I know that smooth Fano varieties over $\mathbb{C}$ may be classified into a finite number of families in each dimension (1 in dimension 1, 10 in dimension 2, 105 in dimension 3 ...). 


I am interested in cases where the Family is non-trivial (i.e. variety is not rigid).


Suppose that we have such a family, I assume that each of the members are anti-canonically polarized, hence each of the underlying varieties $X$ inherits a Kähler structure with Kähler form $\omega\_{X}$, such that $[\omega\_{X}] = c\_{1}(TX)$.


I have two (probably very naïve) questions:


**Question 1**: Are the underlying Kähler forms $\omega\_{X}$ abstractly symplectomorphic?


**Question 2**: Can/are these symplectic forms be induced from a symplectic form on (say the smooth locus of) the family?


Examples would also be appreciated.



symplectic form on an algebraic family
Let $(X\_\alpha, \mathcal{L}\_\alpha)\_{\alpha\in A}=(M, L, J\_\alpha, \overline{\partial}\_\alpha)\_{\alpha\in A}$ be a family of polarized varieties, with $A$ the complex manifold parametrizing the family (assumed to be connected). Here is $M$ the common underlying smooth manifold of the varieties $X\_\alpha$, $J\_\alpha$ are the integrable almost-complex structures corresponding to the different varieties $X\_\alpha$, $L$ is the common underlying smooth complex line bundle of the $J\_\alpha$-holomorphic line bundles $\mathcal{L}\_\alpha$, and $\overline{\partial}\_{\alpha}$ are the integrable delbar-operators on $L$ corresponding to the different holomorphic line bundles $\mathcal{L}\_\alpha$.


**Lemma:** For each $\alpha\_0\in A$, there is a neighbourhood $U\subseteq A$ of $\alpha\_0$, and a smoothly varying family $(\omega\_\alpha)\_{\alpha\in U}$ of symplectic forms in $c\_1(L)$, such that for each $\alpha$, the form $\omega\_\alpha$ is $J\_\alpha$-Kähler.


*Sketch proof:* For each $\alpha$ we have a map
$$F\_\alpha: \{\text{hermitian metrics on $L$}\}\to \{\text{real 2-forms in $c\_1(L)$}\}$$ given by sending a hermitian metric on $L$ to $-i$ times the curvature of its $(J\_\alpha,\overline{\partial}\_\alpha)$-Chern connection. The maps $F\_\alpha$ vary smoothly with $\alpha$. If $\omega\_0$ is a $J\_{\alpha\_0}$-Kähler form, there is some hermitian metric $h$ on $L$ such that $F\_{\alpha\_0}(h)=\omega\_0$. For $\alpha$ sufficiently close to $\alpha\_0$, the $J\_\alpha-(1,1)$-form $\omega\_\alpha:=F\_\alpha(h)$ is also nondegenerate, and so is $J\_\alpha$-Kähler. $\square$


It follows that the space of 2-forms in $c\_1(L)\in H^2(M,\mathbb{Z})$ which are Kähler for some $J\_\alpha$ is connected. By the Moser trick, all these 2-forms are isotopic to each other. This answers your **first question**.


It also follows that one can smoothly select elements $(\omega\_\alpha)\_{\alpha\in a}$ of $c\_1(L)\in H^2(M,\mathbb{Z})$, which are respectively $J\_\alpha$-Kähler.


**Edit:** For the second question, I had previously proposed the pullback $\pi\_1^\*\omega\_\alpha+\pi\_2^\*\widetilde\omega$ as a potential symplectic form on $M\times A$, where $\widetilde\omega$ is some symplectic form on $A$. But, now that I think about it, this pullback is not necessarily closed.


On the other hand, by the arguments for the first question, we can select smoothly varying diffeomorphisms $\psi:A\to\operatorname{Diff}(M)$ such that for all $\alpha$, $\psi\_\alpha^\*\omega\_\alpha=\omega\_{\alpha\_0}$. (Maybe this requires simple connectedness of $A$?) Write $\Psi:M\times A\to M\times A$ for the induced diffeomorphism $\Psi(x,\alpha)=(\psi\_\alpha(x),\alpha)$. Then $(\Psi^{-1})^\*(\pi\_1^\*\omega\_{\alpha\_0}+\pi\_2^\*\widetilde\omega)$ is a symplectic form which restricts on $\alpha$-slices to $\omega\_\alpha$, which should answer the **second question**.


I had also mentioned some general references on moduli spaces of Fano varieties (because of the issue of finding a symplectic form $\widetilde\omega$ on $A$), which I will preserve: [1](https://arxiv.org/abs/1411.0761), [2](https://arxiv.org/abs/1210.0858).


