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Is there any stability notion that led to an algebraic variety be smoothable in general for Fano varieties or for Calabi-Yau varieties?

Note that Friedman found a nesessary condition that $X$ to be smoothable

For $K3$ surfaces and Del-Pezzo surfaces we have semi-stable notion as follows

Friedman proved that every such semistable $K3$ surface can be smoothed into a smooth $K3$ surface, under a flat deformation with a smooth total space

Y. Namikawa showed the following result

Let $X$ be a Calabi–Yau threefold with terminal singularities.

(1) If $X$ is $\mathbb Q$-factorial; then $X$ is smoothable.

(2) If every singularity of $X$ is different from an ordinary double point; then $X$ is smoothable

Note that not any variety can be smoothable, for example there exists a CY variety which remain singular under any flat deformation.

Kachi proved that every semi-stable Del Pezzo surface is smoothed into a smooth Del Pezzo surface, under a flat deformation with a smooth total space

Some old results:

A reduced complex analytic space $X$ of dimension $n$ is a normal crossing variety (or n.c.variety) if for each point $p \in X$, $$\mathcal O_{X,p}\cong \frac{\mathbb C\{x_0,x_1,...,x_n\}}{(x_0x_1...x_r)}\;\;\;\; 0\leq r=r(p)\leq n $$.

In addition, if every component $X_i$ of $X$ is smooth, then $X$ is called a, simple normal crossing variety (or s.n.c.variety).

Let $D=\text{Sing}(X)$, and $X_i$ be a component of $X$ and let $I_X$, (resp. $I_D$) be the defining ideal of $X_i$ (resp. $D$) in $X$. Then define

$$\mathcal O_D(-X)=I_{X_1}/I_{X_1}I_D\otimes_{\mathcal O_D}...\otimes_{\mathcal O_D} I_{X_m}/I_{X_m}I_D$$ and take $\mathcal O_D(X):=\mathcal O_D(-X)^\vee$

A normal crossing variety $X$ is called $d$-semistable if its infinitesmal normal bundle $\mathcal O_D(X)$ be trivial.

Kawamata and Namikawa proved the following theorem for normal crossing varieties with some additional assumption to get smoothing of a flat degeneration

Let $X$ be compact K\"ahler d-semi-stabte n.c.variety of dimension $n\geq 3$ and let $X^{[0]}$ be the normalization of $X$. Assume the following conditions:

(a) $\omega_X\cong \mathcal O_X$,

(b) $H^{n-1}(X, \mathcal O_X) = 0$, and

(c) $H^{n-2}(X^{[0]},\mathcal O_{X^{[0]}}) = 0$

Then $X$ is smoothable by a flat deformation.

Improve this question
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  • $\begingroup$ Take a look at a paper of Kawamata and Namikawa "Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties". Some sufficient conditions for normal crossing varieties to be smoothed to Calabi-Yau manifolds are given there. $\endgroup$
    – Basics
    Feb 15, 2017 at 17:03
  • $\begingroup$ Yahh, I know that paper you mentioned, they proved that a compact K\"ahler normal crossing variety $X$ of arbitrary dimension $d$ with a logarithmic structure such that $K_X\cong 0$, $H^{d-1}(X,\mathcal O_X) = 0$, and $H^{d-2}(X^{[0]},\mathcal O_{X^{[0]}})= 0$ is smoothable, where $X^{[0]}$ is the normalization of $X$ $\endgroup$
    – user21574
    Feb 15, 2017 at 17:19
  • $\begingroup$ A simple normal crossing, variety $X$ admits a logarithmic structure if and only if it is d-semi-stable, so in fact an special casefor d-semi-stable Calabi-Yau varieties with some additional cohomological assumption they solved the question. But I think we can refine the notion of d-semi-stability and remove such assumptions $\endgroup$
    – user21574
    Feb 15, 2017 at 17:23
  • $\begingroup$ For Fano varieties there is a cohomological result cambridge.org/core/journals/… $\endgroup$
    – user21574
    Feb 15, 2017 at 18:22

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