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I'm going to assume reader is aware of semi-stable log structures either in Kawamata-Namikawa version or later approaches.

Anyway, let $X$ be a d-semistable variety. I want to know whether I can smooth it. Typically, line of arguing is "somehow prove that $h^2 (T_X (log)) = 0$ and then use log-deformation theory." This works well in CY and Fano cases, but my setting is more about general type, so I typically gonna prove that h1 > h2 (for the same logarithmic tangent sheaf).

And here I'm kinda stuck, because log-deformation theory seems to have some kind of "zero order obstruction" (obstruction to lifting log-semistable scheme to scheme over $\mathbb{C} [x] / x^2$). In regular deformation theory I could pick any element of $H^1$ and it worked without obstructions.

Is something known about this condition? Is it, say, equivalent to d-semistability to have order 1 smooth thickening? If not, how can I check this additional condition?

Thanks in advance.

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