On the relative Connections

TL;DR

This paper studies relative connections on sheaves of modules, providing conditions for their existence and showing that certain Chern classes vanish under specific geometric conditions.

Contribution

It introduces a sufficient condition for the existence of relative holomorphic connections and proves the vanishing of relative Chern classes for compact Kähler fibers.

Findings

Existence condition for relative holomorphic connections.

Vanishing of relative Chern classes on compact Kähler fibers.

Application to complex analytic families.

Abstract

We investigate relative connections on a sheaf of modules. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic vector bundle over a complex analytic family. We show that the relative Chern classes of a holomorphic vector bundle admitting relative holomorphic connection vanish, if each of the fiber of the complex analytic family is compact and K\"ahler.

Equations

$$0\to \mathcal{H}{om}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{G})\stackrel{ı}{\to }\mathcal{D}{iff}_S^1(\mathcal{F},\mathcal{G})\stackrel{{\sigma }_1}{\to }\mathcal{D}{er}_S({\mathcal{O}}_X,\mathcal{H}{om}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{G}))\to 0$$

$${\mathrm{H}}^1(S,{\pi }_{\ast }({\Omega }_{X/S}^1(\mathcal{E}{nd}_{{\mathcal{O}}_X}(E))))=0.$$

$$\alpha ({\rho }_{U,{\pi }^{-1}(V)}({\pi }_V^{♯}(s))t)={\rho }_{U,{\pi }^{-1}(V)}({\pi }_V^{♯}(s))\alpha (t),$$

$$\delta :{\mathcal{O}}_X\to \mathcal{F}$$

$${\delta }_U(ab)=a{\delta }_U(b)+{\delta }_U(a)b.$$

$${\mathrm{Der}}_S({\mathcal{O}}_X,\mathcal{F}).$$

$$[\xi ,\eta ]=\xi \circ \eta -\eta \circ \xi$$

$$d_{X/S}:{\mathcal{O}}_X⟶{\Omega }_{X/S}^1$$

$$\delta :{\mathcal{O}}_X⟶\mathcal{F},$$

$$\alpha :{\Omega }_{X/S}^1⟶\mathcal{F}$$

$$\delta =\alpha \circ d_{X/S}.$$

$$D:\mathcal{D}{er}_S({\mathcal{O}}_X,{\mathcal{O}}_X)⟶\mathcal{E}{nd}_S(\mathcal{F})$$

$$D_U:\mathcal{D}{er}_S({\mathcal{O}}_X,{\mathcal{O}}_X)(U)⟶\mathcal{E}{nd}_S(\mathcal{F})(U),\xi ⟼(D_U{)}_{\xi }$$

$$((D_U{)}_{\xi }{)}_V(ag)=\xi {|}_V(a)g+a((D_U{)}_{\xi }{)}_V(g)$$

$$ϵ:\mathcal{D}{er}_S({\mathcal{O}}_X,{\mathcal{O}}_X)\hookrightarrow \mathcal{E}{nd}_S({\mathcal{O}}_X)$$

$$\mathcal{F}=\underset{i\in I}{⨁}{\mathcal{F}}_i,$$

$$D_{\xi }(u)=(D_{\xi }^i(u_i){)}_{i\in I}$$

$$D_{\xi }(s_1{\otimes }_{{\mathcal{O}}_X}\cdots {\otimes }_{{\mathcal{O}}_X}{s}_p)=\sum _{i=1}^p{s}_1{\otimes }_{{\mathcal{O}}_X}\cdots {\otimes }_{{\mathcal{O}}_X}{s}_{i-1}{\otimes }_{{\mathcal{O}}_X}{D}_{\xi }^i(s_i){\otimes }_{{\mathcal{O}}_X}{s}_{i+1}{\otimes }_{{\mathcal{O}}_X}\cdots {\otimes }_{{\mathcal{O}}_X}{s}_p$$

$$T_{{\mathcal{O}}_X}(\mathcal{F})=\underset{p\in \mathbb{N}}{⨁}{T}_{{\mathcal{O}}_X}^p(\mathcal{F}).$$

$$D_{\xi }(s\otimes t)=D_{\xi }(s)\otimes t+s\otimes D_{\xi }(t),$$

$$Sy{m}_{{\mathcal{O}}_X}(\mathcal{F})=T_{{\mathcal{O}}_X}(\mathcal{F})/\mathcal{I}$$

$${\Lambda }_{{\mathcal{O}}_X}(\mathcal{F})=T_{{\mathcal{O}}_X}(\mathcal{F})/\mathcal{J}$$

$$D_{\xi }(s{s}^{\prime })=D_{\xi }(s)s^{\prime }+s{D}_{\xi }(s^{\prime })$$

$$D_{\xi }(t\wedge t^{\prime })=D_{\xi }(t)\wedge t^{\prime }+t\wedge D_{\xi }(t^{\prime })$$

$$D_{\xi }(h)=D_{\xi }^{\mathcal{G}}\circ h-h\circ D_{\xi }^{\mathcal{F}},$$

$$D=\xi \mapsto D_{\xi }:\mathcal{D}{er}_S({\mathcal{O}}_X,{\mathcal{O}}_X)\to \mathcal{E}{nd}_S(\mathcal{H}{om}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{G}))$$

$$D_{\xi }(h)=[D_{\xi }^{\mathcal{F}},h]=D_{\xi }^{\mathcal{G}}\circ h-h\circ D_{\xi }^{\mathcal{G}}$$

$$D_{\xi }(f)=\xi \circ f-f\circ D_{\xi }^{\mathcal{F}}$$

$${\mathcal{L}}_{{\mathcal{O}}_X}({\mathcal{F}}_1,\cdots ,{\mathcal{F}}_p;\mathcal{G})(U):=L_{{\mathcal{O}}_X{|}_U}({\mathcal{F}}_1{|}_U,\cdots ,{\mathcal{F}}_p{|}_U;\mathcal{G}{|}_U),$$

$$D_{\xi }(\omega )(u_1,u_2,\dots ,u_p)=D_{\xi }^{\mathcal{G}}(\omega (u_1,u_2,\dots ,u_p))-\sum _{i=1}^p\omega (u_1,\dots ,u_{i-1},D_{\xi }^i(u_i),u_{i+1},\dots ,u_p),$$

Full text

1. Introduction

In [1], Atiyah introduced the notion of holomorphic connections in principal bundles over a complex manifold. A theorem due to Atiyah and Weil, [1], [11] which is known as the Atiyah-Weil criterion, says that a holomorphic vector bundle over a compact Riemann surface admits a holomorphic connection if and only if the degree of each indecomposable component of the holomorphic vector bundle is zero (see [3] for an exposition of the Atiyah-Weil criterion); this criterion generalizes to holomorphic principal bundles over a compact Riemann surface [2].

We can ask following two basic questions:

Question 1.1**.**

Let $\pi\,:\,X\,\longrightarrow\,S$ be a surjective holomorphic proper submersion with connected fibers, and let $\varpi\,:\,E\,\longrightarrow\,X$ be a holomorphic vector bundle.

1. (1)

Suppose $E$ admits a relative holomorphic connection. Do the relative Chern classes vanish? 2. (2)

Is there a good criterion for the existence of a relative holomorphic connection on $E$?

Here we shall answer the (first) Question 1.1(1), when each fiber of the complex analytic family is compact and Kähler (See Theorem 4.13). We are unable to answer the (second) Question 1.1(2); instead we will answer the following question (see Theorem 4.14):

Question 1.2**.**

Let $\pi\,:\,X\,\longrightarrow\,S$ be a surjective holomorphic proper submersion such that $\pi^{-1}(t)\,=\,X_{t}$ are connected. Let $\varpi\,:\,E\,\longrightarrow\,X$ be a holomorphic vector bundle. Suppose that for every $t\,\in\,S$, there is a holomorphic connection on holomorphic vector bundle $\varpi|_{E_{t}}\,:\,E_{t}\,\longrightarrow\,X_{t}$. Does $E$ admit a relative holomorphic connection?

To answer the above questions, we shall develop theory of the relative connections on sheaf of modules. Let $(\pi,\,\pi^{\sharp})\,:\,(X,\,\mathcal{O}_{X})\,\longrightarrow\,(S,\,\mathcal{O}_{S})$ be a morphism of ringed spaces and $\mathcal{F}$ be an $\mathcal{O}_{X}$-module. In [6], Grothendieck described the notion of $S$-derivation or relative derivation from $\mathcal{O}_{X}$ to $\mathcal{F}$. Also, in [8], Koszul did “Differential Calculus” in the frame work of commutative algebra, which can be reformulated in sheaf theoretic manner. We recall in Section 2, from [6] and [8], some preliminaries regarding relative derivations, relative connections, relative connections on associated $\mathcal{O}_{X}$-modules, covariant derivative and connection-curvature matrices in the setup of ringed spaces. In the third section, we define the relative differential operator between sheaves of $\mathcal{O}_{X}$-modules and symbol map of a first order relative differential operator. In Section 4, we recall the notion of complex analytic family, which is done in [9] in great detail. We establish the symbol exact sequence (see Proposition 4.2).

Proposition 1.3** (Symbol exact sequence).**

Let $\pi\,:\,X\,\longrightarrow\,S$ be a holomorphic proper submersion of complex manifolds with connected fibers, and let $\mathcal{F}$, $\mathcal{G}$ be two locally free $\mathcal{O}_{X}$-modules of rank $r$ and $p$ respectively. Then the sequence

[TABLE]

is an exact sequence of $\mathcal{O}_{X}$-modules.

This (symbol) exact sequence is used in defining relative Atiyah algebra. The notion of relative holomorphic connections on an analytic coherent sheaf was introduced by Deligne [4], which coincides with the definition given here if it is formulated in the holomorphic setting. Given a relative holomorphic connection on a holomorphic vector bundle, there is an induced family of holomorphic connections; more precisely we prove the following (see Proposition 4.5):

Proposition 1.4**.**

Let $\pi\,:\,X\,\longrightarrow\,S$ be a surjective holomorphic proper submersion with connected fibers, and let $E\xrightarrow{\varpi}X$ be a holomorphic vector bundle. Suppose that we have a holomorphic $S$-connection $D$ on $E$. Then for every $t\in S$, we have a holomorphic connection $D_{t}$ on the holomorphic vector bundle $E_{t}\to X_{t}$. In other words, we have a family $\{D_{t}\,\mid\,t\,\in\,S\}$ of holomorphic connections on the holomorphic family $\{\varpi\,:\,E_{t}\,\to\,X_{t}\,\mid\,t\,\in\,S\}$ of vector bundles.

Next, we define the relative Chern classes of a complex vector bundle over a complex analytic family, and prove the following (see Theorem 4.13):

Theorem 1.5**.**

Let $\pi\,:\,X\,\longrightarrow\,S$ be a surjective holomorphic proper submersion, such that for each $t\in S$, $\pi^{-1}(t)=X_{t}$ is compact connected Kähler manifold. Let $E\,\xrightarrow{\varpi}\,X$ be a holomorphic vector bundle. Suppose that $E$ admits a holomorphic $S$-connection. Then all the relative Chern classes $C_{p}^{S}(E)\,\in\,\mathcal{H}^{2p}_{dR}(X/S)(S)$ of $E$ over $S$, are zero.

In last subsection 4.G, we prove a sufficient condition for the existence of relative holomorphic connection. More precisely, we prove the following (see Theorem 4.14):

Theorem 1.6**.**

Let $\pi\,:\,X\,\longrightarrow\,S$ be a surjective holomorphic proper submersion such that $\pi^{-1}(t)\,=\,X_{t}$ are connected. Let $\varpi\,:\,E\,\longrightarrow\,X$ be a holomorphic vector bundle. Suppose that for every $t\in S$, there is a holomorphic connection on holomorphic vector bundle $\varpi|_{E_{t}}:E_{t}\to X_{t}$, and

[TABLE]

Then, $E$ admits a holomorphic $S$-connection.

.

2. S-Connections

In this section, we shall introduce the notion of $S$-derivation following [6] and relative connection (or $S$-connection). Throughout this section we shall assume that $(X,\,\mathcal{O}_{X})$, $(S,\,\mathcal{O}_{S})$ are two ringed spaces, while $(\pi,\,\pi^{\sharp})\,:\,(X,\,\mathcal{O}_{X})\,\longrightarrow\,(S,\,\mathcal{O}_{S})$ is a morphism between them.

2.A. S-Linear morphism of sheaves

Let $\mathcal{F}$, $\mathcal{G}$ be two $\mathcal{O}_{X}$-modules. A morphism $\alpha:\mathcal{F}\to\mathcal{G}$ of sheaves of abelian groups is said to be $S$-linear if for every open subset $V\subset S$, for every open subset $U\,\subset\,\pi^{-1}(V)$, for every $t\,\in\,\mathcal{F}(U)$ and for every $s\,\in\,\mathcal{O}_{S}(V)$, we have

[TABLE]

where $\rho_{U,\pi^{-1}(V)}\,:\,\mathcal{O}_{X}(f^{-1}(V))\,\longrightarrow\,\mathcal{O}_{X}(U)$ is the restriction map. We denote by $\mathcal{H}\mathit{om}_{S}(\mathcal{F},\,\mathcal{G})$ the sheaf of $S$-linear morphism from $\mathcal{F}$ to $\mathcal{G}$. We denote $\rho_{U,\pi^{-1}(V)}(\pi^{\sharp}_{V}(s))$ by $s|_{U}$.

2.B. S-derivation

For the following definition see [6] (Chapitre IV, 16.5).

Let $\mathcal{F}$ be an $\mathcal{O}_{X}$-module. An $S$-derivation from $\mathcal{O}_{X}$ to $\mathcal{F}$ is a morphism

[TABLE]

of sheaves of abelian groups which satisfies the following conditions:

1. (1)

$\delta$ is an $S$-linear morphism. 2. (2)

(Leibniz rule.) For every open subset $U\subset X$, and for every $a,b\in\mathcal{O}_{X}(U)$, we have

[TABLE]

The set of all $S$-derivation from $\mathcal{O}_{X}$ to $\mathcal{F}$ form a left $\mathcal{O}_{X}(X)$-module denoted by

[TABLE]

For every open subset $U\,\subset\,X$, we note that $\mathrm{Der}_{S}(\mathcal{O}_{X}|_{U},\mathcal{F}|_{U})$ is a left $\mathcal{O}_{X}(U)$-module. For every open subset $U\,\subset\,X$, the assignment $U\,\longmapsto\,\mathrm{Der}_{S}(\mathcal{O}_{X}|_{U},\mathcal{F}|_{U})$ is a sheaf of $\mathcal{O}_{X}$-module and it is denoted by $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{F})$.

Let $\mathcal{E}\mathit{nd}_{S}(\mathcal{F})$ denote the sheaf of $S$-linear endomorphism on $\mathcal{F}$. Then $\mathcal{E}\mathit{nd}_{S}(\mathcal{F})$ is an $\mathcal{O}_{X}$-module. In particular, if we take $\mathcal{F}\,=\,\mathcal{O}_{X}$, then $\mathcal{E}\mathit{nd}_{S}(\mathcal{O}_{X})$ is a sheaf of Lie algebras with respect to the bracket operation defined as follows:

[TABLE]

for every open subset $U\,\subset\,X$ and for all $\xi,\,\eta\,\in\,\mathcal{E}\mathit{nd}_{S}(\mathcal{O}_{X})(U)$. We note that $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$ is a Lie subalgebra of $\mathcal{E}\mathit{nd}_{S}(\mathcal{O}_{X})$.

Theorem 2.1**.**

Let $(\pi,\,\pi^{\sharp})\,:\,(X,\,\mathcal{O}_{X})\,\longrightarrow\,(S,\,\mathcal{O}_{S})$ be a morphism of ringed spaces. Then there exists a unique $\mathcal{O}_{X}$–module, denoted by $\Omega^{1}_{X/S}$, and endowed with an (universal) $S$-derivation

[TABLE]

satisfying following universal property: For any $\mathcal{O}_{X}$-module $\mathcal{F}$, and for any $S$-derivation

[TABLE]

there exists a unique $\mathcal{O}_{X}$-module homomorphism

[TABLE]

such that

[TABLE]

Proof.

Cf. [6] (Proposition:16.5.3 on p. 28). ∎

2.C. S-Connections

Let $\mathcal{F}$ be an $\mathcal{O}_{X}$-module. An $S$-connection or relative connection on $\mathcal{F}$ is an $\mathcal{O}_{X}$–module homomorphism

[TABLE]

such that for every open subset $U$ of $X$ and for every $\xi\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$, the $\mathcal{O}_{X}(U)$–module homomorphism

[TABLE]

satisfies the Leibniz rule which says that

[TABLE]

for every open subset $V$ of $U$, for all $a\,\in\,\mathcal{O}_{X}(V)$ and $g\,\in\,\mathcal{F}(V)$.

If $\pi\,:\,X\,\longrightarrow\,S$ is a holomorphic map of complex manifolds with connected fibers, and $\mathcal{F}$ is a holomorphic vector bundle over $X$, we call $D$ a holomorphic $S$-connection. Similarly, If $\pi\,:\,X\,\longrightarrow\,S$ is a smooth map of smooth manifolds and $\mathcal{F}$ is a smooth vector bundle, we call $D$ a smooth $S$-connection.

Remark 2.2*.*

1. (1)

We note that $(D_{U})_{\xi}\,:\,\mathcal{F}|_{U}\,\longrightarrow\,\mathcal{F}|_{U}$ is an $S$-linear endomorphism, where $S$-linearity is with respect to $\pi|_{U}\,:\,U\,\longrightarrow\,S$. 2. (2)

The inclusion map

[TABLE]

is an $S$-connection on the $\mathcal{O}_{X}$-module $\mathcal{O}_{X}$, and it is called the canonical $S$-connection on $\mathcal{O}_{X}$. To avoid the cumbersome notation $(D_{U})_{\xi}$, we shall simply denote it by $D_{\xi}$.

2.D. S-Connections on the associated $\mathcal{O}_{X}$-modules

Let $(\mathcal{F}_{i})_{i\in I}$ be a family of $\mathcal{O}_{X}$-modules, and for each $i\in I$, let $D^{i}$ be an $S$-connection on $\mathcal{F}_{i}$. Then the various $\mathcal{O}_{X}$-modules obtained from $(\mathcal{F}_{i})_{i\in I}$ by functorial construction has natural $S$-connections.

2.D.1. Direct Sum

If

[TABLE]

and if we define

[TABLE]

for all sections $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$ and all $u\,=\,(u_{i})_{i\in I}$ of $\mathcal{F}$, then we get an $S$-connection $D$ on $\mathcal{F}$. In particular, if we take each $\mathcal{F}_{i}$ to be $\mathcal{O}_{X}$ and each $D^{i}$ to be $\epsilon$, the canonical $S$-connection on $\mathcal{O}_{X}$, then every free $\mathcal{O}_{X}$-module has a canonical $S$-connection.

2.D.2. Tensor products

Suppose $I\,=\,\{1,2,\cdots,p\}$, where $p$ is an integer $\geq 1$. Then for every open subset $U$ of $X$, and for each $\xi\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$, there exists a unique $S$-linear endomorphism $D_{\xi}$ of $\mathcal{F}_{1}\bigotimes_{\mathcal{O}_{X}}\cdots\bigotimes_{\mathcal{O}_{X}}\mathcal{F}_{p}$ such that on the presheaf level it is given by the formula

[TABLE]

for every $s_{1}\otimes_{\mathcal{O}_{X}}\cdots\otimes_{\mathcal{O}_{X}}s_{p}\,\in\,\mathcal{F}_{1}(U)\bigotimes_{\mathcal{O}_{X}(U)}\cdots\bigotimes_{\mathcal{O}_{X}(U)}\mathcal{F}_{p}(U)$. This gives an $S$-connection on $\mathcal{F}_{1}\bigotimes_{\mathcal{O}_{X}}\cdots\bigotimes_{\mathcal{O}_{X}}\mathcal{F}_{p}$.

Suppose that $\mathcal{F}_{1}\,=\,\mathcal{F}_{2}\,=\,\cdots\,=\,\mathcal{F}_{p}\,=\,\mathcal{F}$, and denote $\mathcal{F}_{1}\bigotimes_{\mathcal{O}_{X}}\cdots\bigotimes_{\mathcal{O}_{X}}\mathcal{F}_{p}$ by $T^{p}_{\mathcal{O}_{X}}(\mathcal{F})$. Equip $T^{0}_{\mathcal{O}_{X}}(\mathcal{F})\,=\,\mathcal{O}_{X}$ with the canonical $S$-connection $\epsilon$ on $\mathcal{O}_{X}$, and for each $p\geq 1$, equip $T^{p}_{\mathcal{O}_{X}}(\mathcal{F})$ with the $S$-connection induced by the $S$-connection $D$ on $\mathcal{F}$; this $S$-connection on $T^{p}_{\mathcal{O}_{X}}(\mathcal{F})$ will be denoted by $D^{p}$. Recall that the tensor algebra of $\mathcal{O}_{X}$-module $\mathcal{F}$ is a graded $\mathcal{O}_{X}$-algebra

[TABLE]

Let $D$ be the connection on the $\mathcal{O}_{X}$-module $T_{\mathcal{O}_{X}}(\mathcal{F})$, which is the direct sum (see Section 2.D.1) of the connections. It is called the induced connection on $T_{\mathcal{O}_{X}}(\mathcal{F})$.

Remark 2.3*.*

On tensor algebra $T_{\mathcal{O}_{X}}(\mathcal{F})$, we have

[TABLE]

for all local sections $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$ and local sections $s,t$ of $T_{\mathcal{O}_{X}}(\mathcal{F})$.

2.D.3. Submodule and quotient module

Let $\mathcal{F}$ be an $\mathcal{O}_{X}$-module with an $S$-connection $D$, and let $\mathcal{G}$ be an $\mathcal{O}_{X}$-submodule of $\mathcal{F}$. Let $\mathcal{H}$ denote the quotient $\mathcal{O}_{X}$-module $\mathcal{F}$/$\mathcal{G}$. Suppose that for every section $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$, we have $D_{\xi}(\mathcal{G})\subset\mathcal{G}$. Then $D$ will induce an $S$-connection on $\mathcal{G}$ and on $\mathcal{H}$.

2.D.4. Symmetric algebra and exterior Algebra

Let $D$ be an $S$-connection on $\mathcal{F}$. The $S$-connection on the tensor algebra $T_{\mathcal{O}_{X}}(\mathcal{F})$ induced by $D$ will also be denoted by $D$. Let $\mathcal{I}$ denote the two sided ideal sheaf of $T_{\mathcal{O}_{X}}(\mathcal{F})$ described as follows:

for every open subset $U$ of $X$, let $\mathcal{I}(U)$ be the two sided ideal in $T_{\mathcal{O}_{X}}(\mathcal{F})(U)$ generated by elements of the form $s\otimes t-t\otimes s$, where $s,\,t\,\in\,\mathcal{F}(U)$. Then $D_{\xi}(\mathcal{I})\,\subset\,\mathcal{I}$ for all sections $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$. Thus, by above Section 2.D.3, we get an $S$-connection $D$ on the symmetric algebra

[TABLE]

of $\mathcal{F}$.

Similarly, let $\mathcal{J}$ denote the two sided ideal sheaf of $T_{\mathcal{O}_{X}}(\mathcal{F})$ generated by the local sections $s\otimes s$ of $T^{2}_{\mathcal{O}_{X}}(\mathcal{F})$, where $s$ is a local section of $\mathcal{F}$. Then we have $D_{\xi}(\mathcal{J})\,\subset\,\mathcal{J}$ for all local sections $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$, and hence a connection on the exterior algebra

[TABLE]

of $\mathcal{F}$ is obtained.

Remark 2.4*.*

1. (1)

For all $p\,\in\,\mathbb{N}$, we have $D_{\xi}(Sym^{p}_{\mathcal{O}_{X}}(\mathcal{F}))\,\subset\,Sym^{p}_{\mathcal{O}_{X}}(\mathcal{F})$, where $Sym^{p}_{\mathcal{O}_{X}}(\mathcal{F})$ is the $p$-th graded component of the symmetric algebra $Sym_{\mathcal{O}_{X}}(\mathcal{F})$. Consequently, we get an $S$-connection on $Sym^{p}_{\mathcal{O}_{X}}(\mathcal{F})$. Similarly, we get an $S$-connection on $\Lambda^{p}_{\mathcal{O}_{X}}(\mathcal{F})$. 2. (2)

We have

[TABLE]

for all local sections $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$ and $s,s^{\prime}$ of $Sym_{\mathcal{O}_{X}}(\mathcal{F})$, and

[TABLE]

for all local sections $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$ and $t,t^{\prime}$ of $\Lambda_{\mathcal{O}_{X}}(\mathcal{F})$.

2.D.5. S-Connection on $\mathcal{H}\mathit{om}_{\mathcal{O}_{X}}(\mathcal{F},\,\mathcal{G})$

Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_{X}$-modules with $S$-connections $D^{\mathcal{F}}$ and $D^{\mathcal{G}}$ respectively. For every local section $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$, let $D_{\xi}$ be the $S$-linear endomorphism of the $\mathcal{O}_{X}$-module $\mathcal{H}\mathit{om}_{\mathcal{O}_{X}}(\mathcal{F},\,\mathcal{G})$,which is defined by

[TABLE]

for all local sections $h$ of $\mathcal{H}\mathit{om}_{\mathcal{O}_{X}}(\mathcal{F},\,\mathcal{G})$. Then the morphism

[TABLE]

is an $S$-connection on $\mathcal{H}\mathit{om}_{\mathcal{O}_{X}}(\mathcal{F},\,\mathcal{G})$.

Remark 2.5*.*

1. (1)

If $\mathcal{F}\,=\,\mathcal{G}$, and $D^{\mathcal{F}}\,=\,D^{\mathcal{G}}$, then the above $S$ -connection $D$ on $\mathcal{E}\mathit{nd}_{S}(\mathcal{F})$ is given by

[TABLE]

for all local sections $h$ of $\mathcal{E}\mathit{nd}_{S}(\mathcal{F})$. 2. (2)

If $\mathcal{G}\,=\,\mathcal{O}_{X}$, and if $D^{\mathcal{G}}$ is the canonical connection on $\mathcal{O}_{X}$, then the above connection on $\mathcal{F}^{*}\,=\,\mathcal{H}\mathit{om}_{\mathcal{O}_{X}}(\mathcal{F},\,\mathcal{O}_{X})$ is given by

[TABLE]

for all local sections $f$ of $\mathcal{F}^{*}$.

2.D.6. S-Connection on $\mathcal{O}_{X}$-module of $\mathcal{O}_{X}$-multilinear maps

Let $p\,\geq\,1$ be an integer, and let $\mathcal{F}_{1},\,\mathcal{F}_{2},\,\cdots,\,\mathcal{F}_{p},\,\mathcal{G}$ be $\mathcal{O}_{X}$-modules with $S$-connections $D^{1},\,D^{2},\,\cdots,\,D^{p},\,D^{\mathcal{G}}$ respectively. For every open subset $U$ of $X$, define

[TABLE]

where $L_{\mathcal{O}_{X}|_{U}}(\mathcal{F}_{1}|_{U},\cdots,\mathcal{F}_{p}|_{U};\mathcal{G}|_{U})$ is the $\mathcal{O}_{X}(U)$-module of $\mathcal{O}_{X}|_{U}$-multilinear maps from $\mathcal{F}_{1}|_{U}\times\mathcal{F}_{2}|_{U}\times\cdots\times\mathcal{F}_{p}|_{U}$ to $\mathcal{G}|_{U}$. The sheaf of $\mathcal{O}_{X}$-multilinear maps is denoted by $\mathcal{L}_{\mathcal{O}_{X}}(\mathcal{F}_{1},\cdots,\mathcal{F}_{p};\mathcal{G}$). For every local section $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$, let $D_{\xi}$ be the $S$-linear endomorphism of the $\mathcal{O}_{X}$-module $\mathcal{L}_{\mathcal{O}_{X}}(\mathcal{F}_{1},\cdots,\mathcal{F}_{p};\mathcal{G})$ defined by

[TABLE]

for all local sections $\omega$ of $\mathcal{L}_{\mathcal{O}_{X}}(\mathcal{F}_{1},\cdots,\mathcal{F}_{p};\mathcal{G})$ and local sections $(u_{1},\,u_{2},\,\cdots,\,u_{p})$ of $\mathcal{F}_{1}\times\mathcal{F}_{2}\times\cdots\times\mathcal{F}_{p}$. Then the morphism

[TABLE]

is an $S$-connection on $\mathcal{L}_{\mathcal{O}_{X}}(\mathcal{F}_{1},\cdots,\mathcal{F}_{p};\mathcal{G})$.

Remark 2.6*.*

Let $\mathcal{L}^{p}_{\mathcal{O}_{X}}(\mathcal{F},\mathcal{G})$ denote the $\mathcal{O}_{X}$-module $\mathcal{L}_{\mathcal{O}_{X}}(\mathcal{F}_{1},\cdots,\mathcal{F}_{p};\mathcal{G})$, where $\mathcal{F}_{1}\,=\,\cdots\,=\,\mathcal{F}_{p}\,=\,\mathcal{F}$. Let $D$ be the $S$-connection on $\mathcal{L}^{p}_{\mathcal{O}_{X}}(\mathcal{F},\mathcal{G})$ induced by $D^{\mathcal{F}}$ and $D^{\mathcal{G}}$. Let $\mathcal{S}ym^{p}_{\mathcal{O}_{X}}(\mathcal{F},\mathcal{G})$ (respectively, $\mathcal{A}lt^{p}_{\mathcal{O}_{X}}(\mathcal{F},\mathcal{G})$) denote the $\mathcal{O}_{X}$-submodule of $\mathcal{L}^{p}_{\mathcal{O}_{X}}(\mathcal{F},\,\mathcal{G})$ consisting of symmetric (respectively, alternating) $\mathcal{O}_{X}$-multilinear maps from $\mathcal{F}^{p}$ to $\mathcal{G}$. Then

[TABLE]

(respectively, $D_{\xi}(\mathcal{A}lt^{p}_{\mathcal{O}_{X}}(\mathcal{F},\mathcal{G}))\,\subset\,\mathcal{A}lt^{p}_{\mathcal{O}_{X}}(\mathcal{F},\mathcal{G})$). Therefore, $D$ induces an $S$-connection on the $\mathcal{O}_{X}$-submodules

[TABLE]

of $\mathcal{L}^{p}_{\mathcal{O}_{X}}(\mathcal{F},\mathcal{G})$.

2.D.7. Compatibility of multilinear maps and $S$-connections

Let $p\,\geq\,1$, and let

[TABLE]

be $\mathcal{O}_{X}$-modules with $S$-connections $D^{1},\,\cdots,\,D^{p},\,D^{\mathcal{G}}$ respectively. Let

[TABLE]

be an $\mathcal{O}_{X}$-multilinear map. We say that $D^{1},\,D^{2},\,\cdots,\,D^{p},\,D^{\mathcal{G}},\,\mu$ are compatible if for every local section $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$, and for all local sections $(u_{1},\,u_{2},\,\cdots,\,u_{p})$ of $\mathcal{F}_{1}\times\mathcal{F}_{2}\times\cdots\times\mathcal{F}_{p}$, we have

[TABLE]

The following proposition is straight-forward to prove.

Proposition 2.7**.**

Let $\mathcal{F}$, $\mathcal{G}$ and $\mathcal{H}$ be $\mathcal{O}_{X}$-modules, and let

[TABLE]

be a $\mathcal{O}_{X}$-bilinear map. Let $\mathcal{K}$ be any $\mathcal{O}_{X}$-module and $p\,\geq\,1$, $q\,\geq\,1$. Then, we have a $\mathcal{O}_{X}$-bilinear map

[TABLE]

defined by

[TABLE]

for all local sections $\alpha$ of $\mathcal{A}lt^{p}_{\mathcal{O}_{X}}(\mathcal{K},\mathcal{F})$, and $\beta$ of $\mathcal{A}lt^{q}_{\mathcal{O}_{X}}(\mathcal{K},\mathcal{G})$, where $S(p,q)$ is the set of all $(p,q)$-shuffles, that is, the set of all permutation $\sigma\in S_{p+q}$ such that $\sigma(1)\,<\,\cdots\,<\,\sigma(p)$ and $\sigma(p+1)\,<\,\cdots\,<\,\sigma(p+q)$.

The construction in Proposition 2.7 produces the following corollary.

Corollary 2.8**.**

Let $D^{\mathcal{F}},D^{\mathcal{G}}$ and $D^{\mathcal{H}}$ be $S$-connections on $\mathcal{F},\mathcal{G}$ and $\mathcal{H}$ respectively, which are compatible with $\mu$. Let $D^{\mathcal{K}}$ be an $S$-connection on $\mathcal{K}$. Denote the induced connections on $\mathcal{A}lt^{p}_{\mathcal{O}_{X}}(\mathcal{K},\mathcal{F})$, $\mathcal{A}lt^{p}_{\mathcal{O}_{X}}(\mathcal{K},\mathcal{G})$ and $\mathcal{A}lt^{p+1}_{\mathcal{O}_{X}}(\mathcal{K},\mathcal{H})$ by $D^{\mathcal{F}}$, $D^{\mathcal{G}}$ and $D^{\mathcal{H}}$ respectively. Then $D^{\mathcal{F}}$, $D^{\mathcal{G}}$, $D^{\mathcal{H}}$ and $\wedge$ are compatible, that is,

[TABLE]

for all local sections $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$, $\alpha$ of $\mathcal{A}lt^{p}_{\mathcal{O}_{X}}(\mathcal{K},\mathcal{F})$ and $\beta$ of $\mathcal{A}lt^{q}_{\mathcal{O}_{X}}(\mathcal{K},\mathcal{G})$.

2.E. The relative Lie derivative

Let $\mathcal{F}$ be an $\mathcal{O}_{X}$-module and $D$ an $S$-connection on $\mathcal{F}$. Let $p\geq 1$ be an integer, $U$ an open subset of $X$, $\xi\in\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$, and $\alpha\in\mathcal{L}^{p}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\mathcal{F})(U)$. Then the map

[TABLE]

defined by

[TABLE]

for all $\eta_{1},\,\cdots,\,\eta_{p}\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$, is an $\mathcal{O}_{X}(U)$-multilinear map. Moreover, the map

[TABLE]

is $S$-linear, because $D_{\xi}$ is $S$-linear.

The $S$-linear morphism

[TABLE]

is called the relative Lie derivation in degree $p$ associated with $D$.

Remark 2.9*.*

1. (1)

The relative Lie derivation satisfies the followings:

• •

$\theta_{\xi}(\alpha+\beta)\,=\,\theta_{\xi}(\alpha)+\theta_{\xi}(\beta)$,

• •

$\theta_{\xi}(a\alpha)\,=\,\xi(a)\alpha+a\theta_{\xi}(\alpha)$,

• •

$\theta_{\xi+\zeta}(\alpha)\,=\,\theta_{\xi}(\alpha)+\theta_{\zeta}(\alpha)$, and

• •

$\theta_{s\xi}(\alpha)\,=\,s\theta_{\xi}(\alpha)$

for all local sections $\alpha,\beta$ of $\mathcal{L}^{p}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\,\mathcal{F})$, $\xi,\,\zeta$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$, $a$ of $\mathcal{O}_{X}$, and $s$ of $\mathcal{O}_{S}$. 2. (2)

If $\alpha$ is alternating (respectively, symmetric), then so is $\theta_{\xi}(\alpha)$, that is,

[TABLE]

(respectively, $\theta_{\xi}(\mathcal{S}ym^{p}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\,\mathcal{F}))\,\subset\,\mathcal{S}ym^{p}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\,\mathcal{F})$).

2.E.1. The Lie derivative and the exterior product

Let $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{H}$ be $\mathcal{O}_{X}$-modules equipped with $S$-connections $D^{\mathcal{F}}$, $D^{\mathcal{G}}$ and $D^{\mathcal{H}}$ respectively. Let $\mu\,:\,\mathcal{F}\times\mathcal{G}\,\longrightarrow\,\mathcal{H}$ be an $\mathcal{O}_{X}$-bilinear map. Take integers $p\,\geq\,1$ and $q\,\geq\,1$. Then, we have an $\mathcal{O}_{X}$-bilinear map

[TABLE]

[TABLE]

Suppose that $D^{\mathcal{F}}$, $D^{\mathcal{G}}$, $D^{\mathcal{H}}$ and $\mu$ are compatible, that is,

[TABLE]

for all local sections $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$, $u$ of $\mathcal{F}$, and $v$ of $\mathcal{G}$.

Then we have

[TABLE]

where the Lie derivations are associated with their respective connections, while $\alpha$ and $\beta$ are local sections of their respective $\mathcal{O}_{X}$-modules.

2.F. Covariant derivative

Let $\mathcal{F}$ be an $\mathcal{O}_{X}$-module and $\xi\in\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$, where $U\,\subset\,X$ is an open subset, and $p\geq 2$ an integer. For each

[TABLE]

define $(\imath_{\xi}(\alpha))_{U}\,\in\,\mathcal{L}^{p-1}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\mathcal{F})(U)$ by

[TABLE]

for all $\eta_{1},\,\cdots,\,\eta_{p-1}\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$. When $\alpha$ is of degree [math], we define $\imath_{\xi}(\alpha)_{U}\,=\,0$. We call $(\imath_{\xi}(\alpha))_{U}$ the relative interior product of $\xi$ and $\alpha$ over $U$. This yields an $\mathcal{O}_{X}$-module homomorphism

[TABLE]

defined by $\imath_{U}(\xi)(\alpha)\,=\,\imath_{\xi}(\alpha)_{U}$, for every open subset $U$ of $X$.

The interior product satisfies the following properties:

1. (1)

$\imath_{\xi+\eta}\,=\,\imath_{\xi}+\imath_{\eta}$, for all local sections $\xi$ and $\eta$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$. 2. (2)

$\imath_{a\xi}\,=\,a\imath_{\xi}$, for all local sections $a$ of $\mathcal{O}_{X}$ and $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$. 3. (3)

If $D$ is an $S$-connection on $\mathcal{F}$, and $\theta$ the associated relative Lie derivative, then for all local sections $\xi,\,\eta$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$,

[TABLE] 4. (4)

If $\alpha$ is a local section of $\mathcal{A}lt^{p}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$, then $\imath_{\xi}(\imath_{\xi}(\alpha))\,=\,0$. 5. (5)

Let $\mathcal{F}$, $\mathcal{G}$ and $\mathcal{H}$ be $\mathcal{O}_{X}$-modules equipped with $S$-connections $D^{\mathcal{F}}$, $D^{\mathcal{G}}$ and $D^{\mathcal{H}}$ respectively. Let $\mu\,:\,\mathcal{F}\times\mathcal{G}\,\longrightarrow\,\mathcal{H}$ be an $\mathcal{O}_{X}$-bilinear map. Let $p\,\geq\,1$ and $q\,\geq\,1$ be integers. Then, we have an $\mathcal{O}_{X}$-bilinear map

[TABLE]

Suppose that $D^{\mathcal{F}}$, $D^{\mathcal{G}}$, $D^{\mathcal{H}}$ and $\mu$ are compatible, that is,

[TABLE]

for all local sections $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$, $u$ of $\mathcal{F}$, and $v$ of $\mathcal{G}$. Then

[TABLE]

where the Lie derivations are associated with their respective connections, while $\alpha$ and $\beta$ are local sections of their respective $\mathcal{O}_{X}$-modules.

Proposition 2.10**.**

Let $D$ be an $S$-connection on an $\mathcal{O}_{X}$-module $\mathcal{F}$. Then, there exists a unique family of $S$-linear morphism

[TABLE]

where $p\,\in\,\mathbb{N}$, such that

[TABLE]

for all local sections $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$.

Proof.

This is proved in [8, p. 11, Chapter I, Theorem 2]. ∎

The $S$-linear morphism $d$ in Proposition 2.10 is called the covariant derivative associated with $D$.

2.F.1. Explicit formula for the covariant derivative

The following proposition is straight-forward.

Proposition 2.11**.**

Let $D$ be an $S$-connection on an $\mathcal{O}_{X}$-module $\mathcal{F}$. Then, the covariant derivative with respect to $D$ is given by

[TABLE]

[TABLE]

for all local sections $\alpha$ of $\mathcal{A}lt^{p}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\mathcal{F})$ and $\xi_{1},\,\cdots,\,\xi_{p+1}$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$.

Proposition 2.11 gives the following:

Corollary 2.12**.**

The following three hold.

1. (1)

$d(\alpha)(\xi)\,=\,D_{\xi}(\alpha)$* for all local sections $\alpha$ of $\mathcal{F}$ and $\xi$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$.* 2. (2)

$d(\alpha)(\xi,\eta)\,=\,D_{\xi}(\alpha(\eta))-D_{\eta}(\alpha(\xi))-\alpha([\xi,\eta])$, for all local sections $\alpha$ of $\mathcal{A}lt^{2}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\,\mathcal{F})$ and $\xi,\,\eta$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$. 3. (3)

$d(\alpha)(\xi,\eta,\nu)\,=\,\sum_{cyclic}(D_{\xi}(\alpha(\eta,\nu))-\alpha([\xi,\eta],\nu))$.

2.F.2. Covariant derivative and exterior product

The following proposition is straight-forward.

Proposition 2.13**.**

Let $\mathcal{F}$, $\mathcal{G}$ and $\mathcal{H}$ be $\mathcal{O}_{X}$-modules equipped with $S$ connections $D^{\mathcal{F}}$, $D^{\mathcal{G}}$ and $D^{\mathcal{H}}$ respectively. Let $\mu\,:\,\mathcal{F}\times\mathcal{G}\,\longrightarrow\,\mathcal{H}$ be an $\mathcal{O}_{X}$-bilinear map. Suppose that $D^{\mathcal{F}}$, $\mathcal{G}$, $D^{\mathcal{H}}$ and $\mu$ are compatible. Then for all local sections $\alpha$ of $\mathcal{A}lt^{p}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\,\mathcal{F})$ and $\beta$ of $\mathcal{A}lt^{\bullet}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\,\mathcal{F})$, we have

[TABLE]

Proposition 2.13 gives the following:

Corollary 2.14**.**

Let $\mathcal{F}$ be an $\mathcal{O}_{X}$-module with connection $D$. Then

[TABLE]

for all local sections $a$ of $\mathcal{O}_{X}$ and $\alpha$ of $\mathcal{A}lt^{\bullet}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\,\mathcal{F})$, where $d(a)$ is the covariant derivative of $a$ with respect to the canonical connection on $\mathcal{O}_{X}$.

2.G. The curvature form

Let $D$ be an $S$-connection on an $\mathcal{O}_{X}$-module $\mathcal{F}$, and let

[TABLE]

be the covariant derivative associated with $D$. Then the map

[TABLE]

is called the curvature operator of $D$, and it will be denoted by $R$.

Let $\alpha$ be a local section of $\mathcal{A}lt^{0}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\,\mathcal{F})\,=\,\mathcal{F}$. Then $R(\alpha)\,=\,d(d(\alpha))$ is a local section of $\mathcal{A}lt^{2}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\,\mathcal{F})$. Let $\xi$ and $\eta$ be local sections of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$. Then

[TABLE]

Thus, for every open subset $U$ of $X$ and for all sections $\xi,\,\eta\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$, we get an $\mathcal{O}_{X}|_{U}$-module homomorphism

[TABLE]

defined by

[TABLE]

Hence these $K_{U}$ together define an $\mathcal{O}_{X}$-bilinear map

[TABLE]

The following is straight-forward.

Proposition 2.15**.**

The above $\mathcal{O}_{X}$-bilinear map $K$ is an alternating map.

The alternating $\mathcal{O}_{X}$-bilinear map $K$ is called the curvature form of $D$. We say the $S$-connection is flat if the curvature form is identically zero.

2.H. Connection and curvature matrices

Let $\mathcal{F}$ be a locally free coherent $\mathcal{O}_{X}$-module of rank $r$. Let $U$ be an open subset of $X$ such that $\mathcal{F}|_{U}$ is a free $\mathcal{O}_{X}|_{U}$-module. Let $s\,=\,(s_{1},\,\cdots,\,s_{r})$ be an $\mathcal{O}_{X}|_{U}$-basis of $\mathcal{F}|_{U}$. For each $\xi\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$, define an $r\times r$ matrix

[TABLE]

of elements of $\mathcal{O}_{X}(U)$ by the equation

[TABLE]

We, thus get, for every $i,\,j\,\in\,\{1,\,\cdots,\,r\}$, an element $\omega_{ij}$ of

[TABLE]

This gives an $r\times r$ matrix $\omega\,=\,(\omega_{ij})_{1\leq i,j\leq r}$, where entries are sections of $\mathcal{A}lt^{1}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\mathcal{O}_{X})$ over $U$. It is called the connection matrix of $D$ with respect to $s$. Considering $s\,=\,(s_{1},\,\cdots,\,s_{r})$ as a row vector, this $\omega$ is the unique $r\times r$ matrix over $\mathcal{A}lt^{1}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\mathcal{O}_{X})(U)$ such that

[TABLE]

for all $\xi\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$. If $u\,=\,\sum^{r}_{j=1}a_{j}s_{j}\,\in\,\mathcal{F}(U)$, where $a_{j}\,\in\,\mathcal{O}_{X}(U)$, then

[TABLE]

for all $\xi\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$. If

[TABLE]

is the covariant derivative associated with canonical connection on $\mathcal{O}_{X}$, then

[TABLE]

for all $\xi\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$.

Let $t\,=\,(t_{1},\,\cdots,\,t_{r})$ be another $\mathcal{O}_{X}|_{U}$-basis of $\mathcal{F}|_{U}$, and $t_{j}\,=\,\sum^{r}_{i=1}a_{ij}s_{i}$, for all $1\,\leq\,j\,\leq\,r$. Then the matrix $a\,=\,(a_{ij})_{1\leq i,j\leq r}$ is an element of ${\rm GL}_{r}(\mathcal{O}_{X}(U))$. Let $\omega^{\prime}$ be the connection matrix of $D$ with respect to $t$. Then we have

[TABLE]

Let $K$ be the curvature form of $D$. For all $\xi,\,\eta\,\,in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$, let

[TABLE]

be the $r\times r$ matrix over $\mathcal{O}_{X}(U)$, defined by

[TABLE]

for $1\,\leq\,j\,\leq\,r$. We thus get for all $i,\,j\,\in\,\{1,\,\cdots,\,r\}$ an element

[TABLE]

This gives a $r\times r$ matrix $\Omega\,=\,(\Omega_{ij})_{1\leq i,j\leq r}$ whose entries are sections of $\mathcal{A}lt^{2}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\,\mathcal{O}_{X})$ over $U$. It is called the curvature matrix of $D$ with respect to $s$. Considering $s\,=\,(s_{1},\,\cdots,\,s_{r})$ as a row vector, this $\Omega$ is the unique $r\times r$ matrix over $\mathcal{A}lt^{2}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\,\mathcal{O}_{X})(U)$ such that

[TABLE]

for all $\xi,\,\eta\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$.

We have $\Omega\,=\,d\omega+\omega\wedge\omega.$ If $t\,=\,(t_{1},\,\cdots,\,t_{r})$ is another $\mathcal{O}_{X}|_{U}$-basis of $\mathcal{F}|_{U}$ as above, and $\Omega^{\prime}$ is the curvature matrix of $D$ with respect to $t$, then

[TABLE]

where $a\,=\,(a_{ij})_{1\leq i,j\leq r}$ as before.

3. S-Differential Operator

3.A. First order S-differential operator

Let $(\pi,\,\pi^{\sharp})\,:\,(X,\,\mathcal{O}_{X})\,\longrightarrow\,(S,\,\mathcal{O}_{S})$ be a morphism of ringed spaces. Let $\mathcal{F}$ and $\mathcal{G}$ be two $\mathcal{O}_{X}$-modules. A first order $S$-differential operator is a morphism

[TABLE]

of sheaves of abelian groups such that

1. (1)

$P$ is an $S$-linear morphism, and 2. (2)

for every open subset $U\,\subset\,X$ and for every $f\,\in\,\mathcal{O}_{X}(U)$, the bracket $[P|_{U},\,f]\,:\,\mathcal{F}|_{U}\,\longrightarrow\,\mathcal{G}|_{U}$ defined as

[TABLE]

is an $\mathcal{O}_{U}$-module homomorphism for every open subset $V\,\subset\,U$ and all $s\,\in\,\mathcal{F}(V)$.

Let $\mathrm{Diff}^{1}_{S}(\mathcal{F},\,\mathcal{G})$ denote the set of all first order $S$-differential operator. Then $\mathrm{Diff}^{1}_{S}(\mathcal{F},\,\mathcal{G})$ is an $\mathcal{O}_{X}(X)$-module. For every open subset $U$ of $X$, $U\,\longmapsto\,\mathrm{Diff}^{1}_{S}(\mathcal{F}|_{U},\,\mathcal{G}|_{U})$ is a sheaf of first order $S$-differential operator from $\mathcal{F}|_{U}$ to $\mathcal{G}|_{U}$. This sheaf is denoted by $\mathcal{D}\mathit{iff}^{1}_{S}(\mathcal{F},\,\mathcal{G})$.

3.B. Symbol of a first order S-differential operator

Given a first order $S$-differential operator $P\,:\,\mathcal{F}\,\longrightarrow\,\mathcal{G}$, define a morphism of abelian sheaves

[TABLE]

by $\theta_{U}(f)\,=\,[P|_{U},\,f]$ for every open subset $U\,\subset\,X$ and $f\,\in\,\mathcal{O}_{X}(U)$. Then $\theta$ is an $S$-derivation. For every $W\,\subset\,S$, $V\,\subset\,\pi^{-1}(W)$, $s\,\in\,\mathcal{O}_{S}(W)$, $t\,\in\,\mathcal{O}_{X}(V)$, and $u\in\mathcal{F}(V)$, we have, by $S$-linearity of $P$,

[TABLE]

Thus $\theta$ is an $S$-linear morphism.

Next, to verify $\theta$ satisfies Leibniz rule, for every $f,\,g\,\in\,\mathcal{O}_{X}(U)$, where $U\,\subset\,X$ is any open subset, and for every $u\,\in\,\mathcal{F}(U)$, we have $[P|_{U},\,f](gu)\,=\,P(fgu)-fP(gu)$ which gives $P(fgu)\,=\,g[P,f](u)+fP(gu)$. Now,

[TABLE]

Thus, $\theta(fg)\,=\,\theta(f)g+f\theta(g)$.

Hence, we have the following:

Proposition 3.1**.**

Let $\mathcal{F}$ and $\mathcal{G}$ be $\mathcal{O}_{X}$-modules and $P\,:\,\mathcal{F}\,\longrightarrow\,\mathcal{G}$ a first order $S$-differential operator. Then there exists a unique $S$-derivation

[TABLE]

such that $\theta_{U}(f)\,=\,[P|_{U},\,f]$ for every open subset $U$ of $X$ and for every $f\,\in\,\mathcal{O}_{X}(U)$.

The $S$-derivation $\theta$ defined above is called the symbol of $P$; it will be denoted by $\sigma_{1}(P)$.

Every $\mathcal{O}_{X}$-module homomorphism is a first order $S$-differential operator. Therefore, $\mathcal{H}\mathit{om}_{\mathcal{O}_{X}}(\mathcal{F},\,\mathcal{G})$ is an $\mathcal{O}_{X}$-submodule of $\mathcal{D}\mathit{iff}^{1}_{S}(\mathcal{F},\,\mathcal{G})$. Let

[TABLE]

be the inclusion morphism. Thus, we have an exact sequence of $\mathcal{O}_{X}$-modules

[TABLE]

In general, $\sigma_{1}$ need not be surjective.

4. Analytic Theory

4.A. Complex analytic families

Let $(S,\mathcal{O}_{S})$ be a complex manifold of dimension $n$. For each $t\,\in\,S$, let there be given a compact connected complex manifold $X_{t}$ of fixed dimension $l$. The set $\{X_{t}\,\mid\,t\in S\}$ of compact connected complex manifolds is called a complex analytic family of compact connected complex manifolds, if there is a complex manifold $(X,\,\mathcal{O}_{X})$ and a surjective holomorphic map $\pi\,:\,X\,\longrightarrow\,S$ of complex manifolds with connected fibers such that the followings hold:

1. (1)

$\pi^{-1}(t)\,=\,X_{t}$ for all $t\,\in\,S$, 2. (2)

$\pi^{-1}(t)$ is a compact connected complex submanifold of $X$ for all $t\,\in\,S$, and 3. (3)

the rank of the Jacobian matrix of $\pi$ is equal to $n$ at each point of $X$.

(See [9] for details.) In other words, $\pi\,:\,X\,\longrightarrow\,S$ is a surjective holomorphic proper submersion, such that $\pi^{-1}(t)\,=\,X_{t}$ is connected for every $t\,\in\,S$.

4.B. Holomorphic relative tangent and cotangent bundles

Let $\pi\,:\,X\,\longrightarrow\,S$ be a surjective holomorphic submersion of complex manifolds with connected fibers such that $\mathrm{dim}(X)\,=\,m$ and $\mathrm{dim}(S)\,=\,n$. For any $t\,\in\,t$, the fiber $\pi^{-1}(t)$ will be denoted by $X_{t}$. Let $d\pi_{S}\,:\,TX\,\longrightarrow\,\pi^{*}TS$ be the differential of $\pi$. The subbundle

[TABLE]

is called the relative tangent bundle for $\pi$. Thus we have a short exact sequence

[TABLE]

of $\mathcal{O}_{X}$-modules and $\mathcal{O}_{X}$-linear maps.

The dual $T(X/S)^{*}$ of the relative tangent bundle is called the relative cotangent bundle and it is denoted by $\Omega^{1}(X/S)$. The sheaf of holomorphic sections of relative cotangent bundle $\Omega^{1}(X/S)$ will also be denoted by $\Omega^{1}_{X/S}$. Dualizing the short exact sequence in (4.1), we get a short exact sequence

[TABLE]

The relative tangent sheaf $\mathcal{T}_{X/S}$ and the relative cotangent sheaf $\Omega^{1}_{X/S}$ are locally free $\mathcal{O}_{X}$-modules of rank $l\,=\,m-n$.

From Theorem 2.1, there exists a unique $S$-derivation $d_{X/S}\,:\,\mathcal{O}_{X}\,\longrightarrow\,\Omega^{1}_{X/S}$. For any integer $r\,\geq\,1$, define $\Omega^{r}_{X/S}\,=\,\Lambda^{r}\Omega^{1}_{X/Y}$, which is called the sheaf of holomorphic relative $r$-forms on $X$ over $S$. We have the short exact sequence

[TABLE]

which is derived from the short exact sequence in (4.2).

Theorem 4.1**.**

There exists canonical $\pi^{-1}\mathcal{O}_{S}$-linear maps $\partial^{r}_{X/S}\,:\,\Omega^{r}_{X/S}\,\longrightarrow\,\Omega^{r+1}_{X/S}$, called the relative exterior derivative, satisfying the following:

1. (1)

$\partial^{0}_{X/S}\,=\,d_{X/S}\,:\,\mathcal{O}_{X}\,\longrightarrow\,\Omega^{1}_{X/S}$, 2. (2)

$\partial^{r+1}_{X/S}\circ\partial^{r}_{X/S}\,=\,0$, and 3. (3)

$\partial^{r+s}_{X/S}(\alpha\wedge\beta)\,=\,\partial^{r}_{X/S}\alpha\wedge\beta+(-1)^{r}\alpha\wedge\partial^{s}_{X/S}\beta$, for all local sections $\alpha$ of $\Omega^{r}_{X/S}$ and $\beta$ of $\Omega^{s}_{X/S}$.

Proof.

This basically follows from Proposition 2.10. First note that the sheaf $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})$ is canonically isomorphic to the holomorphic relative tangent sheaf $\mathcal{T}_{X/S}$ as an $\mathcal{O}_{X}$-module, and hence $\mathcal{A}lt^{r}_{\mathcal{O}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X}),\,\mathcal{O}_{X})$ is canonically isomorphic to the sheaf $\Omega^{r}_{X/S}$ as an $\mathcal{O}_{X}$-module. In view of the canonical $S$-connection in $\mathcal{O}_{X}$, by Proposition 2.10, canonical $S$-linear map exists satisfying (1), and by Proposition 2.13, it satisfies (3). Finally, (2) follows from Corollary 2.12(2). ∎

4.C. Relative Atiyah algebra

Proposition 4.2** (Symbol exact sequence).**

Let $\pi\,:\,X\,\longrightarrow\,S$ be a holomorphic proper submersion of complex manifolds with connected fibers, and let $\mathcal{F}$ and $\mathcal{G}$ be two locally free $\mathcal{O}_{X}$-modules of rank $r$ and $p$ respectively. Then

[TABLE]

is an exact sequence of $\mathcal{O}_{X}$-modules.

Proof.

It is enough to show that $\sigma_{1}$ is surjective. Let $\theta\,\in\,(\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{H}\mathit{om}_{\mathcal{O}_{X}}(\mathcal{F},\,\mathcal{G})))_{x}$ with $x\,\in\,X$. We have to show that there exists a first order $S$-differential operator $P$ defined near $x$, such that $(\sigma_{1})_{x}(P_{x})\,=\,\theta$, where

[TABLE]

is the germ of $P$ at $x$. Let $(U,\,\phi\,=\,(z_{1},\,\cdots,\,z_{l},\,z_{l+1},\,\cdots,\,z_{l+n}))$ be a holomorphic chart on $X$ around $x$, and let $s\,=\,(s_{1},\,\cdots,\,s_{l})$ and $t\,=\,(t_{1},\,\cdots,\,t_{p})$ be holomorphic frames of $\mathcal{F}$ and $\mathcal{G}$ respectively, on $U$. We may assume that $\theta$ is the germ at $x$ of a section $u$ of $\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{H}\mathit{om}_{\mathcal{O}_{X}}(\mathcal{F},\,\mathcal{G}))$ over $U$. Since

[TABLE]

$u$ can be considered as a section of $\mathcal{H}\mathit{om}_{\mathcal{O}_{X}}(\Omega^{1}_{X/S},\,\mathcal{H}\mathit{om}_{\mathcal{O}_{X}}(\mathcal{F},\,\mathcal{G}))$ over $U$, that is, $u\,:\,\Omega^{1}_{X/S}|_{U}\,\longrightarrow\,\mathcal{H}\mathit{om}_{\mathcal{O}_{U}}(\mathcal{F}|_{U},\,\mathcal{G}|_{U})$ is an $\mathcal{O}_{U}$-module homomorphism. As $\{dz_{\alpha}\,\mid\,1\,\leq\,\alpha\,\leq\,l\}$ is an $\mathcal{O}_{U}$-basis of $\Omega^{1}_{X/S}|_{U}$, there exists a uniquely determined function $b^{\alpha}_{ij}\,\in\,\mathcal{O}_{X}(U)$, where $1\,\leq\,i\,\leq\,p$, $1\,\leq\,j\,\leq\,r$ and $1\,\leq\,\alpha\,\leq\,l$, such that

[TABLE]

Define $P\,:\,\mathcal{F}|_{U}\,\longrightarrow\,\mathcal{G}|_{U}$ by

[TABLE]

Then $P$ is $S$-linear, because for any $V\,\subset\,\pi^{-1}(W)\bigcap U$, where $W\,\subset\,S$ is any open subset, and any $g\,\in\,\mathcal{O}_{S}(W)$, we have

[TABLE]

for all $\alpha\,=\,1,\,\cdots,\,l$. The bracket operation $[P,\,f]$ is $\mathcal{O}_{U}$-linear. Thus $P$ is a first order $S$-differential operator, that is, $P\,\in\,\mathcal{D}\mathit{iff}^{1}_{S}(\mathcal{F},\,\mathcal{G})(U)$. Let $V\,\subset\,U$ and $\xi\,\in\,\Omega^{1}_{X/S}(V)$. Then $\xi\,=\,\sum^{l}_{\alpha=1}\xi_{\alpha}dz_{\alpha}$, where $\xi_{\alpha}\,\in\,\mathcal{O}_{U}(V)$. Then by the construction of the symbol map, and the universal property of $(\Omega^{1}_{X/S},\,d_{X/S})$, we have

[TABLE]

From the definition of $P$ it follows that $P(s_{j})\,=\,0$ for all $j$, and hence

[TABLE]

Therefore, $[P,\,z_{\alpha}]\,=\,u(dz_{\alpha})$, and hence (4.4) becomes

[TABLE]

This proves that $\sigma_{1}(P)_{U}\,=\,u$, that is, $(\sigma_{1})_{x}(P_{x})\,=\,\theta$. ∎

Let $E$ be a locally free $\mathcal{O}_{X}$-module. By Proposition 4.2, we have a short exact sequence of $\mathcal{O}_{X}$-modules

[TABLE]

For any $S$-derivation $\xi\,:\,\mathcal{O}_{X}\,\longrightarrow\,\mathcal{O}_{X}$, let $\widetilde{\xi}\,:\,\mathcal{O}_{X}\,\longrightarrow\,\mathcal{E}\mathit{nd}_{\mathcal{O}_{X}}(E)$ be the map defined by $a\,\longmapsto\,\xi(a)\mathbf{1}_{E}$, where $a$ is a local sections of $\mathcal{O}_{X}$. Then $\widetilde{\xi}$ is an $S$-derivation. Thus, we have an $\mathcal{O}_{X}$-module homomorphism

[TABLE]

defined by $\xi\,\longmapsto\,\widetilde{\xi}$. Note that $\Psi$ is an injective homomorphism.

Define

[TABLE]

which is an $\mathcal{O}_{X}$-module and for every open subset $U$ of $X$. Note that $\mathcal{A}t_{S}(E)(U)$ consists of first order $S$-differential operator $P\,\in\,\mathcal{D}\mathit{iff}^{1}_{S}(E,\,E)(U)$ such that $(\sigma_{1})_{U}(P)\,=\,\Psi(\xi)$ for some $\xi\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$, which is equivalent to the assertion that $\sigma_{1}(P)(a)\,=\,\xi(a)\mathbf{1}_{E}$ or $[P,\,a]\,=\,\xi(a)\mathbf{1}_{E}$, for all $a\,\in\,\mathcal{O}_{X}(U)$ and for some $\xi\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$. Hence, we get a short exact sequence

[TABLE]

of $\mathcal{O}_{X}$-modules, which is called the Atiyah sequence while $\mathcal{A}t_{S}(E)$ is called the relative Atiyah algebra of $E$.

Proposition 4.3**.**

Let $\pi\,:\,X\,\longrightarrow\,S$ be a holomorphic proper submersion of complex manifolds with connected fibers, and let $E$ be a holomorphic vector bundle over $X$. Then $E$ admits an holomorphic $S$-connection if and only if the Atiyah sequence in (4.5) splits holomorphically.

Proof.

Suppose that the Atiyah sequence in (4.5) splits holomorphically, that is, there exists an $\mathcal{O}_{X}$-module homomorphism

[TABLE]

such that $\sigma_{1}\circ\nabla\,=\,\mathbf{1}_{\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})}$. Then, for every open subset $U\,\subset\,X$ and for every $\xi\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$, this $\nabla_{U}(\xi)$ is a first order $S$-differential operator such that $\sigma_{1}(\nabla_{U}(\xi))(a)\,=\,\xi^{\prime}(a)\mathbf{1}_{E}$ for some $\xi^{\prime}\,\in\,\mathcal{D}\mathit{er}_{S}(\mathcal{O}_{X},\,\mathcal{O}_{X})(U)$ and for every $a\,\in\,\mathcal{O}_{X}(U)$. This implies that $\xi\,=\,\xi^{\prime}$, because the Atiyah sequence splits. We have $[\nabla_{U}(\xi),\,a]\,=\,\xi(a)\mathbf{1}_{E}$, which can be expressed as

[TABLE]

for every $s\,\in\,E(U)$. Thus $\nabla_{U}(\xi)$ satisfies Leibniz rule, and since $\mathcal{A}t_{S}(E)$ is an $\mathcal{O}_{X}$-submodule of $\mathcal{E}\mathit{nd}_{S}(\mathcal{E})$, it follows that $\nabla$ is actually an $S$-connection on $E$.

The converse follows from the fact that any $S$-connection satisfies Leibniz rule, because it gives an splitting of Atiyah exact sequence. ∎

The extension class of the Atiyah exact sequence (4.5) of a holomorphic vector bundle $E$ over $X$ is an element of $\mathrm{H}^{1}(X,\,\mathcal{H}\mathit{om}_{\mathcal{O}_{X}}(\mathcal{T}_{X/S},\,\mathcal{E}\mathit{nd}_{\mathcal{O}_{X}}(E)))$. This extension class is called the relative Atiyah class of $E$, and it is denoted by at${}_{S}(E)$. Note that

[TABLE]

Proposition 4.3 has the following corollary:

Corollary 4.4**.**

A holomorphic vector bundle $E$ on $E$ admits a holomorphic $S$-connection if and only if its relative Atiyah class $\mathrm{at}_{S}(E)\,\in\,\mathrm{H}^{1}(X,\,\Omega^{1}_{X/S}(\mathcal{E}\mathit{nd}_{\mathcal{O}_{X}}(E)))$ is zero.

4.D. Induced family of holomorphic connections

As before, $\pi\,:\,X\,\longrightarrow\,S$ is a surjective holomorphic proper submersion with connected fibers. Let $\varpi\,:\,E\,\longrightarrow\,X$ be a holomorphic vector bundle. For every $t\,\in\,S$, the restriction of $E$ to $X_{t}\,=\,\pi^{-1}(t)$ is denoted by $E_{t}$. Let $U$ be an open subset of $X$ and $s\,:\,U\,\longrightarrow\,E$ a holomorphic section. We denote by $r_{t}(s)$ the restriction of $s$ to $X_{t}\cap U$, whenever $U\cap X_{t}\,\neq\,\emptyset$. Clearly, $r_{t}(s)$ is a holomorphic section of $E_{t}$ over $U\cap X_{t}$. The map $r_{t}\,:\,s\,\longmapsto\,r_{t}(s)$ induces, therefore, a homomorphism of $\mathbb{C}$-vector spaces from $E$ to $E_{t}$, which is denoted by the same symbol $r_{t}$ (the restriction map $r_{t}$ is discussed in[9, p. 343] and [10, p. 58]). Also, $X_{t}$ is a complex submanifold of $X$, so $\mathcal{O}_{X}|_{X_{t}}\,=\,\mathcal{O}_{X_{t}}$. We also have the restriction map $r_{t}\,:\,\mathcal{E}\mathit{nd}_{\mathcal{O}_{X}}(E)\,\longrightarrow\,\mathcal{E}\mathit{nd}_{\mathcal{O}_{X_{t}}}(E_{t})$.

Similarly, if $P\,:\,E\,\longrightarrow\,F$ is a first order $S$-differential operator, where $F$ is a holomorphic vector bundle over $X$, then the restriction map $r_{t}\,:\,E_{t}\,\longrightarrow\,F_{t}$ gives rise to a first order differential operator $P_{t}\,:\,E_{t}\,\longrightarrow\,F_{t}$ for every $t\,\in\,S$. Thus, we have the restriction map $r_{t}\,:\,\mathcal{D}\mathit{iff}^{1}_{S}(E,\,F)\,\longrightarrow\,\mathcal{D}\mathit{iff}^{1}_{\mathbb{C}}(E_{t},\,F_{t})$. In particular, for $E\,=\,F$, we have the restriction map $r_{t}\,:\,\mathcal{D}\mathit{iff}^{1}_{S}(E,\,E)\,\longrightarrow\,\mathcal{D}\mathit{iff}^{1}_{\mathbb{C}}(E_{t},\,E_{t})$ for every $t\,\in\,S$. Since, the restriction of the relative tangent bundle $T(X/S)$ to each fiber $X_{t}$ of $\pi$ is the tangent bundle $T(X_{t})$ of the fiber $X_{t}$, we have the restriction map $r_{t}\,:\,\mathcal{T}_{X/S}\,\longrightarrow\,\mathcal{T}_{X_{t}}$.

Now, for every $t\,\in\,S$, the restriction maps gives a commutative diagram

[TABLE]

where the bottom sequence is the Atiyah sequence of the holomorphic vector bundle $E_{t}$ over $X_{t}$ (see (4.5)) and $\sigma_{1t}$ is the restriction of the symbol map $\sigma_{1}$.

Suppose that $E$ admits a holomorphic $S$-connection, which is equivalent to saying that the relative Atiyah sequence in (4.5) splits holomorphically. If $\nabla\,:\,\mathcal{T}_{X/S}\,\longrightarrow\,\mathcal{A}t_{S}(E)$ is a holomorphic splitting of the relative Atiyah sequence in (4.5), then for every $t\,\in\,S$, the restriction of $\nabla$ to $\mathcal{T}_{X_{t}}$ gives an $\mathcal{O}_{X_{t}}$-module homomorphism $\nabla_{t}\,:\,\mathcal{T}_{X_{t}}\,\longrightarrow\,\mathcal{A}t(E_{t})$. Now, $\nabla_{t}$ is a holomorphic splitting of the Atiyah sequence of the holomorphic vector bundle $E_{t}$, which follows from the fact that the restriction maps $r_{t}$ defined above are surjective. Thus, we have the following:

Proposition 4.5**.**

Let $\pi\,:\,X\,\longrightarrow\,S$ be a surjective holomorphic proper submersion with connected fibers and $\varpi\,:\,E\,\longrightarrow\,X$ a holomorphic vector bundle. Let $D$ be a holomorphic $S$-connection on $E$. Then for every $t\,\in\,S$, we have a holomorphic connection $D_{t}$ on the holomorphic vector bundle $E_{t}\,\longrightarrow\,X_{t}$. In other words, we have a family $\{D_{t}\,\mid\,t\in S\}$ of holomorphic connections on the holomorphic family of vector bundles $\{E_{t}\,\longrightarrow\,X_{t}\,\mid\,t\in S\}$.

4.E. Smooth relative tangent bundle and smooth relative r-forms

As before, $\pi\,:\,X\,\longrightarrow\,S$ is a complex analytic family of complex manifolds. Consider $\pi$ as a $C^{\infty}$ map between real manifolds. We denote the smooth relative tangent bundle by $T^{\mathbb{R}}(X/S)$, while its sheaf of smooth sections is denoted by $\mathcal{T}^{\mathbb{R}}_{X/S}$. Similarly, there is a smooth relative cotangent bundle denoted by $A^{1}_{\mathbb{R}}(X/S)$ and its sheaf of smooth sections, which is denoted by $\mathcal{A}^{1}_{\mathbb{R}}(X/S)$. Define

[TABLE]

which is nothing but the complexification of the smooth relative cotangent bundle $A^{1}_{\mathbb{R}}(X/S)$.

A smooth section of $A^{1}_{\mathbb{R}}(X/S)_{\mathbb{C}}$ is called a complex valued smooth $1$-form on $X$ relative to $S$, or a complex valued smooth relative $1$-form on $X$ over $S$. We denote the sheaf of smooth sections of $A^{1}_{\mathbb{R}}(X/S)_{\mathbb{C}}$ by $\mathcal{A}^{1}_{X/S}$; also, denote the sheaf of complex valued smooth function on $X$ by $\mathcal{C}^{\infty}_{X}$. Then $\mathcal{A}^{1}_{X/S}$ is an $\mathcal{C}^{\infty}_{X}$-module, and there exists a unique $S$-derivation $d_{X/S}\,:\,\mathcal{C}^{\infty}_{X}\,\longrightarrow\,\mathcal{A}^{1}_{X/S}$. The kernel $\mathcal{K}er(d_{X/S})$ of $d_{X/S}$ is the sheaf of complex valued smooth functions on $X$, which are constant along the fibers $X_{t}$, for all $t\in S$, that is, $\mathcal{K}er(d_{X/S})\,=\,\pi^{-1}{\mathcal{C}^{\infty}_{S}}$, where ${\mathcal{C}^{\infty}_{S}}$ is the sheaf of complex valued smooth functions on $S$.

Similarly, we can define the complex valued smooth relative $r$-forms on $X$ over $S$. A smooth section of $\Lambda^{r}T^{*}(X/S)_{\mathbb{C}}$ is called a complex valued smooth relative $r$-form on $X$ over $S$. Denote the sheaf of smooth sections of $\Lambda^{r}T^{*}(X/S)_{\mathbb{C}}$ by $\mathcal{A}^{r}_{X/S}$. The following analog Theorem 4.1 is derived using Proposition 2.10 again.

Theorem 4.6**.**

There exist canonical $S$-linear maps $\delta^{r}_{X/S}\,:\,\mathcal{A}^{r}_{X/S}\,\longrightarrow\,\mathcal{A}^{r+1}_{X/S}$ called relative exterior derivative satisfying the following:

1. (1)

$\delta^{0}_{X/S}\,=\,d_{X/S}\,:\,\mathcal{C}^{\infty}_{X}\mathcal{A}^{1}_{X/S}$, 2. (2)

$\delta^{r+1}_{X/S}\circ\delta^{r}_{X/S}\,=\,0$, and 3. (3)

$\delta_{X/S}(\alpha\wedge\beta)\,=\,\delta_{X/S}\alpha\wedge\beta+(-1)^{r}\alpha\wedge\delta_{X/S}\beta$* for all local sections $\alpha$ of $\mathcal{A}^{r}_{X/S}$ and $\beta$ of $\mathcal{A}^{s}_{X/S}$.*

Proof.

First note that the sheaf $\mathcal{D}\mathit{er}_{S}(\mathcal{C}^{\infty}_{X},\,\mathcal{C}^{\infty}_{X})$ is canonically isomorphic to the relative tangent sheaf $\mathcal{T}^{\mathbb{R}}_{X/S}$ as an $\mathcal{C}^{\infty}_{X}$-module, and hence $\mathcal{A}lt^{r}_{\mathcal{C}^{\infty}_{X}}(\mathcal{D}\mathit{er}_{S}(\mathcal{C}^{\infty}_{X},\,\mathcal{C}^{\infty}_{X}),\,\mathcal{C}^{\infty}_{X})$ is canonically isomorphic to the sheaf $\mathcal{A}^{r}_{X/S}$ as an $\mathcal{C}^{\infty}_{X}$-module. Considering the canonical $S$-connection in $\mathcal{C}^{\infty}_{X}$, by Proposition 2.10, there is a canonical $S$-linear map that satisfies (1), and by Proposition 2.13, it satisfies (3). Finally, (2) follows using By Corollary 2.12(2). ∎

Henceforth, we shall denote $\delta^{r}_{X/S}$ by $d_{X/S}$, for all $r\,\geq\,0$. By the relative Poincaré lemma, there is an exact sequence

[TABLE]

of $\mathcal{C}^{\infty}_{X}$-module and $S$-linear maps. Thus we have a smooth relative de Rham complex $(\mathcal{A}^{\bullet}_{X/S},\,d_{X/S})$, which is a resolution of the sheaf $\pi^{-1}\mathcal{C}^{\infty}_{S}$.

For every integer $p\,\geq\,0$ and for every open subset $V\,\subset\,S$, the assignment

[TABLE]

is a presheaf of $\pi_{\ast}\mathcal{C}^{\infty}_{X}(V)\,=\,\mathcal{C}^{\infty}_{X}(\pi^{-1}(V))$-module, where $\mathbb{H}^{q}(\pi^{-1}(V),\mathcal{A}^{\bullet}_{X/S}|_{\pi^{-1}(V)})$ denotes the hypercohomolgy group of $\pi^{-1}(V)\,\subset\,X$ with values in $\mathcal{A}^{\bullet}_{X/S}|_{\pi^{-1}(V)}$. The sheafification of this presheaf is a $\mathcal{C}^{\infty}_{S}$-module, and it is denoted by $\mathbb{R}^{p}\pi_{\ast}(\mathcal{A}^{\bullet}_{X/S})$.

The sheaf $\mathbb{R}^{p}\pi_{\ast}(\mathcal{A}^{\bullet}_{X/S})$ of $\mathcal{C}^{\infty}_{S}$-module is called the sheaf of relative de Rham cohomology, and it is denoted by $\mathcal{H}^{p}_{dR}(X/S)$. Since, $\mathcal{A}^{\bullet}_{X/S}$ is an acyclic resolution of $\pi^{-1}\mathcal{C}^{\infty}_{S}$, we have the following:

Proposition 4.7**.**

Let $\pi\,:\,X\,\longrightarrow\,S$ be a holomorphic proper submersion of complex manifolds with connected fibers. Then

[TABLE]

where $R^{p}\pi_{\ast}(\pi^{-1}\mathcal{C}^{\infty}_{S})$ is the higher direct image sheaf of $\pi^{-1}\mathcal{C}^{\infty}_{S}$ on $S$.

Proof.

Since, for each $p\,\geq\,0$, the sheaf $\mathcal{A}^{p}_{X/S}$ is fine, from the definition of hypercohomology, we have

[TABLE]

for every open subset $V\,\subset\,S$. Also, $\mathcal{A}^{\bullet}_{X/S}$ is an acyclic resolution of $\pi^{-1}\mathcal{C}^{\infty}_{S}$. Thus, we have

[TABLE]

for every open subset $V\,\subset\,S$. Now, the proposition follows from the definition of higher direct image sheaves. ∎

Note that $\mathcal{H}^{p}_{dR}(X/S)$ is locally free $\mathcal{C}^{\infty}_{S}$-module.

Proposition 4.8** (Pullback of smooth relative forms).**

Suppose we have the following commutative diagram

[TABLE]

of complex manifolds and holomorphic maps, where $\pi$, and $\pi^{\prime}$ are surjective holomorphic proper submersions. Then, for every open subset $U\,\subset\,X$, and every smooth relative differential form $\omega\,\in\,\mathcal{A}^{r}_{X/S}(U)$, the pullback $\widetilde{f^{*}}(\omega)$ is an element of $\mathcal{A}^{r}_{Y/T}(f^{-1}(U))$.

Proof.

Given the commutative diagram in (4.7), we have the following commutative diagrams:

[TABLE]

[TABLE]

Thus, given any $\omega\,\in\,\mathcal{A}^{1}_{X/S}(U)$, from (4.8), we get that $\widetilde{f^{*}}(\omega)$ is an element of $\mathcal{A}^{1}_{Y/T}(f^{-1}(U))$, and similarly, from (4.9) it follows that for any smooth relative $r$-form $\omega\,\in\,\mathcal{A}^{r}_{X/S}(U)$, we have $\widetilde{f^{*}}(\omega)\,\in\,\mathcal{A}^{r}_{Y/T}(f^{-1}(U))$. ∎

4.F. Smooth relative connection and relative Chern class

In this section, we define the relative Chern class of a differentiable family of complex vector bundles $\varpi\,:\,E\,\longrightarrow\,X$ of rank $r$. For each $t\,\in\,S$, the restriction of $E$ to $X_{t}\,=\,\pi^{-1}(t)$ will be denoted by $E_{t}$.

We follow Section 2.H and substitute $E$ in place of $\mathcal{F}$ there. Let $D$ be a smooth $S$-connection on $E$. Let $(U_{\alpha},\,h_{\alpha})$ be a trivialization of $E$ over $U_{\alpha}\,\subset\,X$. Let $R$ be the $S$-curvature form for $D$, and let $\Omega_{\alpha}\,=\,(\Omega_{ij\alpha})$ be the curvature matrix of $D$ over $U_{\alpha}$, as defined in Section 2.H, so $\Omega_{ij\alpha}\,\in\,\mathcal{A}^{2}_{X/S}(U_{\alpha})$. We have $\Omega_{\beta}\,=\,g_{\alpha\beta}^{-1}\Omega_{\alpha}g_{\alpha\beta}$, where $g_{\alpha\beta}\,:\,U_{\alpha}\cap U_{\beta}\,\longrightarrow\,{\rm GL}_{r}(\mathbb{C})$ is the change of frame matrix (transition function), which is a smooth map.

Consider the adjoint action of ${\rm GL}_{r}(\mathbb{C})$ on it Lie algebra $\mathfrak{gl}_{r}(\mathbb{C})\,=\,{\rm M}_{r}(\mathbb{C})$. Let $f$ be a ${\rm GL}_{r}(\mathbb{C})$-invariant homogeneous polynomial on $\mathfrak{gl}_{r}(\mathbb{C})$ of degree $p$. Then, we can associate a unique $p$-multilinear symmetric map $\widetilde{f}$ on $\mathfrak{gl}_{r}(\mathbb{C})$ such that $f(X)\,=\,\widetilde{f}(X,\cdots,X)$, for all $X\in\mathfrak{gl}_{r}(\mathbb{C})$. Define

[TABLE]

Since $f$ is ${\rm GL}_{r}(\mathbb{C})$-invariant, it follows that $\gamma_{\alpha}$ is independent of the choice of frame, and hence it defines a global smooth relative differential form of degree $2p$, which we denote by the symbol $\gamma\in\mathcal{A}^{2p}_{X/S}(X)$.

Theorem 4.9**.**

Let $\pi\,:\,X\,\longrightarrow\,S$ be a surjective holomorphic proper submersion of complex manifolds with connected fibers and $\varpi\,:\,E\,\longrightarrow\,X$ a differentiable family of complex vector bundle. Let $D$ be a smooth $S$-connection on $E$. Suppose that $f$ is a ${\rm GL}_{r}(\mathbb{C})$-invariant polynomial function on $\mathfrak{gl}_{r}(\mathbb{C})$ of degree $p$. Then the following hold:

1. (1)

$\gamma\,=\,f(\Omega_{\alpha})$* is $d_{X/S}$-closed, that is, $d_{X/S}(\gamma)\,=\,0$.* 2. (2)

The image $[\gamma]$ of $\gamma$ in the relative de Rham cohomology group

[TABLE]

is independent of the smooth $S$-connection $D$ on $E$.

Proof.

This is proved in [7] (Chapter II, Section 2, p. 36). ∎

Define homogeneous polynomials $f_{p}$ on $\mathfrak{gl}_{r}(\mathbb{C})$, of degree $p\,=\,1,\,2,\,\cdots,\,r$, to be the coefficient of $\lambda^{p}$ in the following expression:

[TABLE]

where $f_{0}(\frac{\sqrt{-1}}{2\pi}A)\,=\,1$ while $f_{r}(\frac{\sqrt{-1}}{2\pi}A)$ is the coefficient of $\lambda^{0}$. These polynomials $f_{1},\,\cdots,\,f_{r}$ are $GL_{r}(\mathbb{C})$-invariant, and they generate the algebra of ${\rm GL}_{r}(\mathbb{C})$-invariant polynomials on $\mathfrak{gl}_{r}(\mathbb{C})$. We now define the p-th cohomology class as follows:

[TABLE]

for $p\,=\,0,\,1,\,\cdots,\,r$.

The relative de Rham cohomology sheaf $\mathcal{H}^{p}_{dR}(X/S)\,\cong\,R^{p}\pi_{\ast}(\pi^{-1}\mathcal{C}^{\infty}_{S})$ on $S$ is by definition the sheafification of the presheaf $V\,\longmapsto\,\mathrm{H}^{p}(\pi^{-1}(V),\,\pi^{-1}\mathcal{C}^{\infty}_{S}|_{\pi^{-1}(V)})$, for open subset $V\,\subset\,S$. Therefore, we have a natural homomorphism

[TABLE]

which maps $c^{S}_{p}(E)$ to $\rho(c^{S}_{p}(E))\,\in\,\mathcal{H}^{2p}_{dR}(X/S)(S)$.

Define $C^{S}_{p}(E)\,=\,\rho(c^{S}_{p}(E))$. We call $C^{S}_{p}(E)$ the p-th relative Chern class of E over S. Let

[TABLE]

be the total relative Chern class of $E$.

Proposition 4.10**.**

Let $E\,\xrightarrow{\varpi}\,X\,\xrightarrow{\pi}\,S$ be as in Theorem 4.9. Let $\pi^{\prime}\,:\,Y\,\longrightarrow\,T$ be a surjective holomorphic proper submersion, such that the following diagram

[TABLE]

is commutative, where $f\,:\,Y\,\longrightarrow\,X$ and $g\,:\,T\,\longrightarrow\,S$ are holomorphic maps. Then

[TABLE]

where $C^{S}(E)$ is the total relative Chern class of $E$ over $S$, and $C^{T}(f^{*}E)$ is the total relative Chern class of $f^{*}E$ over $T$.

Proof.

Let $D$ be smooth $S$-connection in $E$. It is enough to define a smooth $T$-connection $D^{*}$ in $f^{*}E$, such that $f^{*}\Omega\,=\,\Omega^{*}$, where $\Omega^{*}$ is the curvature matrix of $D^{*}$. Let $e\,=\,(e_{1},\,\cdots,\,e_{r})$ be a frame of $E$ over an open subset $U$ of $X$. Then, we have $e^{*}\,=\,(e^{*}_{1},\,\cdots,\,e^{*}_{r})$, where $e^{*}_{i}\,=\,f\circ e_{i}^{*}\,:\,f^{-1}(U)\,\longrightarrow\,E$ is a frame of $f^{*}E$ over $f^{-1}(U)$. If $a\,:\,U\,\longrightarrow\,{\rm GL}_{r}(\mathbb{C})$ is a change of frame over $U$, then

[TABLE]

is a change of frame in $f^{*}E$ over $f^{-1}(U)$. Now, we define $S$-connection matrix

[TABLE]

where, $\omega_{ij}\,\in\,\mathcal{A}_{X/S}^{1}(U)$, and $\widetilde{f^{*}}\,:\,\mathcal{A}_{X/S}^{1}\,\longrightarrow\,\mathcal{A}_{Y/T}^{1}$ is the pullback map of the relative forms as in Proposition 4.8. Moreover, if $\omega^{\prime}$ is the connection matrix with respect to the frame $e^{\prime}\,=\,e.a$ and $\omega^{\prime*}$ is the pullback of $\omega^{\prime}$ under $f^{*}$, then

[TABLE]

Thus, if we consider $D^{*}\,=\,d_{Y/T}+\omega^{*}$, then from above compatibility condition, this defines a smooth $T$-connection in $f^{*}E$. Let $\Omega^{*}$ be the curvature form of $D^{*}$. Then

[TABLE]

Now, consider the homogeneous polynomial $f_{p}$ of degree $p$ as defined above. The $p$-th cohomology class is

[TABLE]

for all $p\,\geq\,0$, which is the pullback of the cohomology class $[c_{p}^{S}(E)]$, where

[TABLE]

is the morphism of $\mathbb{C}$-vector spaces induced by the commutative diagram (4.10). Further, we have the following commutative diagram

[TABLE]

which implies that $f^{*}(C_{p}^{S}(E))\,=\,C_{p}^{T}(f^{*}E)$. This completes the proof. ∎

In particular, taking $T\,=\,\{t\}\,\subset\,S$, $g$ to be the inclusion map $t\,\hookrightarrow\,S$, $Y\,=\,X_{t}$, $\pi^{\prime}\,=\,\pi|_{X_{t}}\,:\,X_{t}\,\longrightarrow\,T$ and $f$ to be the inclusion map $j\,:\,X_{t}\,\hookrightarrow\,X$, by Proposition 4.10, we have the following:

Corollary 4.11**.**

For every $t\,\in\,S$, there is a natural map

[TABLE]

which maps the $p$-th relative Chern class of $E$ to the $p$-th Chern class of the smooth vector bundle $E_{t}\,\longrightarrow\,X_{t}$, that is, $j^{*}(C_{p}^{S}(E))\,=\,c_{p}(E_{t})$.

The following topological proper base change theorem is given in [5, p. 202, Remark 4.17.1] and [4, p. 19, Corollary 2.25]:

Theorem 4.12** (Topological proper base change).**

Let $f\,:\,X\,\longrightarrow\,S$ be a proper continuous map of Hausdorff topological spaces. Suppose that $S$ is locally compact, and $\mathcal{F}$ is a sheaf of abelian groups on $X$. Then for all $t\,\in\,S$, we have a canonical isomorphism

[TABLE]

of abelian groups.

Note that $\mathcal{H}^{p}_{dR}(X/S)$ is a locally free $\mathcal{C}^{\infty}_{S}$-module, and from Theorem 4.12 we have a $\mathbb{C}$-vector space isomorphism

[TABLE]

for every $t\,\in\,S$.

Theorem 4.13**.**

Let $\pi\,:\,X\,\longrightarrow\,S$ be a surjective holomorphic proper submersion with connected fibers, such that $\pi^{-1}(t)\,=\,X_{t}$ is compact Kähler manifold for every $t\,\in\,S$. Let $\varpi\,:\,E\,\longrightarrow\,X$ be a holomorphic vector bundle. Suppose that $E$ admits a holomorphic $S$-connection. Then all the relative Chern classes $C_{p}^{S}(E)\,\in\,\mathcal{H}^{2p}_{dR}(X/S)(S)$ of $E$ over $S$ are zero.

Proof.

Let $D$ be a holomorphic $S$-connection on $E$. From Proposition 4.5 it follows that for every $t\,\in\,S$, there is a holomorphic connection $D_{t}$ in $E_{t}$. Since $X_{t}$ is a compact complex manifold of Kähler type, from Theorem 4 in [1, p. 192] it follows that all the Chern classes $c_{p}(E_{t})$ of $E_{t}$ are zero. From Corollary 4.11 and the isomorphism in (4.12) we have the following commutative diagram;

[TABLE]

Now,

[TABLE]

which implies that $C_{p}^{S}(E)_{t}\otimes 1\,=\,0$, for every $t\,\in\,S$, because $\eta$ is an isomorphism. Thus, we have $C_{p}^{S}(E)\,=\,0$. This completes the proof. ∎

4.G. A sufficient condition for holomorphic connection

Given a surjective holomorphic proper submersion $\pi\,:X\,\longrightarrow\,S$ with connected fibers and a holomorphic vector bundle $\varpi\,:\,E\,\longrightarrow\,X$, Proposition 4.5 gives a necessary condition for the existence of a holomorphic $S$-connection on $E$, namely the vector bundle $E_{t}\,=\,E|_{X_{t}}\,\longrightarrow\,X_{t}$ should admit a holomorphic connection for every $t\,\in\,S$.

If for every each $t\,\in\,S$ the vector bundle $E_{t}$ admits a holomorphic connection, it is natural to ask whether $E$ admits a holomorphic $S$-connection. We will present a sufficient condition for the existence of holomorphic $S$-connection on $E$.

Theorem 4.14**.**

Let $E\,\stackrel{{\scriptstyle\varpi}}{{\longrightarrow}}\,X$ be a holomorphic vector bundle. Suppose that for every $t\in S$, there is a holomorphic connection on the holomorphic vector bundle $\varpi|_{E_{t}}\,:\,E_{t}\,\longrightarrow\,X_{t}$, and

[TABLE]

Then, $E$ admits a holomorphic $S$-connection.

Proof.

Consider the relative Atiyah exact sequence in (4.5). Tensoring it by $\Omega^{1}_{X/S}$ produces the exact sequence

[TABLE]

Note that $\mathcal{O}_{X}\cdot\text{Id}\,\subset\,\text{End}(\mathcal{T}_{X/S})\,=\,\Omega^{1}_{X/S}\otimes\mathcal{T}_{X/S}$. Define

[TABLE]

where $q$ is the projection in (4.13). So we have the short exact sequence of sheaves

[TABLE]

on $X$, where $\Omega^{1}_{X/S}(\mathcal{A}t^{\prime}_{S}(E))$ is constructed above. Let

[TABLE]

be the homomorphism in the long exact sequence of cohomologies associated to the exact sequence in (4.14). The relative Atiyah class $\mathrm{at}_{S}(E)$ (see Corollary 4.4) coincides with $\Phi(1)\,\in\,{\rm H}^{1}(X,\,\Omega^{1}_{X/S}(\mathcal{E}\mathit{nd}_{\mathcal{O}_{X}}(E)))$. Therefore, from Corollary 4.4 it follows that $E$ admits a holomorphic $S$-connection if and only if

[TABLE]

To prove the vanishing statement in (4.16), first note that ${\rm H}^{1}(X,\,\Omega^{1}_{X/S}(\mathcal{E}\mathit{nd}_{\mathcal{O}_{X}}(E)))$ fits in the exact sequence

[TABLE]

where $\pi$ is the projection of $X$ to $S$.

The given condition that for every $t\in S$, there is a holomorphic connection on the holomorphic vector bundle $\varpi|_{E_{t}}\,:\,E_{t}\,\longrightarrow\,X_{t}$, implies that

[TABLE]

where $q_{1}$ is the homomorphism in (4.17). Therefore, from the exact sequence in (4.17) we conclude that

[TABLE]

Finally, the given condition that ${\rm H}^{1}(S,\,\pi_{*}(\Omega^{1}_{X/S}(\mathcal{E}\mathit{nd}_{\mathcal{O}_{X}}(E))))\,=\,0$. implies that $\Phi(1)\,=\,0$. Since (4.16) holds, the vector bundle $E$ admits a holomorphic $S$-connection. ∎

Take $\pi\,:\,X\,\longrightarrow\,S$ to be a surjective holomorphic proper submersion of relative dimension one, so $\pi^{-1}(t)$ is a compact connected Riemann surface, for every $t\in S$. Then, by the Atiyah-Weil criterion given in [1], [11] and [3] (Theorem 6.12), we have the following:

Corollary 4.15**.**

Let $\pi\,:\,X\,\longrightarrow\,S$ be a surjective holomorphic proper submersion such that $\pi^{-1}(t)\,=\,X_{t}$ is a compact connected Riemann surface for every $t\,\in\,S$. Let $\varpi\,:\,E\,\longrightarrow\,X$ be a holomorphic vector bundle. Suppose that for every $t\,\in\,S$, the degree of the indecomposable components of $E_{t}$ are zero and

[TABLE]

Then, $E$ admits a holomorphic $S$-connection.

Acknowledgement

We thank the referee for helpful advice. The first author is supported by a J. C. Bose Fellowship. The second author is very grateful to his Ph.D. advisor Prof. N. Raghavendra for introducing this topic and useful discussions. The second author would like to acknowledge the Department of Atomic Energy, Government of India for providing the research grant.