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Aug 5, 2021 at 19:09 vote accept CommunityBot
Mar 15, 2020 at 16:03 comment added user149914 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.
Feb 20, 2020 at 8:12 comment added user149914 X_0 has normal crossing singularity, we can take the normlisation of X_0, and consider the Log cotangent bundle with logarithmic poles along the preimage of the singular locus of X_0. Then the log cotangent bundle of X_0 is the subsheaf of the log cotangent bundle above whose elements have opposite residues along the pullback divisors..this is mentioned in the famous paper of Mumford and Deligne on the irreducibility of moduli of curves..
Feb 20, 2020 at 7:33 history answered Piotr Achinger CC BY-SA 4.0