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.

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    $\begingroup$ Let $\pi:X \to S$ be a family of complex compact manifolds. Then a (strong) relative K\"ahler form $\omega_{X/S}$ is a relative $(1,1)$-form , which is given by $\omega_{X/S}|_{U_j}=\sqrt[]{-1}\partial_s\bar{\partial_s}u_j$ with respect to a suitable open covering $\{U_j\}$ of $X$. The functions $u_j$ have to be strictly plurisubharmonic on $U_j\cap X_s$, $s\in S$ and $u_j-u_k$ are harmonic on $U_j\cap U_k$. $\endgroup$ – user21574 Oct 30 '17 at 23:20
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    $\begingroup$ A weak relative K\"ahler form is given in a similar way, where the $u_j-u_k$ need only be harmonic on the fibers $X_s\cap U_j\cap U_k$ ( $\partial_s$ and $\bar{\partial_s}$ denote derivatives in fiber direction) $\endgroup$ – user21574 Oct 30 '17 at 23:20
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    $\begingroup$ positivity of such fiberwise Kahler-Einstein metric is highly non-trivial. It is still open question. When fibers are of general type there is a paper of Schumacher,in Invent math(with two erratam ) but at the moment I couldn't yet accept his proof. So such forms are not Kahler in general . In my point of view Kontsevich-Soibelman program (which they have copy pasted from Barlet formula(in Invent Math) and Grifiths)is equivalent with positivity of such semi ricci flat forms . See mathoverflow.net/questions/261281/… $\endgroup$ – user21574 Oct 31 '17 at 0:07
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    $\begingroup$ My letter to Gang Tian can give more effectiveness of it hal.archives-ouvertes.fr/hal-01551080/document $\endgroup$ – user21574 Oct 31 '17 at 1:00
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    $\begingroup$ @HassanJolany In general, if you feel like writing a long sequence of comments, you should consider writing an answer instead. In fact, it might be better to write an answer to this question, and delete these comments. $\endgroup$ – S. Carnahan Oct 31 '17 at 5:08

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, 2.

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    $\begingroup$ I have deleted some comments that push the boundaries of professionalism. $\endgroup$ – S. Carnahan Oct 31 '17 at 5:13
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    $\begingroup$ Let $φ : X → B$ be a family of complex manifolds, $0 ∈ B$. If the central fibre $X_0$ is Kahler, then so is $X_b$ for all $b$ sufficiently near $0.$, For proof see Theorem 9.23 of the book of (Claire Voisin Hodge Theory due to Kodaira)wisdom.weizmann.ac.il/~dnovikov/Manifolds5775/… $\endgroup$ – user21574 Nov 6 '17 at 5:09
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    $\begingroup$ Moreover, In symplectic geometry we have two notions of Inflation and Deflation which developed by McDuff and V.Shevchishin see Theorem 0.3, and Theorem of 0.4 arxiv.org/pdf/1708.01518.pdf , Let $J$ be an $ω_0-$ tamed almost complex structure on a symplectic 4-manifold $(X,ω_0)$ that admits an embedded $J$-holomorphic curve $C$ with $[C]·[C]>0$. Then there is a family $ω_s$, for $s≥0$, of symplectic forms that all tame $J$ and have cohomology class $$[ω_s] = [ω_0] +sPD([C]),$$ where $PD([C])$ is Poincare dual to $[C]$ $\endgroup$ – user21574 Nov 17 '17 at 16:58
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    $\begingroup$ @macbeth : Anyway, I don't think your "Sketch proof" is fine, you may look into the theorem of Kodaira I mentioned in my previous comment and edit your proof. $\endgroup$ – user21574 Nov 17 '17 at 17:01
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    $\begingroup$ More comment: If central fibre $X_0$ be Kahler on $π:X→Δ $, then from Kodaira theorem the fibers $X_s$ when $s→0$ are Kahler also, but Kodaira didn't claim that when such $X_s$ could be projective. There are some theorems related it developed by Junyan Cao in his Ph.D. thesis. For instance, if central fiber be compact Kahler Calai-Yau manifold, or tangent bundle $T_{X_0}$ be "nef", or the hermitian metric of $−K_{X_0}$ be semi-positive then $X_s$ are projective when $s→0$(as a sequence) see hal.inria.fr/file/index/docid/749923/filename/Def_pub.pdf $\endgroup$ – user21574 Nov 19 '17 at 8:54

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