L \to M$ and $g:N\to M$ is the differential graded Lie algebra defined as

$$\begin{align*}L \times_M N :=\{(l,n,m(t,dt)) \in L\times N \times M[t,dt] \mid m(0)=h(l), \ m(1)=g(n) \}. \end{align*}$$

Here, we denote by $M[t,dt]$ the dgLa $M\otimes K[t,dt]$ , where t has degree zero, $dt$ has degree $1$ and $d^2 t=0$ as well as $(dt)^2=0$ . Moreover, $m(0)$ and $m(1)$ denotes the evaluation in $t=0$ and $t=1$ , respectively.

Definition 2.4 Let L be a nilpotent differential graded Lie algebra, we define:

$$ \begin{align*}\operatorname{\mathrm{Def}}(L)=\frac{\operatorname{\mathrm{MC}}(L)}{\sim_{\textrm{gauge}}},\end{align*} $$

where:

$$ \begin{align*}\operatorname{\mathrm{MC}}(L)=\left\{x\in L^1 \mid dx+\frac{1}{2}[x,x]=0\right\}\end{align*} $$

is the set of the Maurer-Cartan elements, and the gauge action is the action of $\exp (L^0)$ on $\operatorname {\mathrm {MC}}(L)$ , given by:

$$ \begin{align*}e^a * x= x+\sum_{n=0}^{+\infty} \frac{([a,-])^n}{(n+1)!}([a,x]-da).\end{align*} $$

If L is any dgLa, we define the deformation functor associated with L as the functor

$$\begin{align*}\operatorname{\mathrm{Def}}_L: \operatorname{\mathrm{\textbf{Art}}}_{\mathbb{K}}\to \operatorname{\mathrm{\textbf{Sets}}},\end{align*}$$

that associates to every $A \in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ the set

$$\begin{align*}\operatorname{\mathrm{Def}}_L(A) := \operatorname{\mathrm{Def}}(L\otimes \mathfrak{m}_A).\end{align*}$$

We recall that the tangent space to the deformation functor $\operatorname {\mathrm {Def}}_L$ is the first cohomology space $H^1(L)$ of the dgLa L. Moreover, a complete obstruction theory for the functor $\operatorname {\mathrm {Def}}_L$ can be naturally defined and its obstruction space is the second cohomology space $H^2(L)$ of the dgLa L.

If the functor of deformations of a geometric object $\mathcal {X}$ is isomorphic to the deformation functor associated with a dgLa L, then we say that L controls the deformations of $\mathcal {X}$ .

Any morphism $\varphi : L \to M$ , induces a morphism $\varphi : \operatorname {\mathrm {Def}}_L \to \operatorname {\mathrm {Def}}_M$ , that is an isomorphism, whenever $\varphi $ is a quasi-isomorphism.

Let L be a nilpotent dgLa, we define $C(L)$ as the groupoid whose set of objects is $\operatorname {\mathrm {MC}}(L)$ and whose morphisms between two objects x and y are defined as the set

$$\begin{align*}\operatorname{\mathrm{Mor}}_{C(L)}(x,y) = \{ e^a \in \exp(L^0) \mid e^a * x = y\}. \end{align*}$$

The irrelevant stabilizer of a Maurer–Cartan element $x\in \operatorname {\mathrm {MC}}(L)$ is the normal subgroup

$$\begin{align*}I(x) = \{ e^{du + [x,u]} \mid u \in L^{-1} \} \subseteq\operatorname{\mathrm{Mor}}_{C(L)}(x,x). \end{align*}$$

Definition 2.6 The Deligne groupoid of a nilpotent differential graded Lie algebra L is the groupoid $\operatorname {\mathrm {Del}}(L)$ having as objects the Maurer–Cartan elements of L and as morphisms

$$\begin{align*}\operatorname{\mathrm{Mor}}_{\operatorname{\mathrm{Del}}(L)}(x,y) = \frac{\operatorname{\mathrm{Mor}}_{C(L)}(x,y)}{I(x)} = \frac{\operatorname{\mathrm{Mor}}_{C(L)}(x,y)}{I(y)}, \end{align*}$$

where the second equality is a natural isomorphism (see [Reference ManettiMan22, Lemma 6.5.5]).

If L is any dgLa, we define the Deligne functor

$$\begin{align*}\operatorname{\mathrm{Del}}_L:\operatorname{\mathrm{\textbf{Art}}}_{\mathbb{K}} \to \operatorname{\mathrm{\textbf{Grpds}}},\end{align*}$$

as the functor that associates to every $A \in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ the groupoid

$$\begin{align*}\operatorname{\mathrm{Del}}_L(A):= \operatorname{\mathrm{Del}}(L\otimes \mathfrak{m}_A).\end{align*}$$

It is immediate from the definitions that there is an equivalence of functors:

$$\begin{align*}\pi_0(\operatorname{\mathrm{Del}}_L)\cong \operatorname{\mathrm{Def}}_L.\end{align*}$$

2.2 Semicosimplical differential graded Lie algebras

Here, we recall some preliminaries on the semicosimplicial dgLas, their total object and the deformation functors associated with them. We mainly follow [Reference Fiorenza, Manetti and MartinengoFMM12, Reference Fiorenza, Iacono and MartinengoFIM12], see also [Reference ManettiMan22].

Definition 2.8 A semicosimplicial differential graded Lie algebra is a covariant functor $\mathbf {\Delta }_{\operatorname {mon}}\to \mathbf {DGLA}$ , from the category $\mathbf {\Delta }_{\operatorname {mon}}$ , whose objects are finite ordinal sets and whose morphisms are order-preserving injective maps between them, to the category of dgLas. Equivalently, a semicosimplicial dgLa ${\mathfrak g}^\Delta $ is a diagram

where each ${\mathfrak g}_i$ is a dgLa, and for each $i>0$ , there are $i+1$ morphisms of dgLas

$$\begin{align*}\partial_{k,i}\colon {\mathfrak g}_{i-1}\to {\mathfrak g}_{i}, \qquad k=0,\dots,i, \end{align*}$$

such that $\partial _{k+1,i+1}\partial _{l,i}=\partial _{l,i+1}\partial _{k,i}$ , for any $k\geq l$ .

A morphism of semicosimplicial differential graded Lie algebras $f:{\mathfrak g}^\Delta \to {\mathfrak h}^\Delta $ , is given by a sequence $\{f_i: {\mathfrak g}_i \to {\mathfrak h}_i$ } of morphisms of dgLas, commuting with the maps $\partial _{k,i}$ .

In particular, every $ {\mathfrak g}_{i}$ is a vector space and so we can consider the graded vector space $\bigoplus _{n\ge 0}\mathfrak {g}_{n}[-n]$ that has two differentials, i.e.,

$$\begin{align*}d=\sum_{n}(-1)^nd_n,\qquad \text{where}\quad d_n \text{ is the differential of }\mathfrak{g}_{n}, \end{align*}$$

and

$$\begin{align*}\partial=\sum_{i}\partial_i,\qquad \text{where} \quad \partial_i=\partial_{0,i}-\partial_{1,i}+\cdots+(-1)^{i} \partial_{i,i}. \end{align*}$$

Note that $d\partial +\partial d=0$ , thus the graded vector space $\bigoplus _{n\ge 0}\mathfrak {g}_n[-n]$ , endowed with the differential $D=d+\partial $ , is a complex, called the total complex, but it can not be endowed with a structure of dgLa.

However, there is a dgLa that is naturally associated with any semicosimplicial dgLa and that is quasi-isomorphic to the total complex. It is constructed as follows. For every $n\ge 0$ , we denote by $\Omega _n$ the differential graded commutative algebra of polynomial differential forms on the standard n-simplex $\Delta ^n$ :

$$\begin{align*}\Omega_n=\frac{\mathbb{K}[t_0,\ldots,t_n,dt_0,\ldots,dt_n]}{(\displaystyle \sum_{i=0}^n t_i-1,\displaystyle \sum_{i=0}^n dt_i)}.\end{align*}$$

Definition 2.9 The Thom–Whitney dgLa associated with the semicosimplicial dgLa $\mathfrak {g}^{\Delta }$ is

$$\begin{align*}\operatorname{Tot}(\mathfrak{g}^\Delta) =\{ (x_n)_{n}\in \prod_n \Omega_n\otimes {\mathfrak g}_n \mid \delta^{k,n}x_n= \partial_{k,n}x_{n-1}\quad \forall\; 0\le k\le n\},\end{align*}$$

where, for $k=0,\ldots ,n$ , $\delta ^{k,n}\colon \Omega _n\to \Omega _{n-1}$ are the face maps and $\partial _{k,n}\colon \mathfrak {g}_{n-1}\to \mathfrak {g}_n$ are the maps of the semicosimplicial dgLa $\mathfrak {g}^{\Delta }.$

As already mentioned, $\operatorname {\mathrm {Tot}}(\mathfrak {g}^\Delta )$ is quasi-isomorphic, as graded vector space, to the total complex $\left (\bigoplus _{n\ge 0}\mathfrak {g}_n[-n], d + \partial \right )$ .

Remark 2.10 Let $h:L \to M$ and $g:N\to M$ be two morphisms of dgLas and consider the semicosimplicial dgLa

In this case, the associated Thom–Whitney dgLa is nothing else that the Thom–Whitney homotopy fibre product of Definition 2.3.

Example 2.11 Given a sheaf ${\mathcal L}$ of dgLas on a topological space X and an open cover $\mathcal {V}=\{V_i\}_i$ of X, the Čech semicosimplicial dgLa ${\mathcal L}({\mathcal V})$ is given by:

where, as usual, $V_{ij}=V_i \cap V_j$ , $V_{ijk}=V_i \cap V_j \cap V_k$ and so on, denote the intersections, $\mathcal {L}(V_i)=\Gamma (V_i, \mathcal {L})$ stays for the sections and the morphisms $\partial _{k,i}$ are the restriction maps. Here, the total complex associated with the Čech semicosimplicial dgLie algebra $\mathcal {L}(\mathcal {V})$ is the Čech complex $\check {C}(\mathcal {V},\mathcal {L})$ of the sheaf $\mathcal {L}$ . In particular, by the quasi isomorphism between the total complex and the Thom–Whitney dgLa, we have $H^k(\operatorname {\mathrm {Tot}}(\mathcal {L}(\mathcal {V}))) \cong \check {H}^k(\mathcal {V}, \mathcal {L})$ , for all $k \in \mathbb {Z}$ .

According to [Reference Fiorenza, Manetti and MartinengoFMM12, Reference Fiorenza, Iacono and MartinengoFIM12], we define the following deformation functor associated with a semicosimplicial dgLa.

Definition 2.12 Let $\mathfrak g^{\Delta }$ be a semicosimplicial dgLa. The functor

$$\begin{align*}Z^1_{\mathrm{sc}}(\exp \mathfrak{g}^{\Delta}) : \mathbf{Art}_{\mathbb K} \to \mathbf{Set}\end{align*}$$

is defined, for all $A\in \mathbf {Art}_{\mathbb K}$ , by

$$\begin{align*}Z^1_{\mathrm{sc}}(\exp \mathfrak{g}^{\Delta})(A)= \left\{ (l,m)\in (\mathfrak{g}_0^1 \oplus \mathfrak{g}^0_1) \otimes \mathfrak m_A \left| \begin{array}{l} dl+\frac{1}{2}[l,l]=0,\\ \partial_{1,1}l=e^{m}*\partial_{0,1}l, \\ {\partial_{0,2}m} \bullet {-\partial_{1,2}m} \bullet {\partial_{2,2}m} =dn+[\partial_{2,2}\partial_{0,1}l,n]\\ \qquad\qquad\qquad\qquad \text{for some } n\in {\mathfrak g}_2^{-1}\otimes{\mathfrak m}_A \end{array} \right.\right\},\end{align*}$$

where the symbol $ \bullet $ stands for the Baker Campbell Hausdorff product in a Lie algebra.

Two elements $(l_0,m_0)$ and $(l_1,m_1) \in Z^1_{\mathrm {sc}} (\exp \mathfrak {g}^{\Delta })(A)$ are equivalent under the relation $\sim $ if and only if there exist elements $a \in \mathfrak {g}^0_0\otimes \mathfrak {m}_A$ and $b\in {\mathfrak g}_1^{-1}\otimes {\mathfrak m}_A$ such that

(1) $$ \begin{align} \begin{cases} e^a * l_0=l_1\\ - m_0\bullet -\partial_{1,1}a \bullet m_1 \bullet \partial_{0,1}a=db+[\partial_{0,1}l_0,b]. \end{cases} \end{align} $$

Definition 2.13 Let $\mathfrak {g}^\Delta $ be a semicosimplicial dgLa, the functor

$$\begin{align*}H^1_{\mathrm{sc}}(\exp \mathfrak{g}^\Delta) : \mathbf{Art}_{\mathbb K} \to \mathbf{Set}\end{align*}$$

is defined, for all $A\in \mathbf {Art}_{\mathbb K}$ , by

$$\begin{align*}H^1_{\mathrm{sc}}(\exp \mathfrak{g}^\Delta)(A)= \frac{Z^1_{\mathrm{sc}} (\exp \mathfrak{g}^\Delta)(A)}{\sim}.\end{align*}$$

Note that any morphism of semicosimplicial dgLas $\mathfrak {g}^{\Delta } \to \mathfrak {h}^{\Delta }$ induces a natural transformation of functors $H^1_{\mathrm {sc}}(\exp \mathfrak {g}^\Delta ) \to H^1_{\mathrm {sc}}(\exp \mathfrak {h}^\Delta )$ .

Remark 2.14 If $\mathfrak {g}^\Delta $ is a semicosimplicial Lie algebra, i.e., every dgLa ${\mathfrak g}_i$ is concentrated in degree zero, then the functor $H^1_{\mathrm {sc}}(\exp \mathfrak {g}^\Delta )$ reduces to the one defined in [Reference Fiorenza, Manetti and MartinengoFMM12].

Moreover, if $\mathfrak {g}^\Delta $ is a semicosimplicial dgLa, such that every dgLa is concentrated in non negative degrees, we can easily view the above functor as a functor in groupoids. More explicitly, we have

$$\begin{align*}\widetilde{H}^1_{\mathrm{sc}}(\exp \mathfrak{g}^\Delta): \operatorname{\mathrm{\textbf{Art}}}_{\mathbb{K}} \to \operatorname{\mathrm{\textbf{Grpds}}}, \end{align*}$$

where, for any $A\in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ , the set of object is

(2) $$ \begin{align} Z^1_{\mathrm{sc}}(\exp \mathfrak{g}^{\Delta})(A)= \left\{ (l,m)\in (\mathfrak{g}_0^1 \oplus \mathfrak{g}^0_1) \otimes \mathfrak m_A \left| \begin{array}{l} dl+\frac{1}{2}[l,l]=0,\\ \partial_{1,1}l=e^{m}*\partial_{0,1}l, \\ {\partial_{0,2}m} \bullet {-\partial_{1,2}m} \bullet {\partial_{2,2}m} =0\\ \end{array} \right.\right\},\end{align} $$

and the isomorphisms between two objects $(l_0,m_0)$ and $(l_1,m_1) \in Z^1_{\mathrm {sc}} (\exp \mathfrak {g}^{\Delta })(A)$ are given by the elements $a \in \mathfrak {g}^0_0\otimes \mathfrak {m}_A$ as above, i.e., such that

(3) $$ \begin{align} \begin{cases} e^a * l_0=l_1\\ - m_0\bullet -\partial_{1,1}a \bullet m_1 \bullet \partial_{0,1}a=0. \end{cases}\end{align} $$

We recall the following result [Reference Fiorenza, Iacono and MartinengoFIM12, Theorem 4.10], that relates these functors with the ones associated with the dgLas.

Theorem 2.15 Let $\mathfrak {g}^\Delta $ be a semicosimplicial dgLa, such that $H^j(\mathfrak {g}^i)=0$ for all $i\geq 0$ and $j <0$ . Then, there is an equivalence of functors:

$$\begin{align*}\operatorname{\mathrm{Def}}_{\operatorname{\mathrm{Tot}}(\mathfrak{g}^{\Delta})} \cong H^1_{\mathrm{sc}}(\exp \mathfrak{g}^\Delta). \end{align*}$$

In particular, the functor $H^1_{\mathrm {sc}}(\exp \mathfrak {g}^\Delta )$ is a deformation functor, its tangent space is $H^1(\operatorname {\mathrm {Tot}}(\mathfrak {g}^{\Delta }))$ and its obstructions are contained in $H^2(\operatorname {\mathrm {Tot}}(\mathfrak {g}^{\Delta }))$ .

In particular, if the functor of deformations of a geometric object $\mathcal {X}$ is isomorphic to the deformation functor $H^1_{\mathrm {sc}}(\exp \mathfrak {g}^\Delta )$ associated with a semicosimiplicial dgLa $\mathfrak {g}^\Delta $ , then the Thom–Whitney dgLa ${\operatorname {\mathrm {Tot}}(\mathfrak {g}^{\Delta }})$ controls the deformations of $\mathcal {X}$ .

Example 2.16 Let $\mathcal {E}$ be a locally free sheaf of $\mathcal {O}_X$ -modules on a projective variety X. Denote by $\mathcal {E} nd (\mathcal {E})$ the sheaf of $\mathcal {O}_X$ -modules endomorphisms of $\mathcal {E}$ . It is a classical fact that $\mathcal {E} nd (\mathcal {E})$ encodes all the information of the infinitesimal deformations of $\mathcal {E}$ . Over the field of complex numbers $\mathbb {C}$ , it can be rephrased saying that the dgLa $A_X^{0,*}(\mathcal {E} nd(\mathcal {E}))$ controls the deformations of $\mathcal {E}$ via the Maurer–Cartan functor modulo the gauge equivalence [Reference FukayaFu03]. Over any algebraically closed field $\mathbb {K}$ of characteristic zero, one can consider the Čech semicosimplicial Lie algebra associated with the sheaf $\mathcal {E} nd (\mathcal {E})$ and with an open affine cover $\mathcal {V}=\{V_i\}_i$ of X:

In [Reference Fiorenza, Manetti and MartinengoFMM12] the authors proved that the functor $H_{sc}^1(\exp \mathcal {E} nd (\mathcal {E})(\mathcal V))$ of Definition 2.13 is isomorphic to the functor of infinitesimal deformations of $\mathcal {E}$ . Moreover, the cohomology of the total complex of the semicosimplicial dgLa above is $H^k(\operatorname {\mathrm {Tot}}(\mathcal {E} nd (\mathcal {E})(\mathcal V))) \cong H^k(X, \mathcal {E} nd (\mathcal {E}))$ for all $k \in \mathbb {Z}$ , according to the classical fact that the tangent space to the functor of the infinitesimal deformations of $\mathcal {E}$ is $H^1(X, \mathcal {E} nd (\mathcal {E}))$ and the obstructions are contained in $H^2(X, \mathcal {E} nd (\mathcal {E}))$ .

2.3 Semicosimplicial groupoid and descent theorem

This subsection is dedicated to semicosimplicial objects in the category of groupoids, their total groupoid and the fundamental Hinich’s theorem on descent of Deligne groupoids. Here, we mainly follow [Reference ManettiMan22].

Definition 2.17 A semicosmplicial groupoid is a covariant functor $\mathbf {\Delta }_{\operatorname {mon}}\to \operatorname {\mathrm {\textbf {Grpds}}}$ , from the category $\mathbf {\Delta }_{\operatorname {mon}}$ , whose objects are finite ordinal sets and whose morphisms are order-preserving injective maps between them, to the category of groupoids. Equivalently, a semicosimplicial groupoid $G^\Delta $ is a diagram

where each ${G}_i$ is a groupoid, and for each $i>0$ , there are $i+1$ morphisms of groupoids

$$\begin{align*}\partial_{k,i}\colon {G}_{i-1}\to {G}_{i}, \qquad k=0,\dots,i, \end{align*}$$

such that $\partial _{k+1,i+1}\partial _{l,i}=\partial _{l,i+1}\partial _{k,i}$ , for any $k\geq l$ . Since every groupoid is a small category, equality of morphisms of groupoids is intended in the strict sense.

A morphism of semicosimplicial groupoids $f:G^\Delta \to H^\Delta $ , is given by a sequence $\{f_i: G_i \to H_i$ } of morphisms of groupoids, commuting with the maps $\partial _{k,i}$ .

Example 2.18 Let ${\mathfrak g}^\Delta $ be a semicosimplicial dgLa

such that every $ {\mathfrak g}_i$ is a nilpotent dgLa. Applying the Deligne groupoid of Definition 2.6, one obtains the semicosimplicial groupoid

Analogously, for any $A \in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ and any semicosimplicial dgLa ${\mathfrak g}^\Delta $ , one defines the semicosimplicial groupoid

We end up this section with the following theorem due to Hinich on descent of Deligne groupoids.

Theorem 2.21 [Reference HinichHin97, Corollary 4.1.1]

Let $\mathfrak {g}^{\Delta }$ be a nilpotent semicosimplicial dgLa, such that every $\mathfrak {g}_n$ is trivial in negative degrees. Then, there exists a natural equivalence of groupoids

$$\begin{align*}\operatorname{\mathrm{Del}}_{\operatorname{\mathrm{Tot}}(\mathfrak{g}^{\Delta})} \to \operatorname{\mathrm{Tot}}(\operatorname{\mathrm{Del}}_{\mathfrak{g}^{\Delta}}). \end{align*}$$

3.1 Geometric deformations

Let X be a smooth projective variety over $\mathbb {K}$ , $\mathcal {F}$ and $\mathcal {G}$ locally free sheaves of $\mathcal {O}_X$ -modules over X and $\alpha \colon \mathcal {F}\to \mathcal {G}$ a morphism of them. First of all we recall some classical definitions.

Definition 3.2 An infinitesimal deformation of the morphism $\alpha \colon \mathcal {F}\to \mathcal {G}$ over $A\in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ is a morphism $\alpha _A: \mathcal {F}_A\to \mathcal {G}_A$ of locally free sheaves of $\mathcal {O}_X \otimes A$ -modules over $X\times \operatorname {\mathrm {Spec}} A$ , where $\mathcal {F}_A$ and $\mathcal {G}_A$ are deformations of $\mathcal {F}$ and $\mathcal {G}$ over A respectively, such that the following diagram is commutative:

Two deformations $\alpha _A: \mathcal {F}_A\to \mathcal {G}_A$ and ${\alpha '}_A: {\mathcal {F}'}_A\to {\mathcal {G}'}_A$ of $\alpha : \mathcal {F} \to \mathcal {G}$ over A are isomorphic, if there exist a pair $(\phi ,\psi )$ of isomorphisms of sheaves $\phi : \mathcal {F}_A \to {\mathcal {F}'}_A$ and $\psi : \mathcal {G}_A \to {\mathcal {G}'}_A$ , such that the following diagram commutes:

We recall that, the trivial deformation of a sheaf $\mathcal {F}$ over A is given by $\mathcal {F}\otimes _{\mathcal {O}_X}\mathcal {O}_ {X \times \operatorname {\mathrm {Spec}} A}=\mathcal {F}\otimes _{\mathbb {K}} A $ and that the trivial deformation of $\alpha \colon \mathcal {F}\to \mathcal {G}$ over $A\in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ is given by the trivial extension $ \alpha \otimes \operatorname {\mathrm {Id}}_A \colon \mathcal {F}\otimes _{\mathbb {K}} A \to \mathcal {G}\otimes _{\mathbb {K}} A $ .

Note that, since $\mathcal {F}$ and $\mathcal {G}$ are locally free sheaves, any infinitesimal deformation of $\alpha \colon \mathcal {F}\to \mathcal {G}$ is locally trivial, i.e., it is locally in X isomorphic to the trivial deformation.

Definition 3.3 The functor of infinitesimal deformations of the morphism $\alpha :\mathcal {F} \to \mathcal {G}$ is the functor

$$\begin{align*}\operatorname{\mathrm{Def}}_{(\mathcal{F},\alpha, \mathcal{G})} : \operatorname{\mathrm{\textbf{Art}}}_{\mathbb{K}} \to \operatorname{\mathrm{\textbf{Grpds}}},\end{align*}$$

that associates, to any $A \in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ , the groupoid $\operatorname {\mathrm {Def}}_{(\mathcal {F},\alpha , \mathcal {G})}(A)$ , whose objects are the deformations of the morphism $\alpha $ over A and whose morphisms are the isomorphisms of them. Note that

$$\begin{align*}\pi_0(\operatorname{\mathrm{Def}}_{(\mathcal{F},\alpha, \mathcal{G}))(A)}) = \{\mbox{isomorphism classes of deformations of the morphism } \alpha \mbox{ over } A\} \end{align*}$$

is the classical functor of deformations in $\operatorname {\mathrm {\textbf {Sets}}}$ .

Let $\alpha \colon \mathcal {F}\to \mathcal {G}$ be a morphism of locally free sheaves and $\gamma $ its graph in $ \mathcal {F}\oplus \mathcal {G}$ . We define $\gamma $ as the image of the morphism of sheaves $(\operatorname {\mathrm {Id}}, \alpha ): \mathcal {F} \to \mathcal {F}\oplus \mathcal {G}$ .

For future use, we would like to observe that a deformation of $\alpha : \mathcal {F} \to \mathcal {G}$ over $A \in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ as in Definition 3.2 can be also seen as the collection of the following data:

• • two deformations $\mathcal {F}_A$ and $\mathcal {G}_A$ of $\mathcal {F}$ and $\mathcal {G}$ over A, respectively;

• • a deformation $\gamma _A \subseteq (\mathcal {F}\oplus \mathcal {G})_A$ of the inclusion $\gamma \subseteq \mathcal {F}\oplus \mathcal {G}$ over A;

• • an isomorphism $f_A$ between the deformations $\mathcal {F}_A \oplus \mathcal {G}_A$ and $(\mathcal {F}\oplus \mathcal {G})_A$ .

Two of such collections of data $\left (\mathcal {F}_A, \mathcal {G}_A, \gamma _A \subseteq (\mathcal {F}\oplus \mathcal {G})_A, f_A \right )$ and $\left (\mathcal {F}^{\prime }_A, \mathcal {G}^{\prime }_A, \gamma ^{\prime }_A \subseteq (\mathcal {F}\oplus \mathcal {G})^{\prime }_A, f^{\prime }_A \right )$ define isomorphic deformations of $\alpha :\mathcal {F} \to \mathcal {G}$ , if there exist:

• • two isomorphisms $\phi : \mathcal {F}_A\to \mathcal {F}^{\prime }_A$ and $\psi :\mathcal {G}_A\to \mathcal {G}^{\prime }_A$ of the deformations of $\mathcal {F}$ and $\mathcal {G}$ over A, respectively,

• • an isomorphism $\chi :(\mathcal {F}\oplus \mathcal {G})_A\to (\mathcal {F}\oplus \mathcal {G})^{\prime }_A$ of the deformations of $\mathcal {F}\oplus \mathcal {G}$ over A, with $\chi (\gamma _A) \subseteq \gamma ^{\prime }_A$ ,

such that the following diagram is commutative

We point out that this approach is similar to the one used in [Reference HorikawaHo73, Reference HorikawaHo74, Reference HorikawaHo76, Reference SernesiSe06] to analyze deformations of holomorphic maps of complex manifolds. In particular, this was applied in [Reference IaconoIa06, Reference IaconoIa08] and in [Reference ManettiMan22] to investigate these deformations via dgLas.

3.2 The local case

First of all, we analyze the infinitesimal deformations of a morphism $\alpha : \mathcal {F} \to \mathcal {G}$ in the local case. We assume that X is a smooth affine variety over $\mathbb {K}$ and $\alpha : \mathcal {F} \to \mathcal {G}$ is a morphism of the free sheaves $\mathcal {F}$ and $\mathcal {G}$ on X. Under this hypothesis, we are able to find out a dgLa that controls the deformations of $\alpha $ .

We denote by $\mathcal {E} nd (\mathcal {F})$ , $\mathcal {E} nd (\mathcal {G})$ and $\mathcal {E} nd (\mathcal {F} \oplus \mathcal {G})$ the free sheaves of the $\mathcal {O}_X$ -modules endomorphisms of the sheaves $\mathcal {F}$ , $\mathcal {G}$ and $\mathcal {F} \oplus \mathcal {G}$ , respectively. Moreover, let $\mathcal {L}$ be the subsheaf of $\mathcal {E} nd (\mathcal {F} \oplus \mathcal {G})$ that preserves the graph $\gamma \subseteq \mathcal {F} \oplus \mathcal {G}$ of the morphism $\alpha $ , i.e.,

$$\begin{align*}\mathcal{L} := \{ \varphi \in \mathcal{E} nd (\mathcal{F} \oplus \mathcal{G}) \mid \varphi(\gamma) \subseteq \gamma \}. \end{align*}$$

Definition 3.5 In the above notation, we define the dgLa $H(X, \mathcal {F}, \alpha , \mathcal {G})$ as the Thom–Whitney homotopy fibre product of the diagram of Lie algebras

where h is the inclusion and g is defined as the map $(\varphi , \psi ) \mapsto \left ( \begin {array}{cc} \varphi & 0 \\ 0 & \psi \end {array}\right ). $ More explicitly, according to Definition 2.3, the elements of $H(X, \mathcal {F}, \alpha , \mathcal {G})$ are of the form $(x, y, z(t,dt))$ , where

$$\begin{align*}x \in \Gamma(X, \mathcal{L}), \quad y \in \Gamma(X, \mathcal{E} nd(\mathcal{F}) \oplus \mathcal{E} nd (\mathcal{G})), \end{align*}$$

$$\begin{align*}z(t, dt) \in \Gamma(X, \mathcal{E} nd (\mathcal{F} \oplus \mathcal{G}))[t, dt], \end{align*}$$

such that $ z(0)= h(x), z(1) = g(y)$ .

Then, we can prove the following result.

Proof By Definition 3.5, $H(X,\mathcal {F}, \alpha , \mathcal {G})$ is the Thom–Whitney dgLa of the following semicosimplicial Lie algebra:

By Hinich’s Theorem on descent of Deligne groupoids (Theorem 2.21), there is an equivalence of functors of groupoids

$$\begin{align*}\operatorname{\mathrm{Del}}_{H(X,\mathcal{F}, \alpha, \mathcal{G})} \cong \operatorname{\mathrm{Tot}}(\operatorname{\mathrm{Del}}_{\mathfrak{h}^\Delta}), \end{align*}$$

i.e., for every $A \in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ , there is an equivalence of the groupoids $\operatorname {\mathrm {Del}}_{H(X,\mathcal {F}, \alpha , \mathcal {G})} (A) \cong \operatorname {\mathrm {Tot}}(\operatorname {\mathrm {Del}}_{\mathfrak {h}^\Delta })(A).$

Let us describe explicitly the objects and the morphisms of the groupoid $\operatorname {\mathrm {Tot}}(\operatorname {\mathrm {Del}}_{\mathfrak {h}^\Delta })(A)$ , for any $A \in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ . Its objects are elements of the form $(x,y,z)$ , where:

$$\begin{align*}x \in \operatorname{\mathrm{Del}}(\Gamma(X, \mathcal{L}) \otimes \mathfrak{m}_A), \quad y \in \operatorname{\mathrm{Del}}(\Gamma(\mathcal{E} nd (\mathcal{F}) \oplus \mathcal{E} nd (\mathcal{G})) \otimes \mathfrak{m}_A), \end{align*}$$

and so $x=y=0$ , since we are dealing with Lie algebras, and

$$\begin{align*}z = e^w \in \exp (\Gamma(X, \mathcal{E} nd (\mathcal{F} \oplus \mathcal{G})) \otimes \mathfrak{m}_A), \end{align*}$$

such that $e^w * 0 = 0 \in \Gamma (X, \mathcal {L})\otimes \mathfrak {m}_A$ . A morphism between two of such objects, $e^w$ and $e^t$ , is of the form $e^a$ , where $a=(a_1,a_2,a_3) \in (\Gamma (X,\mathcal {E} nd (\mathcal {F}) \oplus \mathcal {E} nd (\mathcal {G})) \oplus \Gamma (X,\mathcal {L})) \otimes \mathfrak {m}_A$ , such that $e^w e^{h(a)} = e^t e^{g(a)}.$

These data corresponds to the data of the objects and morphisms of the groupoid $\operatorname {\mathrm {Def}}_{(\mathcal {F},\alpha ,\mathcal {G})}(A)$ . Indeed, when X is an affine smooth variety, any deformation of a free sheaf on X is trivial and the only datum of a deformation of $\alpha $ is an isomorphism of the direct sum of the trivial deformations of $\mathcal {F}$ and $\mathcal {G}$ to the trivial deformation of $\mathcal {F} \oplus \mathcal {G}$ , that preserves $\gamma $ , this is $e^w$ .

A morphism between two of such deformations, $e^w$ and $e^t$ , is given by an isomorphism of the trivial deformation of $\mathcal {F}$ , an isomorphism of the trivial deformation of $\mathcal {G}$ , an isomorphism of the trivial deformation of $\mathcal {F} \oplus \mathcal {G}$ that preserve $\gamma $ . These data are $(a_1,a_2,a_3)$ , respectively, and they have to respect compatibilities with the isomorphisms above, that are expressed by the equation $e^w e^{h(a)} = e^t e^{g(a)}$ .

3.3 The global case

Let, now, X be a smooth projective variety over $\mathbb {K}$ , $\alpha : \mathcal {F} \to \mathcal {G}$ a morphism of locally free sheaves on X and $\mathcal {V} = \{ V_i \}_i$ an open affine cover of X trivialising both $\mathcal {F}$ and $\mathcal {G}$ .

Consider the following semicosimplicial dgLa

where $H(V_i,\mathcal {F}, \alpha , \mathcal {G})$ is the Thom–Whitney homotopy fibre of Definition 3.5.

Then, we can prove the following result.

Proof Applying Hinich’s Theorem on descent of Deligne groupoids (Theorem 2.21), there exists an equivalence of groupoids:

$$\begin{align*}\operatorname{\mathrm{Del}}_{\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta)} \cong \operatorname{\mathrm{Tot}}(\operatorname{\mathrm{Del}}_{H(\mathcal{V})^\Delta}).\end{align*}$$

By Proposition 2.20 and by the local case, analyzed in Theorem 3.6, we have

$$\begin{align*}\operatorname{\mathrm{Tot}}(\operatorname{\mathrm{Del}}_{H(\mathcal{V})^\Delta}) \cong \operatorname{\mathrm{Tot}}(\operatorname{\mathrm{Def}}_{(\mathcal{F}, \alpha, \mathcal{G})(\mathcal{V})^\Delta}). \end{align*}$$

Here $\operatorname {\mathrm {Def}}_{(\mathcal {F}, \alpha , \mathcal {G})(\mathcal {V})^\Delta }$ is the semicosimplicial functor in groupoids

where $\operatorname {\mathrm {Def}}_{(\mathcal {F}, \alpha , \mathcal {G})(V)}$ is the functor of infinitesimal deformations of $\alpha _{|V}: \mathcal {F}_{|V} \to \mathcal {G}_{|V}$ , for any $V\subset X$ affine open subset that trivialises both $\mathcal {F}$ and $\mathcal {G}$ . Moreover, by a classical argument, global deformations are given by gluing local ones, then

$$\begin{align*}\operatorname{\mathrm{Tot}}(\operatorname{\mathrm{Def}}_{(\mathcal{F}, \alpha, \mathcal{G})(\mathcal{V})^\Delta}) \cong \operatorname{\mathrm{Def}}_{(\mathcal{F}, \alpha, \mathcal{G})},\end{align*}$$

as desired.

Remark 3.10 Let $\alpha : \mathcal {F} \to \mathcal {G}$ be a morphism of locally free sheaves of $\mathcal {O}_X$ -modules on X and let $\mathcal {V}=\{V_i\}_i$ be an affine open cover of X, we can construct a bisemicosimplicial object associated with these data as follows. On every open set $V_i \in \mathcal {V}$ , we have

Since h and g are morphisms of sheaves, they commute with restrictions of every open subsets, inducing morphisms of the Čech semicosimplicial objects

In Theorem 3.7, we define, for every open set $V_i$ , the dgLas $H(V_i,\mathcal {F}, \alpha , \mathcal {G})$ as the Thom–Withney homotopy fiber products; they form a semicosimplicial dgLas $H(\mathcal {V})^\Delta $ and we consider the associated Thom–Withney dgLa $\operatorname {\mathrm {Tot}}(H(\mathcal {V})^\Delta )$ . This construction does not depend on the order [Reference IaconoIa10]. We could first consider the Thom–Withney dgLas of the Čech semicosimplicial sheaves of Lie algebras $\mathcal {L}(\mathcal {V})$ , $\mathcal {E} nd(\mathcal {F})(\mathcal {V})$ , $\mathcal {E} nd (\mathcal {G})(\mathcal {V})$ and $\mathcal {E} nd (\mathcal {F} \oplus \mathcal {G})(\mathcal {V})$ and obtain the semicosimplicial dgLa

(4)

and then apply Thom–Withney homotopy fibre product. This dgLa is quasi isomorphic to $\operatorname {\mathrm {Tot}}(H(\mathcal {V})^\Delta )$ .

Since the cohomology of the total complex of the semicosimplicial dgLa $H(\mathcal {V})^\Delta $ is the same as the one of the Thom–Whitney homotopy fibre, according to [Reference IaconoIa08, Section 4], we get the following exact sequence:

$$\begin{align*}\ldots \to H^i(\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta) \to H^i( X, \mathcal{E} nd (\mathcal{F}) \oplus \mathcal{E} nd (\mathcal{G}) \oplus \mathcal{L}) \to \end{align*}$$

$$\begin{align*}\to H^i(X, \mathcal{E} nd (\mathcal{F} \oplus \mathcal{G})) \to H^{i+1}(\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta) \to \ldots \end{align*}$$

Moreover, since the map $h: \mathcal {L} \to \mathcal {E} nd (\mathcal {F} \oplus \mathcal {G}) $ is injective, thanks to [Reference IaconoIa08, Lemma 3.1], the following sequence is also exact:

(5) $$ \begin{align} \cdots H^i(\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta) \to H^i( X, \mathcal{E} nd (\mathcal{F}) \oplus \mathcal{E} nd (\mathcal{G})) \to H^i(X,\operatorname{\mathrm{coker}} h) \to H^{i+1}(X,\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta) \cdots. \end{align} $$

Note that, since $\operatorname {\mathrm {Tot}}(H(\mathcal {V})^\Delta )$ controls the infinitesimal deformations of $\alpha :\mathcal {F} \to \mathcal {G}$ , its first cohomology space is isomorphic to the tangent space of the functor $\operatorname {\mathrm {Def}}_{(\mathcal {F}, \alpha , \mathcal {G})}$ , and its second cohomology space is an obstruction space.

Remark 3.11 If we work over the field of complex numbers, we can consider the Dolbeault resolutions of the endomorphisms of sheaves, and avoid the Čech semicosimplicial dgLas. More precisely, we can consider the two morphisms of dgLas

analogous to the previous h and g. Then, the associated Thom–Whitney homotopy fibre product of Definition 2.3 is a dgLa that controls infinitesimal deformations of $\alpha : \mathcal {F} \to \mathcal {G}$ . Note that this construction is the analogous of the one in Equation (4) of Remark 3.10.

4.1 Geometric deformations

Let X be a smooth projective variety over $\mathbb {K}$ , $\mathcal {E}$ be a locally free sheaf of $\mathcal {O}_X$ -modules on X and $U\subseteq H^0(X,\mathcal {E})$ be a subspace of global sections of $\mathcal {E}$ . We study the infinitesimal deformations of $\mathcal {E}$ which preserve the subspace U. We start with some definitions.

Definition 4.1 Let $A\in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ . An infinitesimal deformation of the pair $(\mathcal {E}, U)$ over A is the data $(\mathcal {E}_A,\pi _A, i_A)$ of:

• • a deformation $(\mathcal {E}_A, \pi _A)$ of $\mathcal {E}$ over A,

• • a morphism $i_A:U\otimes A \rightarrow H^0(X\times \operatorname {\mathrm {Spec}} A, \mathcal {E}_A)$ ,

such that the following diagram commutes

(6)

Two deformations $(\mathcal {E}_A,\pi _A,i_A)$ , $(\mathcal {E}^{\prime }_A,\pi ^{\prime }_A,i^{\prime }_A)$ are isomorphic if there exist an isomorphism $\phi :\mathcal {E}_A \rightarrow \mathcal {E}^{\prime }_{A}$ of sheaves of $\mathcal {O}_X \otimes A$ -modules, such that $\pi ^{\prime }_A \circ \phi =\pi _A$ and an isomorphism $\psi :U\otimes A \to U\otimes A$ , that makes the diagram commutative:

Note that, this implies that $\phi $ induces an isomorphism $\phi : i_A(U\otimes A) \rightarrow i^{\prime }_A(U\otimes A)$ . In the following, we will often shorten the notation of such deformations with $(\mathcal {E}_A, i_A)$ .

Definition 4.2 The functor of infinitesimal deformations of $(\mathcal {E},U)$ is

$$\begin{align*}\operatorname{\mathrm{Def}}_{(\mathcal{E},U)}: \operatorname{\mathrm{\textbf{Art}}}_{\mathbb{K}} \to \operatorname{\mathrm{\textbf{Grpds}}},\end{align*}$$

that associates to any $A \in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ the groupoid $\operatorname {\mathrm {Def}}_{(\mathcal {E},U)}(A)$ , whose objects are the infinitesimal deformations of the pair $(\mathcal {E}, U)$ over A and whose morphisms are the isomorphisms among them. Note that

$$\begin{align*}\pi_0(\operatorname{\mathrm{Def}}_{(\mathcal{E}, U))}(A)) = \{\mbox{isomorphism classes of deformations of the pair } (\mathcal{E}, U) \mbox{ over } A\} \end{align*}$$

is the classical functor of deformations in $\operatorname {\mathrm {\textbf {Sets}}}$ .

Remark 4.3 Let X be a smooth projective variety over $\mathbb {K}$ with $H^1(X, \mathcal {O}_X)=0$ . Let $\mathcal {E}$ be a locally free sheaf of $\mathcal {O}_X$ -modules on X and $U\subseteq H^0(X,\mathcal {E})$ a subspace of global sections of it. Let $s: U \otimes \mathcal {O}_X \to \mathcal {E}$ be the morphism of sheaves of $\mathcal {O}_X$ -modules associated with $i: U \hookrightarrow H^0(\mathcal {E})$ via the obvious correspondence between global sections and morphisms from $\mathcal {O}_X$ to $\mathcal {E}$ .

According to Definition 3.3, we denote by

$$\begin{align*}\operatorname{\mathrm{Def}}_{(U\otimes \mathcal{O}_X,s, \mathcal{E})} : \operatorname{\mathrm{\textbf{Art}}}_{\mathbb{K}} \to \operatorname{\mathrm{\textbf{Grpds}}},\end{align*}$$

the functor of infinitesimal deformations of the morphism s, that associates, to any $A \in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ , the groupoid $\operatorname {\mathrm {Def}}_{(U\otimes \mathcal {O}_X,s, \mathcal {E})}(A)$ , whose objects are the infinitesimal deformations of the morphism s over A and whose morphisms are the isomorphisms between them.

We claim that, for any $A\in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ there is a 1-1 correspondence between the objects of the groupoids

$$\begin{align*}\operatorname{\mathrm{Def}}_{(\mathcal{E}, U)} (A)\to \operatorname{\mathrm{Def}}_{(U\otimes \mathcal{O}_X,s, \mathcal{E})}(A).\end{align*}$$

Indeed, given a deformation $(\mathcal {E}_A, i_A)$ of $(\mathcal {E}, U)$ over A as in Definition 4.1, there is just one morphism of sheaves of $\mathcal {O}_X\otimes A$ -modules $s_A:U\otimes \mathcal {O}_X\otimes A \to \mathcal {E}_A$ associated with $i_A$ , it makes the following diagram commutative

(7)

and it defines an object of $\operatorname {\mathrm {Def}}_{(U\otimes \mathcal {O}_X, s, \mathcal {E})}(A)$ . On the other hand, since $H^1(X, \mathcal {O}_X)=0$ , every infinitesimal deformation of the domain $U\otimes \mathcal {O}_X$ is trivial. A deformation of s over A is just the data of a deformation $\mathcal {E}_A$ of $\mathcal {E}$ and of a map $s_A:U\otimes \mathcal {O}_X\otimes A \to \mathcal {E}_A$ , such that the diagram (7) is commutative. Taking global sections, one gets the required deformed map $i_A: U\otimes A \to H^0(X \times \operatorname {\mathrm {Spec}} A, \mathcal {E}_A)$ and the commutative diagram (6).

Analogously, we can prove that isomorphisms of deformations correspond each others. This assures that, under the hypothesis $H^1(X, \mathcal {O}_X)=0$ , the functors $\operatorname {\mathrm {Def}}_{(\mathcal {E}, U)}$ and $\operatorname {\mathrm {Def}}_{(U\otimes \mathcal {O}_X,s, \mathcal {E})}$ are are equivalent.

In [Reference MartinengoMar08, Reference MartinengoMar09], the second author introduced the problem of deformations of the pair $(\mathcal {E},U)$ and found a dgLa that controls it on a variety X over the field of complex numbers. In [Reference Iacono and MartinengoIM23], we studied these deformations in more details and we generalized to vector bundles over a smooth complex projective variety some classical results known for line bundles over curves, concerning the description of the tangent space, the smoothness of the functor $\operatorname {\mathrm {Def}}_{(\mathcal {E},U)}$ and the liftings of a section. Here, we would like to study the infinitesimal deformations of the pair $(\mathcal {E},U)$ over an algebraically closed field $\mathbb {K}$ of characteristic zero using the analysis of the previous sections.

4.2 The dgLa that controls deformations of $(\mathcal {E},U)$

Let X be a smooth projective variety over $\mathbb {K}$ , we assume here $H^1(X, \mathcal {O}_X)=0$ . Let $\mathcal {E}$ be a locally free sheaf of $\mathcal {O}_X$ -modules on X and $U\subseteq H^0(X,\mathcal {E})$ be a subspace of global sections of it.

Let $s: U \otimes \mathcal {O}_X \to \mathcal {E}$ be the morphism of sheaves of $\mathcal {O}_X$ -modules associated with $i: U \hookrightarrow H^0(\mathcal {E})$ . As observed in Remark 4.3, the infinitesimal deformations of the pair $(\mathcal {E}, U)$ are equivalent to the infinitesimal deformations of the morphism s. We indicate with $\gamma $ the graph of the morphism s, that is a subsheaf of $(U\otimes \mathcal {O}_X) \oplus \mathcal {E}$ .

Let $\mathcal {V}=\{ V_i \}_i$ be an affine open cover of X, we denote by $ \mathcal {E} nd (\mathcal {E}), \mathcal {E} nd(U\otimes \mathcal {O}_X)$ and $\mathcal {E} nd \left ((U\otimes \mathcal {O}_X) \oplus \mathcal {E}\right )$ the sheaves of $O_X $ -modules endomorphisms of $\mathcal {E}, U\otimes \mathcal {O}_X$ and $(U\otimes \mathcal {O}_X) \oplus \mathcal {E}$ , respectively. Let $H(\mathcal {V})^\Delta $ be the Cech semicosimplicial dgLa associated with the Thom–Whitney homotopy fibres of the diagram of sheaves of Lie algebras

where $h: \mathcal {L} = \{ \varphi \in \mathcal {E} nd \left ((U\otimes \mathcal {O}_X) \oplus \mathcal {E}\right ) \mid \varphi (\gamma ) \subseteq \gamma \} \to \mathcal {E} nd \left ((U\otimes \mathcal {O}_X) \oplus \mathcal {E}\right )$ is the inclusion and g is defined as the map $(\varphi , \psi ) \mapsto \left ( \begin {array}{cc} \varphi & 0 \\ 0 & \psi \end {array}\right ). $ Applying the results of the previous section, we get the following result.

Theorem 4.4 In the above notation and under the hypothesis $H^1(X,\mathcal {O}_X)=0$ , the functor $\operatorname {\mathrm {Def}}_{(\mathcal {E}, U)}$ of deformations of $(\mathcal {E},U)$ is equivalent to the Deligne functor associated with the Thom–Whitney dgLa of the semicosimplicial dgLa $H(\mathcal {V})^\Delta $ . Moreover, under the additional hypothesis that $H^2(X,\mathcal {O}_X)=0$ , the cohomology of the dgLa $\operatorname {\mathrm {Tot}}(H(\mathcal {V})^\Delta )$ fits into the exact sequence

(8) $$ \begin{align} \begin{aligned} 0 \to H^0(\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta)) \to H^0(X, \mathcal{E} nd (\mathcal{E})) \to \operatorname{\mathrm{Hom}}(U, H^0(\mathcal{E})/U) \to \\ \to H^1(\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta)) \to H^1(X,\mathcal{E} nd (\mathcal{E})) \stackrel{\alpha}{\to} \operatorname{\mathrm{Hom}} (U, H^{1}(X,\mathcal{E})) \to \\ \to H^2(\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta)) \to H^2(X,\mathcal{E} nd (\mathcal{E})) \to\operatorname{\mathrm{Hom}}(U, H^2(X,\mathcal{E})). \end{aligned} \end{align} $$

Proof The equivalence

$$\begin{align*}\operatorname{\mathrm{Def}}_{(\mathcal{E},U)} \cong \operatorname{\mathrm{Del}}_{\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta)} \end{align*}$$

is a direct consequence of Theorem 3.7 and Remark 4.3. According to the sequence (5), the cohomology of the dgLa $\operatorname {\mathrm {Tot}}(H(\mathcal {V})^\Delta )$ fits in the exact sequence

(9) $$ \begin{align} \cdots H^i(\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta) &\to H^i( X, \mathcal{E} nd (U\otimes \mathcal{O}_X) \oplus \mathcal{E} nd (\mathcal{E})) \to H^i(X, \operatorname{\mathrm{coker}} h)\nonumber\\ &\to H^{i+1}(\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta) \cdots. \end{align} $$

Observe that, since X is projective,

$$\begin{align*}H^0( X, \mathcal{E} nd(U\otimes \mathcal{O}_X) \oplus \mathcal{E} nd (\mathcal{E}))= \operatorname{\mathrm{Hom}}(U,U) \oplus H^0(X, \mathcal{E} nd (\mathcal{E})); \end{align*}$$

while for $i=1,2$ , under the hypothesis $H^i(X, \mathcal {O}_X)=0$ , we get

$$\begin{align*}H^i( X, \mathcal{E} nd (U\otimes \mathcal{O}_X) \oplus \mathcal{E} nd (\mathcal{E}))&=\left(H^i(X, \mathcal{O}_X)\otimes U^\vee \otimes U\right) \oplus H^i(X, \mathcal{E} nd (\mathcal{E}))\nonumber\\ &=H^i(X, \mathcal{E} nd (\mathcal{E})).\end{align*}$$

Moreover, we can describe explicitly the $\operatorname {\mathrm {coker}} h$ via the following isomorphisms

$$\begin{align*}\operatorname{\mathrm{coker}} h \cong \mathcal{H} om\left(\gamma, \frac{(U\otimes \mathcal{O}_X) \oplus \mathcal{E}}{\gamma}\right) \cong \mathcal{H} om(U\otimes \mathcal{O}_X, \mathcal{E}). \end{align*}$$

The first one is given by restricting an endomorphism of $(U\otimes \mathcal {O}_X) \oplus \mathcal {E} $ to $\gamma $ and projecting it to the quotient. The second one is obtained just observing that $U\otimes \mathcal {O}_X \cong \gamma $ and $\displaystyle \frac {(U\otimes \mathcal {O}_X) \oplus \mathcal {E}}{\gamma } \cong \mathcal {E}$ . Thus, for $i=0$ , we have

$$\begin{align*}H^0(\operatorname{\mathrm{coker}} h)&= H^0( \mathcal{H} om(U\otimes \mathcal{O}_X, \mathcal{E})) =\operatorname{\mathrm{Hom}}(U, H^0(X, \mathcal{E}))= \operatorname{\mathrm{Hom}}(U, U)\\ &\quad \oplus \operatorname{\mathrm{Hom}}\left(U, \frac{H^0(X,\mathcal{E})}{U}\right), \end{align*}$$

where the last equality follows applying $\operatorname {\mathrm {Hom}}(U, -)$ to the exact sequence of vector spaces

$$\begin{align*}0\to U \to H^0(X, \mathcal{E}) \to H^0(X,\mathcal{E})/U \to 0. \end{align*}$$

While, for $i=1,2$ , we have

$$\begin{align*}H^i(\operatorname{\mathrm{coker}} h)= H^i(\mathcal{H} om(U\otimes \mathcal{O}_X, \mathcal{E})) = \operatorname{\mathrm{Hom}}(U, H^i(X, \mathcal{E})).\end{align*}$$

Thus, the first terms of the exact sequence (9) become

(10) $$ \begin{align} 0 &\to H^0(\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta)) \to \operatorname{\mathrm{Hom}}(U,U) \oplus H^0(X, \mathcal{E} nd (\mathcal{E})) \stackrel{f}{\to} \operatorname{\mathrm{Hom}}(U, U)\nonumber\\ &\quad\oplus \operatorname{\mathrm{Hom}}\left(U, \frac{H^0(X,\mathcal{E})}{U}\right) \to \ \end{align} $$

$$\begin{align*}&\to H^1(\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta)) \to H^1(X,\mathcal{E} nd (\mathcal{E})) \to \operatorname{\mathrm{Hom}} (U, H^{1}(X,\mathcal{E}))\\ &\to H^2(\operatorname{\mathrm{Tot}}(H(\mathcal{V})^\Delta)) \to \end{align*}$$

$$\begin{align*}\to H^2(X,\mathcal{E} nd (\mathcal{E})) \to\operatorname{\mathrm{Hom}}(U, H^2(X,\mathcal{E})).\end{align*}$$

The last step is to make explicit the morphism f of (10). It is immediate to see that f is the morphism

$$\begin{align*}\operatorname{\mathrm{Hom}}(U,U) \oplus H^0(X, \mathcal{E} nd (\mathcal{E})) \stackrel{f}{\to} \operatorname{\mathrm{Hom}}(U, U) \oplus \operatorname{\mathrm{Hom}}\left(U, \frac{H^0(X,\mathcal{E})}{U}\right)\!, \end{align*}$$

whose first component is the identity on $ \operatorname {\mathrm {Hom}}(U, U)$ . Thus the exact sequence (10) induces the sequence (8).

The exact sequence (8) of Theorem 4.4 is a generalization over any algebraically closed field of characteristic zero of the exact sequence (7) in [Reference Iacono and MartinengoIM23], from which we get most of our results, that now can be immediately generalized to any algebraically closed field $\mathbb {K}$ of characteristic zero.

Let us denote by $r_U: \operatorname {\mathrm {Def}}_{(\mathcal {E},U)} \to \operatorname {\mathrm {Def}}_{\mathcal {E}}$ , the forgetful functor, i.e., the functor that ignores all information about U. Then, under the hypothesis $H^i(X,\mathcal {O}_X)=0$ for $i=1,2$ , we have the following results, that holds for over any algebraically closed field $\mathbb {K}$ of characteristic zero.

Corollary 4.6 [Reference Iacono and MartinengoIM23, Corollary 3.13]

In the notation above, we have

$$\begin{align*}\dim t_{\operatorname{\mathrm{Def}}_{(\mathcal{E},U)}} \geq \dim t_{\operatorname{\mathrm{Def}}_{\mathcal{E}}} - m \cdot \dim H^1(X,\mathcal{E}),\end{align*}$$

where m is the dimension of $U \subseteq H^0(X,\mathcal {E})$ .

Corollary 4.8 [Reference Iacono and MartinengoIM23, Corollary 3.15]

The tangent space to the deformations of the pair $(\mathcal {E}, H^0(\mathcal {E}))$ can be identified with

$$\begin{align*}t_{\operatorname{\mathrm{Def}}_{(\mathcal{E},H^0(\mathcal{E}))}} = \{ a \in H^1(X,\mathcal{E} nd (\mathcal{E})) \ \mid \ a \cup s =0, \ \forall \ s \in H^0(X,\mathcal{E}) \}. \end{align*}$$

4.3 An explicit model for the semicosimplicial dgla that controls deformations of $(\mathcal {E}, U)$ .

In this section, we describe another semicosimplicial dgLa whose total complex controls deformations of the pairs $(\mathcal {E},U)$ over a field $\mathbb {K}$ . The construction is more explicit and it avoids the Theorem on descent of Deligne groupoids (Theorem 2.21).

In the previous notation, given the morphism of sheaves $s: U \otimes \mathcal {O}_X \to \mathcal {E}$ , we consider the complex of sheaves

$$\begin{align*}{\mathcal Q}: U\otimes \mathcal{O}_X \stackrel{s}{\to} \mathcal{E}, \end{align*}$$

where the sheaves $U\otimes \mathcal {O}_X$ and $\mathcal {E}$ are settled in degree zero and one, respectively. As above, s is the morphism associated with $i: U \hookrightarrow H^0(\mathcal {E})$ and it is a differential, $s^2=0$ , since there is no degree 2. A similar construction is used in [Reference ManettiMan22, pp. 234–235] to analyze the embedded deformations. Consider the following complex of sheaves concentrated in degree 0 and 1:

$$\begin{align*}\mathcal{H} om^{\geq 0}({\mathcal Q},{\mathcal Q}) = \left\{\!\!\begin{array}{cll} \mathcal{H} om^0({\mathcal Q},{\mathcal Q}) &=& \mathcal{E} nd (U\otimes \mathcal{O}_X) \oplus \mathcal{E} nd(\mathcal{E}) \\ \mathcal{H} om^1({\mathcal Q},{\mathcal Q}) &= & \mathcal{H} om(U\otimes \mathcal{O}_X, \mathcal{E}) \\ \mathcal{H} om^k({\mathcal Q},{\mathcal Q}) &= & 0 \quad \forall \ k \neq 0,1, \end{array} \right. \end{align*}$$

This is a sheaf of dgLas. The differential is given by s, the bracket is the usual one in $\mathcal {E} nd (U\otimes \mathcal {O}_X)$ and $ \mathcal {E} nd(\mathcal {E})$ , it is the trivial bracket between an element in $\mathcal {E} nd (U\otimes \mathcal {O}_X)$ and an element in $ \mathcal {E} nd(\mathcal {E})$ and it is given by the composition between elements in degree 0 and 1.

Next, fix an open affine cover $\mathcal {V}=\{ V_i \}_i$ of X trivialising $\mathcal {E}$ and let $\mathfrak {g}^\Delta $ be the Čech semicosimplicial dgLa $\mathfrak {g}^\Delta $ associated with the sheaf $ \mathcal {H} om^{\geq 0}({\mathcal Q},{\mathcal Q})$ and to the open cover $\mathcal {V}$ . The aim of this section is to prove the following result.

Proof For any $A \in \operatorname {\mathrm {\textbf {Art}}}_{\mathbb {K}}$ , we define a 1-1 correspondence between the objects of the groupoids

$$\begin{align*}\widetilde{H}^1_{sc}(\exp \mathfrak{g}^{\Delta})(A)\to \operatorname{\mathrm{Def}}_{(U\otimes \mathcal{O}_X, \mathcal{E}, s)}(A). \end{align*}$$

An element in $Z^1_{sc}(\exp \mathfrak {g}^{\Delta })(A)$ is a pair $(l,m) \in \left (\mathfrak {g}_0^1 \oplus \mathfrak {g}_1^0\right )\otimes \mathfrak {m}_A$ , where the element $l=\{l_i\}_i \in \prod _i \mathcal {H} om(U\otimes \mathcal {O}_X, \mathcal {E})(V_i) \otimes \mathfrak {m}_A$ and the element $m=\{\nu _{ij}, \mu _{ij}\}_{ij} \in \prod _{i<j} \left (\mathcal {E} nd(U\otimes \mathcal {O}_X)(V_{ij})\oplus \mathcal {E} nd(\mathcal {E})(V_{ij})\right ) \otimes \mathfrak {m}_A$ , and it has to satisfy the equations in (2).

Using these data, we can define a deformation of the morphism $s: U\otimes \mathcal {O}_X \to \mathcal {E}$ over A as follows.

• • The deformation $\mathcal {E}_A$ of the locally free sheaf $\mathcal {E}$ over A is obtained by gluing the local trivial deformations $\{\mathcal {E}|_{V_i}\otimes A\}_i$ by the isomorphisms $\{e^{ \mu _{ij}}: \mathcal {E}_{|_{V_{ij}}}\otimes A \to \mathcal {E}_{|_{V_{ij}}}\otimes A\}_{ij}$ . Note that the third equation of (2) gives rise in our case to the equation ${\partial _{0,2} \mu } \bullet {-\partial _{1,2} \mu } \bullet {\partial _{2,2} \mu } = 0$ , that assures that the given gluing functions are compatible on the triple intersections.

• • Similarly, the data $\{e^{\nu _{ij}}: U\otimes \mathcal {O}_X|_{V_{ij}} \otimes A \to U\otimes \mathcal {O}_X|_{V_{ij}} \otimes A\}_{ij}$ are gluing isomorphisms of the local trivial deformations of the sheaf $U\otimes \mathcal {O}_X$ over A in the open sets ${V_{ij}}$ . The third equation in (2) gives rise to the equation ${\partial _{0,2}\nu } \bullet {-\partial _{1,2}\nu } \bullet {\partial _{2,2}\nu } = 0$ and it assures that the gluing isomorphisms are compatible on the triple intersections.

• • The morphism $s_A: U\otimes \mathcal {O}_X \otimes A \to \mathcal {E}_A$ is locally defined as $\{s|_{V_i} + l_i: U\otimes \mathcal {O}_X|_{V_{i}}\otimes A \to \mathcal {E}|_{V_i} \otimes A\}_i$ . Note that the second equation of (2) is $\delta _{11} l = e^m * \delta _{01} l$ and it can be rewritten as $s + \delta _{11} l = e^\mu (s + \delta _{01} l ) e^{-\nu }$ (see for example [Reference Iacono and MartinengoIM23, Formula 8]). It says that the local maps $\{s|_{V_i} + l_i\}_i$ glue to a global one.

Note that the first equation $dl + \frac {1}{2}[l,l]=0$ is trivial in our case, because our dgLas are zero in degree $2$ .

Thus every element in $Z^1_{sc}(\exp \mathfrak {g}^{\Delta })(A)$ exactly defines an infinitesimal deformation of the morphism $s:U\otimes \mathcal {O}_X \to \mathcal {E}$ on A. Let now $(l_0,m_0), (l_1,m_1) \in Z^1_{sc}(\exp \mathfrak {g}^{\Delta })(A)$ be isomorphic elements. We prove that they define two isomorphic deformations in $\operatorname {\mathrm {Def}}_{(U\otimes \mathcal {O}_X, s,\mathcal {E})}(A)$ .

• • By the first equation in (3), there exists $a \in \mathfrak {g}_0^0 \otimes \mathfrak {m}_A$ such that $e^a *l_0=l_1$ . Note that $a =(\beta ,\alpha )= \{ \beta _i, \alpha _i \}_i \in \prod _i \left ( \mathcal {E} nd(U\otimes \mathcal {O}_X)(V_i) \oplus \mathcal {E} nd(\mathcal {E})(V_i) \right ) \otimes \mathfrak {m}_A$ and as usual, the equation can be rewritten as $s +l_1 =e^\alpha (s +l_0) e^{-\beta } $ . Then, for every i, $e^{\alpha _i}: \mathcal {E}_{|_{V_i}} \otimes A\to \mathcal {E}_{|_{V_i}} \otimes A $ and $e^{\beta _i}: U\otimes {\mathcal {O}_X}|_{V_i} \otimes A \to U\otimes {\mathcal {O}_X}|_{V_i} \otimes A$ define local isomorphisms that make the following diagram commutative:

  

• • The second equation in (3) gives rise to the following two equations $ - \mu _0\bullet -\partial _{1,1}\alpha \bullet \mu _1 \bullet \partial _{0,1}\alpha =0$ and $- \nu _0\bullet -\partial _{1,1}\beta \bullet \nu _1 \bullet \partial _{0,1}\beta =0$ that assure that the isomorphisms $e^{\alpha _i}$ and $e^{\beta _i}$ glue to a global one.

Thus, as wanted, the two deformations defined by $(l_0,m_0), (l_1,m_1)$ are isomorphic.

In this way, we obtain every infinitesimal deformation of the morphism $s:U\otimes \mathcal {O}_X \to \mathcal {E}$ . Indeed, on a fix open affine set $V_i$ , the infinitesimal deformations of $\mathcal {E}$ and $U\otimes \mathcal {O}_X$ are trivial, while a deformation of $s:U \otimes \mathcal {O}_X \to \mathcal {E}$ over $V_i$ , is given by $s |_{V_i} + l_i: U \otimes \mathcal {O}_X|_{V_{i}}\otimes A \to \mathcal {E}|_{V_i}\otimes A $ , for $l_i \in \mathcal {H} om(U\otimes \mathcal {O}_X, \mathcal {E})(V_i) \otimes \mathfrak {m}_A$ . Then, we have to glue these data along double intersections to give a deformation of the morphism s on X. Therefore, we need isomorphisms $ \tilde {\mu }_{ij}: \mathcal {E}_{|_{V_{ij}}}\otimes A \to \mathcal {E}_{|_{V_{ij}}}\otimes A $ for all $i, j$ , that satisfy the cocycle condition on triple intersections: $ \tilde {\mu }_{jk}\circ {\tilde {\mu }_{ik}}^{-1}\circ \tilde {\mu }_{ij}=\operatorname {\mathrm {Id}}$ , to glue the trivial deformations of $\mathcal {E}$ to a global one. Moreover, we need isomorphisms $ \tilde {\nu }_{ij}: U\otimes \mathcal {O}_X|_{V_{ij}}\otimes A \to U\otimes \mathcal {O}_X|_{V_{ij}}\otimes A $ such that the following diagram commutes

i.e., $ {( s |_{V_j} + l_j)}_{|_{V_{ij}}} \circ \tilde {\nu }_{ij} = \tilde {\mu }_{ij}\circ {( s |_{V_i} + l_i)}_{|_{V_{ij}}}.$ Then, the local trivial deformations will glue to a global non trivial deformation of s.

Since we are in characteristic zero, we can take the logarithm to obtain $(\tilde {\mu }_{ij}, \tilde {\nu }_{ij} )= (e^{ \mu _{ij}}, e^{ \nu _{ij}})$ with $\{(\nu _{ij}, \mu _{ij})\}_{ij} \in \prod _{i<j} \left ( \mathcal {E} nd( U\otimes \mathcal {O}_X)(V_{ij})\oplus \mathcal {E} nd( \mathcal {E})(V_{ij})\right ) \otimes \mathfrak {m}_A$ .

Therefore, any deformation of the morphism s over A is given by an element $(l,m) \in \left (\mathfrak {g}_0^1 \oplus \mathfrak {g}_1^0\right )\otimes \mathfrak {m}_A$ , where $l=\{l_i\}_i \in \prod _{i}\mathcal {H} om(U\otimes \mathcal {O}_X, \mathcal {E})(V_i) \otimes \mathfrak {m}_A$ and the element $m=\{\nu _{ij}, \mu _{ij}\}_{ij} \in \prod _{i<j} \left ( \mathcal {E} nd(U\otimes \mathcal {O}_X) (V_{ij})\oplus \mathcal {E} nd( \mathcal {E})(V_{ij})\right ) \otimes \mathfrak {m}_A$ , that has to satisfy the conditions in (2).

Analogously, we can prove that isomorphisms of deformations correspond to the existence of an element $a \in \mathfrak {g}^0_0 \otimes \mathfrak {m}_A $ that satisfy the condition in Equation (3).

As a direct consequence of Remark 4.3, Theorems 4.10 and 2.15, we get the following result.

Corollary 4.11 Under the hypothesis $H^1(X, \mathcal {O}_X)=0$ , there are natural isomorphisms of functors

$$\begin{align*}\operatorname{\mathrm{Def}}_{(\mathcal{E},U)} \cong \operatorname{\mathrm{Def}}_{(U\otimes \mathcal{O}_X, s, \mathcal{E})} \cong H^1_{sc}(\exp \mathfrak{g}^\Delta) \cong \operatorname{\mathrm{Def}}_{\operatorname{\mathrm{Tot}}(\mathfrak{g}^{\Delta})}. \end{align*}$$

This corollary provides another explicit description of a dgLa controlling the infinitesimal deformations of the pair $(\mathcal {E},U)$ .

Fix an open affine cover $\{ V_i \}_{i}$ of X. As above, let $\mathfrak {g}^\Delta $ and $\mathfrak {h}^\Delta $ be the Čech semicosimplicial dgLas associated with the sheaf $\mathcal {H} om^{\geq 0}({\mathcal Q}, {\mathcal Q})$ and to the sheaf $\mathcal {E} nd(\mathcal {E})$ , respectively. There is an obvious surjective morphism of semicosimplicial dgLas $ \mathfrak {g}^\Delta \to \mathfrak {h}^\Delta $ and, denoting with $\mathfrak {m}^\Delta $ the kernel of it, we get the following exact sequence of semicosimplicial dgLas:

(11) $$ \begin{align} 0 \to \mathfrak{m}^\Delta \to \mathfrak{g}^\Delta \to \mathfrak{h}^\Delta \to 0. \end{align} $$

Lemma 4.12 Under the hypothesis that $H^i(X,\mathcal {O}_X)=0$ for $i=1,2$ , the cohomology of the total complex of the semicosimplicial dgLa $\mathfrak {m}^\Delta $ is given by:

$$\begin{align*}H^0\left(\operatorname{\mathrm{Tot}}(\mathfrak{m}^\Delta)\right)&=0, H^1\left(\operatorname{\mathrm{Tot}}(\mathfrak{m}^\Delta)\right) = \operatorname{\mathrm{Hom}}\left(U, H^0(\mathcal{E})/U\right) \ \mbox{and}\ H^2\left(\operatorname{\mathrm{Tot}}(\mathfrak{m}^\Delta)\right)\\ &= \operatorname{\mathrm{Hom}}(U, H^1(\mathcal{E})). \end{align*}$$

Proof The double complex associated with the semicosimplicial dgLa $\mathfrak {m}^\Delta $ is

(12)

where the horizontal differential $\check {\delta }$ is the Čech one, while the vertical d is the differential of the dgLa involved.

Let $\{E_k^{p,q}\}$ be the spectral sequence associated with the double complex (12). Calculating the Čech cohomology of it, we get the first page:

(13)

where we use that $\check {H}^0( X, \mathcal {E} nd(U\otimes \mathcal {O}_X))= \operatorname {\mathrm {Hom}} (U,U)$ and $ \check {H}^k( X, \mathcal {E} nd(U\otimes \mathcal {O}_X))= 0$ for $k=1,2$ and $\check {H}^k(X,\mathcal {H} om(U\otimes \mathcal {O}_X, \mathcal {E}))= \operatorname {\mathrm {Hom}} (U, \check {H}^k(X,\mathcal {E}))$ for all $k\geq 0$ . Since the differential from the second page is always zero, the spectral sequence abuts to $E_2^{p,q}=E_{\infty }^{p,q}$ . This gives the expected cohomology of the total complex $\operatorname {\mathrm {Tot}}(\mathfrak {m}^\Delta )$ .

From the exact sequence (11) and thanks to Lemma 4.12, we recover the exact sequence (8):

(14) $$ \begin{align} \begin{aligned} 0 \to H^0(\operatorname{\mathrm{Tot}}(\mathfrak{g}^\Delta)) \to H^0( X,\mathcal{E} nd (\mathcal{E})) \to \operatorname{\mathrm{Hom}} (U, H^0(X,\mathcal{E})/U )\\ \to H^1(\operatorname{\mathrm{Tot}}(\mathfrak{g}^\Delta)) \to H^1(X,\mathcal{E} nd (\mathcal{E})) \stackrel{\alpha}{\to} \operatorname{\mathrm{Hom}} (U, H^{1}(X,\mathcal{E})) \stackrel{\beta}{\to} \\ \to H^2(\operatorname{\mathrm{Tot}}(\mathfrak{g}^\Delta)) \stackrel{\gamma}{\to} H^2(X,\mathcal{E} nd (\mathcal{E})) \to \operatorname{\mathrm{Hom}} (U, H^{2}(X,\mathcal{E})). \\ \end{aligned} \end{align} $$

Thus, the two dgLas $\operatorname {\mathrm {Tot}}(H({\mathcal V})^\Delta )$ and $\operatorname {\mathrm {Tot}}(\mathfrak {g}^\Delta )$ , constructed to control the infinitesimal deformations of the pair $(\mathcal {E}, U)$ , have actually the same tangent and obstructions space.

Remark 4.13 Note that, if we strengthen the hypothesis of Theorem 4.4 and of Lemma 4.12, assumining that $H^i(X, \mathcal {O}_X)=0$ for all $i>0$ , both the exact sequences (8) and (14) continue in higher degrees giving rise to the exact sequences:

$$\begin{align*}\cdots \to H^i(\operatorname{\mathrm{Tot}}(\mathfrak{g}^\Delta)) \to H^i( X,\mathcal{E} nd (\mathcal{E})) \to \operatorname{\mathrm{Hom}} (U, H^i(X,\mathcal{E}) ) \to H^{i+1}(\operatorname{\mathrm{Tot}}(\mathfrak{g}^\Delta)) \to \cdots \end{align*}$$

and

$$\begin{align*}\cdots \to H^i(\operatorname{\mathrm{Tot}}(H({\mathcal V})^\Delta) \to H^i( X,\mathcal{E} nd (\mathcal{E})) \to \operatorname{\mathrm{Hom}} (U, H^i(X,\mathcal{E}) ) \to H^{i+1}(\operatorname{\mathrm{Tot}}(H({\mathcal V})^\Delta)) \to \cdots \end{align*}$$

respectively. Thus, in this case the two dgLas $\operatorname {\mathrm {Tot}}(H({\mathcal V})^\Delta )$ and $\operatorname {\mathrm {Tot}}(\mathfrak {g}^\Delta )$ are quasi-isomorphic.