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Let $\pi:X \to D^2$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and $D^2 \subset \mathbb{C}$ is a complex disk. Let $X_0:= \pi^{-1}(0)$ and let $f:X_0 \to \mathbb{C}$ be a holomorphic map.

I am looking for obstructions to extend $f$ to a smooth family of holomorphic maps $f_t:X_t \to \mathbb{C}$ with $t \in B$. That is, to find a complex-valued smooth function $F:X \to \mathbb{C}$ such that $F|_{X_t}$ is holomorphic for all $t \in D^2$.

I have seen some works on the literature about extending $f$ to a holomorphic map on all $X$ (or equivalently, that ask for holomorphicity on $t$) which is a lot stronger and in general not possible at all.

A good answer could also be a reference that deals with this kind of framework even if not exactly the same.

Another question is if the following is the right approach:

Take $T_R B$ to be the sheaf of germs of real analytic (or smooth) vector fields on $B$ and take the short exact sequence $$0 \to T_R X_0 \to T_RX_{|_{X_0}} \to \pi^*(T_R B)_{|_{X_0}} \to 0$$ then take the corresponding cohomology exact sequence and look for the connecting homomorphism $$\rho:H^0(X, \pi^*(T_R B)_{|_{X_0}}) \to H^1(X, T_RX).$$ Now if for $f \in H^0(X_0, \mathcal{O}_{X_0})$, one can find a vector $v \in H^0(X, \pi^*(T_R B)_{|_{X_0}})$ that doesn't kill $f$ then the extension that I am looking for exists. Is this true?

I am an amateur in deformation theory so sorry in advance if this is just all nonsense.

Let $\pi:X \to D^2$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and $D^2 \subset \mathbb{C}$ is a complex disk. Let $X_0:= \pi^{-1}(0)$ and let $f:X_0 \to \mathbb{C}$ be a holomorphic map.

I am looking for obstructions to extend $f$ to a smooth family of holomorphic maps $f_t:X_t \to \mathbb{C}$ with $t \in D^2$. That is, to find a complex-valued smooth function $F:X \to \mathbb{C}$ such that $F|_{X_t}$ is holomorphic for all $t \in D^2$.

I have seen some works on the literature about extending $f$ to a holomorphic map on all $X$ (or equivalently, that ask for holomorphicity on $t$) which is a lot stronger and in general not possible at all.

A good answer could also be a reference that deals with this kind of framework even if not exactly the same.

Another question is if the following is the right approach:

Take $T_R B$ to be the sheaf of germs of real analytic (or smooth) vector fields on $B$ and take the short exact sequence $$0 \to T_R X_0 \to T_RX_{|_{X_0}} \to \pi^*(T_R D^2)_{|_{X_0}} \to 0$$ then take the corresponding cohomology exact sequence and look for the connecting homomorphism $$\rho:H^0(X, \pi^*(T_R D^2)_{|_{X_0}}) \to H^1(X, T_RX).$$ Now if for $f \in H^0(X_0, \mathcal{O}_{X_0})$, one can find a vector $v \in H^0(X, \pi^*(T_R D^2)_{|_{X_0}})$ that doesn't kill $f$ then the extension that I am looking for exists. Is this true?

I am an amateur in deformation theory so sorry in advance if this is just all nonsense.

Let $\pi:X \to B$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and you can think of $B$ as a small disk $D^2 \subset \mathbb{C}$. Let $X_0:= \pi^{-1}(0)$ and let $f:X_0 \to \mathbb{C}$ be a holomorphic map.

I am looking for obstructions to extend $f$ to a smooth family of holomorphic maps $f_t:X_t \to \mathbb{C}$ with $t \in B$. That is, the family should be smooth on $t$. What if I only ask to extend it over a real arc $(- \epsilon, \epsilon) \hookrightarrow B$.

I have seen some works on the literature about extending $f$ to a holomorphic map on all $X$ (or equivalently, that ask for holomorphicity on $t$) which is a lot stronger and in general not possible at all.

A good answer could also be a reference that deals with this kind of framework even if not exactly the same.

Is there a real analytic or smooth version of the Kodaira-Spencer map? That is, a map that measures the change of the holomorphic structure in the direction of a real vector?. Naively one would do something like:

Take $T_R B$ to be the sheaf of germs of real analytic (or smooth) vector fields on $B$ and take the short exact sequence $$0 \to T_R X_0 \to T_RX_{|_{X_0}} \to \pi^*(T_R B)_{|_{X_0}} \to 0$$ then take the corresponding cohomology exact sequence and look for the connecting homomorphism $$\rho:H^0(X, \pi^*(T_R B)_{|_{X_0}}) \to H^1(X, T_RX)$$ and see if for a fixed element $f \in H^0(X_0, \mathcal{O}_{X_0})$ one can find a vector $v \in H^0(X, \pi^*(T_R B)_{|_{X_0}})$ that doesn't kill $f$.

I am an amateur in deformation theory so sorry in advance if this is just all nonsense.

Let $\pi:X \to D^2$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and $D^2 \subset \mathbb{C}$ is a complex disk. Let $X_0:= \pi^{-1}(0)$ and let $f:X_0 \to \mathbb{C}$ be a holomorphic map.

I am looking for obstructions to extend $f$ to a smooth family of holomorphic maps $f_t:X_t \to \mathbb{C}$ with $t \in B$. That is, to find a complex-valued smooth function $F:X \to \mathbb{C}$ such that $F|_{X_t}$ is holomorphic for all $t \in D^2$.

I have seen some works on the literature about extending $f$ to a holomorphic map on all $X$ (or equivalently, that ask for holomorphicity on $t$) which is a lot stronger and in general not possible at all.

A good answer could also be a reference that deals with this kind of framework even if not exactly the same.

Another question is if the following is the right approach:

Take $T_R B$ to be the sheaf of germs of real analytic (or smooth) vector fields on $B$ and take the short exact sequence $$0 \to T_R X_0 \to T_RX_{|_{X_0}} \to \pi^*(T_R B)_{|_{X_0}} \to 0$$ then take the corresponding cohomology exact sequence and look for the connecting homomorphism $$\rho:H^0(X, \pi^*(T_R B)_{|_{X_0}}) \to H^1(X, T_RX).$$ Now if for $f \in H^0(X_0, \mathcal{O}_{X_0})$, one can find a vector $v \in H^0(X, \pi^*(T_R B)_{|_{X_0}})$ that doesn't kill $f$ then the extension that I am looking for exists. Is this true?

I am an amateur in deformation theory so sorry in advance if this is just all nonsense.

Let $\pi:X \to B$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and you can think of $B$ as a small disk $D^2 \subset \mathbb{C}$. Let $X_0:= \pi^{-1}(0)$ and let $f:X_0 \to \mathbb{C}$ be a holomorphic map.

I am looking for obstructions to extend $f$ to a smooth family of holomorphic maps $f_t:X_t \to \mathbb{C}$ with $t \in B$. That is, the family should be smooth on $t$. What if I only ask to extend it over a real arc $(- \epsilon, \epsilon) \hookrightarrow B$.

I have seen some works on the literature about extending $f$ to a holomorphic map on all $X$ (or equivalently, that ask for holomorphicity on $t$) which is a lot stronger and in general not possible at all.

A good answer could also be a reference that deals with this kind of framework even if not exactly the same.

Is there a real analytic or smooth version of the Kodaira-Spencer map? That is, a map that measures the change of the holomorphic structure in the direction of a real vector?. Naively one would do something like:

Take $T_R B$ to be the sheaf of germs of real analytic (or smooth) vector fields on $B$ and take the short exact sequence $$0 \to T_R X_0 \to T_RX_{|_{X_0}} \to \pi^*(T_R B)_{|_{X_0}} \to 0$$ then take the corresponding cohomology exact sequence and look for the connecting homomorphism $$\rho:H^0(X, \pi^*(T_R B)_{|_{X_0}}) \to H^1(X, T_RX)$$ and see if for a fixed element $f \in H^0(X, \mathcal{O}_X)$ one can find a vector $v \in H^0(X, \pi^*(T_R B)_{|_{X_0}})$ that doesn't kill $f$.

I am an amateur in deformation theory so sorry in advance if this is just all nonsense.

Let $\pi:X \to B$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and you can think of $B$ as a small disk $D^2 \subset \mathbb{C}$. Let $X_0:= \pi^{-1}(0)$ and let $f:X_0 \to \mathbb{C}$ be a holomorphic map.

I am looking for obstructions to extend $f$ to a smooth family of holomorphic maps $f_t:X_t \to \mathbb{C}$ with $t \in B$. That is, the family should be smooth on $t$. What if I only ask to extend it over a real arc $(- \epsilon, \epsilon) \hookrightarrow B$.

I have seen some works on the literature about extending $f$ to a holomorphic map on all $X$ (or equivalently, that ask for holomorphicity on $t$) which is a lot stronger and in general not possible at all.

A good answer could also be a reference that deals with this kind of framework even if not exactly the same.

Is there a real analytic or smooth version of the Kodaira-Spencer map? That is, a map that measures the change of the holomorphic structure in the direction of a real vector?. Naively one would do something like:

Take $T_R B$ to be the sheaf of germs of real analytic (or smooth) vector fields on $B$ and take the short exact sequence $$0 \to T_R X_0 \to T_RX_{|_{X_0}} \to \pi^*(T_R B)_{|_{X_0}} \to 0$$ then take the corresponding cohomology exact sequence and look for the connecting homomorphism $$\rho:H^0(X, \pi^*(T_R B)_{|_{X_0}}) \to H^1(X, T_RX)$$ and see if for a fixed element $f \in H^0(X_0, \mathcal{O}_{X_0})$ one can find a vector $v \in H^0(X, \pi^*(T_R B)_{|_{X_0}})$ that doesn't kill $f$.

I am an amateur in deformation theory so sorry in advance if this is just all nonsense.