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First remark. I don't know a name for such a structure, but I can try to spell out the definition (that is maybe understood in your question).

Let $\pi: E \to M$ be a smooth fiber bundle with even-dimensional fibers $E_x=\pi^{-1}(x)$, $x \in M$, and let

$\mathcal{V}=\mathcal{V}_\pi=\mathrm{Ker}(\mathrm{d}\pi)\subseteq TE$

be the vertical distribution of the bundle, so that $\mathcal{V}|_{E_x}=TE_x$.

Suppose that there is a smooth section

$J\in \Gamma (E,\mathtt{End}(\mathcal{V}))$, where $\mathtt{End}(\mathcal{V})$ is the endomorphism bundle, such that $J^2=-\mathrm{id}_{\mathcal{V}}$. Now your $(E,J)$ is a bundle of almost-complex manifolds.

Define the Nijenhuis tensor $N_J\in \Gamma (E,\mathcal{V}^{\otimes 2}\otimes \mathcal{V})$ of $J$ in the usual way

$N_J(X,Y) = [X,Y]+J([JX,Y]+J[X,JY]) -[JX,JY]$

(I hope it's correct) for vertical vector fields $X,Y$ in terms of $J$ and Lie brackets. If $N_J$ happens to be zero, then you have a bundle $(E,J)$ of honest complex manifolds $E_x$, $x\in M$.

Now, to get a "$\mathcal{C}^{\infty}$ -bundle of algebraic varieties" you can put the condition that each fiber $E_x$ actually be an algebraic variety. Another way to do it, in case fibers are compact, could be to first endow $(\mathcal{V},J)$ with a Hermitian metric, and then impose the condition that its "associated 2-form"

$\omega\in\Gamma(E, \bigwedge^2\mathcal{V}^*)$

restricts to closed 2-forms $\omega|_x$ on the fibers, so you get a bundle of (compact) Kahler manifolds; then you require that each de Rham cohomology class $[\omega|_x]$ should come from integer cohomology, so you get a bundle of compact Hodge manifolds, hence, from Kodaira embedding theorem, of (smooth) projective varieties.

Second remark. The concept of a $\mathcal{C}^\infty$-varying family of complex manifolds has actually been studied, but in a more general context than bundles, namely that of foliations. The keyword that you should look for is trasversely holomorphic foliation, which means a differentiably-varying family of manifolds -the "leaves"- where each leaf is equipped with a complex structure. There are e.g. theorems about the topology and holonomy of the leaves when they are Kahler. A case of interest is that of one-dimensional compact leaves, cause in that case they're automatically Riemann surfaces (algebraic), and it has been studied even in the more general case in which the total space is just a topological space (the key word is holomorphic laminations, a structure that appears in holomorphic dynamics).

Edit: I should say that the case of a bundle is a special case of foliation, and the structure you address in your question (the one spelled out in my first remark) should be a particular case of transversely holomorphic foliation, i.e.:

1. it happens to be a bundle

2. and has algebraic leaves.

For some reasons, from the point of view of smooth foliations the particular case of a bundle is not considered very interesting: a bundle is somehow a "trivial" foliation. But I don't know in the transversally holomorphic case. As far as I remember, in the holomorphic case (i.e. when the total space of the foliation is a complex manifold, together with the leaves, and the latter vary holomorphically) in some cases a bundle structure is the only foliation that can be found on a given total space.

First remark. I don't know a name for such a structure (though I know a context in which it may appear, see remark 2), but I can try to spell out the definition (that is maybe understood in your question).

Let $\pi: E \to M$ be a smooth fiber bundle with even-dimensional fibers $E_x=\pi^{-1}(x)$, $x \in M$, and let

$\mathcal{V}=\mathcal{V}_\pi=\mathrm{Ker}(\mathrm{d}\pi)\subseteq TE$

be the vertical distribution of the bundle, so that $\mathcal{V}|_{E_x}=TE_x$.

Suppose that there is a smooth section

$J\in \Gamma (E,\mathtt{End}(\mathcal{V}))$, where $\mathtt{End}(\mathcal{V})$ is the endomorphism bundle, such that $J^2=-\mathrm{id}_{\mathcal{V}}$. Now your $(E,J)$ is a bundle of almost-complex manifolds.

Define the Nijenhuis tensor $N_J\in \Gamma (E,\mathcal{V}^{\otimes 2}\otimes \mathcal{V})$ of $J$ in the usual way

$N_J(X,Y) = [X,Y]+J([JX,Y]+J[X,JY]) -[JX,JY]$

(I hope it's correct) for vertical vector fields $X,Y$ in terms of $J$ and Lie brackets. If $N_J$ happens to be zero, then you have a bundle $(E,J)$ of honest complex manifolds $E_x$, $x\in M$.

Now, to get a "$\mathcal{C}^{\infty}$ -bundle of algebraic varieties" you can put the condition that each fiber $E_x$ actually be an algebraic variety. Another way to do it, in case fibers are compact, could be to first endow $(\mathcal{V},J)$ with a Hermitian metric, and then impose the condition that its "associated 2-form"

$\omega\in\Gamma(E, \bigwedge^2\mathcal{V}^*)$

restricts to closed 2-forms $\omega|_x$ on the fibers, so you get a bundle of (compact) Kahler manifolds; then you require that each de Rham cohomology class $[\omega|_x]$ should come from integer cohomology, so you get a bundle of compact Hodge manifolds, hence, from Kodaira embedding theorem, of (smooth) projective varieties.

Second remark. The concept of a $\mathcal{C}^\infty$-varying family of complex manifolds has actually been studied, but in a more general context than bundles, namely that of foliations. The keyword that you should look for is trasversely holomorphic foliation, which means a differentiably-varying family of manifolds -the "leaves"- where each leaf is equipped with a complex structure. There are e.g. theorems about the topology and holonomy of the leaves when they are Kahler. A case of interest is that of one-dimensional compact leaves, cause in that case they're automatically Riemann surfaces (algebraic), and it has been studied even in the more general case in which the total space is just a topological space (the key word is holomorphic laminations, a structure that appears in holomorphic dynamics).

Edit: I should say that the case of a bundle is a special case of foliation, and the structure you address in your question (the one spelled out in my first remark) should be a particular case of transversely holomorphic foliation, i.e.:

1. it happens to be a bundle

2. and has algebraic leaves.

For some reasons, from the point of view of smooth foliations the particular case of a bundle is not considered very interesting: a bundle is somehow a "trivial" foliation. But I don't know in the transversally holomorphic case. As far as I remember, in the holomorphic case (i.e. when the total space of the foliation is a complex manifold, together with the leaves, and the latter vary holomorphically) in some cases a bundle structure is the only foliation that can be found on a given total space.

First remark. I don't know a name for such a structure, but I can try to spell out the definition (that is maybe understood in your question).

Let $\pi: E \to M$ be a smooth fiber bundle with even-dimensional fibers $E_x=\pi^{-1}(x)$, $x \in M$, and let

$\mathcal{V}=\mathcal{V}_\pi=\mathrm{Ker}(\mathrm{d}\pi)\subseteq TE$

be the vertical distribution of the bundle, so that $\mathcal{V}|_{E_x}=TE_x$.

Suppose that there is a smooth section

$J\in \Gamma (E,\mathtt{End}(\mathcal{V}))$, where $\mathtt{End}(\mathcal{V})$ is the endomorphism bundle, such that $J^2=-\mathrm{id}_{\mathcal{V}}$. Now your $(E,J)$ is a bundle of almost-complex manifolds.

Define the Nijenhuis tensor $N_J\in \Gamma (E,\mathcal{V}^{\otimes 2}\otimes \mathcal{V})$ of $J$ in the usual way

$N_J(X,Y) = [X,Y]+J([JX,Y]+J[X,JY]) -[JX,JY]$

(I hope it's correct) for vertical vector fields $X,Y$ in terms of $J$ and Lie brackets. If $N_J$ happens to be zero, then you have a bundle $(E,J)$ of honest complex manifolds $E_x$, $x\in M$.

Now, to get a "$\mathcal{C}^{\infty}$ -bundle of algebraic varieties" you can put the condition that each fiber $E_x$ actually be an algebraic variety. Another way to do it, in case fibers are compact, could be to first endow $(\mathcal{V},J)$ with a Hermitian metric, and then impose the condition that its "associated 2-form"

$\omega\in\Gamma(E, \bigwedge^2\mathcal{V}^*)$

restricts to closed 2-forms $\omega|_x$ on the fibers, so you get a bundle of (compact) Kahler manifolds; then you require that each de Rham cohomology class $[\omega|_x]$ should come from integer cohomology, so you get a bundle of compact Hodge manifolds, hence, from Kodaira embedding theorem, of (smooth) projective varieties.

Second remark. The concept of a $\mathcal{C}^\infty$-varying family of complex manifolds has actually been studied, but in a more general context than bundles, namely that of foliations (which has a relatively vast literature). The keyword that you should look for is trasversely holomorphic foliation, which means a differentiably-varying family of manifolds -the "leaves"- where each leaf is equipped with a complex structure. There are e.g. theorems about the topology and holonomy of the leaves when they are Kahler. A case of interest is that of one-dimensional compact leaves, cause in that case they're automatically Riemann surfaces (algebraic), and it has been studied even in the more general case in which the total space is just a topological space (the key word is holomorphic laminations, a structure that appears in holomorphic dynamics).

Edit: I should say that the case of a bundle is a special case of foliation, and the structure you address in your question (the one spelled out in my first remark) should be a particular case of transversely holomorphic foliation, i.e.:

1. it happens to be a bundle

2. and has algebraic leaves.

For some reasons, from the point of view of smooth foliations the particular case of a bundle is not considered very interesting: a bundle is somehow a "trivial" foliation. But I don't know in the transversally holomorphic case. As far as I remember, in the holomorphic case (i.e. when the total space of the foliation is a complex manifold, together with the leaves, and the latter vary holomorphically) in some cases a bundle structure is the only foliation that can be found on a given total space.

First remark. I don't know a name for such a structure, but I can try to spell out the definition (that is maybe understood in your question).

Let $\pi: E \to M$ be a smooth fiber bundle with even-dimensional fibers $E_x=\pi^{-1}(x)$, $x \in M$, and let

$\mathcal{V}=\mathcal{V}_\pi=\mathrm{Ker}(\mathrm{d}\pi)\subseteq TE$

be the vertical distribution of the bundle, so that $\mathcal{V}|_{E_x}=TE_x$.

Suppose that there is a smooth section

$J\in \Gamma (E,\mathtt{End}(\mathcal{V}))$, where $\mathtt{End}(\mathcal{V})$ is the endomorphism bundle, such that $J^2=-\mathrm{id}_{\mathcal{V}}$. Now your $(E,J)$ is a bundle of almost-complex manifolds.

Define the Nijenhuis tensor $N_J\in \Gamma (E,\mathcal{V}^{\otimes 2}\otimes \mathcal{V})$ of $J$ in the usual way

$N_J(X,Y) = [X,Y]+J([JX,Y]+J[X,JY]) -[JX,JY]$

(I hope it's correct) for vertical vector fields $X,Y$ in terms of $J$ and Lie brackets. If $N_J$ happens to be zero, then you have a bundle $(E,J)$ of honest complex manifolds $E_x$, $x\in M$.

Now, to get a "$\mathcal{C}^{\infty}$ -bundle of algebraic varieties" you can put the condition that each fiber $E_x$ actually be an algebraic variety. Another way to do it, in case fibers are compact, could be to first endow $(\mathcal{V},J)$ with a Hermitian metric, and then impose the condition that its "associated 2-form"

$\omega\in\Gamma(E, \bigwedge^2\mathcal{V}^*)$

restricts to closed 2-forms $\omega|_x$ on the fibers, so you get a bundle of (compact) Kahler manifolds; then you require that each de Rham cohomology class $[\omega|_x]$ should come from integer cohomology, so you get a bundle of compact Hodge manifolds, hence, from Kodaira embedding theorem, of (smooth) projective varieties.

Second remark. The concept of a $\mathcal{C}^\infty$-varying family of complex manifolds has actually been studied, but in a more general context than bundles, namely that of foliations. The keyword that you should look for is trasversely holomorphic foliation, which means a differentiably-varying family of manifolds -the "leaves"- where each leaf is equipped with a complex structure. There are e.g. theorems about the topology and holonomy of the leaves when they are Kahler. A case of interest is that of one-dimensional compact leaves, cause in that case they're automatically Riemann surfaces (algebraic), and it has been studied even in the more general case in which the total space is just a topological space (the key word is holomorphic laminations, a structure that appears in holomorphic dynamics).

Edit: I should say that the case of a bundle is a special case of foliation, and the structure you address in your question (the one spelled out in my first remark) should be a particular case of transversely holomorphic foliation, i.e.:

1. it happens to be a bundle

2. and has algebraic leaves.

For some reasons, from the point of view of smooth foliations the particular case of a bundle is not considered very interesting: a bundle is somehow a "trivial" foliation. But I don't know in the transversally holomorphic case. As far as I remember, in the holomorphic case (i.e. when the total space of the foliation is a complex manifold, together with the leaves, and the latter vary holomorphically) in some cases a bundle structure is the only foliation that can be found on a given total space.

First remark. I don't know a name for such a structure, but I can try to spell out the definition (that is maybe understood in your question).

Let $\pi: E \to M$ be a smooth fiber bundle with even-dimensional fibers $E_x=\pi^{-1}(x)$, $x \in M$, and let

$\mathcal{V}=\mathcal{V}_\pi=\mathrm{Ker}(\mathrm{d}\pi)\subseteq TE$

be the vertical distribution of the bundle, so that $\mathcal{V}|_{E_x}=TE_x$.

Suppose that there is a smooth section

$J\in \Gamma (E,\mathtt{End}(\mathcal{V}))$, where $\mathtt{End}(\mathcal{V})$ is the endomorphism bundle, such that $J^2=-\mathrm{id}_{\mathcal{V}}$. Now your $(E,J)$ is a bundle of almost-complex manifolds.

Define the Nijenhuis tensor $N_J\in \Gamma (E,\mathcal{V}^{\otimes 2}\otimes \mathcal{V})$ of $J$ in the usual way

$N_J(X,Y) = [X,Y]+J([JX,Y]+J[X,JY]) -[JX,JY]$

(I hope it's correct) for vertical vector fields $X,Y$ in terms of $J$ and Lie brackets. If $N_J$ happens to be zero, then you have a bundle $(E,J)$ of honest complex manifolds $E_x$, $x\in M$.

Now, to get a "$\mathcal{C}^{\infty}$ -bundle of algebraic varieties" you can put the condition that each fiber $E_x$ actually be an algebraic variety. Another way to do it, in case fibers are compact, could be to first endow $(\mathcal{V},J)$ with a Hermitian metric, and then impose the condition that its "associated 2-form"

$\omega\in\Gamma(E, \bigwedge^2\mathcal{V}^*)$

restricts to closed 2-forms $\omega|_x$ on the fibers, so you get a bundle of (compact) Kahler manifolds; then you require that each de Rham cohomology class $[\omega|_x]$ should come from integer cohomology, so you get a bundle of compact Hodge manifolds, hence, from Kodaira embedding theorem, of (smooth) projective varieties.

Second remark. The concept of a $\mathcal{C}^\infty$-varying family of complex manifolds has actually been studied, but in a more general context than bundles, namely that of foliations (which has a relatively vast literature). The keyword that you should look for is trasversely holomorphic foliation, which means a differentiably-varying family of manifolds -the "leaves"- where each leaf is equipped with a complex structure. There are e.g. theorems about the topology and holonomy of the leaves when they are Kahler. A case of interest is that of one-dimensional compact leaves, cause in that case they're automatically Riemann surfaces (algebraic), and it has been studied even in the more general case in which the total space is just a topological space (the key word is holomorphic laminations, a structure that appears in holomorphic dynamics).

First remark. I don't know a name for such a structure, but I can try to spell out the definition (that is maybe understood in your question).

Let $\pi: E \to M$ be a smooth fiber bundle with even-dimensional fibers $E_x=\pi^{-1}(x)$, $x \in M$, and let

$\mathcal{V}=\mathcal{V}_\pi=\mathrm{Ker}(\mathrm{d}\pi)\subseteq TE$

be the vertical distribution of the bundle, so that $\mathcal{V}|_{E_x}=TE_x$.

Suppose that there is a smooth section

$J\in \Gamma (E,\mathtt{End}(\mathcal{V}))$, where $\mathtt{End}(\mathcal{V})$ is the endomorphism bundle, such that $J^2=-\mathrm{id}_{\mathcal{V}}$. Now your $(E,J)$ is a bundle of almost-complex manifolds.

Define the Nijenhuis tensor $N_J\in \Gamma (E,\mathcal{V}^{\otimes 2}\otimes \mathcal{V})$ of $J$ in the usual way

$N_J(X,Y) = [X,Y]+J([JX,Y]+J[X,JY]) -[JX,JY]$

(I hope it's correct) for vertical vector fields $X,Y$ in terms of $J$ and Lie brackets. If $N_J$ happens to be zero, then you have a bundle $(E,J)$ of honest complex manifolds $E_x$, $x\in M$.

Now, to get a "$\mathcal{C}^{\infty}$ -bundle of algebraic varieties" you can put the condition that each fiber $E_x$ actually be an algebraic variety. Another way to do it, in case fibers are compact, could be to first endow $(\mathcal{V},J)$ with a Hermitian metric, and then impose the condition that its "associated 2-form"

$\omega\in\Gamma(E, \bigwedge^2\mathcal{V}^*)$

restricts to closed 2-forms $\omega|_x$ on the fibers, so you get a bundle of (compact) Kahler manifolds; then you require that each de Rham cohomology class $[\omega|_x]$ should come from integer cohomology, so you get a bundle of compact Hodge manifolds, hence, from Kodaira embedding theorem, of (smooth) projective varieties.

Second remark. The concept of a $\mathcal{C}^\infty$-varying family of complex manifolds has actually been studied, but in a more general context than bundles, namely that of foliations (which has a relatively vast literature). The keyword that you should look for is trasversely holomorphic foliation, which means a differentiably-varying family of manifolds -the "leaves"- where each leaf is equipped with a complex structure. There are e.g. theorems about the topology and holonomy of the leaves when they are Kahler. A case of interest is that of one-dimensional compact leaves, cause in that case they're automatically Riemann surfaces (algebraic), and it has been studied even in the more general case in which the total space is just a topological space (the key word is holomorphic laminations, a structure that appears in holomorphic dynamics).

Edit: I should say that the case of a bundle is a special case of foliation, and the structure you address in your question (the one spelled out in my first remark) should be a particular case of transversely holomorphic foliation, i.e.:

1. it happens to be a bundle

2. and has algebraic leaves.

For some reasons, from the point of view of smooth foliations the particular case of a bundle is not considered very interesting: a bundle is somehow a "trivial" foliation. But I don't know in the transversally holomorphic case. As far as I remember, in the holomorphic case (i.e. when the total space of the foliation is a complex manifold, together with the leaves, and the latter vary holomorphically) in some cases a bundle structure is the only foliation that can be found on a given total space.