Relative logarithmic cotangent bundle

Question

Let $\mathcal X \rightarrow S$ be a flat family of projective varieties over a discrete valuation ring $S$ such that the generic fibre $\mathcal X_{\eta}$ (say) is smooth projective variety and the special fibre $\mathcal X_0$ (say) is a normal crossing divisor in $\mathcal X$.

Question: Does there exist a vector bundle $\mathcal E$ over $\mathcal X$ such that $\mathcal E|_{_{\mathcal X_{\eta}}}\cong \Omega^1_{_{\mathcal X_{\eta}}}$ and $\mathcal E|_{_{\mathcal X_0}}\cong \Omega^1_{_{\mathcal X_0}}(\mathrm{Log} D)$, where $D$ is the singular locus of $\mathcal X_0$?

It will be very helpful if anyone can explain this construction or provide a reference. Thank you..

Answer 1

First of all, it's unclear what you mean by $\Omega^1_{X_0}(\log D)$ since $X_0$ is singular.

Second, if you make up such a definition then most probably such a vector bundle will not exist. Note that the Euler characteristics $\chi(X_\eta, \Omega^1_\eta)$ and $\chi(X_0, \Omega^1_{X_0}(\log D)$ have to agree. I suggest checking this for an elliptic curve degenerating to a node.

Finally, what people usually consider in your situation is the vector bundle $\Omega^1_{X/S}(\log X_0)$, so differentials on $X$ with log poles along the components of $X_0$, divided by the pull-back $t^{-1}dt$ where $t$ is a uniformizer of $S$. This is indeed locally free and restricts to $\Omega^1_{X_\eta}$.