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I was looking at a paper of Farkas and the following confusing point came up.

Let $\mathscr{M}_g$ be the moduli stack of smooth genus $g$ curves and let $\pi: \mathscr{C} \to \mathscr{M}_g$ be the universal curve. Let $\mathscr{F}$ be $\Omega^1_\pi \otimes \Omega^1_\pi$, where $\Omega^1_\pi$ is the sheaf of relative differentials of $\pi$. Then the pushforward $\pi_* \mathscr{F}$ is isomorphic $\Omega^1_{\mathscr{M}_g}$.

Why is this true? Farkas says this follows from Kodaira-Spencer theory. I googled for a while and asked a few students, but couldn't figure this out.

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up vote 6 down vote accepted

By standard deformation theory (see e.g., Hartshorne III Ex 4.10, but there are probably better references), the tangent sheaf of $\mathscr{M}_g$ is $R^1\pi_{\ast}(\mathscr{C}, T_{\mathscr{C}/\mathscr{M}_g})$, which is Serre dual to $\pi_{\ast}\mathscr{F}$. The tangent sheaf is dual to what you wanted.

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