# Relative logarithmic cotangent bundle

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..

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 $$\Omega^1_{X_\eta}$$.
• If we consider the elliptic curve case then it is just the relative dualising sheaf. Therefore It is actually $\Omega^1_{X_0} (Log D)$. This supports my question. Mar 15, 2020 at 16:03