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Let $X$ be a normal projective surface with only quotient singularities. I have a question about the proof of Theorem 10.8 (p.119) in Megyesi's Chern Classes of $\mathbb{Q}$-sheaves. It goes as follows:

Fix a projective embedding of $X$. A generic pencil of hyperplane sections has reduced elements only, and its base locus is reduced and disjoint from $\text{Sing}(X)$. Blowing up this base locus we obtain a morphism $f:\hat{X}\to \mathbb{P}^1$ with reduced fibers. Let $g$ be the genus of the general fiber. Then there exists an exact sequence $$0\to f^*\Omega^1_{\mathbb{P}^1}\to \Omega^1_{\hat{X}}\to \hat{\omega}_{\hat{X}/\mathbb{P}^1}\to \mathcal{O}_Z\to 0, $$ where $Z$ is a $0$-dimensional scheme supported on $\text{Sing}(\hat{X})=\text{Sing}(X)$ together with the nonsingular points where $df(x)=0$.

  1. What is $\hat{\omega}_{\hat{X}/\mathbb{P}^1}$? I cannot find this elsewhere in the paper.

  2. What is $Z$? It seems that it is just a set of finitely many points including the points where $df(x)=0$ and some singular points of $X$, but I can't see how it is defined.

  3. How is this exact sequence obtained? Maybe the answer of 1 and 2 might handle this..

blancket
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  • Probably $\hat\omega$ is the reflexive hull of the relative dualizing sheaf. – Sasha Apr 03 '25 at 13:32
  • @Sasha Thanks. Does the notion of relative dualizing sheaf of a morphism $f:X\to Y$ have an analytic analog, for example for a holomorphic map between complex manifolds? – blancket Apr 03 '25 at 14:43
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    Or is this a typo, with $\omega_{\hat X/\Bbb P^1}$ actually intended? Note that it's $\hat X$, and not $X$, that maps to $\Bbb P^1$. – Ted Shifrin Apr 03 '25 at 22:52
  • @TedShifrin Thanks, actually that was my typo.. – blancket Apr 07 '25 at 00:34

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