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Let $\mathcal X$ be a smooth complex manifold of dimension $n+1$. We say $\mathcal X \to ∆$ is a large complex structure limit if and only if it’s maximal unipotent degeneration .

$T: H^n(\mathcal X_s, \mathbb Z) → H^n(\mathcal X_s, \mathbb Z)$ is maximal quasi-unipotent, i.e.,

$∃s$ s.t. $(T^s − id)^{n+1} = 0$, $(T^s − id)^n\neq 0$

Here $T$ is the monodromy operator. Take $\mathcal X_s\cong X$ are Calabi-Yau varieties. We call $\mathcal X\to \Delta$ is the CY degeneration of $X$. This definition is due to the work of Griffiths–Landman–Grothendieck-Katz and P.Deligne

Let $X ⊂ \mathbb P^n$ be a CY variety and there exists large complex structure degeneration of $X$ then when the limit $\mathcal X_0$ is unique?

For example the uniqueness of the $\mathcal X_0$ may not be true if $X$ is complete intersection in toric orbifold.

I have a conjecture that the uniqueness of the solution of canonical metric(relative Kahler-Einstein metric ) on degeneration of CY family correspond to uniqueness of $\mathcal X_0$. In fact this is the motivation of my question

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  • $\begingroup$ Is your conjecture true for K3 surfaces? $\endgroup$ Commented Jul 22, 2017 at 16:49
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    $\begingroup$ uniqueness of canonical metric when $0<kod(X)<\dim X$ is mysterious and it correspond to rich information. Unlike Kahler-Einstein which is unique up to constant or finite Aut . Here the life is not easy. Send me email , I will give you more information $\endgroup$
    – user21574
    Commented Jul 22, 2017 at 16:57

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In Physics, the large complex structure limit of $X$ is dual to a large volume limit of the mirror Calabi-Yau $X^v$. More precisely, different phases of $X^v$ connected by flops correspond to different large complex structure limits on $X$. So I would conjecture that this is true if and only if the extended Kaehler cone on $X^v$ is just the ordinary Kaehler cone (as is true for all (I think) cases with $h^{1,1}(X^v) = 1$ but typically fails, e.g. for most toric hypersurfaces).

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