The following appears in Claire Voisin's Hodge Theory an Complex Algebraic Geometry I. Consider a family $\phi: \mathcal X \to B$ of complex manifolds, and assume that $X_0, 0 \in B$ is a Kähler manifold (I don't know if we need the Kähler hypothesis for the following).
Propostion 9.22 (See Kodaira 1986) Let $\Delta = (\Delta_b)_{b \in B}$ be a relative differential operator acting on a vector bundle $F \to \mathcal X$, such that each induced operator $\Delta_b$ on $F_b$ is elliptic of fixed order. Then if $\dim \operatorname{Ker}(\Delta_b)$ is independent of $b$, the subspace $\operatorname{Ker}(\Delta_b) \subset C^\infty(F_b)$ varies in a $C^\infty$ way with b.
This means that up to shrinking $B$ near $b$, there exist $C^\infty$ sections $(\eta_b^i)_{b \in B}$ of $F$ over $\mathcal X$ whose restrictions to $X_b$ for fixed $b$ form a basis of $\operatorname{Ker} \Delta_b$.
I don't understand what is going on here. Unfortunately, Voisin only refers to Kodaira, and currently I don't have access to his book.
- What is a relative differential operator?
- In the proposition, $\operatorname{Ker}(\Delta_b)$ is a subspace of $C^\infty(F_b)$, but below it is a subspace of $F_b$. The latter seems to make a bit more sense, since $C^\infty(F_b)$ is not a manifold, so I don't know what it means to "vary differentially" there. Though that might be pure ignorance on my part.
Question: "I don't understand what is going on here. Unfortunately, Voisin only refers to Kodaira, and currently I don't have access to his book. What is a relative differential operator?"
Answer: PS: There appears to be some "spelling mistakes" in your "Proposition 9.22". What does the following sentence mean?
"Then if dimKer(Δb)⊂C∞(Fb) varies in a C∞ way with b."
Let $K$ be the complex numbers and assume your manifolds $\pi:X\rightarrow B$ are complex projective manifolds (hence they are algebraic). It follows there are open affine subschemes $U:=Spec(B) \subseteq X, V:=Spec(A) \subseteq B$ with $\pi_V: U \rightarrow V$ the induced map You may cover $\pi$ with such open sets. Let $\phi: A \rightarrow B$.
Locally a "relative differential operator of order $l$" is an element $\Delta$ of the group
$$\Delta \in Diff^l_A(B) \subseteq End_A(B)$$
where $Diff^1_A(B):= B \oplus Der_A(B)$. Inductively an element $\partial \in Diff^l_A(B)$ iff
$$[\partial, \phi_a]:= \partial \circ \phi_a -\phi_a \circ \partial \in Diff^{l-1}_A(B).$$
Example: If $B:=K[x]$ and $A:=K$ it follows
$Diff^l_A(B)$ is the $l$'th piece of the canonical filtration of the Weyl algebra:
$$Diff^l_A(B) \cong K[x]\{1, \partial, \partial^2,..,\partial^l\}$$
where $\partial:=\partial/\partial_x$ is partial derivative wrto the $x$-variable.
Example: Hence if $\Delta \in Diff^l_A(B)$ it follows
$$ker(\Delta) \subseteq B$$
is a left $A$-module and this module may be a finite rank projective $A$-module in some cases. In this case there is an associated finite rank algebraic vector bundle $\mathbb{V}(\Delta):=Spec(Sym_A^*(ker(\Delta)^*))$
$$ \pi: \mathbb{V}(\Delta) \rightarrow Spec(A)$$
If $\mathfrak{p} \subseteq A$ is any prime ideal you get a canonical map
$$S^{-1}A \rightarrow T^{-1}B$$
with $S:=A-\mathfrak{p}$ and $T:=\phi(S)\subseteq B$. You get canonically a map
$$T^{-1}\Delta \in Diff^l_{S^{-1}A}(T^{-1}B)$$
and an induced map
$$\Delta_{\mathfrak{p}} \in Diff^l_{\kappa(\mathfrak{p}}(\kappa(\mathfrak{p})\otimes_A B).$$
You get an induced differential operator on the fiber (in your notation $\Delta_b$):
$$\pi_U^{-1}(\mathfrak{p}):=Spec(\kappa(\mathfrak{p})\otimes_A B).$$
Hence
$$ker(\Delta_{\mathfrak{p}}) \subseteq \kappa(\mathfrak{p})\otimes_A B:= H^0(\pi_U^{-1}(\mathfrak{p}), \mathcal{O}_{\pi_U^{-1}(\mathfrak{p})}).$$
This operator is "algebraic" but the complex manifolds $X,B$ have the structure as real smooth manifolds and you may study differential operators that are "holomorphic" and "smooth".
Example: If $X=\mathbb{C}^n$ with holomorphic coordinates $z_j$ and $z_j:=x_j+iy_j$ a holomorphic differential operator is an operator
$$ \partial:= \sum f_{j_1,..,j_n}\partial_1^{j_1}\cdots \partial_n^{j_n}$$
with $f_{j_1,..,j_n}$ a holomorphic function and $\partial_j:=\partial/\partial_{z_j}$.
If $A:=\mathbb{C}[z_1,..,z_m]$ with $m<n$, it follows a relative differential operator (relative to $A$) is an operator with $l_1=\cdots =l_m=0$.
A "smooth differential operator" is an operator
$$ \partial:= \sum g_{a_1,..,a_n, b_1,..,b_n}\partial_1^{a_1}\cdots \partial_n^{a_n}\eta_1^{b_1}\cdots \eta_n^{b_n}$$
with $g$ a smooth function and $\partial_j:=\partial/\partial_{x_j}, \eta_j:=\partial/\partial_{y_j}$. Here you view $X\cong \mathbb{R}^{2n}$ as a real smooth manifold of dimension $2n$.
If the differential operator is "smooth", its kernel will be a subspace of the space of smooth functions on the fiber $\pi^{-1}_U(b)$ at a point $b\in B$.
In your language: If you view $F_b$ as a real smooth manifold with structure sheaf $\mathcal{O}^{\infty}_{F_b}$ it follows
$$ker(\Delta_b) \subseteq H^0(F_b, \mathcal{O}^{\infty}_{F_b}).$$
Note: A differential operator acts on a ring of functions $R$, hence its kernel will be a sub-vector space of the ring $R$.
"In the proposition, Ker(Δb) is a subspace of C∞(Fb), but below it is a subspace of Fb. The latter seems to make a bit more sense, since C∞(Fb) is not a manifold, so I don't know what it means to "vary differentially" there. Though that might be pure ignorance on my part."
Answer: In the case when $F_b$ is a manifold you may consider the ring of realvalued smooth functions $C^{\infty}(F_b)$ on $F_b$, and the kernel $ker(\Delta_b) \subseteq C^{\infty}(F_b)$ is a real sub-vector space. The Serre-Swan theorem says there is an equivalence of categories between the category of finite rank real vector bundles on $F_b$ and the category of finite rank projective $C^{\infty}(F_b)$-modules. But in this case your kernel $ker(\Delta_b)$ is not a $C^{\infty}(F_b)$-module. If it was, it would correspond to a finite rank vector bundle on $F_b$.
If $b\in U$ is a local trivialization of $\pi: F \rightarrow \mathcal{X}$ with
$$\pi^{-1}(U) \cong U \times F_b$$
it follows $C^{\infty}(\pi^{-1}(U))$ is a $C^{\infty}(U)$-module. The kernel $ker(\Delta_U)\subseteq C^{\infty}(\pi^{-1}(U))$ will be a $C^{\infty}(U)$-module. If $ker(\Delta_U)$ is a finite rank locally trivial module it corresponds by the Serre-Swan theorem to a finite rank vector bundle. The sections $\eta^i_b$ you speak of may be sections forming a local basis for the projective module $ker(\Delta_U)$.