What is the definition of relative differential forms of a family $\pi: X \to B$ of (nodal) curves, where $B$ is the base space.
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I learnt the following from the paper On the relative de Rham sequence (MathSciNet JSTOR) by Nick Buchdahl. It is more general than the OP situation, but should easily specialise to it. Let $f:X \to B$ be a smooth map between manifolds. Then the sheaf $\Omega^1_f$ of germs of relative 1-forms on $X$ is defined by the following exact sequence $$ f^*\Omega^1_B \to \Omega^1_X \to \Omega^1_f \to 0 $$ Then the relative differential $k$-forms are defined by $$ \Omega^k_f = \Lambda^k \Omega^1_f $$ as usual. Differentiating along the fibres gives a differential $d_f: \Omega^k_f \to \Omega^{k+1}_f$. and $(\Omega^\bullet_f, d_f)$ is called the relative de Rham complex. When $f$ is a submersion, the relative de Rham complex gives a resolution $$ 0 \to f^{-1}\mathcal{E}_B \to \Omega^\bullet_f $$ of the sheaf of functions of $X$ which are constant on the fibres. |
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