## Kodaira-Spencer Theory and moduli of curves.

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

Let M_g be the moduli stack of smooth genus g curves and let pi:C \to M_g be the universal curve. Let 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_* F is isomorphic \Omega^1_{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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